psychTools/0000755000176200001440000000000014153452126012424 5ustar liggesuserspsychTools/NAMESPACE0000744000176200001440000000512114101357774013651 0ustar liggesusers#last modified December, 2019 by William Revelle #added the various imports from stats, graphics, etc. #importFrom(mnormt,rmnorm,sadmvn,dmnorm) #importFrom(parallel,mclapply,mcmapply) #importFrom(lattice,xyplot,strip.custom) #importFrom(nlme,lme,VarCorr) importFrom(graphics,plot,pairs,points,abline,arrows,axis,barplot,box,curve,hist,image,layout,legend, lines,mtext,par,persp,plot.new,plot.window, polygon,rect,segments,strheight,strwidth,text,axTicks,title,smoothScatter) importFrom(stats,aov,cov,cor,var,sd,median,mad,cov2cor,biplot,loess,predict,predict.lm,rnorm,dnorm,rbinom,density, kmeans, lm,lm.fit,loadings,complete.cases, na.omit,na.fail,nlminb,optim, quantile,qnorm, pnorm,qqnorm,qqline,qqplot,pchisq,qchisq,qt,pt,dt,pf,qf,ppoints,p.adjust,optimize,residuals,spline,symnum,terms,weighted.mean,promax,varimax,uniroot) #importFrom(datasets,USArrests,attitude,Harman23.cor,Harman74.cov,ability.cov,iris) importFrom(utils,head,tail,read.table,write.table,read.fwf,stack,example,download.file,getFromNamespace,untar,unzip,View) importFrom(grDevices,colorRampPalette,topo.colors,devAskNewPage,dev.flush,dev.hold, palette, grey,rainbow,rgb,col2rgb,trans3d,adjustcolor) #importFrom(methods,new) importFrom(tools,file_ext) importFrom(foreign,read.spss,read.xport,read.systat) importFrom(psych,statsBy,cs,setCor, mediate,corPlot, omega, scatterHist) #S3method(print,psych) export( #acs, # autoR, # bassAckward.diagram, # cs, # char2numeric, # chi2r, # cor2cov, cor2latex, df2latex, # cor2, # cor2dist, #cta.15, #cta, dfOrder, # d2r, # d2t, # diagram, # dia.shape, # dia.rect, # dia.ellipse, # dia.ellipse1, # dia.triangle, # dia.arrow, # dia.curve, # dia.curved.arrow, # dia.self, # dia.cone, # extension.diagram, # ellipses, # error.bars, # error.bars.by, # error.bars.tab, # error.crosses, # error.dots, # errorCircles, # fa.diagram, # fa.graph, # # fa.sort, fa2latex, fileCreate, filesInfo, filesList, fileScan, # fisherz, # fisherz2r, # fromTo, # g2r, # headtail, # headTail, ICC2latex, # iclust.diagram, irt2latex, # kurtosi, # isCorrelation, # lavaan.diagram, # levels2numeric, # lowerCor, # lowerMat, # lowerUpper, # mardia, # "%+%", # minkowski, # mssd, # multi.hist, # omega.diagram, omega2latex, # progressBar, # quickView, read.clipboard, read.clipboard.csv, read.clipboard.fwf, read.clipboard.tab, read.clipboard.lower, read.clipboard.upper, read.file, read.file.csv, read.https, # r2c, # r2d, # r2chi, # rmssd, #scaling.fits, # shannon, # table2df, # table2matrix, # topBottom, # tr, # skew, # winsor, # winsor.means, # winsor.mean, # winsor.sd, # winsor.var, write.file, write.file.csv ) psychTools/data/0000755000176200001440000000000014153443073013336 5ustar liggesuserspsychTools/data/spi.rda0000644000176200001440000061714013605124116014625 0ustar liggesusers7zXZi"6!Xs])TW"nRʟxq5(БhҰd)!e#m^#YCXE/NF՚9͈S=Se xa_#bY?G,#xnwtY'Mc|3 qȃ?Xp dQi;i]{ptb{%4oo n/du]fcXVyޭkg|YO:@gvc/i"/ Ȕ,[zD) &Am2SndJ},7ͼC:6z6i&,9=LF}g*]0"%X`jB5t3ͪiͦ=Jchl@܁I:> JI}m@6ֺ./}ļx*2: 앶rua t5V3iUtPP{b]fAAgߧ>38"b5!Gko`/ r=ł{wlu/R' ;K>Gw }C=b.ʜ}3$ʩmԾ\Gn)̓S~c΁S6Ojv?893s&J7yN/P(UxXZv&׫F5XB$bkzԒmG3IʳXŝ4W(l I[\v@!wM- \s_-97%SC5>>R0SD/Ah j35@&~Û! 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E‹R@6O~񎖝a%FRvVD-JȄ-)R%)37z2MC*@f=oEA̍gwT)dWUf4'2W\zReUq"?Efɜ>!DiYK"Q`L?2 Gf4Y-"/E&Y><=Q}&SŪdA##"EdtE]-G;'HfA%2Qu+*G&t(WON 2娖زbDHiN,гUOd4xW䠃i [UzQ t"CO(̴rɞ9q2?O0+-R"ֳ˪R$f,D۶UK >$>[D.L!n='3c/'1vuH.:a?psychTools/man/0000755000176200001440000000000014153443073013200 5ustar liggesuserspsychTools/man/msqR.rd0000644000176200001440000004532713501465754014472 0ustar liggesusers\name{msqR} \alias{msqR} \docType{data} \title{75 mood items from the Motivational State Questionnaire for 3032 unique participants} \description{Emotions may be described either as discrete emotions or in dimensional terms. The Motivational State Questionnaire (MSQ) was developed to study emotions in laboratory and field settings. The data can be well described in terms of a two dimensional solution of energy vs tiredness and tension versus calmness. Alternatively, this space can be organized by the two dimensions of Positive Affect and Negative Affect. Additional items include what time of day the data were collected and a few personality questionnaire scores. 3032 unique participants took the MSQ at least once, 2753 at least twice, 446 three times, and 181 four times. The 3032 participants also took the \code{\link{sai}} state anxiety inventory at the same time. Some studies manipulated arousal by caffeine, others manipulations included affect inducing movies. } \usage{data("msqR")} \format{ A data frame with 6411 observations on the following 88 variables. \describe{ \item{\code{active}}{a numeric vector} \item{\code{afraid}}{a numeric vector} \item{\code{alert}}{a numeric vector} \item{\code{alone}}{a numeric vector} \item{\code{angry}}{a numeric vector} \item{\code{aroused}}{a numeric vector} \item{\code{ashamed}}{a numeric vector} \item{\code{astonished}}{a numeric vector} \item{\code{at.ease}}{a numeric vector} \item{\code{at.rest}}{a numeric vector} \item{\code{attentive}}{a numeric vector} \item{\code{blue}}{a numeric vector} \item{\code{bored}}{a numeric vector} \item{\code{calm}}{a numeric vector} \item{\code{clutched.up}}{a numeric vector} \item{\code{confident}}{a numeric vector} \item{\code{content}}{a numeric vector} \item{\code{delighted}}{a numeric vector} \item{\code{depressed}}{a numeric vector} \item{\code{determined}}{a numeric vector} \item{\code{distressed}}{a numeric vector} \item{\code{drowsy}}{a numeric vector} \item{\code{dull}}{a numeric vector} \item{\code{elated}}{a numeric vector} \item{\code{energetic}}{a numeric vector} \item{\code{enthusiastic}}{a numeric vector} \item{\code{excited}}{a numeric vector} \item{\code{fearful}}{a numeric vector} \item{\code{frustrated}}{a numeric vector} \item{\code{full.of.pep}}{a numeric vector} \item{\code{gloomy}}{a numeric vector} \item{\code{grouchy}}{a numeric vector} \item{\code{guilty}}{a numeric vector} \item{\code{happy}}{a numeric vector} \item{\code{hostile}}{a numeric vector} \item{\code{inspired}}{a numeric vector} \item{\code{intense}}{a numeric vector} \item{\code{interested}}{a numeric vector} \item{\code{irritable}}{a numeric vector} \item{\code{jittery}}{a numeric vector} \item{\code{lively}}{a numeric vector} \item{\code{lonely}}{a numeric vector} \item{\code{nervous}}{a numeric vector} \item{\code{placid}}{a numeric vector} \item{\code{pleased}}{a numeric vector} \item{\code{proud}}{a numeric vector} \item{\code{quiescent}}{a numeric vector} \item{\code{quiet}}{a numeric vector} \item{\code{relaxed}}{a numeric vector} \item{\code{sad}}{a numeric vector} \item{\code{satisfied}}{a numeric vector} \item{\code{scared}}{a numeric vector} \item{\code{serene}}{a numeric vector} \item{\code{sleepy}}{a numeric vector} \item{\code{sluggish}}{a numeric vector} \item{\code{sociable}}{a numeric vector} \item{\code{sorry}}{a numeric vector} \item{\code{still}}{a numeric vector} \item{\code{strong}}{a numeric vector} \item{\code{surprised}}{a numeric vector} \item{\code{tense}}{a numeric vector} \item{\code{tired}}{a numeric vector} \item{\code{unhappy}}{a numeric vector} \item{\code{upset}}{a numeric vector} \item{\code{vigorous}}{a numeric vector} \item{\code{wakeful}}{a numeric vector} \item{\code{warmhearted}}{a numeric vector} \item{\code{wide.awake}}{a numeric vector} \item{\code{anxious}}{a numeric vector} \item{\code{cheerful}}{a numeric vector} \item{\code{idle}}{a numeric vector} \item{\code{inactive}}{a numeric vector} \item{\code{tranquil}}{a numeric vector} \item{\code{kindly}}{a numeric vector} \item{\code{scornful}}{a numeric vector} \item{\code{Extraversion}}{Extraversion from the EPI} \item{\code{Neuroticism}}{Neuroticism from the EPI} \item{\code{Lie}}{Lie from the EPI} \item{\code{Sociability}}{Sociability from the EPI} \item{\code{Impulsivity}}{Impulsivity from the EPI} \item{\code{gender}}{1= male, 2 = female (coded on presumed x chromosome). Slowly being added to the data set.} \item{\code{TOD}}{Time of day that the study was run} \item{\code{drug}}{1 if given placebo, 2 if given caffeine} \item{\code{film}}{1-4 if given a film: 1=Frontline, 2= Halloween, 3=Serengeti, 4 = Parenthood} \item{\code{time}}{Measurement occasion (1 and 2 are same session, 3 and 4 are the same, but a later session)} \item{\code{id}}{a numeric vector} \item{\code{form}}{msq versus msqR} \item{\code{study}}{a character vector of the experiment name} } } \details{The Motivational States Questionnaire (MSQ) is composed of 75 items, which represent the full affective space (Revelle & Anderson, 1998). The MSQ consists of 20 items taken from the Activation-Deactivation Adjective Check List (Thayer, 1986), 18 from the Positive and Negative Affect Schedule (PANAS, Watson, Clark, & Tellegen, 1988) along with the affective circumplex items used by Larsen and Diener (1992). The response format was a four-point scale that corresponds to Russell and Carroll's (1999) "ambiguous--likely-unipolar format" and that asks the respondents to indicate their current standing (``at this moment") with the following rating scale:\cr 0----------------1----------------2----------------3 \cr Not at all A little Moderately Very much \cr The original version of the MSQ included 70 items. Intermediate analyses (done with 1840 subjects) demonstrated a concentration of items in some sections of the two dimensional space, and a paucity of items in others. To begin correcting this, 3 items from redundantly measured sections (alone, kindly, scornful) were removed, and 5 new ones (anxious, cheerful, idle, inactive, and tranquil) were added. Thus, the correlation matrix is missing the correlations between items anxious, cheerful, idle, inactive, and tranquil with alone, kindly, and scornful. 2605 individuals took Form 1 version, 3806 the Form 2 version. 3032 people (1218 form 1, 1814 form 2) took the MSQ at least once. 2086 at least twice, 1112 three times, and 181 four times. To see the relative frequencies by time and form, see the first example. Procedure. The data were collected over nine years in the Personality, Motivation and Cognition laboratory at Northwestern, as part of a series of studies examining the effects of personality and situational factors on motivational state and subsequent cognitive performance. In each of 38 studies, prior to any manipulation of motivational state, participants signed a consent form and in some studies, consumed 0 or 4mg/kg of caffeine. In caffeine studies, they waited 30 minutes and then filled out the MSQ. (Normally, the procedures of the individual studies are irrelevant to this data set and could not affect the responses to the MSQ at time 1, since this instrument was completed before any further instructions or tasks. However, caffeine does have an effect.) The MSQ post test following a movie manipulation) is available in \code{\link{affect}} as well as here. The XRAY study crossed four movie conditions with caffeine. The first MSQ measures are showing the effects of the movies and caffeine, but after an additional 30 minutes, the second MSQ seems to mainly show the caffeine effects. The movies were 9 minute clips from 1) a BBC documentary on British troops arriving at the Bergen-Belsen concentration camp (sad); 2) an early scene from Halloween in which the heroine runs around shutting doors and windows (terror); 3) a documentary about lions on the Serengeti plain, and 4) the "birthday party" scene from Parenthood. The FLAT study measured affect before, immediately after, and then after 30 minutes following a movie manipulation. See the \code{\link{affect}} data set. To see which studies used which conditions, see the second and third examples. The EA and TA scales are from Thayer, the PA and NA scales are from Watson et al. (1988). Scales and items: Energetic Arousal: active, energetic, vigorous, wakeful, wide.awake, full.of.pep, lively, -sleepy, -tired, - drowsy (ADACL) Tense Arousal: Intense, Jittery, fearful, tense, clutched up, -quiet, -still, - placid, - calm, -at rest (ADACL) Positive Affect: active, alert, attentive, determined, enthusiastic, excited, inspired, interested, proud, strong (PANAS) Negative Affect: afraid, ashamed, distressed, guilty, hostile, irritable , jittery, nervous, scared, upset (PANAS) The PA and NA scales can in turn can be thought of as having subscales: (See the PANAS-X) Fear: afraid, scared, nervous, jittery (not included frightened, shaky) Hostility: angry, hostile, irritable, (not included: scornful, disgusted, loathing guilt: ashamed, guilty, (not included: blameworthy, angry at self, disgusted with self, dissatisfied with self) sadness: alone, blue, lonely, sad, (not included: downhearted) joviality: cheerful, delighted, energetic, enthusiastic, excited, happy, lively, (not included: joyful) self-assurance: proud, strong, confident, (not included: bold, daring, fearless ) attentiveness: alert, attentive, determined (not included: concentrating) The next set of circumplex scales were taken from Larsen and Diener (1992). High activation: active, aroused, surprised, intense, astonished Activated PA: elated, excited, enthusiastic, lively Unactivated NA : calm, serene, relaxed, at rest, content, at ease PA: happy, warmhearted, pleased, cheerful, delighted Low Activation: quiet, inactive, idle, still, tranquil Unactivated PA: dull, bored, sluggish, tired, drowsy NA: sad, blue, unhappy, gloomy, grouchy Activated NA: jittery, anxious, nervous, fearful, distressed. Keys for these separate scales are shown in the examples. In addition to the MSQ, there are 5 scales from the Eysenck Personality Inventory (Extraversion, Impulsivity, Sociability, Neuroticism, Lie). The Imp and Soc are subsets of the the total extraversion scale based upon a reanalysis of the EPI by Rocklin and Revelle (1983). This information is in the \code{\link{msq}} data set as well. } \note{In December, 2018 the caffeine, film and personality conditions were added. In the process of doing so, it was discovered that the EMIT data had been incorrectly entered. This has been fixed. } \source{Data collected at the Personality, Motivation, and Cognition Laboratory, Northwestern University. } \references{ Larsen, R. J., & Diener, E. (1992). Promises and problems with the circumplex model of emotion. In M. S. Clark (Ed.), Review of personality and social psychology, No. 13. Emotion (pp. 25-59). Thousand Oaks, CA, US: Sage Publications, Inc. Rafaeli, Eshkol and Revelle, William (2006), A premature consensus: Are happiness and sadness truly opposite affects? Motivation and Emotion, 30, 1, 1-12. Revelle, W. and Anderson, K.J. (1998) Personality, motivation and cognitive performance: Final report to the Army Research Institute on contract MDA 903-93-K-0008. (\url{https://www.personality-project.org/revelle/publications/ra.ari.98.pdf}). Smillie, Luke D. and Cooper, Andrew and Wilt, Joshua and Revelle, William (2012) Do Extraverts Get More Bang for the Buck? Refining the Affective-Reactivity Hypothesis of Extraversion. Journal of Personality and Social Psychology, 103 (2), 206-326. Thayer, R.E. (1989) The biopsychology of mood and arousal. Oxford University Press. New York, NY. Watson,D., Clark, L.A. and Tellegen, A. (1988) Development and validation of brief measures of positive and negative affect: The PANAS scales. Journal of Personality and Social Psychology, 54(6):1063-1070. } \seealso{\code{\link{msq}} for 3896 participants with scores on five scales of the EPI. \code{\link{affect}} for an example of the use of some of these adjectives in a mood manipulation study. \code{\link{make.keys}}, \code{\link{scoreItems}} and \code{\link{scoreOverlap}} for instructions on how to score multiple scales with and without item overlap. Also see \code{\link{fa}} and \code{\link{fa.extension}} for instructions on how to do factor analyses or factor extension. Given the temporal ordering of the \code{\link{sai}} data and the \code{\link{msqR}} data, these data are useful for demonstrations of \code{\link{testRetest}} reliability. See the examples in \code{\link{testRetest}} for how to combine the \code{\link{sai}} \code{\link{tai}} and \code{\link{msqR}} datasets. } \examples{ data(msqR) table(msqR$form,msqR$time) #which forms? table(msqR$study,msqR$drug) #Drug studies table(msqR$study,msqR$film) #Film studies table(msqR$study,msqR$TOD) #To examine time of day #score them for 20 short scales -- note that these have item overlap #The first 2 are from Thayer #The next 2 are classic positive and negative affect #The next 9 are circumplex scales #the last 7 are msq estimates of PANASX scales (missing some items) keys.list <- list( EA = c("active", "energetic", "vigorous", "wakeful", "wide.awake", "full.of.pep", "lively", "-sleepy", "-tired", "-drowsy"), TA =c("intense", "jittery", "fearful", "tense", "clutched.up", "-quiet", "-still", "-placid", "-calm", "-at.rest") , PA =c("active", "excited", "strong", "inspired", "determined", "attentive", "interested", "enthusiastic", "proud", "alert"), NAf =c("jittery", "nervous", "scared", "afraid", "guilty", "ashamed", "distressed", "upset", "hostile", "irritable" ), HAct = c("active", "aroused", "surprised", "intense", "astonished"), aPA = c("elated", "excited", "enthusiastic", "lively"), uNA = c("calm", "serene", "relaxed", "at.rest", "content", "at.ease"), pa = c("happy", "warmhearted", "pleased", "cheerful", "delighted" ), LAct = c("quiet", "inactive", "idle", "still", "tranquil"), uPA =c( "dull", "bored", "sluggish", "tired", "drowsy"), naf = c( "sad", "blue", "unhappy", "gloomy", "grouchy"), aNA = c("jittery", "anxious", "nervous", "fearful", "distressed"), Fear = c("afraid" , "scared" , "nervous" , "jittery" ) , Hostility = c("angry" , "hostile", "irritable", "scornful" ), Guilt = c("guilty" , "ashamed" ), Sadness = c( "sad" , "blue" , "lonely", "alone" ), Joviality =c("happy","delighted", "cheerful", "excited", "enthusiastic", "lively", "energetic"), Self.Assurance=c( "proud","strong" , "confident" , "-fearful" ), Attentiveness = c("alert" , "determined" , "attentive" )) #acquiscence = c("sleepy" , "wakeful" , "relaxed","tense") #Yik Russell and Steiger list the following items Yik.keys <- list( pleasure =psych::cs(happy,content,satisfied, pleased), act.pleasure =psych::cs(proud,enthusiastic,euphoric), pleasant.activation = psych::cs(energetic,full.of.pep,excited,wakeful,attentive, wide.awake,active,alert,vigorous), activation = psych::cs(aroused,hyperactivated,intense), unpleasant.act = psych::cs(anxious,frenzied,jittery,nervous), activated.displeasure =psych::cs(scared,upset,shaky,fearful,clutched.up,tense, ashamed,guilty,agitated,hostile), displeaure =psych::cs(troubled,miserable,unhappy,dissatisfied), Ueactivated.Displeasure = psych::cs(sad,down,gloomy,blue,melancholy), Unpleasant.Deactivation = psych::cs(droopy,drowsy,dull,bored,sluggish,tired), Deactivation =psych::cs( quiet,still), pleasant.deactivation = psych::cs(placid,relaxed,tranquil, at.rest,calm), deactived.pleasure =psych::cs( serene,soothed,peaceful,at.ease,secure) ) #of these 60 items, 46 appear in the msqR Yik.msq.keys <- list( Pleasure =psych::cs(happy,content,satisfied, pleased), Activated.Pleasure =psych::cs(proud,enthusiastic), Pleasant.Activation = psych::cs(energetic,full.of.pep,excited,wakeful,attentive, wide.awake,active,alert,vigorous), Activation = psych::cs(aroused,intense), Unpleasant.Activation = psych::cs(anxious,jittery,nervous), Activated.Displeasure =psych::cs(scared,upset,fearful, clutched.up,tense,ashamed,guilty,hostile), Displeasure = psych::cs(unhappy), Deactivated.Displeasure = psych::cs(sad,gloomy,blue), Unpleasant.Deactivation = psych::cs(drowsy,dull,bored,sluggish,tired), Deactivation =psych::cs( quiet,still), Pleasant.Deactivation = psych::cs(placid,relaxed,tranquil, at.rest,calm), Deactivated.Pleasure =psych::cs( serene,at.ease) ) yik.scores <- psych::scoreItems(Yik.msq.keys,msqR) yik <- yik.scores$scores f2.yik <- psych::fa(yik,2) #factor the yik scores psych::fa.plot(f2.yik,labels=colnames(yik),title="Yik-Russell-Steiger circumplex",cex=.8, pos=(c(1,1,2,1,1,1,3,1,4,1,2,4))) msq.scores <- psych::scoreItems(keys.list,msqR) #show a circumplex structure for the non-overlapping items fcirc <- psych::fa(msq.scores$scores[,5:12],2) psych::fa.plot(fcirc,labels=colnames(msq.scores$scores)[5:12]) \donttest{ #now, find the correlations corrected for item overlap msq.overlap <- psych::scoreOverlap(keys.list,msqR) f2 <- psych::fa(msq.overlap$cor,2) psych::fa.plot(f2,labels=colnames(msq.overlap$cor), title="2 dimensions of affect, corrected for overlap") #extend this solution to EA/TA NA/PA space fe <- psych::fa.extension(cor(msq.scores$scores[,5:12],msq.scores$scores[,1:4]),fcirc) psych::fa.diagram(fcirc,fe=fe,main="Extending the circumplex structure to EA/TA and PA/NA ") #show the 2 dimensional structure f2 <- psych::fa(msqR[1:72],2) psych::fa.plot(f2,labels=colnames(msqR)[1:72],title="2 dimensions of affect at the item level") #sort them by polar coordinates round(psych::polar(f2),2) } #the msqR and sai data sets have 10 overlapping items which can be used for #testRetest analysis. We need to specify the keys, and then choose the appropriate #data sets sai.msq.keys <- list(pos =c( "at.ease" , "calm" , "confident", "content","relaxed"), neg = c("anxious", "jittery", "nervous" ,"tense" , "upset"), anx = c("anxious", "jittery", "nervous" ,"tense", "upset","-at.ease" , "-calm" , "-confident", "-content","-relaxed")) select <- psych::selectFromKeys(sai.msq.keys$anx) #The following is useful for examining test retest reliabilities msq.control <- subset(msqR,is.element( msqR$study , c("Cart", "Fast", "SHED", "SHOP"))) msq.film <- subset(msqR,(is.element( msqR$study , c("FIAT", "FILM","FLAT","MIXX","XRAY")) & (msqR$time < 3) )) msq.film[((msq.film$study == "FLAT") & (msq.film$time ==3)) ,] <- NA msq.drug <- subset(msqR,(is.element( msqR$study , c("AGES","SALT", "VALE", "XRAY"))) &(msqR$time < 3)) msq.day <- subset(msqR,is.element( msqR$study , c("SAM", "RIM"))) } \keyword{datasets} psychTools/man/big5.100.adjectives.Rd0000644000176200001440000000420513767767265016763 0ustar liggesusers\name{big5.100.adjectives} \alias{big5.100.adjectives} \alias{bfi.adjectives} \alias{big5.adjectives.keys} \alias{bfi.adjectives.keys} \docType{data} \title{100 adjectives describing the "big 5" for 502 subjects} \description{Lew Goldberg organized 100 adjectives to measure 5 factors of personality (The Big5). 500 hundred participants were given these adjectives along with other personality measures in the Personality, Motivation and Cognition (PMC) lab. This data set is for demonstrations of factor and cluster analysis. } \usage{data("big5.100.adjectives")} \format{ A data frame with 554 observations on the following 102 variables. \describe{ \item{\code{study}}{a character vector} \item{\code{id}}{a numeric vector} \item{\code{V1}}{numeric vector (see big5.adjectives.dictionary) } \item{\code{V100}}{A numeric vector. (see big5.adjectives.dictionary)} \item{bfi.adjectives.keys}{a key list} } } \details{ Procedure. The data were collected over nine years in the Personality, Motivation and Cognition laboratory at Northwestern, as part of a series of studies examining the effects of personality and situational factors on motivational state and subsequent cognitive performance. In each of 38 studies, prior to any manipulation of motivational state, participants signed a consent form and in some studies, consumed 0 or 4mg/kg of caffeine. In caffeine studies, they waited 30 minutes and then filled out the MSQ as well as other personality trait measures (e.g. the Big 5 adjectives) } \source{Data collected at the Personality, Motivation, and Cognition Laboratory, Northwestern University.} \references{ Lewis R. Goldberg,(1992) The development of markers for the Big-Five factor structure, Psychological Assessment, 4 (1) 26-42. Revelle, W. and Anderson, K.J. (1998) Personality, motivation and cognitive performance: Final report to the Army Research Institute on contract MDA 903-93-K-0008. (\url{https://www.personality-project.org/revelle/publications/ra.ari.98.pdf}). } \examples{ data(big5.100.adjectives) five.scores <- psych::scoreItems(big5.adjectives.keys,big5.100.adjectives) summary(five.scores) } \keyword{datasets} psychTools/man/holzinger.swineford.Rd0000644000176200001440000002410513723304444017471 0ustar liggesusers\name{holzinger.swineford} \alias{holzinger.swineford} \alias{holzinger.raw} \alias{holzinger.dictionary} \docType{data} \title{ The raw and transformed data from Holzinger and Swineford, 1939 } \description{ A classic data set in psychometrics is that from Holzinger and Swineford (1939). A 4 and 5 factor solution to 24 of these variables problem is presented by Harman (1976), and 9 of these are used by the lavaan package. The two data sets were supplied by Keith Widaman. } \usage{data(holzinger.swineford) data(holzinger.raw) data(holzinger.dictionary) } \format{ A data frame with 301 observations on the following 33 variables. Longer descriptions taken from Thompson, (1998). \describe{ \item{\code{case}}{a numeric vector} \item{\code{school}}{School Pasteur or Grant-White} \item{\code{grade}}{Grade (7 or 8)} \item{\code{female}}{male = 1, female = 2} \item{\code{ageyr}}{age in years} \item{\code{mo}}{months over year} \item{\code{agemo}}{Age in months } \item{\code{t01_visperc}}{Visual perception test from Spearman VPT Part I} \item{\code{t02_cubes}}{Cubes, Simplification of Brighams Spatial Relations Test} \item{\code{t03_frmbord}}{Paper formboard-Shapes that can be combined to form a target} \item{\code{t04_lozenges}}{Lozenges from Thorndike-Shapes flipped over then identify target} \item{\code{t05_geninfo}}{General Information Verbal Test} \item{\code{t06_paracomp}}{Paragraph Comprehension Test} \item{\code{t07_sentcomp}}{Sentence Completion Test} \item{\code{t08_wordclas}}{Word clasification-Which word not belong in set} \item{\code{t09_wordmean}}{Word Meaning Test} \item{\code{t10_addition}}{Speeded addition test} \item{\code{t11_code}}{Speeded codetest-Transform shapes into alpha with code} \item{\code{t12_countdot}}{Speeded counting of dots in shap} \item{\code{t13_sccaps}}{Speeded discrimation of straight and curved caps} \item{\code{t14_wordrecg}}{Memory of Target Words} \item{\code{t15_numbrecg}}{Memory of Target Numbers} \item{\code{t16_figrrecg}}{Memory of Target Shapes} \item{\code{t17_objnumb}}{Memory of object-Number association targets} \item{\code{t18_numbfig}}{Memory of number-Object association targets} \item{\code{t19_figword}}{Memory of figure-Word association target} \item{\code{t20_deduction}}{Deductive Math Ability} \item{\code{t21_numbpuzz}}{Math number puzzles} \item{\code{t22_probreas}}{Math word problem reasoning} \item{\code{t23_series}}{Completion of a Math Number Series} \item{\code{t24_woody}}{Woody-McCall mixed math fundamentals test} \item{\code{t25_frmbord2}}{Revision of t3-Paper form board} \item{\code{t26_flags}}{Flags-possible substitute for t4 lozenges} } } \details{The following commentary was provided by Keith Widaman: ``The Holzinger and Swineford (1939) data have been used as a model data set by many investigators. For example, Harman (1976) used the ``24 Psychological Variables" example prominently in his authoritative text on multiple factor analysis, and the data presented under this rubric consisted of 24 of the variables from the Grant-White school (N = 145). Meredith (1964a, 1964b) used several variables from the Holzinger and Swineford study in his work on factorial invariance under selection. Joreskog (1971) based his work on multiple-group confirmatory factor analysis using the Holzinger and Swineford data, subsetting the data into four groups. Rosseel, who developed the `lavaan' package for R, included 9 of the manifest variables from Holzinger and Swineford (1939) as a ``resident" data set when one downloads the `lavaan' package. Several background variables are included in this ``resident" data set in addition to 9 of the psychological tests (which are named x1 -- x9 in the data set). When analyzing these data, I found the distributions of the variables (means, SDs) did not match the sample statistics from the original article. For example, in the ``resident" data set in `lavaan', scores on all manifest variables ranged between 0 and 10, sample means varied between 3 and 6, and sample SDs varied between 1.0 and 1.5. In the original data set, scores ranges were rather different across tests, with some variables having scores that ranged between 0 and 20, but other manifest variables having scores ranging from 50 to over 300 -- with obvious attendant differences in sample means and SDs. After a bit of snooping (i.e., data analysis), I discovered that the 9 variables in the ``resident" data set in `lavaan' had been rescored through ratio transformations. The ratio transformations involved dividing the raw score for each person on a given test by a particular constant for that test that transformed scores on the test to have the desired range. I decided to perform transformations of all 26 variables so that two data sets could be available to interested researchers:" holzinger.raw are the raws scores on all variables from Holzinger & Swineford (1939) holzinger.swineford are rescaled scores on all variables from Holzinger & Swineford. holzinger.dictionary is a list of the variable names in short and long form. ... Widaman continues: ``As several persons have noted, Harman (1976) used data only from the Grant-White school (N = 145) for his 24 Psychological Variables data set. In doing so, Harman replaced t03_frmbord and t04_lozenges with t25_frmbord2 and t26_flags, because the latter two tests were experimental tests that were designed to be more appropriate for this age level. This substitution is fine, as long as one analyzes data from only the Grant- White school. If one wishes to perform multiple-group analyses and uses school as a grouping variable (as Meredith, 1964a, 1964b, and Joreskog, 1971, did), then tests 25 and 26 should not be used." ``As have others, Gorsuch (1983) mentioned that analyses based on the raw data reported by Holzinger and Swineford (1939) will not produce statistics (means, SDs, correlations) that match precisely the values reported by Holzinger and Swineford or Harman (1976). Following Gorsuch, I have assumed that the raw data are correct. Applying factor analytic techniques to the raw data from the Grant-White school and to the summary data reported by Harman (1976) will produce slightly different results, but results that differ in only minor, unimportant details." These data are interesting not just for the historical completeness of having the orinal data, but also as an example of suppressor variables. Age and grade are positively correlated, and scores are higher in the 8th grade than in the 7th grade. But age (particularly in months) is negatively correlated with many of the cognitive tasks, and when grade and age are both entered into regression, this negative correlation is enhanced. That is, although increasing grade increases cognitive perforamnce, younger children in both grades do better than the older children. } \source{ Keith Widaman (2019, personal communication). Original data from Holzinger and Swineford (1939). } \references{ Gorsuch, R. L. (1983). Factor analysis (2nd ed.). Hillsdale, NJ: Erlbaum. Harman, Harry Horace (1967), Modern factor analysis. Chicago, University of Chicago Press. Holzinger, K. J., & Swineford, F. (1939). A study in factor analysis: The stability of a bi-factor solution. Supplementary Educational Monographs, no. 48. Chicago: University of Chicago, Department of Education. Joreskog, K. G. (1971). Simultaneous factor analysis in several populations. Psychometrika, 36, 409-426. Meredith, W. (1964a). Notes on factorial invariance. Psychometrika, 29, 177-185. Meredith, W. (1964b). Rotation to achieve factorial invariance. Psychometrika, 29, 177-206. Meredith, W. (1977). On weighted Procrustes and hyperplane fitting in factor analytic rotation. Psychometrika, 42, 491-522. Thompson, Bruce. Five Methodology Errors in Educational Research:The Pantheon of Statistical Significance and Other Faux Pas. Paper presented at the Annual Meeting of the American Educational Research Association(San Diego, CA, April 13-17,1998) } \note{As discussed by Widaman, the descriptive values reported in Harman (1967) (p 124) do not quite match the descriptive statistics in \code{\link{holzinger.raw}}. Further note that the correlation matrix and factor loadings are trivially different from the Harman.24 factor loadings in the GPA rotation package. The purpose behind presenting both the raw and transformed data is to show that the fit statistics from factor analysis are identical for these two data sets. The variables v1 ... v9 in the lavaan package correspond to tests 1, 2, 4, 6, 7, 9, 10, 12 and 13. } \examples{ data(holzinger.raw) psych::describe(holzinger.raw) data(holzinger.dictionary) holzinger.dictionary #to see the longer names for these data (taken from Thompson) #Compare these to the lavaan correlation matrix psych::lowerCor(holzinger.swineford[ 7+ c(1, 2, 4, 6, 7, 9, 10, 12, 13)]) psych::setCor(t01_visperc + t05_geninfo + t08_wordclas ~ grade + agemo,data = holzinger.raw) psych::setCor( t06_paracomp ~ grade + agemo, data=holzinger.swineford) psych::mediate(t06_paracomp ~ grade + (agemo),data = holzinger.raw,std=TRUE) #show the omega structure of the 24 variables om4 <- psych::omega(holzinger.swineford[8:31],4) psych::omega.diagram(om4,sl=FALSE,main="26 variables from Holzinger-Swineford") #these data also show an interesting suppression effect psych::lowerCor(holzinger.swineford[c(3,7,12:14)]) psych::setCor( t06_paracomp ~ grade + agemo, data=holzinger.swineford) #or show as a mediation effect mod <- psych::mediate(t06_paracomp ~ grade + (agemo),data = holzinger.raw,std=TRUE,n.iter=50) summary(mod) #now, show a plot of these effets plot(t07_sentcomp ~ agemo, col=c("red","blue")[holzinger.swineford$grade -6], pch=26-holzinger.swineford$grade,data=holzinger.swineford, ylab="Sentence Comprehension",xlab="Age in Months", main="Sentence Comprehension varies by age and grade") by(holzinger.swineford,holzinger.swineford$grade -6,function(x) abline( lm(t07_sentcomp ~ agemo,data=x),lty=c("dashed","solid")[x$grade-6])) text(190,3.3,"grade = 8") text(190,2,"grade = 7") } \keyword{datasets} psychTools/man/eminence.Rd0000644000176200001440000000452214152717631015260 0ustar liggesusers\name{eminence} \alias{eminence} \docType{data} \title{Eminence of 69 American Psychologists} \description{ Marco Del Giudice criticized an earlier study by Simonton for using partial regression weights to estimate the importance of various predictors of rated eminence. This is a nice example of the (mis)interpretation of beta weights of highly correlated predictors.} \usage{data("eminence")} \format{ A data frame with 69 observations on the following 9 variables. \describe{ \item{\code{name}}{a character vector} \item{\code{reputation}}{Log of rated reputation} \item{\code{birth.year}}{Year of birth} \item{\code{first.year}}{Year of first cited publicatin} \item{\code{last.year}}{Year of last cited publication} \item{\code{works}}{Log of number of publications} \item{\code{citations}}{Log of number of citations} \item{\code{composite}}{A composite index of publications} \item{\code{h}}{The 'h' index of citations} } } \details{Simonton (1997, 2014) discusses various estimates of eminence among 69 psychologists born between 1842 and 1912 and reports that the regression weights are small and interprets this as meaning number of publications and citations are not very important. Del Giudice (2020) points out that citations and the number of publications are highly collinear and thus while their independent contributioons are small, their joint effect is quite large (R= .69 ). These data are given here as an example of multiple correlation and partial correlation } \source{ Del Giudice (2020) links to a web page with the data. } \references{ Marco Del Giudice (2020). How Well Do Bibliometric Indicators Correlate With Scientific Eminence? A Comment on Simonton (2016). Perspective in Psychological Science, 15, 202-203. Simonton, D. K. (1992). Leaders of American psychology, 1879-1967: Career development, creative output, and professional achievement. Journal of Personality and Social Psychology, 62, 5-17. Simonton, D. K. (2016). Giving credit where credit is due: Why it's so hard to do in psychological science. Perspectives on Psychological Science, 11, 888-892. } \examples{ data(eminence) psych::lowerCor(eminence) cs <- psych::cs psych::partial.r(eminence, x= cs(reputation, works, citations),y=cs(birth.year)) psych::setCor(reputation ~ works + h + first.year,data=eminence) } \keyword{datasets}psychTools/man/cubits.Rd0000644000176200001440000000451213464310220014751 0ustar liggesusers\name{cubits} \alias{cubits} \docType{data} \title{Galton's example of the relationship between height and 'cubit' or forearm length} \description{Francis Galton introduced the 'co-relation' in 1888 with a paper discussing how to measure the relationship between two variables. His primary example was the relationship between height and forearm length. The data table (cubits) is taken from Galton (1888). Unfortunately, there seem to be some errors in the original data table in that the marginal totals do not match the table. The data frame, \code{\link{heights}}, is converted from this table. } \usage{data(cubits)} \format{ A data frame with 9 observations on the following 8 variables. \describe{ \item{\code{16.5}}{Cubit length < 16.5} \item{\code{16.75}}{16.5 <= Cubit length < 17.0} \item{\code{17.25}}{17.0 <= Cubit length < 17.5} \item{\code{17.75}}{17.5 <= Cubit length < 18.0} \item{\code{18.25}}{18.0 <= Cubit length < 18.5} \item{\code{18.75}}{18.5 <= Cubit length < 19.0} \item{\code{19.25}}{19.0 <= Cubit length < 19.5} \item{\code{19.75}}{19.5 <= Cubit length } } } \details{Sir Francis Galton (1888) published the first demonstration of the correlation coefficient. The regression (or reversion to mediocrity) of the height to the length of the left forearm (a cubit) was found to .8. There seem to be some errors in the table as published in that the row sums do not agree with the actual row sums. These data are used to create a matrix using \code{\link{table2matrix}} for demonstrations of analysis and displays of the data. } \seealso{ \code{\link[psych]{table2matrix}}, \code{\link[psych]{table2df}}, \code{\link[psych]{ellipses}}, \code{\link{heights}}, \code{\link{peas}},\code{\link{galton}}} \source{Galton (1888) } \references{Galton, Francis (1888) Co-relations and their measurement. Proceedings of the Royal Society. London Series,45,135-145, } \examples{ data(cubits) cubits heights <- psych::table2df(cubits,labs = c("height","cubit")) psych::ellipses(heights,n=1,main="Galton's co-relation data set") psych::ellipses(jitter(heights$height,3),jitter(heights$cubit,3),pch=".", main="Galton's co-relation data set",xlab="height", ylab="Forearm (cubit)") #add in some noise to see the points psych::pairs.panels(heights,jiggle=TRUE,main="Galton's cubits data set") } \keyword{datasets} psychTools/man/spengler.Rd0000644000176200001440000000663313730223474015317 0ustar liggesusers\name{Spengler} \alias{Spengler} \alias{spengler} \alias{Damian} \alias{Spengler.stat} \docType{data} \title{Project Talent data set from Marion Spengler and Rodica Damian } \description{Project Talent gave 440,000 US high school students a number of personality and ability tests. Of these, the data fror 346,000 were available for followup. Subsequent followups were collected 11 and 50 years later. Marion Spengler and her colleagues Rodica Damian, and Brent Roberts reported on the stability and change across 50 years of personality and ability. Here is the correlation matrix of 25 of their variables (Spengler) as well as a slightly different set of 19 variables (Damian). This is a nice example of mediation and regression from a correlation matrix. } \usage{data("Damian")} \format{ A 25 x 25 correlation matrix of demographic, personality, and ability variables, based upon 346,660 participants. \describe{ \item{\code{Race/Ethnicity}}{1 = other, 2 = white/caucasian} \item{\code{Sex}}{1=Male, 2=Female} \item{\code{Age}}{Cohort =9th grade, 10th grade, 11th grade, 12th grade} \item{\code{Parental}}{Parental SES based upon 9 questions of home value, family income, etc.} \item{\code{IQ}}{Standardized composite of Verbal, Spatial and Mathematical} \item{\code{Sociability etc.}}{10 scales based upon prior work by Damian and Roberts} \item{\code{Maturity}}{A higher order factor from the prior 10 scales} \item{\code{Extraversion}}{The second higher order factor} \item{\code{Interest}}{Self reported interest in school} \item{\code{Reading}}{Self report reading skills} \item{\code{Writing}}{Self report writing skills } \item{\code{Responsible}}{Self reported responsibility scale} \item{\code{Ed.11}}{Education level at 11 year followup} \item{\code{Educ.50}}{Education level at 50 year followup} \item{\code{OccPres.11}}{Occupational Prestige at 11 year followup} \item{\code{OccPres.50}}{Occupational Prestige at 50 year followup} \item{\code{Income.11}}{Income at 11 year followup} \item{\code{Income.50}}{Income at 50 year followup} } } \details{ Data from Project Talent was collected in 1960 on a representative sample of American high school students. Subsequent follow up 11 and 50 years later are reported by Spengler et al (2018) and others. } \source{ Marion Spengler, supplementary material to Damian et al. and Spengler et al. } \references{ Rodica Ioana Damian and Marion Spengler and Andreea Sutu and Brent W. Roberts, 2019, Sixteen going on sixty-six: A longitudinal study of personality stability and change across 50 years Journal of Personality and Social Psychology, 117, (3) 274-695. Marian Spengler and Rodica Ioana Damian and Brent W. Roberts (2018), How you behave in school predicts life success above and beyond family background, broad traits, and cognitive ability Journal of Personality and Social Psychology, 114 (4) 600-636 } \examples{ data(Damian) Spengler.stat #show the basic descriptives of the original data set psych::lowerMat(Spengler[psych::cs(IQ,Parental,Ed.11,OccPres.50), psych::cs(IQ,Parental,Ed.11,OccPres.50)]) psych::setCor(OccPres.50 ~ IQ + Parental + (Ed.11),data=Spengler) #we reduce the number of subjects for faster replication in this example mod <- psych::mediate(OccPres.50 ~ IQ + Parental + (Ed.11),data=Spengler, n.iter=50,n.obs=1000) #for speed summary(mod) } \keyword{datasets} psychTools/man/ability.Rd0000644000176200001440000001231014152204376015121 0ustar liggesusers\name{ability} \alias{ability} \alias{ability.keys} \docType{data} \title{16 ability items scored as correct or incorrect.} \description{ 16 multiple choice ability items 1525 subjects taken from the Synthetic Aperture Personality Assessment (SAPA) web based personality assessment project are saved as \code{\link{iqitems}}. Those data are shown as examples of how to score multiple choice tests and analyses of response alternatives. When scored correct or incorrect, the data are useful for demonstrations of tetrachoric based factor analysis \code{\link{irt.fa}} and finding tetrachoric correlations. } \usage{data(iqitems)} \format{ A data frame with 1525 observations on the following 16 variables. The number following the name is the item number from SAPA. \describe{ \item{\code{reason.4}}{Basic reasoning questions } \item{\code{reason.16}}{Basic reasoning question} \item{\code{reason.17}}{Basic reasoning question} \item{\code{reason.19}}{Basic reasoning question } \item{\code{letter.7}}{In the following alphanumeric series, what letter comes next?} \item{\code{letter.33}}{In the following alphanumeric series, what letter comes next?} \item{\code{letter.34}}{In the following alphanumeric series, what letter comes next} \item{\code{letter.58}}{In the following alphanumeric series, what letter comes next?} \item{\code{matrix.45}}{A matrix reasoning task} \item{\code{matrix.46}}{A matrix reasoning task} \item{\code{matrix.47}}{A matrix reasoning task} \item{\code{matrix.55}}{A matrix reasoning task} \item{\code{rotate.3}}{Spatial Rotation of type 1.2} \item{\code{rotate.4}}{Spatial Rotation of type 1.2} \item{\code{rotate.6}}{Spatial Rotation of type 1.1} \item{\code{rotate.8}}{Spatial Rotation of type 2.3} } } \details{16 items were sampled from 80 items given as part of the SAPA (\url{https://www.sapa-project.org/}) project (Revelle, Wilt and Rosenthal, 2009; Condon and Revelle, 2014) to develop online measures of ability. These 16 items reflect four lower order factors (verbal reasoning, letter series, matrix reasoning, and spatial rotations. These lower level factors all share a higher level factor ('g'). This data set may be used to demonstrate item response functions, \code{\link{tetrachoric}} correlations, or \code{\link{irt.fa}} as well as \code{\link{omega}} estimates of of reliability and hierarchical structure. In addition, the data set is a good example of doing item analysis to examine the empirical response probabilities of each item alternative as a function of the underlying latent trait. When doing this, it appears that two of the matrix reasoning problems do not have monotonically increasing trace lines for the probability correct. At moderately high ability (theta = 1) there is a decrease in the probability correct from theta = 0 and theta = 2. } \source{ The example data set is taken from the Synthetic Aperture Personality Assessment personality and ability test at \url{https://www.sapa-project.org/}. The data were collected with David Condon from 8/08/12 to 8/31/12. Similar data are available from the International Cognitive Ability Resource at \url{https://www.icar-project.org/}. } \references{ Condon, David and Revelle, William, (2014) The International Cognitive Ability Resource: Development and initial validation of a public-domain measure. Intelligence, 43, 52-64. Revelle, William, Dworak, Elizabeth M. and Condon, David (2020) Cognitive ability in everyday life: the utility of open-source measures. Current Directions in Psychological Science, 29, (4) 358-363. Open access at \doi{10.1177/0963721420922178}. Dworak, Elizabeth M., Revelle, William, Doebler, Philip and Condon, David (2021) Using the International Cognitive Ability Resource as an open source tool to explore individual differences in cognitive ability. Personality and Individual Differences, 169. Open access at \doi{10.1016/j.paid.2020.109906}. Revelle, William, Wilt, Joshua, and Rosenthal, Allen (2010) Personality and Cognition: The Personality-Cognition Link. In Gruszka, Alexandra and Matthews, Gerald and Szymura, Blazej (Eds.) Handbook of Individual Differences in Cognition: Attention, Memory and Executive Control, Springer. } \examples{ data(ability) cs<- psych::cs keys <- list(ICAR16=colnames(ability),reasoning = cs(reason.4,reason.16,reason.17,reason.19), letters=cs(letter.7, letter.33,letter.34,letter.58), matrix=cs(matrix.45,matrix.46,matrix.47,matrix.55), rotate=cs(rotate.3,rotate.4,rotate.6,rotate.8)) psych::scoreOverlap(keys,ability) \donttest{ #this next step takes a few seconds to run and demonstrates IRT approaches ability.irt <- psych::irt.fa(ability) ability.scores <- psych::scoreIrt(ability.irt,ability) ability.sub.scores <- psych::scoreIrt.2pl(keys,ability) #demonstrate irt scoring } #It is sometimes asked how to handle missing data when finding scores #this next example compares 3 ways of scoring ability items from icar #Just sum the items #Sum the means for the items #IRT score the items total <- rowSums(ability, na.rm=TRUE) means <- rowMeans(ability, na.rm=TRUE) irt <- psych::scoreIrt(items=ability)[1] df <- data.frame(total, means,irt) psych:: pairs.panels(df) } \keyword{datasets} psychTools/man/iqitems.Rd0000644000176200001440000001241714101334701015135 0ustar liggesusers\name{iqitems} \alias{iqitems} \docType{data} \title{16 multiple choice IQ items} \description{16 multiple choice ability items taken from the Synthetic Aperture Personality Assessment (SAPA) web based personality assessment project. The data from 1525 subjects are included here as a demonstration set for scoring multiple choice inventories and doing basic item statistics. For more information on the development of an open source measure of cognitive ability, consult the readings available at the \url{https://personality-project.org/}. } \usage{data(iqitems)} \format{ A data frame with 1525 observations on the following 16 variables. The number following the name is the item number from SAPA. \describe{ \item{\code{reason.4}}{Basic reasoning questions } \item{\code{reason.16}}{Basic reasoning question} \item{\code{reason.17}}{Basic reasoning question} \item{\code{reason.19}}{Basic reasoning question } \item{\code{letter.7}}{In the following alphanumeric series, what letter comes next?} \item{\code{letter.33}}{In the following alphanumeric series, what letter comes next?} \item{\code{letter.34}}{In the following alphanumeric series, what letter comes next} \item{\code{letter.58}}{In the following alphanumeric series, what letter comes next?} \item{\code{matrix.45}}{A matrix reasoning task} \item{\code{matrix.46}}{A matrix reasoning task} \item{\code{matrix.47}}{A matrix reasoning task} \item{\code{matrix.55}}{A matrix reasoning task} \item{\code{rotate.3}}{Spatial Rotation of type 1.2} \item{\code{rotate.4}}{Spatial Rotation of type 1.2} \item{\code{rotate.6}}{Spatial Rotation of type 1.1} \item{\code{rotate.8}}{Spatial Rotation of type 2.3} } } \details{16 items were sampled from 80 items given as part of the SAPA (\url{https://www.sapa-project.org/}) project (Revelle, Wilt and Rosenthal, 2009; Condon and Revelle, 2014) to develop online measures of ability. These 16 items reflect four lower order factors (verbal reasoning, letter series, matrix reasoning, and spatial rotations. These lower level factors all share a higher level factor ('g'). Similar data are available from the International Cognitive Abiity Resource at \url{https://www.icar-project.org/} . This data set and the associated data set (\code{\link{ability}} based upon scoring these multiple choice items and converting them to correct/incorrect may be used to demonstrate item response functions, \code{\link{tetrachoric}} correlations, or \code{\link{irt.fa}} as well as \code{\link{omega}} estimates of of reliability and hierarchical structure. In addition, the data set is a good example of doing item analysis to examine the empirical response probabilities of each item alternative as a function of the underlying latent trait. When doing this, it appears that two of the matrix reasoning problems do not have monotonically increasing trace lines for the probability correct. At moderately high ability (theta = 1) there is a decrease in the probability correct from theta = 0 and theta = 2. } \source{ The example data set is taken from the Synthetic Aperture Personality Assessment personality and ability test at \url{https://www.sapa-project.org/}. The data were collected with David Condon from 8/08/12 to 8/31/12. } \references{ Condon, David and Revelle, William, (2014) The International Cognitive Ability Resource: Development and initial validation of a public-domain measure. Intelligence, 43, 52-64. Revelle, William, Dworak, Elizabeth M. and Condon, David (2020) Cognitive ability in everyday life: the utility of open-source measures. Current Directions in Psychological Science, 29, (4) 358-363. Open access at \doi{10.1177/0963721420922178}. Dworak, Elizabeth M., Revelle, William, Doebler, Philip and Condon, David (2021) Using the International Cognitive Ability Resource as an open source tool to explore individual differences in cognitive ability. Personality and Individual Differences, 169. Open access at \doi{10.1016/j.paid.2020.109906}. Revelle, W., Wilt, J., and Rosenthal, A. (2010) Individual Differences in Cognition: New Methods for examining the Personality-Cognition Link In Gruszka, A. and Matthews, G. and Szymura, B. (Eds.) Handbook of Individual Differences in Cognition: Attention, Memory and Executive Control, Springer. Revelle, W, Condon, D.M., Wilt, J., French, J.A., Brown, A., and Elleman, L.G. (2016) Web and phone based data collection using planned missing designs. In Fielding, N.G., Lee, R.M. and Blank, G. (Eds). SAGE Handbook of Online Research Methods (2nd Ed), Sage Publcations. } \examples{ \donttest{ data(iqitems) iq.keys <- c(4,4,4, 6, 6,3,4,4, 5,2,2,4, 3,2,6,7) psych::score.multiple.choice(iq.keys,iqitems) #this just gives summary statisics #convert them to true false iq.scrub <- psych::scrub(iqitems,isvalue=0) #first get rid of the zero responses iq.tf <- psych::score.multiple.choice(iq.keys,iq.scrub,score=FALSE) #convert to wrong (0) and correct (1) for analysis psych::describe(iq.tf) #see the ability data set for these analyses #now, for some item analysis iq.irt <- psych::irt.fa(iq.tf) #do a basic irt iq.sc <- psych::scoreIrt(iq.irt,iq.tf) #find the scores op <- par(mfrow=c(4,4)) psych::irt.responses(iq.sc[,1], iq.tf) op <- par(mfrow=c(1,1)) } } \keyword{datasets} psychTools/man/read.clipboard.Rd0000644000176200001440000003074513663336003016350 0ustar liggesusers\name{read.file} \alias{read.clipboard} \alias{read.clipboard.csv} \alias{read.clipboard.tab} \alias{read.clipboard.lower} \alias{read.clipboard.upper} \alias{read.clipboard.fwf} \alias{read.file} \alias{read.file.csv} \alias{write.file} \alias{write.file.csv} \alias{read.https} \title{Shortcuts for reading from the clipboard or a file} \description{Input from a variety of sources may be read. data.frames may be read from files with suffixes of .txt, .text, .TXT, .dat, .DATA,.data, .csv, .rds, rda, .xpt, XPT, or .sav (i.e., data from SPSS sav files may be read as can files saved by SAS using the .xpt option). Data exported by JMP or EXCEL in the csv format are also able to be read. Fixed Width Files saved in .txt mode may be read if the widths parameter is specified. Files saved with writeRDS have suffixes of .rds or Rds, and are read using readRDS. Files associated with objects with suffixes .rda and .Rda are loaded (following a security prompt). The default values for read.spss are adjusted for more standard input from SPSS files. Input from the clipboard is easy but a bit obscure, particularly for Mac users. \code{\link{read.clipboard}} and its variations are just an easier way to do so. Data may be copied to the clipboard from Excel spreadsheets, csv files, or fixed width formatted files and then into a data.frame. Data may also be read from lower (or upper) triangular matrices and filled out to square matrices. Writing text files may be done using \code{\link{write.file}} which will prompt for a file name (if not given) and then write or save to that file depending upon the suffix (text, txt, or csv will call write.table, R, or r will dput, rda, Rda will save, Rds,rds will saveRDS). } \usage{ read.file(file=NULL,header=TRUE,use.value.labels=FALSE,to.data.frame=TRUE,sep=",", quote="\"", widths=NULL,f=NULL, filetype=NULL,...) #for .txt, .text, TXT, .csv, .sav, .xpt, XPT, R, r, Rds, .rds, or .rda, # .Rda, .RData, .Rdata, .dat and .DAT files read.clipboard(header = TRUE, ...) #assumes headers and tab or space delimited read.clipboard.csv(header=TRUE,sep=',',...) #assumes headers and comma delimited read.clipboard.tab(header=TRUE,sep='\t',...) #assumes headers and tab delimited #read in a matrix given the lower off diagonal read.clipboard.lower(diag=TRUE,names=FALSE,...) read.clipboard.upper(diag=TRUE,names=FALSE,...) #read in data using a fixed format width (see read.fwf for instructions) read.clipboard.fwf(header=FALSE,widths=rep(1,10),...) read.https(filename,header=TRUE) read.file.csv(file=NULL,header=TRUE,f=NULL,...) #For output: write.file(x,file=NULL,row.names=FALSE,f=NULL,...) write.file.csv(x,file=NULL,row.names=FALSE,f=NULL,...) } \arguments{ \item{header}{Does the first row have variable labels (generally assumed to be TRUE). } \item{sep}{What is the designated separater between data fields? For typical csv files, this will be a comma, but if commas designate decimals, then a ; can be used to designate different records. } \item{quote}{Specified to } \item{diag}{for upper or lower triangular matrices, is the diagonal specified or not} \item{names}{for read.clipboard.lower or upper, are colnames in the the first column} \item{widths}{how wide are the columns in fixed width input. The default is to read 10 columns of size 1. } \item{filename}{Name or address of remote https file to read.} \item{\dots}{ Other parameters to pass to read } \item{f}{A file name to read from or write to. If omitted, \code{\link{file.choose}} is called to dynamically get the file name.} \item{file}{A file name to read from or write to. (same as f, but perhaps more intuitive) If omitted and if f is omitted,then \code{\link{file.choose}} is called to dynamically get the file name.} \item{x}{The data frame or matrix to write to f} \item{row.names}{Should the output file include the rownames? By default, no.} \item{to.data.frame}{Should the spss input be converted to a data frame?} \item{use.value.labels}{Should the SPSS input values be converted to numeric?} \item{filetype}{If specified the reading will use this term rather than the suffix.} } \details{A typical session of R might involve data stored in text files, generated online, etc. Although it is easy to just read from a file (particularly if using \code{\link{read.file}}, an alternative is to use one's local system to copy from the file to the clipboard and then read from the clipboard \code{\link{read.clipboard}}. This is very convenient (and somewhat more intuitive to the naive user). This is particularly convenient when copying from a text book or article and just moving a section of text into R.) However, copying from a file and then reading the clipboard is hard to automate in a script. Thus, code{\link{read.file}} will read from a file. The \code{\link{read.file}} function combines the \code{\link{file.choose}} and either \code{\link{read.table}}, \code{\link{read.fwf}}, \code{\link{read.spss}} or \code{\link{read.xport}}(from foreign) or \code{\link{load}} or \code{\link{readRDS}} commands. By examining the file suffix, it chooses the appropriate way to read the file. For more complicated file structures, see the foreign package. For even more complicated file structures, see the rio or haven packages. Note that \code{\link{read.file}} assumes by default that the first row has column labels (header =TRUE). If this is not true, then make sure to specify header = FALSE. If the file is fixed width, the assumption is that it does not have a header field. In the unlikely case that a fwf file does have a header, then you probably should try fn <- file.choose() and then my.data <- read.fwf(fn,header=TRUE,widths= widths) Further note: If the file is a .Rda, .rda, etc. file, the read.file command will return the name and location of the file. It will prompt the user to load this file. In this case, it is necessary to either assign the output (the file name) to an object that has a different name than any of the objects in the file, or to call read.file() without any specification. Notice that loading an .Rda file can overwrite existing objects. Thus the warning and the need to do the second step. If the file has no suffix the default action is to quit with a warning. However, if the filetype is specified, it will use that type in the reading (e.g. filetype="txt" will read as text file, even if there is no suffix.) If the file is specified and has a prefix of http:// https:// it will be downloaded and then read. Currently supported input formats are \tabular{ll}{ .sav \tab SPSS.sav files\cr .csv \tab A comma separated file (e.g. from Excel or Qualtrics)\cr .txt \tab A typical text file \cr .TXT \tab A typical text file \cr .text \tab A typical text file \cr .data \tab A data file \cr .dat \tab A data file \cr .rds \tab A R data file \cr .Rds \tab A R data file (created by a write) \cr .Rda \tab A R data structure (created using save) \cr .rda \tab A R data structure (created using save) \cr .RData \tab A R data structure (created using save) \cr .rdata \tab A R data structure (created using save) \cr .R \tab A R data structure created using dput \cr .r \tab A R data structure created using dput \cr .xpt \tab A SAS data file in xport format \cr .XPT \tab A SAS data file in XPORT format \cr } Some data files have an extra ' in the data ( e.g. the NYT covid data base). These files can be read specifying quote "" The foreign function \code{\link{read.spss}} is used to read SPSS .sav files using the most common options. Just as \code{\link{read.spss}} issues various warnings, so does \code{\link{read.file}}. In general, these can be ignored. For more detailed information about using \code{\link{read.spss}}, see the help pages in the foreign package. If you have a file written by JMP, you must first export to a csv or text file. The \code{\link{write.file}} function combines the \code{\link{file.choose}} and either \code{\link{write.table}} or \code{\link{saveRDS}}. By examining the file suffix, it chooses the appropriate way to write. For more complicated file structures, see the foreign package, or the save function in R Base. If no suffix is added, it will write as a .txt file. \code{\link{write.file.csv}} will write in csv format to an arbitrary file name. Currently supported output formats are \tabular{ll}{ .csv \tab A comma separated file (e.g. for reading into Excel)\cr .txt \tab A typical text file \cr .text \tab A typical text file \cr .rds \tab A R data file \cr .Rds \tab A R data file (created by a write) \cr .Rda \tab A R data structure (created using save) \cr .rda \tab A R data structure (created using save) \cr .R \tab A R data structure created using dput \cr .r \tab A R data structure created using dput \cr } Many Excel based files specify missing values as a blank field. When reading from the clipboard, using \code{\link{read.clipboard.tab}} will change these blank fields to NA. Sometimes missing values are specified as "." or "999", or some other values. These can be converted by the read.file command specifying what values are missing (e.g., na ="."). See the example for the reading from the remote mtcars.csv file. \code{\link{read.clipboard}} was based upon a suggestion by Ken Knoblauch to the R-help listserve. If the input file that was copied into the clipboard was an Excel file with blanks for missing data, then read.clipboard.tab() will correctly replace the blanks with NAs. Similarly for a csv file with blank entries, read.clipboard.csv will replace empty fields with NA. \code{\link{read.clipboard.lower}} and \code{\link{read.clipboard.upper}} are adapted from John Fox's read.moments function in the sem package. They will read a lower (or upper) triangular matrix from the clipboard and return a full, symmetric matrix for use by factanal, \code{\link{fa}} , \code{\link{ICLUST}}, \code{\link{pca}}. \code{\link{omega}} , etc. If the diagonal is false, it will be replaced by 1.0s. These two function were added to allow easy reading of examples from various texts and manuscripts with just triangular output. Many articles will report lower triangular matrices with variable labels in the first column. read.clipboard.lower will handle this case. Names must be in the first column if names=TRUE is specified. Other articles will report upper triangular matrices with variable labels in the first row. read.clipboard.upper will handle this. Note that labels in the first column will not work for read.clipboard.upper. The names, if present, must be in the first row. read.clipboard.fwf will read fixed format files from the clipboard. It includes a patch to read.fwf which will not read from the clipboard or from remote file. See read.fwf for documentation of how to specify the widths. } \value{the contents of the file to be read or of the clipboard. } \author{ William Revelle} \examples{ #All of these functions are meant for interactive Input #Because these are dynamic functions, they need to be run interactively and # can not be run as examples. #Thus they are not to be tested by CRAN \donttest{ if(interactive()) { my.data <- read.file() #search the directory for a file and then read it. #return the result into an object #or, if the file is a rda, etc. file my.data <- read.file() #return the path and instructions of how to load # without assigning a value. filesList() #search the system for a particular file and then list all the files in that directory fileCreate() #search for a particular directory and create a file there. write.file(Thurstone) #open the search window, choose a location and name the output file, # write the data file (e.g., Thurstone ) to the file chosen #the example data set from read.delim in the readr package to read a remote csv file my.data <-read.file( "https://github.com/tidyverse/readr/raw/master/inst/extdata/mtcars.csv", na=".") #the na option is used for an example, but is not needed for these data #These functions read from the local clipboard and thus are interactive my.data <- read.clipboard() #space delimited columns my.data <- read.clipboard.csv() # , delimited columns my.data <- read.clipboard.tab() #typical input if copied from a spreadsheet my.data <- read.clipboad(header=FALSE) #data start on line 1 my.matrix <- read.clipboard.lower() } } } \keyword{ multivariate } \keyword{ IO } psychTools/man/peas.Rd0000644000176200001440000000340413464307413014421 0ustar liggesusers\name{peas} \alias{peas} \docType{data} \title{Galton`s Peas} \description{Francis Galton introduced the correlation coefficient with an analysis of the similarities of the parent and child generation of 700 sweet peas. } \usage{data(peas)} \format{ A data frame with 700 observations on the following 2 variables. \describe{ \item{\code{parent}}{The mean diameter of the mother pea for 700 peas} \item{\code{child}}{The mean diameter of the daughter pea for 700 sweet peas} } } \details{Galton's introduction of the correlation coefficient was perhaps the most important contribution to the study of individual differences. This data set allows a graphical analysis of the data set. There are two different graphic examples. One shows the regression lines for both relationships, the other finds the correlation as well. } \source{Stanton, Jeffrey M. (2001) Galton, Pearson, and the Peas: A brief history of linear regression for statistics intstructors, Journal of Statistics Education, 9. (retrieved from the web from https://www.amstat.org/publications/jse/v9n3/stanton.html) reproduces the table from Galton, 1894, Table 2. The data were generated from this table. } \references{Galton, Francis (1877) Typical laws of heredity. paper presented to the weekly evening meeting of the Royal Institution, London. Volume VIII (66) is the first reference to this data set. The data appear in Galton, Francis (1894) Natural Inheritance (5th Edition), New York: MacMillan). } \seealso{The other Galton data sets: \code{\link{heights}}, \code{\link{galton}},\code{\link{cubits}}} \examples{ data(peas) psych::pairs.panels(peas,lm=TRUE,xlim=c(14,22),ylim=c(14,22),main="Galton's Peas") psych::describe(peas) psych::pairs.panels(peas,main="Galton's Peas") } \keyword{datasets} psychTools/man/neo.Rd0000644000176200001440000000744313466571504014267 0ustar liggesusers\name{neo} \Rdversion{1.1} \alias{neo} \docType{data} \title{NEO correlation matrix from the NEO_PI_R manual} \description{The NEO.PI.R is a widely used personality test to assess 5 broad factors (Neuroticism, Extraversion, Openness, Agreeableness and Conscientiousness) with six facet scales for each factor. The correlation matrix of the facets is reported in the NEO.PI.R manual for 1000 subjects. } \usage{data(neo)} \format{ A data frame of a 30 x 30 correlation matrix with the following 30 variables. \describe{ \item{N1}{Anxiety} \item{N2}{AngryHostility} \item{ N3}{Depression } \item{ N4}{Self-Consciousness } \item{ N5}{Impulsiveness } \item{ N6}{Vulnerability } \item{ E1}{Warmth } \item{ E2}{Gregariousness } \item{ E3}{Assertiveness } \item{ E4}{Activity } \item{ E5}{Excitement-Seeking } \item{ E6}{PositiveEmotions } \item{ O1}{Fantasy } \item{ O2}{Aesthetics } \item{ O3}{Feelings } \item{ O4}{Ideas } \item{ O5}{Actions } \item{ O6}{Values } \item{ A1}{Trust } \item{ A2}{Straightforwardness } \item{ A3}{Altruism } \item{ A4}{Compliance } \item{ A5}{Modesty } \item{ A6}{Tender-Mindedness } \item{ C1}{Competence } \item{ C2}{Order } \item{ C3}{Dutifulness } \item{ C4}{AchievementStriving } \item{ C5}{Self-Discipline } \item{ C6}{Deliberation } } } \details{The past thirty years of personality research has led to a general consensus on the identification of major dimensions of personality. Variously known as the ``Big 5" or the ``Five Factor Model", the general solution represents 5 broad domains of personal and interpersonal experience. Neuroticism and Extraversion are thought to reflect sensitivity to negative and positive cues from the environment and the tendency to withdraw or approach. Openness is sometimes labeled as Intellect and reflects an interest in new ideas and experiences. Agreeableness and Conscientiousness reflect tendencies to get along with others and to want to get ahead. The factor structure of the NEO suggests five correlated factors as well as two higher level factors. The NEO was constructed with 6 ``facets" for each of the five broad factors. For a contrasting structure, examine the items of the \code{link{spi}} data set (Condon, 2017). } \source{Costa, Paul T. and McCrae, Robert R. (1992) (NEO PI-R) professional manual. Psychological Assessment Resources, Inc. Odessa, FL. (with permission of the author and the publisher) } \references{ Condon, D. (2017) The SAPA Personality Inventory:An empirically-derived, hierarchically-organized self-report personality assessment model Digman, John M. (1990) Personality structure: Emergence of the five-factor model. Annual Review of Psychology. 41, 417-440. John M. Digman (1997) Higher-order factors of the Big Five. Journal of Personality and Social Psychology, 73, 1246-1256. McCrae, Robert R. and Costa, Paul T., Jr. (1999) A Five-Factor theory of personality. In Pervin, Lawrence A. and John, Oliver P. (eds) Handbook of personality: Theory and research (2nd ed.) 139-153. Guilford Press, New York. N.Y. Revelle, William (1995), Personality processes, Annual Review of Psychology, 46, 295-328. Joshua Wilt and William Revelle (2009) Extraversion and Emotional Reactivity. In Mark Leary and Rick H. Hoyle (eds). Handbook of Individual Differences in Social Behavior. Guilford Press, New York, N.Y. Joshua Wil and William Revelle (2016) Extraversion. In Thomas Widiger (ed) The Oxford Handbook of the Five Factor Model. Oxford University Press. } \examples{ data(neo) n5 <- psych::fa(neo,5) neo.keys <- psych::make.keys(30,list(N=c(1:6),E=c(7:12),O=c(13:18),A=c(19:24),C=c(25:30))) n5p <- psych::target.rot(n5,neo.keys) #show a targeted rotation for simple structure n5p } \keyword{datasets} psychTools/man/GERAS.Rd0000644000176200001440000001164314106537463014342 0ustar liggesusers\name{GERAS} \alias{GERAS} \alias{GERAS.items} \alias{GERAS.keys} \alias{GERAS.dictionary} \alias{GERAS.scales} \docType{data} \title{Data from Gruber et al, 2020, Study 2: Gender Related Attributes Survey } \description{ Gruber et al. (2020) report on the psychometric properties of a multifaceted Gender Related Attributes Survey. Here are the data from their 3 domains (Personality, Cognition and Activities and Interests from their study 2. Eagly and Revelle (in press) include these data in their review of the power of aggregation. The data are included here as demonstrations of the \code{\link{cohen.d}} and \code{\link{scatterHist}} functions in the psych package and may be used to show the power of aggregation. } \usage{data("GERAS") #These other objects are included in the file # data("GERAS.scales") # data("GERAS.dictionary") # data("GERAS.items") # data("GERAS.keys") } \format{ A data frame with 471 observations on the following 51 variables (selected from the original 93) The code numbers are item numbers from the bigger set. \describe{ \item{\code{V15}}{ reckless} \item{\code{V22}}{ willing to take risks} \item{\code{V11}}{ courageous} \item{\code{V6}}{a adventurous} \item{\code{V19}}{ dominant} \item{\code{V14}}{ controlling} \item{\code{V20}}{ boastful} \item{\code{V21}}{ rational} \item{\code{V23}}{ analytical} \item{\code{V9}}{ pragmatic} \item{\code{V44}}{ to find an address for the first time} \item{\code{V45}}{ to find a way again} \item{\code{V46}}{ to understand equations} \item{\code{V50}}{ to follow directions} \item{\code{V51}}{ to understand equations} \item{\code{V53}}{ day-to-day calculations} \item{\code{V48}}{ to write a computer program} \item{\code{V69}}{ paintball} \item{\code{V73}}{ driving go-cart} \item{\code{V71}}{ drinking beer} \item{\code{V68}}{ watching action movies} \item{\code{V75}}{ playing cards (poker)} \item{\code{V72}}{ watching sports on TV} \item{\code{V67}}{ doing certain sports (e.g. soccer, ...)} \item{\code{V74}}{ Gym (weightlifting)} \item{\code{V27}}{ warm-hearted} \item{\code{V28}}{ loving} \item{\code{V29}}{ caring} \item{\code{V26}}{ compassionate} \item{\code{V32}}{ delicate} \item{\code{V30}}{ tender} \item{\code{V24}}{ familiy-oriented} \item{\code{V40}}{ anxious} \item{\code{V39}}{ thin-skinned} \item{\code{V41}}{ careful} \item{\code{V55}}{ to explain foreign words} \item{\code{V58}}{to find the right words to express certain content} \item{\code{V59}}{ synonyms for a word in order to avoid repetitions} \item{\code{V60}}{ to phrase a text} \item{\code{V54}}{ remembering events from your own life} \item{\code{V63}}{ to notice small changes} \item{\code{V57}}{ to remember names and faces} \item{\code{V89}}{ shopping} \item{\code{V92}}{ gossiping} \item{\code{V81}}{ watching a romantic movie} \item{\code{V80}}{ talking on the phone with a friend} \item{\code{V90}}{ yoga} \item{\code{V83}}{ rhythmic gymnastics} \item{\code{V84}}{ going for a walk} \item{\code{V86}}{ dancing} \item{\code{gender}}{gender (M=1 F=2)} } } \details{These 50 items (+ gender) may be formed into scales using the GERAS.keys The first 10 items are Male Personality, the next 10 are Female Personality, then 7 and 7 M and F Cognition, then 8 and 8 M and F Activity items. The Pers, Cog and Act scales are formed from the M-F scales for the three domains. M and F are the composites of the Male and then the Female scales. MF.all is the composite of the M - F scales. See the GERAS.keys object for scoring directions. "M.pers" "F.pers" "M.cog" "F.cog" "M.act" "F.act" "Pers" "Cog" "Act" "M" "F" "MF.all" "gender" } \source{Study 2 data downloaded from the Open Science Framework https://osf.io/42jhr/ Used by kind permission of Freya M. Gruber, Tullia Ortner, and Belinda A. Pletzer.} \references{ Alice H. Eagly and William Revelle, Understanding the Magnitude of Psychological Differences Between Women and Men Requires Seeing the Forest and the Tree. Perspectives in Psychological Science (in press) Gruber, Freya M. and Distlberger, Eva and Scherndl, Thomas and Ortner, Tuulia M. and Pletzer, Belinda (2020) Psychometric properties of the multifaceted Gender-Related Attributes Survey {(GERAS)} European Journal of Psychological Assessment, 36, (4) 612-623. } \examples{ data(GERAS) GERAS.keys #show the keys #show the items from the dictionary psych::lookupFromKeys(GERAS.keys, GERAS.dictionary[,1,drop=FALSE]) #now, use the GERAS.scales to show a scatterHist plot showing univariate d and bivariate # Mahalanobis D. psych::scatterHist(F ~ M + gender, data=GERAS.scales, cex.point=.3,smooth=FALSE, xlab="Masculine Scale",ylab="Feminine Scale",correl=FALSE, d.arrow=TRUE,col=c("red","blue"), bg=c("red","blue"), lwd=4, title="Combined M and F scales",cex.cor=2,cex.arrow=1.25, cex.main=2) } \keyword{datasets} psychTools/man/Schutz.Rd0000644000176200001440000000347213472246613014761 0ustar liggesusers\name{Schutz} \alias{Schutz} \docType{data} \title{ The Schutz correlation matrix example from Shapiro and ten Berge} \description{Shapiro and ten Berge use the Schutz correlation matrix as an example for Minimum Rank Factor Analysis. The Schutz data set is also a nice example of how normal minres or maximum likelihood will lead to a Heywood case, but minrank factoring will not. } \usage{data("Schutz")} \format{ The format is: num [1:9, 1:9] 1 0.8 0.28 0.29 0.41 0.38 0.44 0.4 0.41 0.8 ... - attr(*, "dimnames")=List of 2 ..$ :1] "Word meaning" "Odd Words" "Boots" "Hatchets" ... ..$ : chr [1:9] "V1" "V2" "V3" "V4" ... } \details{ These are 9 cognitive variables of importance mainly because they are used as an example by Shapiro and ten Berge for their paper on Minimum Rank Factor Analysis. The solution from the \code{\link{fa}} function with the fm='minrank' option is very close (but not exactly equal) to their solution. This example is used to show problems with different methods of factoring. Of the various factoring methods, fm = "minres", "uls", or "mle" produce a Heywood case. Minrank, alpha, and pa do not. See the blant data set for another example of differences across methods. } \source{ Richard E. Schutz,(1958) Factorial Validity of the Holzinger-Crowdeer Uni-factor tests. Educational and Psychological Measurement, 48, 873-875. } \references{ Alexander Shapiro and Jos M.F. ten Berge (2002) Statistical inference of minimum rank factor analysis. Psychometrika, 67. 70-94 } \examples{ data(Schutz) psych::corPlot(Schutz,numbers=TRUE,upper=FALSE) \donttest{ f4min <- psych::fa(Schutz,4,fm="minrank") #for an example of minimum rank factor Analysis #compare to f4 <- psych::fa(Schutz,4,fm="mle") #for the maximum likelihood solution which has a Heywood case } } \keyword{datasets} psychTools/man/BFI.adjectives.dictionary.Rd0000644000176200001440000000422213767766605020376 0ustar liggesusers\name{BFI.adjectives.dictionary} \alias{BFI.adjectives.dictionary} \alias{bfi.adjectives.dictionary} \alias{big5.adjectives.dictionary} \docType{data} \title{Dictionary for the 100 Big Five Adjectives } \description{Lew Goldberg organized 100 adjectives to measure 5 factors of personality (The Big5). 500 hundred participants were given these adjectives along with other personality measures. This dictionary allows for easy item labeling of the results. ~ } \usage{data("BFI.adjectives.dictionary") } \format{ A data frame with 100 observations on the following 2 variables. \describe{ \item{\code{numer}}{a character vector of the item label} \item{\code{Item}}{a character vector of the actual adjectives} } } \details{ Keying information for the 100 adjectives: } \source{Data collected at the Personality, Motivation, and Cognition Laboratory, Northwestern University.} \references{Lewis R. Goldberg,(1992) The development of markers for the Big-Five factor structure, Psychological Assessment, 4 (1) 26-42. } \seealso{\code{\link{big5.100.adjectives}} for examples of the data. \code{\link{msqR}} for 3896 participants with scores on five scales of the EPI. \code{\link{affect}} for an example of the use of some of these adjectives in a mood manipulation study.} \examples{ data(BFI.adjectives.dictionary) #this includes the bfi.adjectives.keys bfi.adjectives.keys <- list( Agreeableness = psych::cs(V2, -V11, V14, V15, -V19, -V21, V29, -V31, V32, V48, V55,-V61, -V63, V69, V76, -V78, -V79, -V90, -V94, V99), Conscientiousness = psych::cs(V9, -V10, V13, -V20, V22, -V30, -V37, -V38, -V39, V50, -V51, V53, V56, V57, -V67, V68, V70, V73, -V82, -V95), Extraversion = psych::cs(V1,V5, -V6,V7, V17, V24, V26, -V40,-V45, -V58, -V60,-V65, V71, -V74, -V77, V92, -V96, V97, V98, -V100), Neuroticism= psych::cs(V3, V23, V25, V27,V28, V33,-V36, V42, V46,V47, V49, V52,-V59,V62, V72, V75, -V81,-V83,-V84, -V85), Openness = psych::cs(V4,V8,V12, V16, V18,V34, -V35,V41, V43, V44, V54, -V64,-V66, -V80, -V86, -V87, -V88, -V89, -V91, -V93) ) psych::lookupFromKeys(bfi.adjectives.keys,bfi.adjectives.dictionary,20) } \keyword{datasets} psychTools/man/income.Rd0000644000176200001440000000301513465312012014731 0ustar liggesusers\name{income} \alias{income} \alias{all.income} \docType{data} \title{US family income from US census 2008 } \description{US census data on family income from 2008 } \usage{data(income)} \format{ A data frame with 44 observations on the following 4 variables. \describe{ \item{\code{value}}{lower boundary of the income group} \item{\code{count}}{Number of families within that income group} \item{\code{mean}}{Mean of the category} \item{\code{prop}}{proportion of families} } } \details{The distribution of income is a nice example of a log normal distribution. It is also an interesting example of the power of graphics. It is quite clear when graphing the data that income statistics are bunched to the nearest 5K. That is, there is a clear sawtooth pattern in the data. The all.income set is interpolates intervening values for 100-150K, 150-200K and 200-250K} \source{US Census: Table HINC-06. Income Distribution to $250,000 or More for Households: 2008 https://www.census.gov/hhes/www/cpstables/032009/hhinc/new06_000.htm } \examples{ data(income) with(income[1:40,], plot(mean,prop, main="US family income for 2008",xlab="income", ylab="Proportion of families",xlim=c(0,100000))) with (income[1:40,], points(lowess(mean,prop,f=.3),typ="l")) psych::describe(income) with(all.income, plot(mean,prop, main="US family income for 2008",xlab="income", ylab="Proportion of families",xlim=c(0,250000))) with (all.income[1:50,], points(lowess(mean,prop,f=.25),typ="l")) } \keyword{datasets} psychTools/man/blot.Rd0000644000176200001440000000414313472313363014432 0ustar liggesusers\name{blot} \alias{blot} \docType{data} \title{Bond's Logical Operations Test -- BLOT } \description{35 items for 150 subjects from Bond's Logical Operations Test. A good example of Item Response Theory analysis using the Rasch model. One parameter (Rasch) analysis and two parameter IRT analyses produce somewhat different results. } \usage{data(blot)} \format{ A data frame with 150 observations on 35 variables. The BLOT was developed as a paper and pencil test for children to measure Logical Thinking as discussed by Piaget and Inhelder. } \details{Bond and Fox apply Rasch modeling to a variety of data sets. This one, Bond's Logical Operations Test, is used as an example of Rasch modeling for dichotomous items. In their text (p 56), Bond and Fox report the results using WINSTEPS. Those results are consistent (up to a scaling parameter) with those found by the rasch function in the ltm package. The WINSTEPS seem to produce difficulty estimates with a mean item difficulty of 0, whereas rasch from ltm has a mean difficulty of -1.52. In addition, rasch seems to reverse the signs of the difficulty estimates when reporting the coefficients and is effectively reporting "easiness". However, when using a two parameter model, one of the items (V12) behaves very differently. This data set is useful when comparing 1PL, 2PL and 2PN IRT models. } \source{The data are taken (with kind permission from Trevor Bond) from the webpage https://www.winsteps.com/BF3/bondfox3.htm and read using read.fwf. } \references{ T.G. Bond. BLOT:Bond's Logical Operations Test. Townsville, Australia: James Cook Univer- sity. (Original work published 1976), 1995. T. Bond and C. Fox. (2007) Applying the Rasch model: Fundamental measurement in the human sciences. Lawrence Erlbaum, Mahwah, NJ, US, 2 edition. } \seealso{ See also the \code{\link{irt.fa}} and associated plot functions. } \examples{ data(blot) #ltm is not required by psychTools, but if available, may be run to show a Rasch model #do the same thing with functions in psych blot.fa <- psych::irt.fa(blot) # a 2PN model plot(blot.fa) } \keyword{datasets} psychTools/man/epi.Rd0000644000176200001440000001341614153433055014250 0ustar liggesusers\name{epi} \alias{epi} \alias{epi.dictionary} \alias{epiR} \alias{epi.keys} \docType{data} \title{Eysenck Personality Inventory (EPI) data for 3570 participants} \description{The EPI is and has been a very frequently administered personality test with 57 measuring two broad dimensions, Extraversion-Introversion and Stability-Neuroticism, with an additional Lie scale. Developed by Eysenck and Eysenck, 1964. Eventually replaced with the EPQ which measures three broad dimensions. This data set represents 3570 observations collected in the early 1990s at the Personality, Motivation and Cognition lab at Northwestern. An additional data set (epiR) has test and retest information for 474 participants. The data are included here as demonstration of scale construction and test-retest reliability. } \usage{data(epi) data(epi.dictionary) data(epiR)} \format{ A data frame with 3570 observations on the following 57 variables. \describe{ \item{\code{id}}{The identification number within the study} \item{\code{time}}{First (group testing) or 2nd time (before a lab experiment) for the epiR data set.} \item{\code{study}}{Four lab based studies and their pretest data} \item{\code{V1}}{a numeric vector} \item{\code{V2}}{a numeric vector} \item{\code{V3}}{a numeric vector} \item{\code{V4}}{a numeric vector} \item{\code{V5}}{a numeric vector} \item{\code{V6}}{a numeric vector} \item{\code{V7}}{a numeric vector} \item{\code{V8}}{a numeric vector} \item{\code{V9}}{a numeric vector} \item{\code{V10}}{a numeric vector} \item{\code{V11}}{a numeric vector} \item{\code{V12}}{a numeric vector} \item{\code{V13}}{a numeric vector} \item{\code{V14}}{a numeric vector} \item{\code{V15}}{a numeric vector} \item{\code{V16}}{a numeric vector} \item{\code{V17}}{a numeric vector} \item{\code{V18}}{a numeric vector} \item{\code{V19}}{a numeric vector} \item{\code{V20}}{a numeric vector} \item{\code{V21}}{a numeric vector} \item{\code{V22}}{a numeric vector} \item{\code{V23}}{a numeric vector} \item{\code{V24}}{a numeric vector} \item{\code{V25}}{a numeric vector} \item{\code{V26}}{a numeric vector} \item{\code{V27}}{a numeric vector} \item{\code{V28}}{a numeric vector} \item{\code{V29}}{a numeric vector} \item{\code{V30}}{a numeric vector} \item{\code{V31}}{a numeric vector} \item{\code{V32}}{a numeric vector} \item{\code{V33}}{a numeric vector} \item{\code{V34}}{a numeric vector} \item{\code{V35}}{a numeric vector} \item{\code{V36}}{a numeric vector} \item{\code{V37}}{a numeric vector} \item{\code{V38}}{a numeric vector} \item{\code{V39}}{a numeric vector} \item{\code{V40}}{a numeric vector} \item{\code{V41}}{a numeric vector} \item{\code{V42}}{a numeric vector} \item{\code{V43}}{a numeric vector} \item{\code{V44}}{a numeric vector} \item{\code{V45}}{a numeric vector} \item{\code{V46}}{a numeric vector} \item{\code{V47}}{a numeric vector} \item{\code{V48}}{a numeric vector} \item{\code{V49}}{a numeric vector} \item{\code{V50}}{a numeric vector} \item{\code{V51}}{a numeric vector} \item{\code{V52}}{a numeric vector} \item{\code{V53}}{a numeric vector} \item{\code{V54}}{a numeric vector} \item{\code{V55}}{a numeric vector} \item{\code{V56}}{a numeric vector} \item{\code{V57}}{a numeric vector} } } \details{ The original data were collected in a group testing framework for screening participants for subsequent studies. The participants were enrolled in an introductory psychology class between Fall, 1991 and Spring, 1995. The actual items may be found in the \code{\link{epi.dictionary}}. The structure of the E scale has been shown by Rocklin and Revelle (1981) to have two subcomponents, Impulsivity and Sociability. These were subsequently used by Revelle, Humphreys, Simon and Gilliland (1980) to examine the relationship between personality, caffeine induced arousal, and cognitive performance. The epiR data include the original group testing data and matched data for 474 participants collected several weeks later. This is useful for showing that internal consistency estimates (e.g. \code{\link{alpha}} or \code{\link{omega}}) can be low even though the test is stable across time. For more demonstrations of the distinction between immediate internal consistency and delayed test-retest reliability see the \code{\link{msqR}} and \code{\link{sai}} data sets and \code{\link{testRetest}}. } \source{Data from the PMC laboratory at Northwestern. } \references{ Eysenck, H.J. and Eysenck, S. B.G. (1968). Manual for the Eysenck Personality Inventory.Educational and Industrial Testing Service, San Diego, CA. Revelle, W. and Humphreys, M. S. and Simon, L. and Gilliland, K. (1980) Interactive effect of personality, time of day, and caffeine: A test of the arousal model, Journal of Experimental Psychology General, 109, 1, 1-31, } \examples{ data(epi) epi.keys <- list(E = c("V1", "V3", "V8", "V10", "V13", "V17", "V22", "V25", "V27", "V39", "V44", "V46", "V49", "V53", "V56", "-V5", "-V15", "-V20", "-V29", "-V32", "-V34","-V37", "-V41", "-V51"), N = c( "V2", "V4", "V7", "V9", "V11", "V14", "V16", "V19", "V21", "V23", "V26", "V28", "V31", "V33", "V35", "V38", "V40","V43", "V45", "V47", "V50", "V52","V55", "V57"), L = c("V6", "V24", "V36", "-V12", "-V18", "-V30", "-V42", "-V48", "-V54"), Imp = c( "V1", "V3", "V8", "V10", "V13", "V22", "V39", "-V5", "-V41"), Soc = c( "V17", "V25", "V27", "V44", "V46", "V53", "-V11", "-V15", "-V20", "-V29", "-V32", "-V37", "-V51") ) scores <- psych::scoreItems(epi.keys,epi) psych::keys.lookup(epi.keys[1:3],epi.dictionary) #show the items and keying information #a variety of demonstrations (not run) of test retest reliability versus alpha versus omega E <- psych::selectFromKeys(epi.keys$E) #look at the testRetest help file for more examples } \keyword{datasets} psychTools/man/blant.Rd0000644000176200001440000000432013464173503014570 0ustar liggesusers\name{blant} \alias{blant} \docType{data} \title{A 29 x 29 matrix that produces weird factor analytic results} \description{Normally, min.res factor analysis and maximum likelihood produce very similar results. This data set (from Alexandra Blant) does not. Warnings are given for the min.res solution, the pa solution, but not the old.min nor the mle solution. Included as a test case for the factor analysis function. } \usage{data("blant")} \format{ The format is: num [1:29, 1:29] 1 0.77 0.813 0.68 0.717 ... - attr(*, "dimnames")=List of 2 ..$ : NULL ..$ : chr [1:29] "V1" "V2" "V3" "V4" ... } \details{ This data matrix was sent by Alexandra Blant as an example of a problem with the minres solution in the \code{\link{fa}} function. The default solution, using fm="minres" issues a warning that the solution has improper factor score weights. This is not the case for the fm="old.min" and fm="mle" options, but is for fm="pa", fm="ols". The residuals are indeed smaller for fm="minres" than for fm="old.min" or fm="mle". "old.min" attempts to find the minimum residual but uses the gradient for mle. This was the approach until version 1.7.5 but was changed (see the help page for fa) following extensive communication with Hao Wu. The problem with this matrix is probably that it is almost singular, with some smcs approaching 1 and the smallest three eigenvalues of .006, .004 and .001. This problem matrix was provided by Alexandra Blant. } \source{Alexandra Blant, personal communication} \examples{ data(blant) #compare f5 <- psych::fa(blant,5,rotate="none") #the default minres f5.old <- psych::fa(blant,5, fm="old.min",rotate="none") #old version of minres f5.mle <- psych::fa(blant,5,fm="mle",rotate= "none") #maximum likelihood #compare solutions psych::factor.congruence(list(f5,f5.old,f5.mle)) #compare sums of squared residuals sum(residuals(f5,diag=FALSE)^2,na.rm=TRUE) # 1.355489 sum(residuals(f5.old,diag=FALSE)^2,na.rm=TRUE) # 1.539757 sum(residuals(f5.mle,diag=FALSE)^2,na.rm=TRUE) # 2.402092 #but, when we divide the squared residuals by the original (squared) correlations, we find #a different ordering of fit f5$fit # 0.9748177 f5.old$fit # 0.9752774 f5.mle$fit # 0.9603324 } \keyword{datasets} psychTools/man/cities.Rd0000644000176200001440000000402213501546553014750 0ustar liggesusers\name{cities} \alias{cities} \alias{city.location} \docType{data} \title{Distances between 11 US cities} \description{Airline distances between 11 US cities may be used as an example for multidimensional scaling or cluster analysis. } \usage{data(cities)} \format{ A data frame with 11 observations on the following 11 variables. \describe{ \item{\code{ATL}}{Atlana, Georgia} \item{\code{BOS}}{Boston, Massachusetts} \item{\code{ORD}}{Chicago, Illinois} \item{\code{DCA}}{Washington, District of Columbia} \item{\code{DEN}}{Denver, Colorado} \item{\code{LAX}}{Los Angeles, California} \item{\code{MIA}}{Miami, Florida} \item{\code{JFK}}{New York, New York} \item{\code{SEA}}{Seattle, Washington} \item{\code{SFO}}{San Francisco, California} \item{\code{MSY}}{New Orleans, Lousianna} } } \details{An 11 x11 matrix of distances between major US airports. This is a useful demonstration of multiple dimensional scaling. city.location is a dataframe of longitude and latitude for those cities. Note that the 2 dimensional MDS solution does not perfectly capture the data from these city distances. Boston, New York and Washington, D.C. are located slightly too far west, and Seattle and LA are slightly too far south. } \source{ \url{https://www.timeanddate.com/worldclock/distance.html} } \examples{ data(cities) city.location[,1] <- -city.location[,1] #included in the cities data set plot(city.location, xlab="Dimension 1", ylab="Dimension 2", main ="Multidimensional scaling of US cities") #do the mds city.loc <- cmdscale(cities, k=2) #ask for a 2 dimensional solution round(city.loc,0) city.loc <- -city.loc #flip the axes city.loc <- psych::rescale(city.loc,apply(city.location,2,mean),apply(city.location,2,sd)) points(city.loc,type="n") #add the date point to the map text(city.loc,labels=names(cities)) \dontrun{ #we need the maps package to be available #an overlay map can be added if the package maps is available if(require(maps)) { map("usa",add=TRUE) } } } \keyword{datasets} psychTools/man/heights.Rd0000644000176200001440000000340113464307337015126 0ustar liggesusers\name{heights} \alias{heights} \docType{data} \title{A data.frame of the Galton (1888) height and cubit data set.} \description{Francis Galton introduced the 'co-relation' in 1888 with a paper discussing how to measure the relationship between two variables. His primary example was the relationship between height and forearm length. The data table (\code{\link{cubits}}) is taken from Galton (1888). Unfortunately, there seem to be some errors in the original data table in that the marginal totals do not match the table. The data frame, \code{\link{heights}}, is converted from this table using \code{\link{table2df}}. } \usage{data(heights)} \format{ A data frame with 348 observations on the following 2 variables. \describe{ \item{\code{height}}{Height in inches} \item{\code{cubit}}{Forearm length in inches} } } \details{Sir Francis Galton (1888) published the first demonstration of the correlation coefficient. The regression (or reversion to mediocrity) of the height to the length of the left forearm (a cubit) was found to .8. The original table \code{\link{cubits}} is taken from Galton (1888). There seem to be some errors in the table as published in that the row sums do not agree with the actual row sums. These data are used to create a matrix using \code{\link{table2matrix}} for demonstrations of analysis and displays of the data. } \seealso{ \code{\link[psych]{table2matrix}}, \code{\link[psych]{table2df}}, \code{\link{cubits}}, \code{\link{ellipses}}, \code{\link{galton}} } \source{Galton (1888) } \references{Galton, Francis (1888) Co-relations and their measurement. Proceedings of the Royal Society. London Series,45,135-145, } \examples{ data(heights) psych::ellipses(heights,n=1,main="Galton's co-relation data set") } \keyword{datasets} psychTools/man/spi.Rd0000644000176200001440000001061614055516677014302 0ustar liggesusers\name{spi} \alias{spi} \alias{spi.dictionary} \alias{spi.keys} \docType{data} \title{A sample from the SAPA Personality Inventory including an item dictionary and scoring keys.} \description{The SPI (SAPA Personality Inventory) is a set of 135 items primarily selected from International Personality Item Pool (ipip.ori.org). This is an example data set collected using SAPA procedures the sapa-project.org web site. This data set includes 10 demographic variables as well. The data set with 4000 observations on 145 variables may be used for examples in scale construction and validation, as well as empirical scale construction to predict multiple criteria. } \usage{data("spi") data(spi.dictionary) data(spi.keys) } \format{ A data frame with 4000 observations on the following 145 variables. (The q numbers are the SAPA item numbers). \describe{ \item{\code{age}}{Age in years from 11 -90} \item{\code{sex}}{Reported biological sex (coded by X chromosones => 1=Male, 2 = Female)} \item{\code{health}}{Self rated health 1-5: poor, fair, good, very good, excellent } \item{\code{p1edu}}{Parent 1 education} \item{\code{p2edu}}{Parent 2 education} \item{\code{education}}{Respondents education: less than 12, HS grad, current univ, some univ, associate degree, college degree, in grad/prof, grad/prof degree } \item{\code{wellness}}{Self rated "wellnes" 1-2} \item{\code{exer}}{Frequency of exercise: very rarely, < 1/month, < 1/wk, 1 or 2 times/week, 3-5/wk, > 5 times/week} \item{\code{smoke}}{never, not last year, < 1/month, <1/week, 1-3 days/week, most days, up to 5 x /day, up to 20 x /day, > 20x/day} \item{\code{ER}}{Emergency room visits none, 1x, 2x, 3 or more times} \item{\code{q_253}}{ see the spi.dictionary for these items (q_253} \item{\code{q_1328}}{see the dictionary for all items q_1328)} } } \details{Using the data contributed by about 125,000 visitors to the \url{https://www.SAPA-project.org/} website, David Condon has developed a hierarchical framework for assessing personality at two levels. The higher level has the familiar five factors that have been studied extensively in personality research since the 1980s -- Conscientiousness, Agreeableness, Neuroticism, Openness, and Extraversion. The lower level has 27 factors that are considerably more narrow. These were derived based on administrations of about 700 public-domain IPIP items to 3 large samples. Condon describes these scales as being "empirically-derived" because relatively little theory was used to select the number of factors in the hierarchy and the items in the scale for each factor (to be clear, he means relatively little personality theory though he relied on quite a lot of sampling and statistical theory). You can read all about the procedures used to develop this framework in his book/manual. If you would like to reproduce these analyses, you can download the data files from Dataverse (links are also provided in the manual) and compile this script in R (he used knitR). Instructions are provided in the Preface to the manual. The content of the spi items may be seen by examining the spi.dictionary. Included in the dictionary are the item_id number from the SAPA project, the wording of the item, the source of the item, which Big 5 scale the item marks, and which "Little 27" scale the item marks. This small subset of the data is provided for demonstration purposes. } \source{ https://sapa-project.org/research/SPI/SPIdevelopment.pdf. } \references{Condon, D. (2017) The SAPA Personality Inventory:An empirically-derived, hierarchically-organized self-report personality assessment model An analysis using the spi data set and various tools from the psych package may be found at Revelle, Dworak and Condon, (2021) Exploring the persome: the power of the item in understanding persnality structure. Personality and Individual Differences, 169, 1. Doi: 10.1016/j.paid.2020.109905. } \examples{ data(spi) data(spi.dictionary) psych::bestScales(spi, criteria="health",dictionary=spi.dictionary) sc <- psych::scoreVeryFast(spi.keys,spi) #much faster scoring for just scores sc <- psych::scoreOverlap(spi.keys,spi) #gives the alpha reliabilities and various stats #these are corrected for overlap psych::corPlot(sc$corrected,numbers=TRUE,cex=.4,xlas=2,min.length=6, main="Structure of SPI (Corrected for overlap) disattenuated r above the diagonal)") } \keyword{datasets} psychTools/man/affect.Rd0000644000176200001440000000600413463645166014731 0ustar liggesusers\name{affect} \alias{affect} \alias{maps} \alias{flat} \docType{data} \title{Two data sets of affect and arousal scores as a function of personality and movie conditions } \description{A recurring question in the study of affect is the proper dimensionality and the relationship to various personality dimensions. Here is a data set taken from two studies of mood and arousal using movies to induce affective states. } \usage{data(affect)} \details{These are data from two studies conducted in the Personality, Motivation and Cognition Laboratory at Northwestern University. Both studies used a similar methodology: Collection of pretest data using 5 scales from the Eysenck Personality Inventory and items taken from the Motivational State Questionnaire (see \code{\link{msq}}. In addition, state and trait anxiety measures were given. In the ``maps" study, the Beck Depression Inventory was given also. Then subjects were randomly assigned to one of four movie conditions: 1: Frontline. A documentary about the liberation of the Bergen-Belsen concentration camp. 2: Halloween. A horror film. 3: National Geographic, a nature film about the Serengeti plain. 4: Parenthood. A comedy. Each film clip was shown for 9 minutes. Following this the MSQ was given again. Data from the MSQ were scored for Energetic and Tense Arousal (EA and TA) as well as Positive and Negative Affect (PA and NA). Study flat had 170 participants, study maps had 160. These studies are described in more detail in various publications from the PMC lab. In particular, Revelle and Anderson, 1997 and Rafaeli and Revelle (2006). An analysis of these data has also appeared in Smillie et al. (2012). For a much more complete data set involving film, caffeine, and time of day manipulations, see the \code{\link{msqR}} data set. } \source{Data collected at the Personality, Motivation, and Cognition Laboratory, Northwestern University. } \references{ Revelle, William and Anderson, Kristen Joan (1997) Personality, motivation and cognitive performance: Final report to the Army Research Institute on contract MDA 903-93-K-0008 Rafaeli, Eshkol and Revelle, William (2006), A premature consensus: Are happiness and sadness truly opposite affects? Motivation and Emotion, 30, 1, 1-12. Smillie, Luke D. and Cooper, Andrew and Wilt, Joshua and Revelle, William (2012) Do Extraverts Get More Bang for the Buck? Refining the Affective-Reactivity Hypothesis of Extraversion. Journal of Personality and Social Psychology, 103 (2), 206-326. } \examples{ data(affect) psych::describeBy(affect[-1],group="Film") psych::pairs.panels(affect[14:17],bg=c("red","black","white","blue")[affect$Film],pch=21, main="Affect varies by movies ") psych::errorCircles("EA2","TA2",data=affect,group="Film",labels=c("Sad","Fear","Neutral","Humor") , main="Enegetic and Tense Arousal by Movie condition") psych::errorCircles(x="PA2",y="NA2",data=affect,group="Film",labels=c("Sad","Fear","Neutral"," Humor"), main="Positive and Negative Affect by Movie condition") } \keyword{datasets} psychTools/man/usaf.Rd0000644000176200001440000000475213545450375014444 0ustar liggesusers\name{usaf} \alias{usaf} \alias{USAF} \docType{data} \title{17 anthropometric measures from the USAF showing a general factor} \description{The correlation matrix of 17 anthropometric measures from the United States Air Force survey of 2420 airmen. The data are taken from the Anthropometry package and included here as a demonstration of a hierarchical factor structure suitable for analysis by the \code{\link{omega}} or \code{\link{omegaSem}}. } \usage{data("USAF")} \format{ The format is: num [1:17, 1:17] 1 0.1148 -0.0309 -0.028 -0.0908 ... - attr(*, "dimnames")=List of 2 ..$ : chr [1:17] "age" "weight" "grip" "height" ... ..$ : chr [1:17] "age" "weight" "grip" "height" ... } \details{ The original data were collected by the USAF and reported in Churchill et al, 1977. They are included as a data file of 2420 participants and 202 variables (the first being an id) in the Anthropometry package. The list of variable names may be found in Churchill et al, on pages 96-99. The three (correlated) factor structure shows a clear height, bulk, and head size structure with an overall general factor (g) which may be interpreted as body size. The variables included (and their variable numbers in Antropometry) are: \tabular{ll}{ age \tab V1\cr weight \tab V2 \cr grip strength \tab V12 \cr height (stature) \tab V13 \cr leg length \tab V26 \cr knee height \tab V37 \cr upper arm \tab V42 \cr thumb tip reach \tab V47 \cr in sleeve \tab V49 \cr chest breadth \tab V52\cr hip breadth \tab V55 \cr waist circumference \tab V71 \cr thigh circumference \tab V97 \cr scye circumference \tab V103\cr head circumference \tab V141 \cr bitragion coronal \tab V145 \cr head length \tab V150 \cr glabella to wall \tab V181 \cr external canthus to wall \tab V183 \cr } Note that these numbers are equivalant to the numbers in Churchill et al. The numbers in Anthropometry are these + 1. } \source{ Guillermo Vinue, Anthropometry: An R Package for Analysis of Anthropometric Data, Journal of Statistical Software, (2017), 77, 6.} \references{ Edmund Churchill, Thomas Churchill, Paul Kikta (1977) The AMRL anthropmetric data bank library, volumes I-V. (Technical report AMRL-TR-77-1) ) https://apps.dtic.mil/dtic/tr/fulltext/u2/a047314.pdf Guillermo Vinue, Anthropometry: An R Package for Analysis of Anthropometric Data, Journal of Statistical Software, (2017), 77, 6. } \examples{ data(USAF) psych::corPlot(USAF,xlas=3) psych::omega(USAF[c(4:8,10:19),c(4:8,10:19)]) #just the size variables } \keyword{datasets} psychTools/man/dfOrder.Rd0000644000176200001440000000351513375423044015061 0ustar liggesusers\name{dfOrder} \alias{dfOrder} \title{Sort (order) a dataframe or matrix by multiple columns } \description{Although \code{\link{order}} will order a vector, and it is possible to order several columns of a data.frame by specifying each column individually in the call to order, \code{\link{dfOrder}} will order a dataframe or matrix by as many columns as desired. } \usage{ dfOrder(object, columns,absolute=FALSE,ascending=TRUE) } \arguments{ \item{object}{The data.frame to be sorted} \item{columns}{Column numbers to use for sorting. If positive, then they will be sorted in increasing order. If negative, then in decreasing order} \item{absolute}{If TRUE, then sort the absolute values} \item{ascending}{By default, order from smallest to largest.} } \details{ This is just a simple helper function to reorder data.frames. Originally developed to organize IRT output from the ltm package. It is a basic add on to the order function. (Completely rewritten for version 1.8.1.) } \value{ The original data frame is now in sorted order. } \author{William Revelle } \seealso{ Other useful file manipulation functions include \code{\link{read.file}} to read in data from a file or \code{\link{read.clipboard}} from the clipboard, \code{\link{fileScan}}, \code{\link{filesList}}, \code{\link{filesInfo}}, and \code{\link{fileCreate}} \code{\link{dfOrder}} code is used in the \code{\link{test.irt}} function to combine ltm and \code{\link{sim.irt}} output. } \examples{ set.seed(42) x <- matrix(sample(1:4,64,replace=TRUE),ncol=4) dfOrder(x) # sort by all columns dfOrder(x,c(1,4)) #sort by the first and 4th column x.df <- data.frame(x) dfOrder(x.df,c(1,-2)) #sort by the first in increasing order, #the second in decreasing order } \keyword{manip }% use one of RShowDoc("KEYWORDS") \keyword{utilities }% __ONLY ONE__ keyword per line psychTools/man/vegetables.Rd0000644000176200001440000000564713714653347015635 0ustar liggesusers\name{vegetables} \alias{vegetables} \alias{veg} \docType{data} \title{ Paired comparison of preferences for 9 vegetables} \description{A classic data set for demonstrating Thurstonian scaling is the preference matrix of 9 vegetables from Guilford (1954). Used by Guiford, Nunnally, and Nunally and Bernstein, this data set allows for examples of basic scaling techniques. } \usage{data(vegetables)} \format{ A data frame with 9 choices on the following 9 vegetables. The values reflect the perecentage of times where the column entry was preferred over the row entry. \describe{ \item{\code{Turn}}{Turnips} \item{\code{Cab}}{Cabbage} \item{\code{Beet}}{Beets} \item{\code{Asp}}{Asparagus} \item{\code{Car}}{Carrots} \item{\code{Spin}}{Spinach} \item{\code{S.Beans}}{String Beans} \item{\code{Peas}}{Peas} \item{\code{Corn}}{Corn} } } \details{Louis L. Thurstone was a pioneer in psychometric theory and measurement of attitudes, interests, and abilities. Among his many contributions was a systematic analysis of the process of comparative judgment (thurstone, 1927). He considered the case of asking subjects to successively compare pairs of objects. If the same subject does this repeatedly, or if subjects act as random replicates of each other, their judgments can be thought of as sampled from a normal distribution of underlying (latent) scale scores for each object, Thurstone proposed that the comparison between the value of two objects could be represented as representing the differences of the average value for each object compared to the standard deviation of the differences between objects. The basic model is that each item has a normal distribution of response strength and that choice represents the stronger of the two response strengths. A justification for the normality assumption is that each decision represents the sum of many independent inputs and thus, through the central limit theorem, is normally distributed. Thurstone considered five different sets of assumptions about the equality and independence of the variances for each item (Thurston, 1927). Torgerson expanded this analysis slightly by considering three classes of data collection (with individuals, between individuals and mixes of within and between) crossed with three sets of assumptions (equal covariance of decision process, equal correlations and small differences in variance, equal variances). This vegetable data set is used by Guilford and by Nunnally to demonstrate Thurstonian scaling. } \source{ Guilford, J.P. (1954) Psychometric Methods. McGraw-Hill, New York. } \references{ Nunnally, J. C. (1967). Psychometric theory., McGraw-Hill, New York.\cr Revelle, W. An introduction to psychometric theory with applications in R. (in preparation), Springer. \url{https://personality-project.org/r/book/} } \seealso{ \code{\link[psych]{thurstone}}} \examples{ data(vegetables) psych::thurstone(veg) } \keyword{datasets} psychTools/man/galton.Rd0000644000176200001440000000357513464307747015000 0ustar liggesusers\name{galton} \alias{galton} \docType{data} \title{Galton's Mid parent child height data} \description{Two of the earliest examples of the correlation coefficient were Francis Galton's data sets on the relationship between mid parent and child height and the similarity of parent generation peas with child peas. This is the data set for the Galton height. } \usage{data(galton)} \format{ A data frame with 928 observations on the following 2 variables. \describe{ \item{\code{parent}}{Mid Parent heights (in inches) } \item{\code{child}}{Child Height} } } \details{Female heights were adjusted by 1.08 to compensate for sex differences. (This was done in the original data set) } \source{This is just the galton data set from UsingR, slightly rearranged. } \references{Stigler, S. M. (1999). Statistics on the Table: The History of Statistical Concepts and Methods. Harvard University Press. Galton, F. (1886). Regression towards mediocrity in hereditary stature. Journal of the Anthropological Institute of Great Britain and Ireland, 15:246-263. Galton, F. (1869). Hereditary Genius: An Inquiry into its Laws and Consequences. London: Macmillan. Wachsmuth, A.W., Wilkinson L., Dallal G.E. (2003). Galton's bend: A previously undiscovered nonlinearity in Galton's family stature regression data. The American Statistician, 57, 190-192. } \seealso{The other Galton data sets: \code{\link{heights}}, \code{\link{peas}},\code{\link{cubits}}} \examples{ data(galton) psych::describe(galton) #show the scatter plot and the lowess fit psych::pairs.panels(galton,main="Galton's Parent child heights") #but this makes the regression lines look the same psych::pairs.panels(galton,lm=TRUE,main="Galton's Parent child heights") #better is to scale them psych::pairs.panels(galton,lm=TRUE,xlim=c(62,74),ylim=c(62,74), main="Galton's Parent child heights") } \keyword{datasets} psychTools/man/epi.bfi.Rd0000644000176200001440000000350113463322304014776 0ustar liggesusers\name{epi.bfi} \alias{epi.bfi} \docType{data} \title{13 personality scales from the Eysenck Personality Inventory and Big 5 inventory} \description{A small data set of 5 scales from the Eysenck Personality Inventory, 5 from a Big 5 inventory, a Beck Depression Inventory, and State and Trait Anxiety measures. Used for demonstrations of correlations, regressions, graphic displays. } \usage{data(epi.bfi)} \format{ A data frame with 231 observations on the following 13 variables. \describe{ \item{\code{epiE}}{EPI Extraversion } \item{\code{epiS}}{EPI Sociability (a subset of Extraversion items} \item{\code{epiImp}}{EPI Impulsivity (a subset of Extraversion items} \item{\code{epilie}}{EPI Lie scale} \item{\code{epiNeur}}{EPI neuroticism} \item{\code{bfagree}}{Big 5 inventory (from the IPIP) measure of Agreeableness} \item{\code{bfcon}}{Big 5 Conscientiousness} \item{\code{bfext}}{Big 5 Extraversion} \item{\code{bfneur}}{Big 5 Neuroticism} \item{\code{bfopen}}{Big 5 Openness} \item{\code{bdi}}{Beck Depression scale} \item{\code{traitanx}}{Trait Anxiety} \item{\code{stateanx}}{State Anxiety} } } \details{Self report personality scales tend to measure the ``Giant 2" of Extraversion and Neuroticism or the ``Big 5" of Extraversion, Neuroticism, Agreeableness, Conscientiousness, and Openness. Here is a small data set from Northwestern University undergraduates with scores on the Eysenck Personality Inventory (EPI) and a Big 5 inventory taken from the International Personality Item Pool. } \source{Data were collected at the Personality, Motivation, and Cognition Lab (PMCLab) at Northwestern by William Revelle) } \references{\url{https://personality-project.org/pmc.html} } \examples{ data(epi.bfi) psych::pairs.panels(epi.bfi[,1:5]) psych::describe(epi.bfi) } \keyword{datasets} psychTools/man/fileUtilities.Rd0000644000176200001440000000614313470536352016312 0ustar liggesusers\name{Utility} \alias{fileScan} \alias{fileCreate} \alias{filesList} \alias{filesInfo} \alias{Utility} \title{Useful utility functions for file/directory exploration and manipulation.} \description{ Wrappers for dirname, file.choose, readLines. file.create, file.path to be called directly for listing directories, creating files, showing the files in a directory, and listing the content of files in a directory. \code{\link{fileCreate}} gives the functionality of \code{\link{file.choose}}(new=TRUE). \code{\link{filesList}} combines file.choose, dirname, and list.files to show the files in a directory, \code{\link{fileScan}} extends this and then returns the first few lines of each readable file } \usage{ fileScan(f = NULL, nlines = 3, max = NULL, from = 1, filter = NULL) filesList(f=NULL) filesInfo(f=NULL,max=NULL) fileCreate(newName="new.file") } \arguments{ \item{f}{File path to use as base path (will use file.choose() if missing. If f is a directory, will list the files in that directory, if f is a file, will find the directory for that file and then list all of those files.) } \item{nlines}{How many lines to display} \item{max}{maximum number of files to display} \item{from}{First file (number) to display} \item{filter}{Just display files with "filter" in the name} \item{newName}{The name of the file to be created.} } \details{ Just a collection of simple wrappers to powerful core R functions. Allows the user more direct control of what directory to list, to create a file, or to display the content of files. The functions called include \code{\link{file.choose}}, \code{\link{file.path}}, \code{\link{file.info}},\code{\link{file.create}}, \code{\link{dirname}}, and \code{\link{dir.exists}}. All of these are very powerful functions, but not easy to call interactively. \code{\link{fileCreate}} will ask to locate a file using file.choose, set the directory to that location, and then prompt to create a file with the new.name. This is a workaround for file.choose(new=TRUE) which only works for Macs not using R.studio. \code{\link{filesInfo}} will interactively search for a file and then list the information (size, date, ownership) of all the files in that directory. \code{\link{filesList}} will interactively search for a file and then list all the files in same directory. } \author{William Revelle} \note{Work arounds for core-R functions for interactive file manipulation } \seealso{\code{\link{read.file}} to read in data from a file or \code{\link{read.clipboard}} from the clipboard. \code{\link{dfOrder}} to sort data.frames. } \examples{ \donttest{ if(interactive()) { #all of these require interactive input and thus are not given as examples fileCreate("my.new.file.txt") filesList() #show the items in the directory where a file is displayed fileScan() #show the content of the files in a directory #or, if you have a file in mind f <- file.choose() #go find it filesList(f) fileScan(f) } } } % Add one or more standard keywords, see file 'KEYWORDS' in the % R documentation directory. \keyword{IO }% use one of RShowDoc("KEYWORDS") \keyword{file}% __ONLY ONE__ keyword per line psychTools/man/cushny.Rd0000644000176200001440000000412613464306355015010 0ustar liggesusers\name{cushny} \alias{cushny} \docType{data} \title{ A data set from Cushny and Peebles (1905) on the effect of three drugs on hours of sleep, used by Student (1908) } \description{The classic data set used by Gossett (publishing as Student) for the introduction of the t-test. The design was a within subjects study with hours of sleep in a control condition compared to those in 3 drug conditions. Drug1 was 06mg of L Hscyamine, Drug 2L and Drug2R were said to be .6 mg of Left and Right isomers of Hyoscine. As discussed by Zabell (2008) these were not optical isomers. The detal1, delta2L and delta2R are changes from the baseline control. } \usage{data(cushny)} \format{ A data frame with 10 observations on the following 7 variables. \describe{ \item{\code{Control}}{Hours of sleep in a control condition} \item{\code{drug1}}{Hours of sleep in Drug condition 1} \item{\code{drug2L}}{Hours of sleep in Drug condition 2} \item{\code{drug2R}}{Hours of sleep in Drug condition 3 (an isomer of the drug in condition 2} \item{\code{delta1}}{Change from control, drug 1} \item{\code{delta2L}}{Change from control, drug 2L} \item{\code{delta2R}}{Change from control, drug 2R} } } \details{The original analysis by Student is used as an example for the t-test function, both as a paired t-test and a two group t-test. The data are also useful for a repeated measures analysis of variance. } \source{Cushny, A.R. and Peebles, A.R. (1905) The action of optical isomers: II hyoscines. The Journal of Physiology 32, 501-510. Student (1908) The probable error of the mean. Biometrika, 6 (1) , 1-25. } \references{See also the data set sleep and the examples for the t.test S. L. Zabell. On Student's 1908 Article "The Probable Error of a Mean" Journal of the American Statistical Association, Vol. 103, No. 481 (Mar., 2008), pp. 1- 20} \examples{ data(cushny) with(cushny, t.test(drug1,drug2L,paired=TRUE)) #within subjects psych::error.bars(cushny[1:4],within=TRUE,ylab="Hours of sleep",xlab="Drug condition", main="95\% confidence of within subject effects") } \keyword{datasets} psychTools/man/msq.Rd0000644000176200001440000003374213501503155014272 0ustar liggesusers\name{msq} \Rdversion{1.1} \alias{msq} \docType{data} \title{75 mood items from the Motivational State Questionnaire for 3896 participants} \description{Emotions may be described either as discrete emotions or in dimensional terms. The Motivational State Questionnaire (MSQ) was developed to study emotions in laboratory and field settings. The data can be well described in terms of a two dimensional solution of energy vs tiredness and tension versus calmness. Additional items include what time of day the data were collected and a few personality questionnaire scores. } \usage{data(msq)} \format{ A data frame with 3896 observations on the following 92 variables. \describe{ \item{\code{active}}{a numeric vector} \item{\code{afraid}}{a numeric vector} \item{\code{alert}}{a numeric vector} \item{\code{angry}}{a numeric vector} \item{\code{anxious}}{a numeric vector} \item{\code{aroused}}{a numeric vector} \item{\code{ashamed}}{a numeric vector} \item{\code{astonished}}{a numeric vector} \item{\code{at.ease}}{a numeric vector} \item{\code{at.rest}}{a numeric vector} \item{\code{attentive}}{a numeric vector} \item{\code{blue}}{a numeric vector} \item{\code{bored}}{a numeric vector} \item{\code{calm}}{a numeric vector} \item{\code{cheerful}}{a numeric vector} \item{\code{clutched.up}}{a numeric vector} \item{\code{confident}}{a numeric vector} \item{\code{content}}{a numeric vector} \item{\code{delighted}}{a numeric vector} \item{\code{depressed}}{a numeric vector} \item{\code{determined}}{a numeric vector} \item{\code{distressed}}{a numeric vector} \item{\code{drowsy}}{a numeric vector} \item{\code{dull}}{a numeric vector} \item{\code{elated}}{a numeric vector} \item{\code{energetic}}{a numeric vector} \item{\code{enthusiastic}}{a numeric vector} \item{\code{excited}}{a numeric vector} \item{\code{fearful}}{a numeric vector} \item{\code{frustrated}}{a numeric vector} \item{\code{full.of.pep}}{a numeric vector} \item{\code{gloomy}}{a numeric vector} \item{\code{grouchy}}{a numeric vector} \item{\code{guilty}}{a numeric vector} \item{\code{happy}}{a numeric vector} \item{\code{hostile}}{a numeric vector} \item{\code{idle}}{a numeric vector} \item{\code{inactive}}{a numeric vector} \item{\code{inspired}}{a numeric vector} \item{\code{intense}}{a numeric vector} \item{\code{interested}}{a numeric vector} \item{\code{irritable}}{a numeric vector} \item{\code{jittery}}{a numeric vector} \item{\code{lively}}{a numeric vector} \item{\code{lonely}}{a numeric vector} \item{\code{nervous}}{a numeric vector} \item{\code{placid}}{a numeric vector} \item{\code{pleased}}{a numeric vector} \item{\code{proud}}{a numeric vector} \item{\code{quiescent}}{a numeric vector} \item{\code{quiet}}{a numeric vector} \item{\code{relaxed}}{a numeric vector} \item{\code{sad}}{a numeric vector} \item{\code{satisfied}}{a numeric vector} \item{\code{scared}}{a numeric vector} \item{\code{serene}}{a numeric vector} \item{\code{sleepy}}{a numeric vector} \item{\code{sluggish}}{a numeric vector} \item{\code{sociable}}{a numeric vector} \item{\code{sorry}}{a numeric vector} \item{\code{still}}{a numeric vector} \item{\code{strong}}{a numeric vector} \item{\code{surprised}}{a numeric vector} \item{\code{tense}}{a numeric vector} \item{\code{tired}}{a numeric vector} \item{\code{tranquil}}{a numeric vector} \item{\code{unhappy}}{a numeric vector} \item{\code{upset}}{a numeric vector} \item{\code{vigorous}}{a numeric vector} \item{\code{wakeful}}{a numeric vector} \item{\code{warmhearted}}{a numeric vector} \item{\code{wide.awake}}{a numeric vector} \item{\code{alone}}{a numeric vector} \item{\code{kindly}}{a numeric vector} \item{\code{scornful}}{a numeric vector} \item{\code{EA}}{Thayer's Energetic Arousal Scale} \item{\code{TA}}{Thayer's Tense Arousal Scale} \item{\code{PA}}{Positive Affect scale} \item{\code{NegAff}}{Negative Affect scale} \item{\code{Extraversion}}{Extraversion from the Eysenck Personality Inventory} \item{\code{Neuroticism}}{Neuroticism from the Eysenck Personality Inventory} \item{\code{Lie}}{Lie from the EPI} \item{\code{Sociability}}{The sociability subset of the Extraversion Scale} \item{\code{Impulsivity}}{The impulsivity subset of the Extraversions Scale} \item{\code{MSQ_Time}}{Time of day the data were collected} \item{\code{MSQ_Round}}{Rounded time of day} \item{\code{TOD}}{a numeric vector} \item{\code{TOD24}}{a numeric vector} \item{\code{ID}}{subject ID} \item{\code{condition}}{What was the experimental condition after the msq was given} \item{\code{scale}}{a factor with levels \code{msq} \code{r} original or revised msq} \item{\code{exper}}{Which study were the data collected: a factor with levels \code{AGES} \code{BING} \code{BORN} \code{CART} \code{CITY} \code{COPE} \code{EMIT} \code{FAST} \code{Fern} \code{FILM} \code{FLAT} \code{Gray} \code{imps} \code{item} \code{knob} \code{MAPS} \code{mite} \code{pat-1} \code{pat-2} \code{PATS} \code{post} \code{RAFT} \code{Rim.1} \code{Rim.2} \code{rob-1} \code{rob-2} \code{ROG1} \code{ROG2} \code{SALT} \code{sam-1} \code{sam-2} \code{SAVE/PATS} \code{sett} \code{swam} \code{swam-2} \code{TIME} \code{VALE-1} \code{VALE-2} \code{VIEW}} } } \details{The Motivational States Questionnaire (MSQ) is composed of 72 items, which represent the full affective space (Revelle & Anderson, 1998). The MSQ consists of 20 items taken from the Activation-Deactivation Adjective Check List (Thayer, 1986), 18 from the Positive and Negative Affect Schedule (PANAS, Watson, Clark, & Tellegen, 1988) along with the items used by Larsen and Diener (1992). The response format was a four-point scale that corresponds to Russell and Carroll's (1999) "ambiguous--likely-unipolar format" and that asks the respondents to indicate their current standing (``at this moment") with the following rating scale:\cr 0----------------1----------------2----------------3 \cr Not at all A little Moderately Very much \cr The original version of the MSQ included 70 items. Intermediate analyses (done with 1840 subjects) demonstrated a concentration of items in some sections of the two dimensional space, and a paucity of items in others. To begin correcting this, 3 items from redundantly measured sections (alone, kindly, scornful) were removed, and 5 new ones (anxious, cheerful, idle, inactive, and tranquil) were added. Thus, the correlation matrix is missing the correlations between items anxious, cheerful, idle, inactive, and tranquil with alone, kindly, and scornful. Procedure. The data were collected over nine years, as part of a series of studies examining the effects of personality and situational factors on motivational state and subsequent cognitive performance. In each of 38 studies, prior to any manipulation of motivational state, participants signed a consent form and filled out the MSQ. (The procedures of the individual studies are irrelevant to this data set and could not affect the responses to the MSQ, since this instrument was completed before any further instructions or tasks). Some MSQ post test (after manipulations) is available in \code{\link{affect}}. The EA and TA scales are from Thayer, the PA and NA scales are from Watson et al. (1988). Scales and items: Energetic Arousal: active, energetic, vigorous, wakeful, wide.awake, full.of.pep, lively, -sleepy, -tired, - drowsy (ADACL) Tense Arousal: Intense, Jittery, fearful, tense, clutched up, -quiet, -still, - placid, - calm, -at rest (ADACL) Positive Affect: active, alert, attentive, determined, enthusiastic, excited, inspired, interested, proud, strong (PANAS) Negative Affect: afraid, ashamed, distressed, guilty, hostile, irritable , jittery, nervous, scared, upset (PANAS) The PA and NA scales can in turn can be thought of as having subscales: (See the PANAS-X) Fear: afraid, scared, nervous, jittery (not included frightened, shaky) Hostility: angry, hostile, irritable, (not included: scornful, disgusted, loathing guilt: ashamed, guilty, (not included: blameworthy, angry at self, disgusted with self, dissatisfied with self) sadness: alone, blue, lonely, sad, (not included: downhearted) joviality: cheerful, delighted, energetic, enthusiastic, excited, happy, lively, (not included: joyful) self-assurance: proud, strong, confident, (not included: bold, daring, fearless ) attentiveness: alert, attentive, determined (not included: concentrating) The next set of circumplex scales were taken (I think) from Larsen and Diener (1992). High activation: active, aroused, surprised, intense, astonished Activated PA: elated, excited, enthusiastic, lively Unactivated NA : calm, serene, relaxed, at rest, content, at ease PA: happy, warmhearted, pleased, cheerful, delighted Low Activation: quiet, inactive, idle, still, tranquil Unactivated PA: dull, bored, sluggish, tired, drowsy NA: sad, blue, unhappy, gloomy, grouchy Activated NA: jittery, anxious, nervous, fearful, distressed. Keys for these separate scales are shown in the examples. In addition to the MSQ, there are 5 scales from the Eysenck Personality Inventory (Extraversion, Impulsivity, Sociability, Neuroticism, Lie). The Imp and Soc are subsets of the the total extraversion scale. } \source{Data collected at the Personality, Motivation, and Cognition Laboratory, Northwestern University. } \references{ Larsen, R. J., & Diener, E. (1992). Promises and problems with the circumplex model of emotion. In M. S. Clark (Ed.), Review of personality and social psychology, No. 13. Emotion (pp. 25-59). Thousand Oaks, CA, US: Sage Publications, Inc. Rafaeli, Eshkol and Revelle, William (2006), A premature consensus: Are happiness and sadness truly opposite affects? Motivation and Emotion, 30, 1, 1-12. Revelle, W. and Anderson, K.J. (1998) Personality, motivation and cognitive performance: Final report to the Army Research Institute on contract MDA 903-93-K-0008. (\url{https://www.personality-project.org/revelle/publications/ra.ari.98.pdf}). Thayer, R.E. (1989) The biopsychology of mood and arousal. Oxford University Press. New York, NY. Watson,D., Clark, L.A. and Tellegen, A. (1988) Development and validation of brief measures of positive and negative affect: The PANAS scales. Journal of Personality and Social Psychology, 54(6):1063-1070. } \seealso{\code{\link{msqR}} for a larger data set with repeated measures for 3032 participants measured at least once, 2753 measured twice, 446 three times and 181 four times. \code{\link{affect}} for an example of the use of some of these adjectives in a mood manipulation study. \code{\link{make.keys}}, \code{\link{scoreItems}} and \code{\link{scoreOverlap}} for instructions on how to score multiple scales with and without item overlap. Also see \code{\link{fa}} and \code{\link{fa.extension}} for instructions on how to do factor analyses or factor extension. } \examples{ data(msq) \donttest{ #in in the interests of time #basic descriptive statistics psych::describe(msq) } #score them for 20 short scales -- note that these have item overlap #The first 2 are from Thayer #The next 2 are classic positive and negative affect #The next 9 are circumplex scales #the last 7 are msq estimates of PANASX scales (missing some items) keys.list <- list( EA = c("active", "energetic", "vigorous", "wakeful", "wide.awake", "full.of.pep", "lively", "-sleepy", "-tired", "-drowsy"), TA =c("intense", "jittery", "fearful", "tense", "clutched.up", "-quiet", "-still", "-placid", "-calm", "-at.rest") , PA =c("active", "excited", "strong", "inspired", "determined", "attentive", "interested", "enthusiastic", "proud", "alert"), NAf =c("jittery", "nervous", "scared", "afraid", "guilty", "ashamed", "distressed", "upset", "hostile", "irritable" ), HAct = c("active", "aroused", "surprised", "intense", "astonished"), aPA = c("elated", "excited", "enthusiastic", "lively"), uNA = c("calm", "serene", "relaxed", "at.rest", "content", "at.ease"), pa = c("happy", "warmhearted", "pleased", "cheerful", "delighted" ), LAct = c("quiet", "inactive", "idle", "still", "tranquil"), uPA =c( "dull", "bored", "sluggish", "tired", "drowsy"), naf = c( "sad", "blue", "unhappy", "gloomy", "grouchy"), aNA = c("jittery", "anxious", "nervous", "fearful", "distressed"), Fear = c("afraid" , "scared" , "nervous" , "jittery" ) , Hostility = c("angry" , "hostile", "irritable", "scornful" ), Guilt = c("guilty" , "ashamed" ), Sadness = c( "sad" , "blue" , "lonely", "alone" ), Joviality =c("happy","delighted", "cheerful", "excited", "enthusiastic", "lively", "energetic"), Self.Assurance=c( "proud","strong" , "confident" , "-fearful" ), Attentiveness = c("alert" , "determined" , "attentive" ) #, acquiscence = c("sleepy" , "wakeful" , "relaxed","tense") #dropped because it has a negative alpha and throws warnings ) msq.scores <- psych::scoreItems(keys.list,msq) #show a circumplex structure for the non-overlapping items fcirc <- psych::fa(msq.scores$scores[,5:12],2) psych::fa.plot(fcirc,labels=colnames(msq.scores$scores)[5:12]) \donttest{#now, find the correlations corrected for item overlap msq.overlap <- psych::scoreOverlap(keys.list,msq) #a warning is thrown by smc because of some NAs in the matrix f2 <- psych::fa(msq.overlap$cor,2) psych::fa.plot(f2,labels=colnames(msq.overlap$cor), title="2 dimensions of affect, corrected for overlap") #extend this solution to EA/TA NA/PA space fe <- psych::fa.extension(cor(msq.scores$scores[,5:12],msq.scores$scores[,1:4]),fcirc) psych::fa.diagram(fcirc,fe=fe, main="Extending the circumplex structure to EA/TA and PA/NA ") #show the 2 dimensional structure f2 <- psych::fa(msq[1:72],2) psych::fa.plot(f2,labels=colnames(msq)[1:72], title="2 dimensions of affect at the item level",cex=.5) #sort them by polar coordinates round(psych::polar(f2),2) } } \keyword{datasets} psychTools/man/bfi.Rd0000644000176200001440000001316713714654277014253 0ustar liggesusers\name{bfi} \alias{bfi} \alias{bfi.dictionary} \alias{bfi.keys} \docType{data} \title{25 Personality items representing 5 factors} \description{25 personality self report items taken from the International Personality Item Pool (ipip.ori.org) were included as part of the Synthetic Aperture Personality Assessment (SAPA) web based personality assessment project. The data from 2800 subjects are included here as a demonstration set for scale construction, factor analysis, and Item Response Theory analysis. Three additional demographic variables (sex, education, and age) are also included. } \usage{data(bfi) data(bfi.dictionary) } \format{ A data frame with 2800 observations on the following 28 variables. (The q numbers are the SAPA item numbers). \describe{ \item{\code{A1}}{Am indifferent to the feelings of others. (q_146)} \item{\code{A2}}{Inquire about others' well-being. (q_1162)} \item{\code{A3}}{Know how to comfort others. (q_1206) } \item{\code{A4}}{Love children. (q_1364)} \item{\code{A5}}{Make people feel at ease. (q_1419)} \item{\code{C1}}{Am exacting in my work. (q_124)} \item{\code{C2}}{Continue until everything is perfect. (q_530)} \item{\code{C3}}{Do things according to a plan. (q_619)} \item{\code{C4}}{Do things in a half-way manner. (q_626)} \item{\code{C5}}{Waste my time. (q_1949)} \item{\code{E1}}{Don't talk a lot. (q_712)} \item{\code{E2}}{Find it difficult to approach others. (q_901)} \item{\code{E3}}{Know how to captivate people. (q_1205)} \item{\code{E4}}{Make friends easily. (q_1410)} \item{\code{E5}}{Take charge. (q_1768)} \item{\code{N1}}{Get angry easily. (q_952)} \item{\code{N2}}{Get irritated easily. (q_974)} \item{\code{N3}}{Have frequent mood swings. (q_1099} \item{\code{N4}}{Often feel blue. (q_1479)} \item{\code{N5}}{Panic easily. (q_1505)} \item{\code{O1}}{Am full of ideas. (q_128)} \item{\code{O2}}{Avoid difficult reading material.(q_316)} \item{\code{O3}}{Carry the conversation to a higher level. (q_492)} \item{\code{O4}}{Spend time reflecting on things. (q_1738)} \item{\code{O5}}{Will not probe deeply into a subject. (q_1964)} \item{\code{gender}}{Males = 1, Females =2} \item{\code{education}}{1 = HS, 2 = finished HS, 3 = some college, 4 = college graduate 5 = graduate degree} \item{\code{age}}{age in years} } } \details{The first 25 items are organized by five putative factors: Agreeableness, Conscientiousness, Extraversion, Neuroticism, and Opennness. The scoring key is created using \code{\link{make.keys}}, the scores are found using \code{\link{score.items}}. These five factors are a useful example of using \code{\link{irt.fa}} to do Item Response Theory based latent factor analysis of the \code{\link{polychoric}} correlation matrix. The endorsement plots for each item, as well as the item information functions reveal that the items differ in their quality. The item data were collected using a 6 point response scale: 1 Very Inaccurate 2 Moderately Inaccurate 3 Slightly Inaccurate 4 Slightly Accurate 5 Moderately Accurate 6 Very Accurate as part of the Synthetic Apeture Personality Assessment (SAPA \url{https://www.sapa-project.org/}) project. To see an example of the data collection technique, visit \url{https://www.SAPA-project.org/} or the International Cognitive Ability Resource at \url{https://icar-project.org}. The items given were sampled from the International Personality Item Pool of Lewis Goldberg using the sampling technique of SAPA. This is a sample data set taken from the much larger SAPA data bank. } \source{The items are from the ipip (Goldberg, 1999). The data are from the SAPA project (Revelle, Wilt and Rosenthal, 2010) , collected Spring, 2010 ( \url{https://www.sapa-project.org/}). } \references{Goldberg, L.R. (1999) A broad-bandwidth, public domain, personality inventory measuring the lower-level facets of several five-factor models. In Mervielde, I. and Deary, I. and De Fruyt, F. and Ostendorf, F. (eds) Personality psychology in Europe. 7. Tilburg University Press. Tilburg, The Netherlands. Revelle, W., Wilt, J., and Rosenthal, A. (2010) Individual Differences in Cognition: New Methods for examining the Personality-Cognition Link In Gruszka, A. and Matthews, G. and Szymura, B. (Eds.) Handbook of Individual Differences in Cognition: Attention, Memory and Executive Control, Springer. Revelle, W, Condon, D.M., Wilt, J., French, J.A., Brown, A., and Elleman, L.G. (2016) Web and phone based data collection using planned missing designs. In Fielding, N.G., Lee, R.M. and Blank, G. (Eds). SAGE Handbook of Online Research Methods (2nd Ed), Sage Publcations. } \seealso{\code{\link{bi.bars}} to show the data by age and gender, \code{\link{irt.fa}} for item factor analysis applying the irt model.} \note{The bfi data set and items should not be confused with the BFI (Big Five Inventory) of Oliver John and colleagues (John, O. P., Donahue, E. M., & Kentle, R. L. (1991). The Big Five Inventory--Versions 4a and 54. Berkeley, CA: University of California,Berkeley, Institute of Personality and Social Research.) } \examples{ data(bfi) psych::describe(bfi) # create the bfi.keys (actually already saved in the data file) bfi.keys <- list(agree=c("-A1","A2","A3","A4","A5"),conscientious=c("C1","C2","C3","-C4","-C5"), extraversion=c("-E1","-E2","E3","E4","E5"),neuroticism=c("N1","N2","N3","N4","N5"), openness = c("O1","-O2","O3","O4","-O5")) scores <- psych::scoreItems(bfi.keys,bfi,min=1,max=6) #specify the minimum and maximum values scores #show the use of the keys.lookup with a dictionary psych::keys.lookup(bfi.keys,bfi.dictionary[,1:4]) } \keyword{datasets} psychTools/man/burt.Rd0000644000176200001440000000436513464056134014455 0ustar liggesusers\name{burt} \alias{burt} \docType{data} \title{11 emotional variables from Burt (1915)} \description{Cyril Burt reported an early factor analysis with a circumplex structure of 11 emotional variables in 1915. 8 of these were subsequently used by Harman in his text on factor analysis. Unfortunately, it seems as if Burt made a mistake for the matrix is not positive definite. With one change from .87 to .81 the matrix is positive definite. } \usage{data(burt)} \format{ A correlation matrix based upon 172 "normal school age children aged 9-12". \describe{ \item{Sociality}{Sociality} \item{Sorrow}{Sorrow} \item{Tenderness}{Tenderness} \item{Joy}{Joy} \item{Wonder}{Wonder} \item{Elation}{Elation} \item{Disgust}{Disgust} \item{Anger}{Anger} \item{Sex}{Sex} \item{Fear}{Fear} \item{Subjection}{Subjection} } } \details{ The Burt data set is interesting for several reasons. It seems to be an early example of the organizaton of emotions into an affective circumplex, a subset of it has been used for factor analysis examples (see \code{\link{Harman.Burt}}, and it is an example of how typos affect data. The original data matrix has one negative eigenvalue. With the replacement of the correlation between Sorrow and Tenderness from .87 to .81, the matrix is positive definite. Alternatively, using \code{\link{cor.smooth}}, the matrix can be made positive definite as well, although cor.smooth makes more (but smaller) changes. } \source{ (retrieved from the web at https://www.biodiversitylibrary.org/item/95822#790) Following a suggestion by Jan DeLeeuw. } \references{ Burt, C.General and Specific Factors underlying the Primary Emotions. Reports of the British Association for the Advancement of Science, 85th meeting, held in Manchester, September 7-11, 1915. London, John Murray, 1916, p. 694-696 (retrieved from the web at https://www.biodiversitylibrary.org/item/95822#790) } \seealso{ \code{\link{Harman.Burt}} in the \code{\link{Harman}} dataset and \code{\link{cor.smooth}} } \examples{ data(burt) eigen(burt)$values #one is negative! burt.new <- burt burt.new[2,3] <- burt.new[3,2] <- .81 eigen(burt.new)$values #all are positive bs <- psych::cor.smooth(burt) round(burt.new - bs,3) } \keyword{datasets} psychTools/man/Pollack.Rd0000644000176200001440000000315513771230255015061 0ustar liggesusers\name{Pollack} \alias{Pollack} \alias{estress} \docType{data} \title{Pollack et al (2012) correlation matrix for mediation example } \description{A correlation matrix taken from Pollack (2012) with 9 variables. Primarily used as an example for setCor and mediation.} \usage{data("Pollack")} \format{A correlation matrix based upon 262 participants. \describe{ \item{\code{sex}}{Male = 1, Female = 0, 62\% male} \item{\code{age}}{mean =33} \item{\code{tenure}}{length of employent, mean = 5.9 years} \item{\code{self.efficacy}}{self ratings} \item{\code{competence}}{self rating of competence} \item{\code{social.ties}}{Contact with business-related social ties} \item{\code{economic.stress}}{mean of two items on economic stress} \item{\code{depression}}{6 items from MAACL measuring depression} \item{\code{withdrawal}}{Withdrawal intentions in domain of entrepreneurship} } } \details{This is the correlation matrix from Pollack et al. (2012) p 797. The raw data are available from the processR package (Keon-Woong Moon, 2020). The data set is used by Hayes in example p 179 in example 3. } \source{Pollack et al. 2012 } \references{Pollack, Jeffrey M. and Vanepps, Eric M. and Hayes, Andrew F. (2012). The moderating role of social ties on entrepreneurs' depressed affect and withdrawal intentions in response to economic stress, Journal of Organizational Behavior 33 (6) 789-810. Hayes, Andrew F. (2013) Introduction to mediation, moderation, and conditional process analysis: A regression-based approach. Guilford Press. } \examples{ psych::lowerMat(Pollack) } \keyword{datasets} psychTools/man/sai.Rd0000644000176200001440000002077113501204371014242 0ustar liggesusers\name{sai} \alias{sai} \alias{tai} \alias{sai.dictionary} \docType{data} \title{State Anxiety data from the PMC lab over multiple occasions. } \description{ State Anxiety was measured two-three times in 11 studies at the Personality-Motivation-Cognition laboratory. Here are item responses for 11 studies (9 repeated twice, 2 repeated three times). In all studies, the first occasion was before a manipulation. In some studies, caffeine, or movies or incentives were then given to some of the participants before the second and third STAI was given. In addition, Trait measures are available and included in the tai data set (3032 subjects). } \usage{data(sai) data(tai) data(sai.dictionary) } \format{ A data frame with 3032 unique observations on the following 23 variables. \describe{ \item{\code{id}}{a numeric vector} \item{\code{study}}{a factor with levels \code{ages} \code{cart} \code{fast} \code{fiat} \code{film} \code{flat} \code{home} \code{pat} \code{rob} \code{salt} \code{shed}\code{shop} \code{xray}} \item{\code{time}}{1=First, 2 = Second, 3=third administration} \item{\code{TOD}}{TOD (time of day 1= 8:50-9:30 am,2 = 1=3 pm, 3= 7:-8pm} \item{\code{drug}}{drug (placebo (0) vs. caffeine (1))} \item{\code{film}}{film (1=Frontline (concentration camp), 2 = Halloween 3= National Geographic (control), 4- Parenthood (humor)} \item{\code{anxious}}{anxious} \item{\code{at.ease}}{at ease} \item{\code{calm}}{calm} \item{\code{comfortable}}{comfortable} \item{\code{confident}}{confident} \item{\code{content}}{content} \item{\code{high.strung}}{high.strung} \item{\code{jittery}}{jittery} \item{\code{joyful}}{joyful} \item{\code{nervous}}{nervous} \item{\code{pleasant}}{pleasant} \item{\code{rattled}}{over-excited and rattled} \item{\code{regretful}}{regretful} \item{\code{relaxed}}{relaxed} \item{\code{rested}}{rested} \item{\code{secure}}{secure} \item{\code{tense}}{tense} \item{\code{upset}}{upset} \item{\code{worried}}{worried} \item{\code{worrying}}{worrying} } } \details{The standard experimental study at the Personality, Motivation and Cognition (PMC) laboratory (Revelle and Anderson, 1997) was to administer a number of personality trait and state measures (e.g. the \code{\link{epi}}, \code{\link{msq}}, \code{\link{msqR}} and \code{\link{sai}}) to participants before some experimental manipulation of arousal/effort/anxiety. Following the manipulation (with a 30 minute delay if giving caffeine/placebo), some performance task was given, followed once again by measures of state arousal/effort/anxiety. Here are the item level data on the \code{\link{sai}} (state anxiety) and the \code{\link{tai}} (trait anxiety). Scores on these scales may be found using the scoring keys. The \code{\link{affect}} data set includes pre and post scores for two studies (flat and maps) which manipulated state by using four types of movies. In addition to being useful for studies of motivational state, these studies provide examples of test-retest and alternate form reliabilities. Given that 10 items overlap with the \code{\link{msqR}} data, they also allow for a comparison of immediate duplication of items with 30 minute delays. Studies CART, FAST, SHED, RAFT, and SHOP were either control groups, or did not experimentally vary arousal/effort/anxiety. AGES, CITY, EMIT, RIM, SALT, and XRAY were caffeine manipulations between time 1 and 2 (RIM and VALE were repeated day 1 and day 2) FIAT, FLAT, MAPS, MIXX, and THRU were 1 day studies with film manipulation between time 1 and time 2. SAM1 and SAM2 were the first and second day of a two day study. The STAI was given once per day. MSQ not MSQR was given. VALE and PAT were two day studies with the STAI given pre and post on both days RIM was a two day study with the STAI and MSQ given once per day. Usually, time of day 1 = 8:50-9am am, and 2 = 7:30 pm, however, in rob, with paid subjects, the times were 0530 and 22:30. } \source{Data collected at the Personality, Motivation, and Cognition Laboratory, Northwestern University, between 1991 and 1999. } \references{ Charles D. Spielberger and Richard L. Gorsuch and R. E. Lushene, (1970) Manual for the State-Trait Anxiety Inventory. Revelle, William and Anderson, Kristen Joan (1997) Personality, motivation and cognitive performance: Final report to the Army Research Institute on contract MDA 903-93-K-0008 Rafaeli, Eshkol and Revelle, William (2006), A premature consensus: Are happiness and sadness truly opposite affects? Motivation and Emotion, 30, 1, 1-12. Smillie, Luke D. and Cooper, Andrew and Wilt, Joshua and Revelle, William (2012) Do Extraverts Get More Bang for the Buck? Refining the Affective-Reactivity Hypothesis of Extraversion. Journal of Personality and Social Psychology, 103 (2), 206-326. } \examples{ data(sai) table(sai$study,sai$time) #show the counts for repeated measures #Here are the keys to score the sai total score, positive and negative items sai.keys <- list(sai = c("tense","regretful" , "upset", "worrying", "anxious", "nervous" , "jittery" , "high.strung", "worried" , "rattled","-calm", "-secure","-at.ease","-rested","-comfortable", "-confident" ,"-relaxed" , "-content" , "-joyful", "-pleasant" ) , sai.p = c("calm","at.ease","rested","comfortable", "confident", "secure" ,"relaxed" , "content" , "joyful", "pleasant" ), sai.n = c( "tense" , "anxious", "nervous" , "jittery" , "rattled", "high.strung", "upset", "worrying","worried","regretful" ) ) tai.keys <- list(tai=c("-pleasant" ,"nervous" , "not.satisfied", "wish.happy", "failure","-rested", "-calm", "difficulties" , "worry" , "-happy" , "disturbing.thoughts","lack.self.confidence", "-secure", "decisive" , "inadequate","-content","thoughts.bother","disappointments" , "-steady" , "tension" ), tai.pos = c("pleasant", "-wish.happy", "rested","calm","happy" ,"secure", "content","steady" ), tai.neg = c("nervous", "not.satisfied", "failure","difficulties", "worry", "disturbing.thoughts" ,"lack.self.confidence","decisive","inadequate" , "thoughts.bother","disappointments","tension" ) ) #using the is.element function instead of the \%in\% function #just get the control subjects control <- subset(sai,is.element(sai$study,c("Cart", "Fast", "SHED", "RAFT", "SHOP")) ) #pre and post drug studies drug <- subset(sai,is.element(sai$study, c("AGES","CITY","EMIT","SALT","VALE","XRAY"))) #pre and post film studies film <- subset(sai,is.element(sai$study, c("FIAT","FLAT", "MAPS", "MIXX") )) #this next set allows us to score those sai items that overlap with the msq item sets msq.items <- c("anxious", "at.ease" ,"calm", "confident","content", "jittery", "nervous" , "relaxed" , "tense" , "upset" ) #these overlap with the msq sai.msq.keys <- list(pos =c( "at.ease" , "calm" , "confident", "content","relaxed"), neg = c("anxious", "jittery", "nervous" ,"tense" , "upset"), anx = c("anxious", "jittery", "nervous" ,"tense", "upset","-at.ease" , "-calm" , "-confident", "-content","-relaxed")) sai.not.msq.keys <- list(pos=c( "secure","rested","comfortable" ,"joyful" , "pleasant" ), neg=c("regretful","worrying", "high.strung","worried", "rattled" ), anx = c("regretful","worrying", "high.strung","worried", "rattled", "-secure", "-rested", "-comfortable", "-joyful", "-pleasant" )) sai.alternate.forms <- list( pos1 =c( "at.ease","calm","confident","content","relaxed"), neg1 = c("anxious", "jittery", "nervous" ,"tense" , "upset"), anx1 = c("anxious", "jittery", "nervous" ,"tense", "upset","-at.ease" , "-calm" , "-confident", "-content","-relaxed"), pos2=c( "secure","rested","comfortable" ,"joyful" , "pleasant" ), neg2=c("regretful","worrying", "high.strung","worried", "rattled" ), anx2 = c("regretful","worrying", "high.strung","worried", "rattled", "-secure", "-rested", "-comfortable", "-joyful", "-pleasant" )) sai.repeated <- c("AGES","Cart","Fast","FIAT","FILM","FLAT","HOME","PAT","RIM","SALT", "SAM","SHED","SHOP","VALE","XRAY") sai12 <- subset(sai,is.element(sai$study, sai.repeated)) #the subset with repeated measures #Choose those studies with repeated measures by : sai.control <- subset(sai,is.element(sai$study, c("Cart", "Fast", "SHED", "SHOP"))) sai.film <- subset(sai,is.element(sai$study, c("FIAT","FLAT") ) ) sai.drug <- subset(sai,is.element(sai$study, c("AGES", "SALT", "VALE", "XRAY"))) sai.day <- subset(sai,is.element(sai$study, c("SAM", "RIM"))) } \keyword{datasets} psychTools/man/df2latex.Rd0000644000176200001440000001366314054510674015213 0ustar liggesusers\name{df2latex} \alias{df2latex} \alias{cor2latex} \alias{fa2latex} \alias{omega2latex} \alias{irt2latex} \alias{ICC2latex} \title{Convert a data frame, correlation matrix, or factor analysis output to a LaTeX table} \description{A set of handy helper functions to convert data frames or matrices to LaTeX tables. Although Sweave is the preferred means of converting R output to LaTeX, it is sometimes useful to go directly from a data.frame or matrix to a LaTeX table. cor2latex will find the correlations and then create a lower (or upper) triangular matrix for latex output. fa2latex will create the latex commands for showing the loadings and factor intercorrelations. As the default option, tables are prepared in an approximation of APA format. } \usage{ df2latex(x,digits=2,rowlabels=TRUE,apa=TRUE,short.names=TRUE,font.size ="scriptsize", big.mark=NULL,drop.na=TRUE, heading="A table from the psych package in R", caption="df2latex",label="default", char=FALSE, stars=FALSE,silent=FALSE,file=NULL,append=FALSE,cut=0,big=0,abbrev=NULL,long=FALSE) cor2latex(x,use = "pairwise", method="pearson", adjust="holm",stars=FALSE, digits=2,rowlabels=TRUE,lower=TRUE,apa=TRUE,short.names=TRUE, font.size ="scriptsize", heading="A correlation table from the psych package in R.", caption="cor2latex",label="default",silent=FALSE,file=NULL,append=FALSE,cut=0,big=0) fa2latex(f,digits=2,rowlabels=TRUE,apa=TRUE,short.names=FALSE,cumvar=FALSE, cut=0,big=.3,alpha=.05,font.size ="scriptsize", heading="A factor analysis table from the psych package in R", caption="fa2latex",label="default",silent=FALSE,file=NULL,append=FALSE) omega2latex(f,digits=2,rowlabels=TRUE,apa=TRUE,short.names=FALSE,cumvar=FALSE,cut=.2, font.size ="scriptsize", heading="An omega analysis table from the psych package in R", caption="omega2latex",label="default",silent=FALSE,file=NULL,append=FALSE) irt2latex(f,digits=2,rowlabels=TRUE,apa=TRUE,short.names=FALSE, font.size ="scriptsize", heading="An IRT factor analysis table from R", caption="fa2latex",label="default",silent=FALSE,file=NULL,append=FALSE) ICC2latex(icc,digits=2,rowlabels=TRUE,apa=TRUE,ci=TRUE, font.size ="scriptsize",big.mark=NULL, drop.na=TRUE, heading="A table from the psych package in R", caption="ICC2latex",label="default",char=FALSE,silent=FALSE,file=NULL,append=FALSE) } \arguments{ \item{x}{A data frame or matrix to convert to LaTeX. If non-square, then correlations will be found prior to printing in cor2latex} \item{digits}{Round the output to digits of accuracy. NULL for formatting character data} \item{abbrev}{How many characters should be used in column names --defaults to digits + 3} \item{rowlabels}{If TRUE, use the row names from the matrix or data.frame} \item{short.names}{Name the columns with abbreviated rownames to save space} \item{apa}{If TRUE formats table in APA style} \item{cumvar}{For factor analyses, should we show the cumulative variance accounted for?} \item{font.size}{e.g., "scriptsize", "tiny" or anyother acceptable LaTeX font size.} \item{heading}{The label appearing at the top of the table} \item{caption}{The table caption} \item{lower}{in cor2latex, just show the lower triangular matrix} \item{f}{The object returned from a factor analysis using \code{\link{fa}} or \code{\link{irt.fa}}. } \item{label}{The label for the table} \item{big.mark}{Comma separate numbers large numbers (big.mark=",")} \item{drop.na}{Do not print NA values} \item{method}{When finding correlations, which method should be used (pearson)} \item{use}{use="pairwise" is the default when finding correlations in cor2latex} \item{adjust}{If showing probabilities, which adjustment should be used (holm)} \item{stars}{Should probability 'magic astericks' be displayed in cor2latex (FALSE)} \item{char}{char=TRUE allows printing tables with character information, but does not allow for putting in commas into numbers} \item{cut}{In omega2latex, df2latex and fa2latex, do not print abs(values) < cut } \item{big}{In fa2latex and df2latex boldface those abs(values) > big} \item{alpha}{If fa has returned confidence intervals, then what values of loadings should be boldfaced?} \item{icc}{Either the output of an ICC, or the data to be analyzed.} \item{ci}{Should confidence intervals of the ICC be displayed} \item{silent}{If TRUE, do not print any output, just return silently -- useful if using Sweave} \item{file}{If specified, write the output to this file} \item{append}{If file is specified, then should we append (append=TRUE) or just write to the file} \item{long}{if TRUE, then do long tables. (requires the longtables package in latex)}} \value{A LaTeX table. Note that if showing "stars" for correlations, then one needs to use the siunitx package in LaTex. The entire LaTeX output is also returned invisibly. If using Sweave to create tables, then the silent option should be set to TRUE and the returned object saved as a file. See the last example.} \author{William Revelle with suggestions from Jason French and David Condon and Davide Morselli} \seealso{ The many LaTeX conversion routines in Hmisc. } \examples{ df2latex(psych::Thurstone,rowlabels=FALSE,apa=FALSE,short.names=FALSE, caption="Thurstone Correlation matrix") df2latex(psych::Thurstone,heading="Thurstone Correlation matrix in APA style") df2latex(psych::describe(psych::sat.act)[2:10],short.names=FALSE) cor2latex(psych::Thurstone) cor2latex(psych::sat.act,short.names=FALSE) fa2latex(psych::fa(psych::Thurstone,3),heading="Factor analysis from R in quasi APA style") #If using Sweave to create a LateX table as a separate file then set silent=TRUE #e.g., #LaTex preamble #.... #<>= #f3 <- fa(Thurstone,3) #fa2latex(f3,silent=TRUE,file='testoutput.tex') #@ # #\input{testoutput.tex} } % Add one or more standard keywords, see file 'KEYWORDS' in the % R documentation directory. \keyword{ utilities } psychTools/man/psychTools.Rd0000644000176200001440000001257013567612163015651 0ustar liggesusers\name{psychTools} \alias{psychTools} \docType{data} \title{psychTools: datasets and utility functions to accompany the psych package } \description{ PsychTools includes the larger data sets used by the \code{\link[psych]{psych}} package and also includes a few general utility functions such as the \code{\link{read.file}} and \code{\link{read.clipboard}} functions. The data sets ara made available for demonstrations of a variety of psychometric functions. } \details{ See the various helpfiles listed in the index or as links from here. Also see the main functions in the psych package \code{\link[psych]{00.psych-package}}. Data sets from the SAPA/ICAR project: \tabular{ll}{ \code{\link{ability}} \tab 16 ICAR ability items scored as correct or incorrect for 1525 participants. \cr \code{\link{iqitems}} \tab multiple choice IQ items (raw responses) \cr \code{\link{affect}} \tab Two data sets of affect and arousal scores as a function of personality and movie conditions \cr \code{\link{bfi}} \tab 25 Personality items representing 5 factors from the SAPA project for 2800 participants \cr bfi.dictionary \tab Dictionary of the bfi \cr \code{\link{epi}} \tab Eysenck Personality Inventory (EPI) data for 3570 participants \cr epi.dictionary \tab The items for the epi \cr \code{\link{epi.bfi}} \tab 13 personality scales from the Eysenck Personality Inventory and Big 5 inventory \cr \code{\link{epiR}} \tab 474 participants took the epi twice \cr \code{\link{msq}} \tab 75 mood items from the Motivational State Questionnaire for 3896 participants \cr \code{\link{msqR}} \tab 75 mood items from the Motivational State Questionnaire for 3032 unique participants \cr \code{\link{tai}} \tab Trait Anxiety data from the PMC lab matching the sai sample. 3032 unique subjects \cr \code{\link{sai}} \tab State Anxiety data from the PMC lab over multiple occasions. 3032 unique subjects. \cr sai.dictionary \tab items used in the sai \cr \code{\link{spi}} \tab 4000 cases from the SAPA Personality Inventory including an item dictionary and scoring keys. \cr spi.dictionary \tab The items for the spi \cr spi.keys \tab Scoring keys for the spi \cr } Historically interesting data sets \tabular{ll}{ \code{\link{burt}} \tab 11 emotional variables from Burt (1915) \cr \code{\link{galton}} \tab Galtons Mid parent child height data \cr \code{\link{heights}} \tab A data.frame of the Galton (1888) height and cubit data set \cr \code{\link{cubits}} \tab Galtons example of the relationship between height and cubit or forearm length \cr \code{\link{peas}} \tab Galtons Peas \cr \code{\link{cushny}} \tab The data set from Cushny and Peebles (1905) on the effect of three drugs on hours of sleep, used by Student (1908) \cr \code{\link{holzinger.swineford}} \tab 26 cognitive variables + 7 demographic variables for 301 cases from Holzinger and Swineford. } Miscellaneous example data sets \tabular{ll}{ \code{\link{blant}} \tab A 29 x 29 matrix that produces weird factor analytic results \cr \code{\link{blot}} \tab Bonds Logical Operations Test - BLOT \cr \code{\link{cities}} \tab Distances between 11 US cities \cr city.location \tab and their geograpical location \cr \code{\link{income}} \tab US family income from US census 2008 \cr all.income \tab US family income from US census 2008 \cr \code{\link{neo}} \tab NEO correlation matrix from the NEO_PI_R manual \cr \code{\link{Schutz}} \tab The Schutz correlation matrix example from Shapiro and ten Berge \cr \code{\link{Spengler}} \tab The Spengler and Damian correlation matrix example from Spengler, Damian and Roberts (2018) \cr \code{\link{Damian}} \tab Another correlation matrix from Spengler, Damian and Roberts (2018) \cr \code{\link{usaf}} \tab A correlation of 17 body size (anthropometric) measures from the US Air Force. Adapted from the Anthropometric package.\cr veg \tab Paired comparison of preferences for 9 vegetables (scaling example) \cr } Functions to convert various objects to latex \tabular{ll}{ \code{\link{fa2latex}} \tab Convert a data frame, correlation matrix, or factor analysis output to a LaTeX table \cr \code{\link{df2latex}} \tab Convert a data frame, correlation matrix, or factor analysis output to a LaTeX table \cr \code{\link{ICC2latex}} \tab Convert an ICC analyssis output to a LaTeX table \cr \code{\link{irt2latex}} \tab Convert an irt analysis output to a LaTeX table \cr \code{\link{cor2latex}} \tab Convert a correlation matrix output to a LaTeX table \cr omega2latex \tab Convert a data frame, correlation matrix, or factor analysis output to a LaTeX table \cr } File manipulation functions \tabular{ll}{ \code{\link{fileCreate}} \tab Create a file \cr fileScan \tab Show the first few lines of multitple files \cr filesInfo \tab Show the information for all files in a directory \cr filesList \tab Show the names of all files in a directory \cr } \code{\link{dfOrder}} Sorts a data frame File input/output functions \tabular{ll}{ \code{\link{read.clipboard}} \tab Shortcuts for reading from the clipboard or a file \cr read.clipboard.csv \tab \cr read.clipboard.fwf \tab \cr read.clipboard.lower \tab \cr read.clipboard.tab \tab \cr read.clipboard.upper \tab \cr \code{\link{read.file}} \tab Read a file according to its suffix \cr read.file.csv \tab \cr read.https \tab \cr \code{\link{write.file}} \tab Write data to a file \cr write.file.csv \tab \cr } } \examples{ psych::describe(ability) } \keyword{datasets} psychTools/man/globalWarm.Rd0000644000176200001440000000253113774157445015574 0ustar liggesusers\name{globalWarm} \alias{globalWarm} \alias{glbwarm} \docType{data} \title{7 attitude items about Global Warming policy from Erik Nisbet } \description{Erik Nisbet reported the relationship between emotions, ideology, and party affiliation as predictors of attitudes towards government action on climate change. The data were used by Hayes (2013) in a discussion of regression. They are available as the glbwarm data set in the processR package. They are copied here for examples of mediation. } \usage{data("globalWarm")} \format{ A data frame with 815 observations on the following 7 variables. \describe{ \item{\code{govact}}{a numeric} \item{\code{posemot}}{a numeric} \item{\code{negemot}}{a numeric} \item{\code{ideology}}{a numeric} \item{\code{age}}{a numeric} \item{\code{sex}}{a numeric} \item{\code{partyid}}{a numeric} } } \details{More details to follow } \source{ The raw data are available from the processR package (Keon-Woong Moon, 2020). as the glbwarm data set. The data set is used by Hayes in several examples. Used here by kind permission of Erik Nisbet. } \references{ Nisbet, E. (?) Hayes, Andrew F. (2013) Introduction to mediation, moderation, and conditional process analysis: A regression-based approach. Guilford Press. } \examples{ data(globalWarm) psych::lowerCor(globalWarm) } \keyword{datasets} psychTools/DESCRIPTION0000644000176200001440000000212714153452125014133 0ustar liggesusersPackage: psychTools Version: 2.1.12 Date: 2021-12-02 Title: Tools to Accompany the 'psych' Package for Psychological Research Authors@R: person("William", "Revelle", role =c("aut","cre"), email="revelle@northwestern.edu", comment=c(ORCID = "0000-0003-4880-9610") ) Description: Support functions, data sets, and vignettes for the 'psych' package. Contains several of the biggest data sets for the 'psych' package as well as one vignette. A few helper functions for file manipulation are included as well. For more information, see the web page. License: GPL (>= 2) Imports: foreign,psych Suggests: parallel, GPArotation, lavaan,knitr Depends: R(>= 2.10) LazyData: yes ByteCompile: TRUE VignetteBuilder: knitr URL: https://personality-project.org/r/psych/ https://personality-project.org/r/psych-manual.pdf NeedsCompilation: no Packaged: 2021-12-06 17:19:55 UTC; WR Author: William Revelle [aut, cre] () Maintainer: William Revelle Repository: CRAN Date/Publication: 2021-12-06 18:20:05 UTC psychTools/build/0000755000176200001440000000000014153443073013524 5ustar liggesuserspsychTools/build/vignette.rds0000644000176200001440000000051314153443073016062 0ustar liggesusersSO0-٤&O,3F6вFhI[!`-M}1zW؅LuGo LF{HqxeeN5<=k!J"KFs޾(S+N BC>9jŔ :$yqQm 2wFwi;В k0F4h1jˁBFS[Š(Xi[Y%IΟqNC oXf{tox9|kR/h@JHlj)&[;,M|VgXkQ `}~ (FSpsychTools/build/partial.rdb0000644000176200001440000002640314153443005015651 0ustar liggesusers}MpG^H H Q34QO˜9 h$Q=+WwUjUUiygu^f2#t]|ٛ=֬O13ˎeP?]Mi3;e2eU|L*N Ox:=DÑia*52,||$hKT|NfwJˬJf3,R'P))p]INѫ]sZW1Ew+JwVeK&$|7oeWѪ&YmQSjMVK Kfid%3[34Y1lpo$khukU(Z񸑣vٝln4wlR5R\Y] kkE%۴ЬlCҳ؜ܥܥwx8U4.#KZxTe^oLѹ&Nt#9n:Νv|Ԙj|>{jC)01ICՍg'˒JOCB$&.9S;\$JT$iSH'5Mr"%skXU$'($<GB()DZjO?E$l+e?l;CvꆥԍLEyBMQ#3bWr4tj` -SN,ib꒭+Vw{[ӌ:-K4RIdAohV!5aRp2ԤMrMP8.c6igjA'!ˀ  {.&`C?p0/0;:O㶕!A?"O.TIeY|F4Jzw|9i6}~TN^3(r-|`-K2QKDjnV@B6hTuΠ*V>!lfJ vOȡ1fhqK*U4vwzQQL! xS@+tS$fep-?\ÄcOu4BT5&+bPZ1X+bjD堽y9?FO^FrlS^qdс pi2hWP^z3vr1pU@&a)6F6A4D>֮i)x+iΏN!qqi r(G#d_i 4,Vi 4,@S5@Sɒi 45X2M!!%Kd::,!LWPN`wt.<8( *5%!u74C9T<'Mm@6 q+(G[+֬KzREXirPl+Tb]yb~Iw^G(?J~? aH20$RP9lC%5Avge4l:YAvP㦑eqh d\ Ȳ 2B_zƅhyP~G- G9ϣ$ ^ N`[Y J$LApq3ȲTC l&5!@ q#$אlS}Y^ ʸȲ2"b_wi Fl цK(/K?(Zڌ%~E@Yl\ k3,f?3f?ri3;X.".!.%˥ˋ?%ϥK?%ϥrqi3(Vgyg.Y.!˥9䏃eD4fd+k}|_r}G ݔlUH j5b6B㋈(ǷcϗOCk)w^|k|(_I&'a8zt9t%Y>]A], O^xIB*2P @i˩^ˀ-(BC<ʡ,;^pPx `z9w1 ;jO Sѹ QP P# yzVLtK@#S * ! 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There are lots. \geometry{letterpaper} % ... or a4paper or a5paper or ... %\geometry{landscape} % Activate for for rotated page geometry \usepackage[parfill]{parskip} % Activate to begin paragraphs with an empty line rather than an indent \usepackage{graphicx} \usepackage{amssymb} \usepackage{epstopdf} \usepackage{mathptmx} \usepackage{helvet} \usepackage{courier} \usepackage{epstopdf} \usepackage{makeidx} % allows index generation \usepackage[authoryear,round]{natbib} \usepackage{gensymb} \usepackage{longtable} %\usepackage{geometry} \usepackage{amssymb} \usepackage{amsmath} %\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \usepackage{Sweave} %\usepackage{/Volumes/'Macintosh HD'/Library/Frameworks/R.framework/Versions/2.13/Resources/share/texmf/tex/latex/Sweave} %\usepackage[ae]{Rd} %\usepackage[usenames]{color} %\usepackage{setspace} \bibstyle{apacite} \bibliographystyle{apa} %this one plus author year seems to work? %\usepackage{hyperref} \usepackage[colorlinks=true,citecolor=blue]{hyperref} %this makes reference links hyperlinks in pdf! \DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \usepackage{multicol} % used for the two-column index \usepackage[bottom]{footmisc}% places footnotes at page bottom \let\proglang=\textsf \newcommand{\R}{\proglang{R}} %\newcommand{\pkg}[1]{{\normalfont\fontseries{b}\selectfont #1}} \newcommand{\Rfunction}[1]{{\texttt{#1}}} \newcommand{\fun}[1]{{\texttt{#1}\index{#1}\index{R function!#1}}} \newcommand{\pfun}[1]{{\texttt{#1}\index{#1}\index{R function!#1}\index{R function!psych package!#1}}}\newcommand{\Rc}[1]{{\texttt{#1}}} %R command same as Robject \newcommand{\Robject}[1]{{\texttt{#1}}} \newcommand{\Rpkg}[1]{{\textit{#1}\index{#1}\index{R package!#1}}} %different from pkg - which is better? \newcommand{\iemph}[1]{{\emph{#1}\index{#1}}} \newcommand{\wrc}[1]{\marginpar{\textcolor{blue}{#1}}} %bill's comments \newcommand{\wra}[1]{\textcolor{blue}{#1}} %bill's comments \newcommand{\ve}[1]{{\textbf{#1}}} %trying to get a vector command \makeindex % used for the subject index \title{An introduction to the psych package: Part II\\Scale construction and psychometrics} \author{William Revelle\\Department of Psychology\\Northwestern University} %\affiliation{Northwestern University} %\acknowledgements{Written to accompany the psych package. Comments should be directed to William Revelle \\ \url{revelle@northwestern.edu}} %\date{} % Activate to display a given date or no date \begin{document} \SweaveOpts{concordance=TRUE} \maketitle \tableofcontents \newpage \subsection{Jump starting the \Rpkg{psych} package--a guide for the impatient} You have installed \Rpkg{psych} (section \ref{sect:starting}) and you want to use it without reading much more. What should you do? \begin{enumerate} \item Activate the \Rpkg{psych} package: @ \begin{scriptsize} \begin{Schunk} \begin{Sinput} library(psych) library(psychTools) \end{Sinput} \end{Schunk} \end{scriptsize} \item Input your data (see the \href{https://personality-project.org/r/psych/intro.pdf}{Introduction to Psych} vignette section 3.1). There are two ways to do this: \begin{itemize} \item Find and read standard files using \pfun{read.file}. This will open a search window for your operating system which you can use to find the file. If the file has a suffix of .text, .txt, .csv, .data, .sav, .r, .R, .rds, .Rds, .rda, .Rda, .rdata, or .RData, then the file will be opened and the data will be read in. \begin{scriptsize} \begin{Schunk} \begin{Sinput} myData <- read.file() # find the appropriate file using your normal operating system \end{Sinput} \end{Schunk} \end{scriptsize} \item Alternatively, go to your friendly text editor or data manipulation program (e.g., Excel) and copy the data to the clipboard. Include a first line that has the variable labels. Paste it into \Rpkg{psych} using the \pfun{read.clipboard.tab} command: \begin{scriptsize} \begin{Schunk} \begin{Sinput} myData <- read.clipboard.tab() # if on the clipboard \end{Sinput} \end{Schunk} \end{scriptsize} Note that there are number of options for \pfun{read.clipboard} for reading in Excel based files, lower triangular files, etc. \end{itemize} \item Make sure that what you just read is right. Describe it (see the \href{https://personality-project.org/r/psych/intro.pdf}{Introduction to Psych} vignette section 3.3) on how to \pfun{describe} data) and perhaps look at the first and last few lines. If you have multiple groups, try \pfun{describeBy}. \begin{scriptsize} \begin{Schunk} \begin{Sinput} dim(myData) #What are the dimensions of the data? describe(myData) # or descrbeBy(myData,groups="mygroups") #for descriptive statistics by groups headTail(myData) #show the first and last n lines of a file \end{Sinput} \end{Schunk} \end{scriptsize} \item Look at the patterns in the data. If you have fewer than about 12 variables, look at the SPLOM (Scatter Plot Matrix) of the data using \pfun{pairs.panels} ( (see the \href{https://personality-project.org/r/psych/intro.pdf}{Introduction to Psych} vignette section 3.4 for a discussion of graphics)) . Then, use the \pfun{outlier} function to detect outliers. \begin{scriptsize} \begin{Schunk} \begin{Sinput} pairs.panels(myData) outlier(myData) \end{Sinput} \end{Schunk} \end{scriptsize} \item Note that you might have some weird subjects, probably due to data entry errors. Either edit the data by hand (use the \fun{edit} command) or just \pfun{scrub} the data). \begin{scriptsize} \begin{Schunk} \begin{Sinput} cleaned <- scrub(myData, max=9) #e.g., change anything great than 9 to NA \end{Sinput} \end{Schunk} \end{scriptsize} \item Graph the data with error bars for each variable ( (see the \href{https://personality-project.org/r/psych/intro.pdf}{Introduction to Psych} vignette section 3.1)). \begin{scriptsize} \begin{Schunk} \begin{Sinput} error.bars(myData) \end{Sinput} \end{Schunk} \end{scriptsize} \item Find the correlations of all of your data. \pfun{lowerCor} will by default find the pairwise correlations, round them to 2 decimals, and display the lower off diagonal matrix. \begin{itemize} \item Descriptively (just the values) (section \ref{sect:lowerCor}) \begin{scriptsize} \begin{Schunk} \begin{Sinput} r <- lowerCor(myData) #The correlation matrix, rounded to 2 decimals \end{Sinput} \end{Schunk} \end{scriptsize} \item Graphically (section \ref{sect:corplot}). Another way is to show a heat map of the correlations with the correlation values included. \begin{scriptsize} \begin{Schunk} \begin{Sinput} corPlot(r) #examine the many options for this function. \end{Sinput} \end{Schunk} \end{scriptsize} \item Inferentially (the values, the ns, and the p values) (section \ref{sect:corr.test}) \begin{scriptsize} \begin{Schunk} \begin{Sinput} corr.test(myData) \end{Sinput} \end{Schunk} \end{scriptsize} \end{itemize} \item Apply various regression models. Several functions are meant to do multiple regressions, either from the raw data or from a variance/covariance matrix, or a correlation matrix. \begin{itemize} \item \pfun{setCor} will take raw data or a correlation matrix and find (and graph the path diagram) for multiple y variables depending upon multiple x variables. \begin{scriptsize} \begin{Schunk} \begin{Sinput} myData <- sat.act colnames(myData) <- c("mod1","med1","x1","x2","y1","y2") setCor(y1 + y2 ~ x1 + x2 , data = myData) \end{Sinput} \end{Schunk} \end{scriptsize} \item \pfun{mediate} will take raw data or a correlation matrix and find (and graph the path diagram) for multiple y variables depending upon multiple x variables mediated through a mediation variable. It then tests the mediation effect using a boot strap. \begin{scriptsize} \begin{Schunk} \begin{Sinput} mediate(y1 + y2 ~ x1 + x2 + (med1) , data = myData) \end{Sinput} \end{Schunk} \end{scriptsize} \item \pfun{mediate} will take raw data and find (and graph the path diagram) a moderated multiple regression model for multiple y variables depending upon multiple x variables mediated through a mediation variable. It then tests the mediation effect using a boot strap. \begin{scriptsize} \begin{Schunk} \begin{Sinput} mediate(y1 + y2 ~ x1 + x2* mod1 +(med1), data = myData) \end{Sinput} \end{Schunk} \end{scriptsize} \end{itemize} \subsection{Psychometric functions are summarized in this vignette} Many additional functions, particularly designed for basic and advanced psychometrics are discussed more fully in this Vignette. A brief review of the functions available is included here. For basic data entry and descriptive statistics, see the Vignette Intro to Psych \url{https://personality-project.org/r}. In addition, there are helpful tutorials for \emph{Finding omega}, \emph{How to score scales and find reliability}, and for \emph{Using psych for factor analysis} at \url{https://personality-project.org/r}. \begin{itemize} \item Test for the number of factors in your data using parallel analysis (\pfun{fa.parallel}, section \ref{sect:fa.parallel}) or Very Simple Structure (\pfun{vss}, \ref{sect:vss}) . \begin{scriptsize} \begin{Schunk} \begin{Sinput} fa.parallel(myData) vss(myData) \end{Sinput} \end{Schunk} \end{scriptsize} \item Factor analyze (see section \ref{sect:fa}) the data with a specified number of factors (the default is 1), the default method is minimum residual, the default rotation for more than one factor is oblimin. There are many more possibilities (see sections \ref{sect:minres}-\ref{sect:wls}). Compare the solution to a hierarchical cluster analysis using the ICLUST algorithm \citep{revelle:iclust} (see section \ref{sect:iclust}). Also consider a hierarchical factor solution to find coefficient $\omega$ (see \ref{sect:omega}). \begin{scriptsize} \begin{Schunk} \begin{Sinput} fa(myData) iclust(myData) omega(myData) \end{Sinput} \end{Schunk} \end{scriptsize} If you prefer to do a principal components analysis you may use the \pfun{principal} function. The default is one component. \begin{scriptsize} \begin{Schunk} \begin{Sinput} principal(myData) \end{Sinput} \end{Schunk} \end{scriptsize} \item Some people like to find coefficient $\alpha$ as an estimate of reliability. This may be done for a single scale using the \pfun{alpha} function (see \ref{sect:alpha}). Perhaps more useful is the ability to create several scales as unweighted averages of specified items using the \pfun{scoreItems} function (see \ref{sect:score}) and to find various estimates of internal consistency for these scales, find their intercorrelations, and find scores for all the subjects. \begin{scriptsize} \begin{Schunk} \begin{Sinput} alpha(myData) #score all of the items as part of one scale. myKeys <- make.keys(nvar=20,list(first = c(1,-3,5,-7,8:10),second=c(2,4,-6,11:15,-16))) my.scores <- scoreItems(myKeys,myData) #form several scales my.scores #show the highlights of the results \end{Sinput} \end{Schunk} \end{scriptsize} \end{itemize} \end{enumerate} At this point you have had a chance to see the highlights of the \Rpkg{psych} package and to do some basic (and advanced) data analysis. You might find reading this entire vignette as well as the Overview Vignette to be helpful to get a broader understanding of what can be done in \R{} using the \Rpkg{psych}. Remember that the help command (?) is available for every function. Try running the examples for each help page. \newpage\newpage \section{Overview of this and related documents} The \Rpkg{psych} package \citep{psych} has been developed at Northwestern University since 2005 to include functions most useful for personality, psychometric, and psychological research. The package is also meant to supplement a text on psychometric theory \citep{revelle:intro}, a draft of which is available at \url{https://personality-project.org/r/book/}. Some of the functions (e.g., \pfun{read.file}, \pfun{read.clipboard}, \pfun{describe}, \pfun{pairs.panels}, \pfun{scatter.hist}, \pfun{error.bars}, \pfun{multi.hist}, \pfun{bi.bars}) are useful for basic data entry and descriptive analyses. Psychometric applications emphasize techniques for dimension reduction including factor analysis, cluster analysis, and principal components analysis. The \pfun{fa} function includes five methods of \iemph{factor analysis} (\iemph{minimum residual}, \iemph{principal axis}, \iemph{weighted least squares}, \iemph{generalized least squares} and \iemph{maximum likelihood} factor analysis). Principal Components Analysis (PCA) is also available through the use of the \pfun{principal} or \pfun{pca} functions. Determining the number of factors or components to extract may be done by using the Very Simple Structure \citep{revelle:vss} (\pfun{vss}), Minimum Average Partial correlation \citep{velicer:76} (\pfun{MAP}) or parallel analysis (\pfun{fa.parallel}) criteria. These and several other criteria are included in the \pfun{nfactors} function. Two parameter Item Response Theory (IRT) models for dichotomous or polytomous items may be found by factoring \pfun{tetrachoric} or \pfun{polychoric} correlation matrices and expressing the resulting parameters in terms of location and discrimination using \pfun{irt.fa}. Bifactor and hierarchical factor structures may be estimated by using Schmid Leiman transformations \citep{schmid:57} (\pfun{schmid}) to transform a hierarchical factor structure into a \iemph{bifactor} solution \citep{holzinger:37}. Higher order models can also be found using \pfun{fa.multi}. Scale construction can be done using the Item Cluster Analysis \citep{revelle:iclust} (\pfun{iclust}) function to determine the structure and to calculate reliability coefficients $\alpha$ \citep{cronbach:51}(\pfun{alpha}, \pfun{scoreItems}, \pfun{score.multiple.choice}), $\beta$ \citep{revelle:iclust,rz:09} (\pfun{iclust}) and McDonald's $\omega_h$ and $\omega_t$ \citep{mcdonald:tt} (\pfun{omega}). Guttman's six estimates of internal consistency reliability (\cite{guttman:45}, as well as additional estimates \citep{rz:09} are in the \pfun{guttman} function. The six measures of Intraclass correlation coefficients (\pfun{ICC}) discussed by \cite{shrout:79} are also available. For data with a a multilevel structure (e.g., items within subjects across time, or items within subjects across groups), the \pfun{describeBy}, \pfun{statsBy} functions will give basic descriptives by group. \pfun{StatsBy} also will find within group (or subject) correlations as well as the between group correlation. \pfun{multilevel.reliability} \pfun{mlr} will find various generalizability statistics for subjects over time and items. \pfun{mlPlot} will graph items over for each subject, \pfun{mlArrange} converts wide data frames to long data frames suitable for multilevel modeling. Graphical displays include Scatter Plot Matrix (SPLOM) plots using \pfun{pairs.panels}, correlation ``heat maps'' (\pfun{corPlot}) factor, cluster, and structural diagrams using \pfun{fa.diagram}, \pfun{iclust.diagram}, \pfun{structure.diagram} and \pfun{het.diagram}, as well as item response characteristics and item and test information characteristic curves \pfun{plot.irt} and \pfun{plot.poly}. This vignette is meant to give an overview of the \Rpkg{psych} package. That is, it is meant to give a summary of the main functions in the \Rpkg{psych} package with examples of how they are used for data description, dimension reduction, and scale construction. The extended user manual at \url{psych_manual.pdf} includes examples of graphic output and more extensive demonstrations than are found in the help menus. (Also available at \url{https://personality-project.org/r/psych_manual.pdf}). The vignette, psych for sem, at \url{psych_for_sem.pdf}, discusses how to use psych as a front end to the \Rpkg{sem} package of John Fox \citep{sem}. (The vignette is also available at \href{"https://personality-project.org/r/book/psych_for_sem.pdf"}{\url{https://personality-project.org/r/book/psych_for_sem.pdf}}). In addition, there are a growing number of ``HowTo"s at the personality project. Currently these include: \begin{enumerate} \item An \href{https://personality-project.org/r/psych/intro.pdf}{introduction} (vignette) of the \Rpkg{psych} package \item An \href{https://personality-project.org/r/psych/overview.pdf}{overview} (vignette) of the \Rpkg{psych} package \item \href{https://personality-project.org/r/psych/HowTo/getting_started.pdf}{Installing} \R{} and some useful packages \item Using \R{} and the \Rpkg{psych} package to find \href{https://personality-project.org/r/psych/HowTo/omega.pdf}{$omega_h$} and $\omega_t$. \item Using \R{} and the \Rpkg{psych} for \href{https://personality-project.org/r/psych/HowTo/factor.pdf}{factor analysis} and principal components analysis. \item Using the \pfun{scoreItems} function to find \href{https://personality-project.org/r/psych/HowTo/scoring.pdf}{scale scores and scale statistics}. \item Using \pfun{mediate} and \pfun{setCor} to do \href{https://personality-project.org/r/psych/HowTo/mediation.pdf}{mediation, moderation and regression analysis}. \end{enumerate} For a step by step tutorial in the use of the psych package and the base functions in R for basic personality research, see the guide for using \R{} for personality research at \url{https://personalitytheory.org/r/r.short.html}. For an \iemph{introduction to psychometric theory with applications in \R{}}, see the draft chapters at \url{https://personality-project.org/r/book}). \section{Getting started} \label{sect:starting} Some of the functions described in this overview require other packages. Particularly useful for rotating the results of factor analyses (from e.g., \pfun{fa}, \pfun{factor.minres}, \pfun{factor.pa}, \pfun{factor.wls}, or \pfun {principal}) or hierarchical factor models using \pfun{omega} or \pfun{schmid}, is the \Rpkg{GPArotation} package. These and other useful packages may be installed by first installing and then using the task views (\Rpkg{ctv}) package to install the ``Psychometrics" task view, but doing it this way is not necessary. \begin{Schunk} \begin{Sinput} install.packages("ctv") library(ctv) task.views("Psychometrics") \end{Sinput} \end{Schunk} The ``Psychometrics'' task view will install a large number of useful packages. To install the bare minimum for the examples in this vignette, it is necessary to install just 3 packages: \begin{Schunk} \begin{Sinput} install.packages(list(c("GPArotation","mnormt","psychTools") \end{Sinput} \end{Schunk} Because of the difficulty of installing the package \Rpkg{Rgraphviz}, alternative graphics have been developed and are available as \iemph{diagram} functions. If \Rpkg{Rgraphviz} is available, some functions will take advantage of it. An alternative is to use ``dot'' output of commands for any external graphics package that uses the dot language. \section{Basic data analysis} A number of \Rpkg{psych} functions facilitate the entry of data and finding basic descriptive statistics. These are described in more detail in the companion vignette: An introduction to the psych package: Part I which is also available from the personality-project site. \url{https://personality-project.org/r/psych/vignettes/intro.pdf}. Please consult that vignette first for information on how to read data (particularly using the \pfun{read.file} and \pfun{read.clipboard} commands), Also, the \pfun{describe} and \pfun{describeBy} functions are described in more detail in the introductory vignette. For even though you probably want to jump immediately to factor analyze your data, this is a mistake. It is very important to first describe them and look for weird responses. It is also useful to \pfun{scrub} your data when removing outliers, to graphically display them using \pfun{pairs.panesl} and \pfun{corPlot}. Basic multiple regression and moderated or mediated regressions may be done from either the raw data or from correlation matrices using \pfun{setCor}, or \pfun{mediation}. Remember, to run any of the \Rpkg{psych} functions, it is necessary to make the package active by using the \fun{library} command: \begin{Schunk} \begin{Sinput} library(psych) \end{Sinput} \end{Schunk} The other packages, once installed, will be called automatically by \Rpkg{psych}. It is possible to automatically load \Rpkg{psych} and other functions by creating and then saving a ``.First" function: e.g., \begin{Schunk} \begin{Sinput} .First <- function(x) {library(psych)} \end{Sinput} \end{Schunk} \section{Item and scale analysis} The main functions in the \Rpkg{psych} package are for analyzing the structure of items and of scales and for finding various estimates of scale reliability. These may be considered as problems of dimension reduction (e.g., factor analysis, cluster analysis, principal components analysis) and of forming and estimating the reliability of the resulting composite scales. \subsection{Dimension reduction through factor analysis and cluster analysis} \label{sect:fa} Parsimony of description has been a goal of science since at least the famous dictum commonly attributed to William of Ockham to not multiply entities beyond necessity\footnote{Although probably neither original with Ockham nor directly stated by him \citep{thornburn:1918}, Ockham's razor remains a fundamental principal of science.}. The goal for parsimony is seen in psychometrics as an attempt either to describe (components) or to explain (factors) the relationships between many observed variables in terms of a more limited set of components or latent factors. The typical data matrix represents multiple items or scales usually thought to reflect fewer underlying constructs\footnote{\cite{cattell:fa78} as well as \cite{maccallum:07} argue that the data are the result of many more factors than observed variables, but are willing to estimate the major underlying factors.}. At the most simple, a set of items can be be thought to represent a random sample from one underlying domain or perhaps a small set of domains. The question for the psychometrician is how many domains are represented and how well does each item represent the domains. Solutions to this problem are examples of \iemph{factor analysis} (\iemph{FA}), \iemph{principal components analysis} (\iemph{PCA}), and \iemph{cluster analysis} (\emph{CA}). All of these procedures aim to reduce the complexity of the observed data. In the case of FA, the goal is to identify fewer underlying constructs to explain the observed data. In the case of PCA, the goal can be mere data reduction, but the interpretation of components is frequently done in terms similar to those used when describing the latent variables estimated by FA. Cluster analytic techniques, although usually used to partition the subject space rather than the variable space, can also be used to group variables to reduce the complexity of the data by forming fewer and more homogeneous sets of tests or items. At the data level the data reduction problem may be solved as a \iemph{Singular Value Decomposition} of the original matrix, although the more typical solution is to find either the \iemph{principal components} or \iemph{factors} of the covariance or correlation matrices. Given the pattern of regression weights from the variables to the components or from the factors to the variables, it is then possible to find (for components) individual \index{component scores} \emph{component} or \iemph{cluster scores} or estimate (for factors) \iemph{factor scores}. Several of the functions in \Rpkg{psych} address the problem of data reduction. \begin{description} \item[\pfun{fa}] incorporates six alternative algorithms: \iemph{minres factor analysis}, \iemph{principal axis factor analysis}, \iemph{alpha factor analysis}, \iemph{weighted least squares factor analysis}, \iemph{generalized least squares factor analysis} and \iemph{maximum likelihood factor analysis}. That is, it includes the functionality of three other functions that are deprecated and will be eventually phased out. \begin{tiny} \item[\pfun{fa.poly} (deprecated) ] is useful when finding the factor structure of categorical items. \pfun{fa.poly} first finds the tetrachoric or polychoric correlations between the categorical variables and then proceeds to do a normal factor analysis. By setting the n.iter option to be greater than 1, it will also find confidence intervals for the factor solution. Warning. Finding polychoric correlations is very slow, so think carefully before doing so. These options are now part of the \iemph{fa} function and can be controlled by setting the cor parameter to `tet' or `poly'. \item [\pfun{factor.minres} (deprecated)] Minimum residual factor analysis is a least squares, iterative solution to the factor problem. minres attempts to minimize the residual (off-diagonal) correlation matrix. It produces solutions similar to maximum likelihood solutions, but will work even if the matrix is singular. \item [\pfun{factor.pa} (deprecated)] Principal Axis factor analysis is a least squares, iterative solution to the factor problem. PA will work for cases where maximum likelihood techniques (\fun{factanal}) will not work. The original communality estimates are either the squared multiple correlations (\pfun{smc}) for each item or 1. \item [\pfun{factor.wls} (deprecated)] Weighted least squares factor analysis is a least squares, iterative solution to the factor problem. It minimizes the (weighted) squared residual matrix. The weights are based upon the independent contribution of each variable. \end{tiny} \item [\pfun{principal}] Principal Components Analysis reports the largest n eigen vectors rescaled by the square root of their eigen values. Note that PCA is not the same as factor analysis and the two should not be confused. \item [\pfun{factor.congruence}] The congruence between two factors is the cosine of the angle between them. This is just the cross products of the loadings divided by the sum of the squared loadings. This differs from the correlation coefficient in that the mean loading is not subtracted before taking the products. \pfun{factor.congruence} will find the cosines between two (or more) sets of factor loadings. \item [\pfun{vss}] Very Simple Structure \cite{revelle:vss} applies a goodness of fit test to determine the optimal number of factors to extract. It can be thought of as a quasi-confirmatory model, in that it fits the very simple structure (all except the biggest c loadings per item are set to zero where c is the level of complexity of the item) of a factor pattern matrix to the original correlation matrix. For items where the model is usually of complexity one, this is equivalent to making all except the largest loading for each item 0. This is typically the solution that the user wants to interpret. The analysis includes the \pfun{MAP} criterion of \cite{velicer:76} and a $\chi^2$ estimate. \item [\pfun{nfactors}] combines VSS, MAP, and a number of other fit statistics. The depressing reality is that frequently these conventional fit estimates of the number of factors do not agree. \item [\pfun{fa.parallel}] The parallel factors technique compares the observed eigen values of a correlation matrix with those from random data. \item [\pfun{fa.plot}] will plot the loadings from a factor, principal components, or cluster analysis (just a call to plot will suffice). If there are more than two factors, then a SPLOM of the loadings is generated. \item[\pfun{fa.diagram}] replaces \pfun{fa.graph} and will draw a path diagram representing the factor structure. It does not require Rgraphviz and thus is probably preferred. \item[\pfun{fa.graph}] requires \fun{Rgraphviz} and will draw a graphic representation of the factor structure. If factors are correlated, this will be represented as well. \item[\pfun{iclust} ] is meant to do item cluster analysis using a hierarchical clustering algorithm specifically asking questions about the reliability of the clusters \citep{revelle:iclust}. Clusters are formed until either coefficient $\alpha$ \cite{cronbach:51} or $\beta$ \cite{revelle:iclust} fail to increase. \end{description} \subsubsection{Minimum Residual Factor Analysis} \label{sect:minres} The factor model is an approximation of a correlation matrix by a matrix of lower rank. That is, can the correlation matrix, $\vec{_nR_n}$ be approximated by the product of a factor matrix, $\vec{_nF_k}$ and its transpose plus a diagonal matrix of uniqueness. \begin{equation} R = FF' + U^2 \end{equation} The maximum likelihood solution to this equation is found by \fun{factanal} in the \Rpkg{stats} package as well as the \pfun{fa} function in \Rpkg{psych}. Seven alternatives are provided in \Rpkg{psych}, all of them are included in the \pfun{fa} function and are called by specifying the factor method (e.g., fm=``minres", fm=``pa", fm=``alpha" fm=`wls", fm=``gls", fm = ``min.rank", and fm=``ml"). In the discussion of the other algorithms, the calls shown are to the \pfun{fa} function specifying the appropriate method. \pfun{factor.minres} attempts to minimize the off diagonal residual correlation matrix by adjusting the eigen values of the original correlation matrix. This is similar to what is done in \fun{factanal}, but uses an ordinary least squares instead of a maximum likelihood fit function. The solutions tend to be more similar to the MLE solutions than are the \pfun{factor.pa} solutions. \iemph{min.res} is the default for the \pfun{fa} function. A classic data set, collected by \cite{thurstone:41} and then reanalyzed by \cite{bechtoldt:61} and discussed by \cite{mcdonald:tt}, is a set of 9 cognitive variables with a clear bi-factor structure \citep{holzinger:37}. The minimum residual solution was transformed into an oblique solution using the default option on rotate which uses an oblimin transformation (Table~\ref{tab:factor.minres}). Alternative rotations and transformations include ``none", ``varimax", ``quartimax", ``bentlerT", ``varimin'' and ``geominT" (all of which are orthogonal rotations). as well as ``promax", ``oblimin", ``simplimax", ``bentlerQ, and ``geominQ" and ``cluster" which are possible oblique transformations of the solution. The default is to do a oblimin transformation. The measures of factor adequacy reflect the multiple correlations of the factors with the best fitting linear regression estimates of the factor scores \citep{grice:01}. Note that if extracting more than one factor, and doing any oblique rotation, it is necessary to have the \Rpkg{GPArotation} installed. This is checked for in the appropriate functions. <>= if(!require('GPArotation')) {stop('GPArotation must be installed to do rotations')} @ \begin{table}[htpb] \caption{Three correlated factors from the Thurstone 9 variable problem. By default, the solution is transformed obliquely using oblimin. The extraction method is (by default) minimum residual.} \begin{scriptsize} \begin{center} <>= if(!require('GPArotation')) {stop('GPArotation must be installed to do rotations')} else { library(psych) library(psychTools) f3t <- fa(Thurstone,3,n.obs=213) f3t } @ \end{center} \end{scriptsize} \label{tab:factor.minres} \end{table}% \subsubsection{Principal Axis Factor Analysis} An alternative, least squares algorithm (included in \pfun{fa} with the fm=pa option or as a standalone function (\pfun{factor.pa}), does a Principal Axis factor analysis by iteratively doing an eigen value decomposition of the correlation matrix with the diagonal replaced by the values estimated by the factors of the previous iteration. This OLS solution is not as sensitive to improper matrices as is the maximum likelihood method, and will sometimes produce more interpretable results. It seems as if the SAS example for PA uses only one iteration. Setting the max.iter parameter to 1 produces the SAS solution. The solutions from the \pfun{fa}, the \pfun{factor.minres} and \pfun{factor.pa} as well as the \pfun{principal} functions can be rotated or transformed with a number of options. Some of these call the \Rpkg{GPArotation} package. Orthogonal rotations include \fun{varimax}, \fun{quartimax}, \pfun{varimin}, \pfun{bifactor} . Oblique transformations include \fun{oblimin}, \fun{quartimin}, \pfun{biquartimin} and then two targeted rotation functions \pfun{Promax} and \pfun{target.rot}. The latter of these will transform a loadings matrix towards an arbitrary target matrix. The default is to transform towards an independent cluster solution. Using the Thurstone data set, three factors were requested and then transformed into an independent clusters solution using \pfun{target.rot} (Table~\ref{tab:Thurstone}). \begin{table}[htpb] \caption{The 9 variable problem from Thurstone is a classic example of factoring where there is a higher order factor, g, that accounts for the correlation between the factors. The extraction method was principal axis. The transformation was a targeted transformation to a simple cluster solution.} \begin{center} \begin{scriptsize} <>= if(!require('GPArotation')) {stop('GPArotation must be installed to do rotations')} else { f3 <- fa(Thurstone,3,n.obs = 213,fm="pa") f3o <- target.rot(f3) f3o} @ \end{scriptsize} \end{center} \label{tab:Thurstone} \end{table} \subsubsection{Alpha Factor Analysis} Introduced by \cite{kaiser:65} and discussed by \cite{loehlin:17}, \emph{alpha factor analysis} factors the correlation matrix of correlations or covariances corrected for their communalities. This has the effect of making all correlations corrected for reliabiity to reflect their true, latent correlations. \emph{alpha factor analysis} was added in August, 2017 to increase the range of EFA options available. This is added more completeness rather than an endorsement of the procedure. It is worth comparing solutions from minres, alpha, and MLE, for they are not the same. \subsubsection{Weighted Least Squares Factor Analysis} \label{sect:wls} Similar to the minres approach of minimizing the squared residuals, factor method ``wls" weights the squared residuals by their uniquenesses. This tends to produce slightly smaller overall residuals. In the example of weighted least squares, the output is shown by using the \pfun{print} function with the cut option set to 0. That is, all loadings are shown (Table~\ref{tab:Thurstone.wls}). \begin{table}[htpb] \caption{The 9 variable problem from Thurstone is a classic example of factoring where there is a higher order factor, g, that accounts for the correlation between the factors. The factors were extracted using a weighted least squares algorithm. All loadings are shown by using the cut=0 option in the \pfun{print.psych} function.} \begin{scriptsize} <>= f3w <- fa(Thurstone,3,n.obs = 213,fm="wls") print(f3w,cut=0,digits=3) @ \end{scriptsize} \label{tab:Thurstone.wls} \end{table} subsection{Displaying factor solutions} The unweighted least squares solution may be shown graphically using the \pfun{fa.plot} function which is called by the generic \fun{plot} function (Figure~\ref{fig:thurstone}). Factors were transformed obliquely using a oblimin. These solutions may be shown as item by factor plots (Figure~\ref{fig:thurstone}) or by a structure diagram (Figure~\ref{fig:thurstone.diagram}). \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= plot(f3t) @ \end{scriptsize} \caption{A graphic representation of the 3 oblique factors from the Thurstone data using \pfun{plot}. Factors were transformed to an oblique solution using the oblimin function from the GPArotation package.} \label{fig:thurstone} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= fa.diagram(f3t) @ \end{scriptsize} \caption{A graphic representation of the 3 oblique factors from the Thurstone data using \pfun{fa.diagram}. Factors were transformed to an oblique solution using oblimin.} \label{fig:thurstone.diagram} \end{center} \end{figure} A comparison of these three approaches suggests that the minres solution is more similar to a maximum likelihood solution and fits slightly better than the pa or wls solutions. Comparisons with SPSS suggest that the pa solution matches the SPSS OLS solution, but that the minres solution is slightly better. At least in one test data set, the weighted least squares solutions, although fitting equally well, had slightly different structure loadings. Note that the rotations used by SPSS will sometimes use the ``Kaiser Normalization''. By default, the rotations used in psych do not normalize, but this can be specified as an option in \pfun{fa}. \subsubsection{Principal Components analysis (PCA)} An alternative to factor analysis, which is unfortunately frequently confused with \iemph{factor analysis}, is \iemph{principal components analysis}. Although the goals of \iemph{PCA} and \iemph{FA} are similar, PCA is a descriptive model of the data, while FA is a structural model. Some psychologists use PCA in a manner similar to factor analysis and thus the \pfun{principal} function produces output that is perhaps more understandable than that produced by \fun{princomp} in the \Rpkg{stats} package. Table~\ref{tab:pca} shows a PCA of the Thurstone 9 variable problem rotated using the \pfun{Promax} function. Note how the loadings from the factor model are similar but smaller than the principal component loadings. This is because the PCA model attempts to account for the entire variance of the correlation matrix, while FA accounts for just the \iemph{common variance}. This distinction becomes most important for small correlation matrices. Also note how the goodness of fit statistics, based upon the residual off diagonal elements, is much worse than the \pfun{fa} solution. \begin{table}[htpb] \caption{The Thurstone problem can also be analyzed using Principal Components Analysis. Compare this to Table~\ref{tab:Thurstone}. The loadings are higher for the PCA because the model accounts for the unique as well as the common variance.The fit of the off diagonal elements, however, is much worse than the \pfun{fa} results.} \begin{center} \begin{scriptsize} <>= p3p <-principal(Thurstone,3,n.obs = 213,rotate="Promax") p3p @ \end{scriptsize} \end{center} \label{tab:pca} \end{table} \subsubsection{Hierarchical and bi-factor solutions} \label{sect:omega} For a long time structural analysis of the ability domain have considered the problem of factors that are themselves correlated. These correlations may themselves be factored to produce a higher order, general factor. An alternative \citep{holzinger:37,jensen:weng} is to consider the general factor affecting each item, and then to have group factors account for the residual variance. Exploratory factor solutions to produce a hierarchical or a bifactor solution are found using the \pfun{omega} function. This technique has more recently been applied to the personality domain to consider such things as the structure of neuroticism (treated as a general factor, with lower order factors of anxiety, depression, and aggression). Consider the 9 Thurstone variables analyzed in the prior factor analyses. The correlations between the factors (as shown in Figure~\ref{fig:thurstone.diagram} can themselves be factored. This results in a higher order factor model (Figure~\ref{fig:omega}). An an alternative solution is to take this higher order model and then solve for the general factor loadings as well as the loadings on the residualized lower order factors using the \iemph{Schmid-Leiman} procedure. (Figure ~\ref{fig:omega.2}). Yet another solution is to use structural equation modeling to directly solve for the general and group factors. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= om.h <- omega(Thurstone,n.obs=213,sl=FALSE) op <- par(mfrow=c(1,1)) @ \end{scriptsize} \caption{A higher order factor solution to the Thurstone 9 variable problem} \label{fig:omega} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= om <- omega(Thurstone,n.obs=213) @ \end{scriptsize} \caption{A bifactor factor solution to the Thurstone 9 variable problem} \label{fig:omega.2} \end{center} \end{figure} Yet another approach to the bifactor structure is do use the \pfun{bifactor} rotation function in either \Rpkg{psych} or in \Rpkg{GPArotation}. This does the rotation discussed in \cite{jennrich:11}. \subsubsection{Item Cluster Analysis: iclust} \label{sect:iclust} An alternative to factor or components analysis is \iemph{cluster analysis}. The goal of cluster analysis is the same as factor or components analysis (reduce the complexity of the data and attempt to identify homogeneous subgroupings). Mainly used for clustering people or objects (e.g., projectile points if an anthropologist, DNA if a biologist, galaxies if an astronomer), clustering may be used for clustering items or tests as well. Introduced to psychologists by \cite{tryon:39} in the 1930's, the cluster analytic literature exploded in the 1970s and 1980s \citep{blashfield:80,blashfield:88,everitt:74,hartigan:75}. Much of the research is in taxonmetric applications in biology \citep{sneath:73,sokal:63} and marketing \citep{cooksey:06} where clustering remains very popular. It is also used for taxonomic work in forming clusters of people in family \citep{henry:05} and clinical psychology \citep{martinent:07,mun:08}. Interestingly enough it has has had limited applications to psychometrics. This is unfortunate, for as has been pointed out by e.g. \citep{tryon:35,loevinger:53}, the theory of factors, while mathematically compelling, offers little that the geneticist or behaviorist or perhaps even non-specialist finds compelling. \cite{cooksey:06} reviews why the \pfun{iclust} algorithm is particularly appropriate for scale construction in marketing. \emph{Hierarchical cluster analysis} \index{hierarchical cluster analysis} forms clusters that are nested within clusters. The resulting \iemph{tree diagram} (also known somewhat pretentiously as a \iemph{rooted dendritic structure}) shows the nesting structure. Although there are many hierarchical clustering algorithms in \R{} (e.g., \fun{agnes}, \fun{hclust}, and \pfun{iclust}), the one most applicable to the problems of scale construction is \pfun{iclust} \citep{revelle:iclust}. \begin{enumerate} \item Find the proximity (e.g. correlation) matrix, \item Identify the most similar pair of items \item Combine this most similar pair of items to form a new variable (cluster), \item Find the similarity of this cluster to all other items and clusters, \item Repeat steps 2 and 3 until some criterion is reached (e.g., typicallly, if only one cluster remains or in \pfun{iclust} if there is a failure to increase reliability coefficients $\alpha$ or $\beta$). \item Purify the solution by reassigning items to the most similar cluster center. \end{enumerate} \pfun{iclust} forms clusters of items using a hierarchical clustering algorithm until one of two measures of internal consistency fails to increase \citep{revelle:iclust}. The number of clusters may be specified a priori, or found empirically. The resulting statistics include the average split half reliability, $\alpha$ \citep{cronbach:51}, as well as the worst split half reliability, $\beta$ \citep{revelle:iclust}, which is an estimate of the general factor saturation of the resulting scale (Figure~\ref{fig:iclust}). Cluster loadings (corresponding to the structure matrix of factor analysis) are reported when printing (Table~\ref{tab:iclust}). The pattern matrix is available as an object in the results. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= data(bfi) ic <- iclust(bfi[1:25]) @ \end{scriptsize} \caption{Using the \pfun{iclust} function to find the cluster structure of 25 personality items (the three demographic variables were excluded from this analysis). When analyzing many variables, the tree structure may be seen more clearly if the graphic output is saved as a pdf and then enlarged using a pdf viewer.} \label{fig:iclust} \end{center} \end{figure} \begin{table}[htpb] \caption{The summary statistics from an iclust analysis shows three large clusters and smaller cluster.} \begin{center} \begin{scriptsize} <>= summary(ic) #show the results @ \end{scriptsize} \end{center} \label{tab:iclust} \end{table}% The previous analysis (Figure~\ref{fig:iclust}) was done using the Pearson correlation. A somewhat cleaner structure is obtained when using the \pfun{polychoric} function to find polychoric correlations (Figure~\ref{fig:iclust.poly}). Note that the first time finding the polychoric correlations some time, but the next three analyses were done using that correlation matrix (r.poly\$rho). When using the console for input, \pfun{polychoric} will report on its progress while working using \pfun{progressBar}. \begin{table}[htpb] \caption{The \pfun{polychoric} and the \pfun{tetrachoric} functions can take a long time to finish and report their progress by a series of dots as they work. The dots are suppressed when creating a Sweave document.} \begin{center} \begin{tiny} <>= data(bfi) r.poly <- polychoric(bfi[1:25],correct=0) #the ... indicate the progress of the function @ \end{tiny} \end{center} \label{tab:bad}1.7.1\end{table}% \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= ic.poly <- iclust(r.poly$rho,title="ICLUST using polychoric correlations") iclust.diagram(ic.poly) @ \end{scriptsize} \caption{ICLUST of the BFI data set using polychoric correlations. Compare this solution to the previous one (Figure~\ref{fig:iclust}) which was done using Pearson correlations. } \label{fig:iclust.poly} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= ic.poly <- iclust(r.poly$rho,5,title="ICLUST using polychoric correlations for nclusters=5") iclust.diagram(ic.poly) @ \end{scriptsize} \caption{ICLUST of the BFI data set using polychoric correlations with the solution set to 5 clusters. Compare this solution to the previous one (Figure~\ref{fig:iclust.poly}) which was done without specifying the number of clusters and to the next one (Figure~\ref{fig:iclust.3}) which was done by changing the beta criterion. } \label{fig:iclust.5} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= ic.poly <- iclust(r.poly$rho,beta.size=3,title="ICLUST beta.size=3") @ \end{scriptsize} \caption{ICLUST of the BFI data set using polychoric correlations with the beta criterion set to 3. Compare this solution to the previous three (Figure~\ref{fig:iclust},~\ref{fig:iclust.poly}, \ref{fig:iclust.5}).} \label{fig:iclust.3} \end{center} \end{figure} \begin{table}[htpb] \caption{The output from \pfun{iclust} includes the loadings of each item on each cluster. These are equivalent to factor structure loadings. By specifying the value of cut, small loadings are suppressed. The default is for cut=0.su } \begin{center} \begin{scriptsize} <>= print(ic,cut=.3) @ \end{scriptsize} \end{center} \label{tab:iclust} \end{table}% A comparison of these four cluster solutions suggests both a problem and an advantage of clustering techniques. The problem is that the solutions differ. The advantage is that the structure of the items may be seen more clearly when examining the clusters rather than a simple factor solution. \subsection{Estimates of fit} Exploratory factoring techniques are sometimes criticized because of the lack of statistical information on the solutions. There are perhaps as many fit statistics as there are psychometricians. When using Maximum Likelihood extraction, many of these various fit statistics are based upon the $\chi^{2}$ which is minimized using ML. If not using ML, these same statistics can be found, but they are no longer maximum likelihood estimates. They are, however, still useful. Overall estimates of goodness of fit including $\chi^{2}$ and RMSEA are found in the \pfun{fa} and \pfun{omega} functions. \subsection{Confidence intervals using bootstrapping techniques} Confidence intervals for the factor loadings may be found by doing multiple bootstrapped iterations of the original analysis. This is done by setting the n.iter parameter to the desired number of iterations. This can be done for factoring of Pearson correlation matrices as well as polychoric/tetrachoric matrices (See Table~\ref{tab:bootstrap}). Although the example value for the number of iterations is set to 20, more conventional analyses might use 1000 bootstraps. This will take much longer. Bootstrapped confidence intervals can also be found for the loadings of a factoring of a polychoric matrix. \pfun{fa.poly} will find the polychoric correlation matrix and if the n.iter option is greater than 1, will then randomly resample the data (case wise) to give bootstrapped samples. This will take a long time for large number of items or interations. \begin{table}[htpb] \caption{An example of bootstrapped confidence intervals on 10 items from the Big 5 inventory. The number of bootstrapped samples was set to 20. More conventional bootstrapping would use 100 or 1000 replications. } \begin{tiny} \begin{center} <>= fa(bfi[1:10],2,n.iter=20) @ \end{center} \end{tiny} \label{tab:bootstrap} \end{table}% \subsection{Comparing factor/component/cluster solutions} Cluster analysis, factor analysis, and principal components analysis all produce structure matrices (matrices of correlations between the dimensions and the variables) that can in turn be compared in terms of Burt's \iemph{congruence coefficient} (also known as Tucker's coefficient) which is just the cosine of the angle between the dimensions $$c_{f_{i}f_{j}} = \frac{\sum_{k=1}^{n}{f_{ik}f_{jk}}} {\sum{f_{ik}^{2}}\sum{f_{jk}^{2}}}.$$ Consider the case of a four factor solution and four cluster solution to the Big Five problem. \begin{scriptsize} <>= f4 <- fa(bfi[1:25],4,fm="pa") factor.congruence(f4,ic) @ \end{scriptsize} A more complete comparison of oblique factor solutions (both minres and principal axis), bifactor and component solutions to the Thurstone data set is done using the \pfun{factor.congruence} function. (See table~\ref{tab:congruence}). \begin{table}[htpb] \caption{Congruence coefficients for oblique factor, bifactor and component solutions for the Thurstone problem.} \begin{scriptsize} <>= factor.congruence(list(f3t,f3o,om,p3p)) @ \end{scriptsize} \label{tab:congruence} \end{table}% \subsection{Determining the number of dimensions to extract.} How many dimensions to use to represent a correlation matrix is an unsolved problem in psychometrics. There are many solutions to this problem, none of which is uniformly the best. Henry Kaiser once said that ``a solution to the number-of factors problem in factor analysis is easy, that he used to make up one every morning before breakfast. But the problem, of course is to find \emph{the} solution, or at least a solution that others will regard quite highly not as the best" \cite{horn:79}. Techniques most commonly used include 1) Extracting factors until the chi square of the residual matrix is not significant. 2) Extracting factors until the change in chi square from factor n to factor n+1 is not significant. 3) Extracting factors until the eigen values of the real data are less than the corresponding eigen values of a random data set of the same size (parallel analysis) \pfun{fa.parallel} \citep{horn:65}. 4) Plotting the magnitude of the successive eigen values and applying the scree test (a sudden drop in eigen values analogous to the change in slope seen when scrambling up the talus slope of a mountain and approaching the rock face \citep{cattell:scree}. 5) Extracting factors as long as they are interpretable. 6) Using the Very Structure Criterion (\pfun{vss}) \citep{revelle:vss}. 7) Using Wayne Velicer's Minimum Average Partial (\pfun{MAP}) criterion \citep{velicer:76}. 8) Extracting principal components until the eigen value < 1. Each of the procedures has its advantages and disadvantages. Using either the chi square test or the change in square test is, of course, sensitive to the number of subjects and leads to the nonsensical condition that if one wants to find many factors, one simply runs more subjects. Parallel analysis is partially sensitive to sample size in that for large samples the eigen values of random factors will all tend towards 1. The scree test is quite appealing but can lead to differences of interpretation as to when the scree ``breaks". Extracting interpretable factors means that the number of factors reflects the investigators creativity more than the data. vss, while very simple to understand, will not work very well if the data are very factorially complex. (Simulations suggests it will work fine if the complexities of some of the items are no more than 2). The eigen value of 1 rule, although the default for many programs, seems to be a rough way of dividing the number of variables by 3 and is probably the worst of all criteria. An additional problem in determining the number of factors is what is considered a factor. Many treatments of factor analysis assume that the residual correlation matrix after the factors of interest are extracted is composed of just random error. An alternative concept is that the matrix is formed from major factors of interest but that there are also numerous minor factors of no substantive interest but that account for some of the shared covariance between variables. The presence of such minor factors can lead one to extract too many factors and to reject solutions on statistical grounds of misfit that are actually very good fits to the data. This problem is partially addressed later in the discussion of simulating complex structures using \pfun{sim.structure} and of small extraneous factors using the \pfun{sim.minor} function. \subsubsection{Very Simple Structure} \label{sect:vss} The \pfun{vss} function compares the fit of a number of factor analyses with the loading matrix ``simplified" by deleting all except the c greatest loadings per item, where c is a measure of factor complexity \cite{revelle:vss}. Included in \pfun{vss} is the MAP criterion (Minimum Absolute Partial correlation) of \cite{velicer:76}. Using the Very Simple Structure criterion for the bfi data suggests that 4 factors are optimal (Figure~\ref{fig:vss}). However, the MAP criterion suggests that 5 is optimal. \begin{figure}[htbp] \begin{center} <>= vss <- vss(bfi[1:25],title="Very Simple Structure of a Big 5 inventory") @ \caption{The Very Simple Structure criterion for the number of factors compares solutions for various levels of item complexity and various numbers of factors. For the Big 5 Inventory, the complexity 1 and 2 solutions both achieve their maxima at four factors. This is in contrast to parallel analysis which suggests 6 and the MAP criterion which suggests 5. } \label{fig:vss} \end{center} \end{figure} \begin{scriptsize} <>= vss @ \end{scriptsize} \subsubsection{Parallel Analysis} \label{sect:fa.parallel} An alternative way to determine the number of factors is to compare the solution to random data with the same properties as the real data set. If the input is a data matrix, the comparison includes random samples from the real data, as well as normally distributed random data with the same number of subjects and variables. For the BFI data, parallel analysis suggests that 6 factors might be most appropriate (Figure~\ref{fig:parallel}). It is interesting to compare \pfun{fa.parallel} with the \fun{paran} from the \Rpkg{paran} package. This latter uses smcs to estimate communalities. Simulations of known structures with a particular number of major factors but with the presence of trivial, minor (but not zero) factors, show that using smcs will tend to lead to too many factors. \begin{figure}[htbp] \begin{scriptsize} \begin{center} <>= fa.parallel(bfi[1:25],main="Parallel Analysis of a Big 5 inventory") @ \caption{Parallel analysis compares factor and principal components solutions to the real data as well as resampled data. Although vss suggests 4 factors, MAP 5, parallel analysis suggests 6. One more demonstration of Kaiser's dictum.} \label{fig:parallel} \end{center} \end{scriptsize} \end{figure} Experience with problems of various sizes suggests that parallel analysis is useful for less than about 1,000 subjects, and that using the number of components greater than a random solution is more robust than using the number of factors greater than random factors. A more tedious problem in terms of computation is to do parallel analysis of \iemph{polychoric} correlation matrices. This is done by \pfun{fa.parallel.poly}. By default the number of replications is 20. This is appropriate when choosing the number of factors from dicthotomous or polytomous data matrices. \subsection{Factor extension} Sometimes we are interested in the relationship of the factors in one space with the variables in a different space. One solution is to find factors in both spaces separately and then find the structural relationships between them. This is the technique of structural equation modeling in packages such as \Rpkg{sem} or \Rpkg{lavaan}. An alternative is to use the concept of \iemph{factor extension} developed by \citep{dwyer:37}. Consider the case of 16 variables created to represent one two dimensional space. If factors are found from eight of these variables, they may then be extended to the additional eight variables (See Figure~\ref{fig:fa.extension}). \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= v16 <- sim.item(16) s <- c(1,3,5,7,9,11,13,15) f2 <- fa(v16[,s],2) fe <- fa.extension(cor(v16)[s,-s],f2) fa.diagram(f2,fe=fe) @ \end{scriptsize} \caption{Factor extension applies factors from one set (those on the left) to another set of variables (those on the right). \pfun{fa.extension} is particularly useful when one wants to define the factors with one set of variables and then apply those factors to another set. \pfun{fa.diagram} is used to show the structure. } \label{fig:fa.extension} \end{center} \end{figure} Another way to examine the overlap between two sets is the use of \iemph{set correlation} found by \pfun{setCor} (discussed later). \subsection{Exploratory Structural Equation Modeling (ESEM)} Generaizing the procedures of factor extension, we can do Exploratory Structural Equation Modeling (ESEM). Traditional Exploratory Factor Analysis (EFA) examines how latent variables can account for the correlations within a data set. All loadings and cross loadings are found and rotation is done to some approximation of simple structure. Traditional Confirmatory Factor Analysis (CFA) tests such models by fitting just a limited number of loadings and typically does not allow any (or many) cross loadings. Structural Equation Modeling then applies two such measurement models, one to a set of X variables, another to a set of Y variables, and then tries to estimate the correlation between these two sets of latent variables. (Some SEM procedures estimate all the parameters from the same model, thus making the loadings in set Y affect those in set X.) It is possible to do a similar, exploratory modeling (ESEM) by conducting two Exploratory Factor Analyses, one in set X, one in set Y, and then finding the correlations of the X factors with the Y factors, as well as the correlations of the Y variables with the X factors and the X variables with the Y factors. Consider the simulated data set of three ability variables, two motivational variables, and three outcome variables: <>= fx <-matrix(c( .9,.8,.6,rep(0,4),.6,.8,-.7),ncol=2) fy <- matrix(c(.6,.5,.4),ncol=1) rownames(fx) <- c("V","Q","A","nach","Anx") rownames(fy)<- c("gpa","Pre","MA") Phi <-matrix( c(1,0,.7,.0,1,.7,.7,.7,1),ncol=3) gre.gpa <- sim.structural(fx,Phi,fy) print(gre.gpa) @ We can fit this by using the \pfun{esem} function and then draw the solution (see Figure~\ref{fig:esem}) using the \pfun{esem.diagram} function (which is normally called automatically by \pfun{esem}. <>= esem.example <- esem(gre.gpa$model,varsX=1:5,varsY=6:8,nfX=2,nfY=1,n.obs=1000,plot=FALSE) esem.example @ \begin{figure}[htpb] \begin{center} <>= esem.diagram(esem.example) @ \caption{An example of a Exploratory Structure Equation Model.} \label{fig:esem} \end{center} \end{figure} \section{Classical Test Theory and Reliability} Surprisingly, 113 years after \cite{spearman:rho} introduced the concept of reliability to psychologists, there are still multiple approaches for measuring it. Although very popular, Cronbach's $\alpha$ \citep{cronbach:51} underestimates the reliability of a test and over estimates the first factor saturation \citep{rz:09}. $\alpha$ \citep{cronbach:51} is the same as Guttman's $\lambda3$ \citep{guttman:45} and may be found by $$ \lambda_3 = \frac{n}{n-1}\Bigl(1 - \frac{tr(\vec{V})_x}{V_x}\Bigr) = \frac{n}{n-1} \frac{V_x - tr(\vec{V}_x)}{V_x} = \alpha $$ Perhaps because it is so easy to calculate and is available in most commercial programs, alpha is without doubt the most frequently reported measure of internal consistency reliability. Alpha is the mean of all possible spit half reliabilities (corrected for test length). For a unifactorial test, it is a reasonable estimate of the first factor saturation, although if the test has any microstructure (i.e., if it is ``lumpy") coefficients $\beta$ \citep{revelle:iclust} (see \pfun{iclust}) and $\omega_h$ (see \pfun{omega}) are more appropriate estimates of the general factor saturation. $\omega_t$is a better estimate of the reliability of the total test. Guttman's $\lambda _6$ (G6) considers the amount of variance in each item that can be accounted for the linear regression of all of the other items (the squared multiple correlation or smc), or more precisely, the variance of the errors, $e_j^2$, and is $$ \lambda_6 = 1 - \frac{\sum e_j^2}{V_x} = 1 - \frac{\sum(1-r_{smc}^2)}{V_x}. $$ The squared multiple correlation is a lower bound for the item communality and as the number of items increases, becomes a better estimate. G6 is also sensitive to lumpiness in the test and should not be taken as a measure of unifactorial structure. For lumpy tests, it will be greater than alpha. For tests with equal item loadings, alpha > G6, but if the loadings are unequal or if there is a general factor, G6 > alpha. G6 estimates item reliability by the squared multiple correlation of the other items in a scale. A modification of G6, G6*, takes as an estimate of an item reliability the smc with all the items in an inventory, including those not keyed for a particular scale. This will lead to a better estimate of the reliable variance of a particular item. Alpha, G6 and G6* are positive functions of the number of items in a test as well as the average intercorrelation of the items in the test. When calculated from the item variances and total test variance, as is done here, raw alpha is sensitive to differences in the item variances. Standardized alpha is based upon the correlations rather than the covariances. More complete reliability analyses of a single scale can be done using the \pfun{omega} function which finds $\omega_h$ and $\omega_t$ based upon a hierarchical factor analysis. Alternative functions \pfun{scoreItems} and \pfun{cluster.cor} will also score multiple scales and report more useful statistics. ``Standardized" alpha is calculated from the inter-item correlations and will differ from raw alpha. Functions for examining the reliability of a single scale or a set of scales include: \begin{description} \item [alpha] Internal consistency measures of reliability range from $\omega_h$ to $\alpha$ to $\omega_t$. The \pfun{alpha} function reports two estimates: Cronbach's coefficient $\alpha$ and Guttman's $\lambda_6$. Also reported are item - whole correlations, $\alpha$ if an item is omitted, and item means and standard deviations. \item [guttman] Eight alternative estimates of test reliability include the six discussed by \cite{guttman:45}, four discussed by ten Berge and Zergers (1978) ($\mu_0 \dots \mu_3$) as well as $\beta$ \citep[the worst split half,][]{revelle:iclust}, the glb (greatest lowest bound) discussed by Bentler and Woodward (1980), and $\omega_h$ and$\omega_t$ (\citep{mcdonald:tt,zinbarg:pm:05}. \item [omega] Calculate McDonald's omega estimates of general and total factor saturation. (\cite{rz:09} compare these coefficients with real and artificial data sets.) \item [cluster.cor] Given a n x c cluster definition matrix of -1s, 0s, and 1s (the keys) , and a n x n correlation matrix, find the correlations of the composite clusters. \item [scoreItems] Given a matrix or data.frame of k keys for m items (-1, 0, 1), and a matrix or data.frame of items scores for m items and n people, find the sum scores or average scores for each person and each scale. If the input is a square matrix, then it is assumed that correlations or covariances were used, and the raw scores are not available. In addition, report Cronbach's alpha, coefficient G6*, the average r, the scale intercorrelations, and the item by scale correlations (both raw and corrected for item overlap and scale reliability). Replace missing values with the item median or mean if desired. Will adjust scores for reverse scored items. \item [score.multiple.choice] Ability tests are typically multiple choice with one right answer. score.multiple.choice takes a scoring key and a data matrix (or data.frame) and finds total or average number right for each participant. Basic test statistics (alpha, average r, item means, item-whole correlations) are also reported. \item [splitHalf] Given a set of items, consider all (if n.items < 17) or 10,000 random splits of the item into two sets. The correlation between these two split halfs is then adjusted by the Spearman-Brown prophecy formula to show the range of split half reliablities. \end{description} \subsection{Reliability of a single scale} \label{sect:alpha} A conventional (but non-optimal) estimate of the internal consistency reliability of a test is coefficient $\alpha$ \citep{cronbach:51}. Alternative estimates are Guttman's $\lambda_6$, Revelle's $\beta$, McDonald's $\omega_h$ and $\omega_t$. Consider a simulated data set, representing 9 items with a hierarchical structure and the following correlation matrix. Then using the \pfun{alpha} function, the $\alpha$ and $\lambda_6$ estimates of reliability may be found for all 9 items, as well as the if one item is dropped at a time. \begin{scriptsize} <>= set.seed(17) r9 <- sim.hierarchical(n=500,raw=TRUE)$observed round(cor(r9),2) alpha(r9) @ \end{scriptsize} Some scales have items that need to be reversed before being scored. Rather than reversing the items in the raw data, it is more convenient to just specify which items need to be reversed scored. This may be done in \pfun{alpha} by specifying a \iemph{keys} vector of 1s and -1s. (This concept of keys vector is more useful when scoring multiple scale inventories, see below.) As an example, consider scoring the 7 attitude items in the attitude data set. Assume a conceptual mistake in that items 2 and 6 (complaints and critical) are to be scored (incorrectly) negatively. \begin{scriptsize} <>= alpha(attitude,keys=c("complaints","critical")) @ \end{scriptsize} Note how the reliability of the 7 item scales with an incorrectly reversed item is very poor, but if items 2 and 6 is dropped then the reliability is improved substantially. This suggests that items 2 and 6 were incorrectly scored. Doing the analysis again with the items positively scored produces much more favorable results. \begin{scriptsize} <>= alpha(attitude) @ \end{scriptsize} It is useful when considering items for a potential scale to examine the item distribution. This is done in \pfun{scoreItems} as well as in \pfun{alpha}. \begin{scriptsize} <>= items <- sim.congeneric(N=500,short=FALSE,low=-2,high=2,categorical=TRUE) #500 responses to 4 discrete items alpha(items$observed) #item response analysis of congeneric measures @ \end{scriptsize} \subsection{Using \pfun{omega} to find the reliability of a single scale} Two alternative estimates of reliability that take into account the hierarchical structure of the inventory are McDonald's $\omega_h$ and $\omega_t$. These may be found using the \pfun{omega} function for an exploratory analysis (See Figure~\ref{fig:omega.9}) or \pfun{omegaSem} for a confirmatory analysis using the \Rpkg{lavaan} package based upon the exploratory solution from \pfun{omega}. McDonald has proposed coefficient omega (hierarchical) ($\omega_h$) as an estimate of the general factor saturation of a test. \cite{zinbarg:pm:05} \url{https://personality-project.org/revelle/publications/zinbarg.revelle.pmet.05.pdf} compare McDonald's $\omega_h$ to Cronbach's $\alpha$ and Revelle's $\beta$. They conclude that $\omega_h$ is the best estimate. (See also \cite{zinbarg:apm:06} and \cite{rz:09} \url{https://personality-project.org/revelle/publications/revelle.zinbarg.08.pdf} ). One way to find $\omega_h$ is to do a factor analysis of the original data set, rotate the factors obliquely, factor that correlation matrix, do a Schmid-Leiman (\pfun{schmid}) transformation to find general factor loadings, and then find $\omega_h$. $\omega_h$ differs slightly as a function of how the factors are estimated. Four options are available, the default will do a minimum residual factor analysis, fm=``pa" does a principal axes factor analysis (\pfun{factor.pa}), fm=``mle" uses the factanal function, and fm=``pc" does a principal components analysis (\pfun{principal}). For ability items, it is typically the case that all items will have positive loadings on the general factor. However, for non-cognitive items it is frequently the case that some items are to be scored positively, and some negatively. Although probably better to specify which directions the items are to be scored by specifying a key vector, if flip =TRUE (the default), items will be reversed so that they have positive loadings on the general factor. The keys are reported so that scores can be found using the \pfun{scoreItems} function. Arbitrarily reversing items this way can overestimate the general factor. (See the example with a simulated circumplex). $\beta$, an alternative to $\omega$, is defined as the worst split half reliability. It can be estimated by using \pfun{iclust} (Item Cluster analysis: a hierarchical clustering algorithm). For a very complimentary review of why the iclust algorithm is useful in scale construction, see \cite{cooksey:06}. The \pfun{omega} function uses exploratory factor analysis to estimate the $\omega_h$ coefficient. It is important to remember that ``A recommendation that should be heeded, regardless of the method chosen to estimate $\omega_h$, is to always examine the pattern of the estimated general factor loadings prior to estimating $\omega_h$. Such an examination constitutes an informal test of the assumption that there is a latent variable common to all of the scale's indicators that can be conducted even in the context of EFA. If the loadings were salient for only a relatively small subset of the indicators, this would suggest that there is no true general factor underlying the covariance matrix. Just such an informal assumption test would have afforded a great deal of protection against the possibility of misinterpreting the misleading $\omega_h$ estimates occasionally produced in the simulations reported here." \citep[][p 137]{zinbarg:apm:06}. Although $\omega_h$ is uniquely defined only for cases where 3 or more subfactors are extracted, it is sometimes desired to have a two factor solution. By default this is done by forcing the \pfun{schmid} extraction to treat the two subfactors as having equal loadings. There are three possible options for this condition: setting the general factor loadings between the two lower order factors to be ``equal" which will be the $\sqrt{r_{ab}}$ where $r_{ab}$ is the oblique correlation between the factors) or to ``first" or ``second" in which case the general factor is equated with either the first or second group factor. A message is issued suggesting that the model is not really well defined. This solution discussed in Zinbarg et al., 2007. To do this in omega, add the option=``first" or option=``second" to the call. Although obviously not meaningful for a 1 factor solution, it is of course possible to find the sum of the loadings on the first (and only) factor, square them, and compare them to the overall matrix variance. This is done, with appropriate complaints. In addition to $\omega_h$, another of McDonald's coefficients is $\omega_t$. This is an estimate of the total reliability of a test. McDonald's $\omega_t$, which is similar to Guttman's $\lambda_6$, (see \pfun{guttman}) uses the estimates of uniqueness $u^2$ from factor analysis to find $e_j^2$. This is based on a decomposition of the variance of a test score, $V_x$ into four parts: that due to a general factor, $\vec{g}$, that due to a set of group factors, $\vec{f}$, (factors common to some but not all of the items), specific factors, $\vec{s}$ unique to each item, and $\vec{e}$, random error. (Because specific variance can not be distinguished from random error unless the test is given at least twice, some combine these both into error). Letting $\vec{x} = \vec{cg} + \vec{Af} + \vec {Ds} + \vec{e} $ then the communality of item$_j$, based upon general as well as group factors, $h_j^2 = c_j^2 + \sum{f_{ij}^2}$ and the unique variance for the item $u_j^2 = \sigma_j^2 (1-h_j^2)$ may be used to estimate the test reliability. That is, if $h_j^2$ is the communality of item$_j$, based upon general as well as group factors, then for standardized items, $e_j^2 = 1 - h_j^2$ and $$ \omega_t = \frac{\vec{1}\vec{cc'}\vec{1} + \vec{1}\vec{AA'}\vec{1}'}{V_x} = 1 - \frac{\sum(1-h_j^2)}{V_x} = 1 - \frac{\sum u^2}{V_x} $$ Because $h_j^2 \geq r_{smc}^2$, $\omega_t \geq \lambda_6$. It is important to distinguish here between the two $\omega$ coefficients of McDonald, 1978 and Equation 6.20a of McDonald, 1999, $\omega_t$ and $\omega_h$. While the former is based upon the sum of squared loadings on all the factors, the latter is based upon the sum of the squared loadings on the general factor. $$\omega_h = \frac{ \vec{1}\vec{cc'}\vec{1}}{V_x}$$ Another estimate reported is the omega for an infinite length test with a structure similar to the observed test. This is found by $$\omega_{\inf} = \frac{ \vec{1}\vec{cc'}\vec{1}}{\vec{1}\vec{cc'}\vec{1} + \vec{1}\vec{AA'}\vec{1}'}$$ \begin{figure}[htbp] \begin{center} <>= om.9 <- omega(r9,title="9 simulated variables") @ \caption{A bifactor solution for 9 simulated variables with a hierarchical structure. } \label{fig:omega.9} \end{center} \end{figure} In the case of these simulated 9 variables, the amount of variance attributable to a general factor ($\omega_h$) is quite large, and the reliability of the set of 9 items is somewhat greater than that estimated by $\alpha$ or $\lambda_6$. \begin{scriptsize} <>= om.9 @ \end{scriptsize} \subsection{Estimating $\omega_h$ using Confirmatory Factor Analysis} The \pfun{omegaSem} function will do an exploratory analysis and then take the highest loading items on each factor and do a confirmatory factor analysis using the \Rpkg{sem} package. These results can produce slightly different estimates of $\omega_h$, primarily because cross loadings are modeled as part of the general factor. \begin{scriptsize} <>= omegaSem(r9,n.obs=500,lavaan=TRUE) @ \end{scriptsize} \subsubsection{Other estimates of reliability} Other estimates of reliability are found by the \pfun{splitHalf} and \pfun{guttman} functions. These are described in more detail in \cite{rz:09} and in \cite{rc:reliability}. They include the 6 estimates from Guttman, four from TenBerge, and an estimate of the greatest lower bound. \begin{scriptsize} <>= splitHalf(r9) @ \end{scriptsize} \subsection{Reliability and correlations of multiple scales within an inventory} \label{sect:score} A typical research question in personality involves an inventory of multiple items purporting to measure multiple constructs. For example, the data set \pfun{bfi} includes 25 items thought to measure five dimensions of personality (Extraversion, Emotional Stability, Conscientiousness, Agreeableness, and Openness). The data may either be the raw data or a correlation matrix (\pfun{scoreItems}) or just a correlation matrix of the items ( \pfun{cluster.cor} and \pfun{cluster.loadings}). When finding reliabilities for multiple scales, item reliabilities can be estimated using the squared multiple correlation of an item with all other items, not just those that are keyed for a particular scale. This leads to an estimate of G6*. \subsubsection{Scoring from raw data} To score these five scales from the 25 items, use the \pfun{scoreItems} function and a list of items to be scored on each scale (a keys.list). Items may be listed by location (convenient but dangerous), or name (probably safer). Make a keys.list by by specifying the items for each scale, preceding items to be negatively keyed with a - sign: \begin{scriptsize} <>= #the newer way is probably preferred keys.list <- list(agree=c("-A1","A2","A3","A4","A5"), conscientious=c("C1","C2","C2","-C4","-C5"), extraversion=c("-E1","-E2","E3","E4","E5"), neuroticism=c("N1","N2","N3","N4","N5"), openness = c("O1","-O2","O3","O4","-O5")) #this can also be done by location-- keys.list <- list(Agree=c(-1,2:5),Conscientious=c(6:8,-9,-10), Extraversion=c(-11,-12,13:15),Neuroticism=c(16:20), Openness = c(21,-22,23,24,-25)) #These two approaches can be mixed if desired keys.list <- list(agree=c("-A1","A2","A3","A4","A5"),conscientious=c("C1","C2","C3","-C4","-C5"), extraversion=c("-E1","-E2","E3","E4","E5"), neuroticism=c(16:20),openness = c(21,-22,23,24,-25)) keys.list @ \end{scriptsize} \begin{tiny}In the past (prior to version 1.6.9, the keys.list was then converted a keys matrix using the helper function \pfun{make.keys}. This is no longer necessary. Logically, scales are merely the weighted composites of a set of items. The weights used are -1, 0, and 1. 0 implies do not use that item in the scale, 1 implies a positive weight (add the item to the total score), -1 a negative weight (subtract the item from the total score, i.e., reverse score the item). Reverse scoring an item is equivalent to subtracting the item from the maximum + minimum possible value for that item. The minima and maxima can be estimated from all the items, or can be specified by the user. There are two different ways that scale scores tend to be reported. Social psychologists and educational psychologists tend to report the scale score as the \emph{average item score} while many personality psychologists tend to report the \emph{total item score}. The default option for \pfun{scoreItems} is to report item averages (which thus allows interpretation in the same metric as the items) but totals can be found as well. Personality researchers should be encouraged to report scores based upon item means and avoid using the total score although some reviewers are adamant about the following the tradition of total scores. The printed output includes coefficients $\alpha$ and G6*, the average correlation of the items within the scale (corrected for item ovelap and scale relliability), as well as the correlations between the scales (below the diagonal, the correlations above the diagonal are corrected for attenuation. As is the case for most of the \Rpkg{psych} functions, additional information is returned as part of the object. First, create keys matrix using the \pfun{make.keys} function. (The keys matrix could also be prepared externally using a spreadsheet and then copying it into \R{}). Although not normally necessary, show the keys to understand what is happening. There are two ways to make up the keys. You can specify the items by \emph{location} (the old way) or by \emph{name} (the newer and probably preferred way). To use the newer way you must specify the file on which you will use the keys. The example below shows how to construct keys either way. Note that the number of items to specify in the \pfun{make.keys} function is the total number of items in the inventory. This is done automatically in the new way of forming keys, but if using the older way, the number must be specified. That is, if scoring just 5 items from a 25 item inventory, \pfun{make.keys} should be told that there are 25 items. \pfun{make.keys} just changes a list of items on each scale to make up a scoring matrix. Because the \pfun{bfi} data set has 25 items as well as 3 demographic items, the number of variables is specified as 28. \end{tiny} Then, use this keys list to score the items. \begin{scriptsize} <>= scores <- scoreItems(keys.list,bfi) scores @ \end{scriptsize} To see the additional information (the raw correlations, the individual scores, etc.), they may be specified by name. Then, to visualize the correlations between the raw scores, use the \pfun{pairs.panels} function on the scores values of scores. (See figure~\ref{fig:scores} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= png('scores.png') pairs.panels(scores$scores,pch='.',jiggle=TRUE) dev.off() @ \end{scriptsize} \includegraphics{scores} \caption{A graphic analysis of the Big Five scales found by using the scoreItems function. The pair.wise plot allows us to see that some participants have reached the ceiling of the scale for these 5 items scales. Using the pch='.' option in pairs.panels is recommended when plotting many cases. The data points were ``jittered'' by setting jiggle=TRUE. Jiggling this way shows the density more clearly. To save space, the figure was done as a png. For a clearer figure, save as a pdf.} \label{fig:scores} \end{center} \end{figure} \subsubsection{Forming scales from a correlation matrix} There are some situations when the raw data are not available, but the correlation matrix between the items is available. In this case, it is not possible to find individual scores, but it is possible to find the reliability and intercorrelations of the scales. This may be done using the \pfun{cluster.cor} function or the \pfun{scoreItems} function. The use of a keys matrix is the same as in the raw data case. Consider the same \pfun{bfi} data set, but first find the correlations, and then use \pfun{scoreItems}. \begin{scriptsize} <>= r.bfi <- cor(bfi,use="pairwise") scales <- scoreItems(keys.list,r.bfi) summary(scales) @ \end{scriptsize} To find the correlations of the items with each of the scales (the ``structure" matrix) or the correlations of the items controlling for the other scales (the ``pattern" matrix), use the \pfun{cluster.loadings} function. To do both at once (e.g., the correlations of the scales as well as the item by scale correlations), it is also possible to just use \pfun{scoreItems}. \subsection{Scoring Multiple Choice Items} Some items (typically associated with ability tests) are not themselves mini-scales ranging from low to high levels of expression of the item of interest, but are rather multiple choice where one response is the correct response. Two analyses are useful for this kind of item: examining the response patterns to all the alternatives (looking for good or bad distractors) and scoring the items as correct or incorrect. Both of these operations may be done using the \pfun{score.multiple.choice} function. Consider the 16 example items taken from an online ability test at the Personality Project: \url{https://www.sapa-project.org/}. This is part of the Synthetic Aperture Personality Assessment (\iemph{SAPA}) study discussed in \cite{rcw:methods,rwr:sapa}. \begin{scriptsize} <>= data(iqitems) iq.keys <- c(4,4,4, 6,6,3,4,4, 5,2,2,4, 3,2,6,7) score.multiple.choice(iq.keys,iqitems) #just convert the items to true or false iq.tf <- score.multiple.choice(iq.keys,iqitems,score=FALSE) describe(iq.tf) #compare to previous results @ \end{scriptsize} Once the items have been scored as true or false (assigned scores of 1 or 0), they made then be scored into multiple scales using the normal \pfun{scoreItems} function. \subsection{Item analysis} Basic item analysis starts with describing the data (\pfun{describe}, finding the number of dimensions using factor analysis (\pfun{fa}) and cluster analysis \pfun{iclust} perhaps using the Very Simple Structure criterion (\pfun{vss}), or perhaps parallel analysis \pfun{fa.parallel}. Item whole correlations may then be found for scales scored on one dimension (\pfun{alpha} or many scales simultaneously (\pfun{scoreItems}). Scales can be modified by changing the keys matrix (i.e., dropping particular items, changing the scale on which an item is to be scored). This analysis can be done on the normal Pearson correlation matrix or by using polychoric correlations. Validities of the scales can be found using multiple correlation of the raw data or based upon correlation matrices using the \pfun{setCor} function. However, more powerful item analysis tools are now available by using Item Response Theory approaches. Although the \pfun{response.frequencies} output from \pfun{score.multiple.choice} is useful to examine in terms of the probability of various alternatives being endorsed, it is even better to examine the pattern of these responses as a function of the underlying latent trait or just the total score. This may be done by using \pfun{irt.responses} (Figure~\ref{fig:irt.response}). \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= data(iqitems) iq.keys <- c(4,4,4, 6,6,3,4,4, 5,2,2,4, 3,2,6,7) scores <- score.multiple.choice(iq.keys,iqitems,score=TRUE,short=FALSE) #note that for speed we can just do this on simple item counts rather than IRT based scores. op <- par(mfrow=c(2,2)) #set this to see the output for multiple items irt.responses(scores$scores,iqitems[1:4],breaks=11) @ \end{scriptsize} \caption{ The pattern of responses to multiple choice ability items can show that some items have poor distractors. This may be done by using the the \pfun{irt.responses} function. A good distractor is one that is negatively related to ability.} \label{fig:irt.response} \end{center} \end{figure} \subsubsection{Exploring the item structure of scales} The Big Five scales found above can be understood in terms of the item - whole correlations, but it is also useful to think of the endorsement frequency of the items. The \pfun{item.lookup} function will sort items by their factor loading/item-whole correlation, and then resort those above a certain threshold in terms of the item means. Item content is shown by using the dictionary developed for those items. This allows one to see the structure of each scale in terms of its endorsement range. This is a simple way of thinking of items that is also possible to do using the various IRT approaches discussed later. \begin{tiny} <>= m <- colMeans(bfi,na.rm=TRUE) item.lookup(scales$item.corrected[,1:3],m,dictionary=bfi.dictionary[1:2]) @ \end{tiny} \subsubsection{Empirical scale construction} There are some situations where one wants to identify those items that most relate to a particular criterion. Although this will capitalize on chance and the results should interpreted cautiously, it does give a feel for what is being measured. Consider the following example from the \pfun{bfi} data set. The items that best predicted gender, education, and age may be found using the \pfun{bestScales} function. This also shows the use of a dictionary that has the item content. \begin{scriptsize} <>= data(bfi) bestScales(bfi,criteria=c("gender","education","age"),cut=.1,dictionary=bfi.dictionary[,1:3]) @ \end{scriptsize} \section{Item Response Theory analysis} The use of Item Response Theory has become is said to be the ``new psychometrics". The emphasis is upon item properties, particularly those of item difficulty or location and item discrimination. These two parameters are easily found from classic techniques when using factor analyses of correlation matrices formed by \pfun{polychoric} or \pfun{tetrachoric} correlations. The \pfun{irt.fa} function does this and then graphically displays item discrimination and item location as well as item and test information (see Figure~\ref{fig:irt}). \subsection{Factor analysis and Item Response Theory} If the correlations of all of the items reflect one underlying latent variable, then factor analysis of the matrix of tetrachoric correlations should allow for the identification of the regression slopes ($\alpha$) of the items on the latent variable. These regressions are, of course just the factor loadings. Item difficulty, $\delta_j$ and item discrimination, $\alpha_j$ may be found from factor analysis of the tetrachoric correlations where $\lambda_j$ is just the factor loading on the first factor and $\tau_j$ is the normal threshold reported by the \pfun{tetrachoric} function. \begin{equation} \delta_j = \frac{D\tau}{\sqrt{1-\lambda_j^2}}, \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\;\; \alpha_j = \frac{\lambda_j}{\sqrt{1-\lambda_j^2}} \label{eq:irt:diff} \end{equation} where D is a scaling factor used when converting to the parameterization of \iemph{logistic} model and is 1.702 in that case and 1 in the case of the normal ogive model. Thus, in the case of the normal model, factor loadings ($\lambda_j$) and item thresholds ($\tau$) are just \begin{equation*} \lambda_j = \frac{\alpha_j}{\sqrt{1+\alpha_j^2}}, \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\;\;\tau_j = \frac{\delta_j}{\sqrt{1+\alpha_j^2}}. \end{equation*} Consider 9 dichotomous items representing one factor but differing in their levels of difficulty \begin{scriptsize} <>= set.seed(17) d9 <- sim.irt(9,1000,-2.0,2.0,mod="normal") #dichotomous items test <- irt.fa(d9$items,correct=0) test @ \end{scriptsize} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= op <- par(mfrow=c(3,1)) plot(test,type="ICC") plot(test,type="IIC") plot(test,type="test") op <- par(mfrow=c(1,1)) @ \end{scriptsize} \caption{A graphic analysis of 9 dichotomous (simulated) items. The top panel shows the probability of item endorsement as the value of the latent trait increases. Items differ in their location (difficulty) and discrimination (slope). The middle panel shows the information in each item as a function of latent trait level. An item is most informative when the probability of endorsement is 50\%. The lower panel shows the total test information. These items form a test that is most informative (most accurate) at the middle range of the latent trait.} \label{fig:irt} \end{center} \end{figure} Similar analyses can be done for polytomous items such as those of the bfi extraversion scale: \begin{scriptsize} <>= data(bfi) e.irt <- irt.fa(bfi[11:15]) e.irt @ \end{scriptsize} The item information functions show that not all of items are equally good (Figure~\ref{fig:e.irt}): \begin{figure}[htbp] \begin{center} <>= e.info <- plot(e.irt,type="IIC") @ \caption{A graphic analysis of 5 extraversion items from the bfi. The curves represent the amount of information in the item as a function of the latent score for an individual. That is, each item is maximally discriminating at a different part of the latent continuum. Print e.info to see the average information for each item.} \label{fig:e.irt} \end{center} \end{figure} These procedures can be generalized to more than one factor by specifying the number of factors in \pfun{irt.fa}. The plots can be limited to those items with discriminations greater than some value of cut. An invisible object is returned when plotting the output from \pfun{irt.fa} that includes the average information for each item that has loadings greater than cut. \begin{scriptsize} <>= print(e.info,sort=TRUE) @ \end{scriptsize} More extensive IRT packages include the \Rpkg{ltm} and \Rpkg{eRm} and should be used for serious Item Response Theory analysis. \subsection{Speeding up analyses} Finding tetrachoric or polychoric correlations is very time consuming. Thus, to speed up the process of analysis, the original correlation matrix is saved as part of the output of both \pfun{irt.fa} and \pfun{omega}. Subsequent analyses may be done by using this correlation matrix. This is done by doing the analysis not on the original data, but rather on the output of the previous analysis. In addition, recent releases of the \Rpkg{psych} take advantage of the \Rpkg{parallels} package and use multi-cores. The default for Macs and Unix machines is to use two cores, but this can be increased using the options command. The biggest step up in improvement is from 1 to 2 cores, but for large problems using polychoric correlations, the more cores available, the better. For example of taking the output from the 16 ability items from the \iemph{SAPA} project when scored for True/False using \pfun{score.multiple.choice} we can first do a simple IRT analysis of one factor (Figure~\ref{fig:iq.irt}) and then use that correlation matrix to do an \pfun{omega} analysis to show the sub-structure of the ability items . We can also show the total test information (merely the sum of the item information. This shows that even with just 16 items, the test is very reliable for most of the range of ability. The \pfun{fa.irt} function saves the correlation matrix and item statistics so that they can be redrawn with other options. \begin{scriptsize} \begin{Schunk} \begin{Sinput} detectCores() #how many are available options("mc.cores") #how many have been set to be used options("mc.cores"=4) #set to use 4 cores \end{Sinput} \end{Schunk} \end{scriptsize} \begin{figure}[htbp] \begin{tiny} \begin{center} <>= iq.irt <- irt.fa(ability) @ \end{center} \end{tiny} \caption{A graphic analysis of 16 ability items sampled from the \iemph{SAPA} project. The curves represent the amount of information in the item as a function of the latent score for an individual. That is, each item is maximally discriminating at a different part of the latent continuum. Print iq.irt to see the average information for each item. Partly because this is a power test (it is given on the web) and partly because the items have not been carefully chosen, the items are not very discriminating at the high end of the ability dimension. } \label{fig:iq.irt} \end{figure} \begin{figure}[htbp] \begin{tiny} \begin{center} <>= plot(iq.irt,type='test') @ \end{center} \end{tiny} \caption{A graphic analysis of 16 ability items sampled from the \iemph{SAPA} project. The total test information at all levels of difficulty may be shown by specifying the type='test' option in the plot function. } \label{fig:iq.irt.test} \end{figure} \begin{scriptsize} <>= iq.irt @ \end{scriptsize} \begin{figure}[htbp] \begin{center} <>= om <- omega(iq.irt$rho,4) @ \caption{An Omega analysis of 16 ability items sampled from the SAPA project. The items represent a general factor as well as four lower level factors. The analysis is done using the tetrachoric correlations found in the previous \pfun{irt.fa} analysis. The four matrix items have some serious problems, which may be seen later when examine the item response functions.} \label{fig:iq.irt} \end{center} \end{figure} \subsection{IRT based scoring} The primary advantage of IRT analyses is examining the item properties (both difficulty and discrimination). With complete data, the scores based upon simple total scores and based upon IRT are practically identical (this may be seen in the examples for \pfun{scoreIrt}). However, when working with data such as those found in the Synthetic Aperture Personality Assessment (\iemph{SAPA}) project, it is advantageous to use IRT based scoring. \iemph{SAPA} data might have 2-3 items/person sampled from scales with 10-20 items. Simply finding the average of the three (classical test theory) fails to consider that the items might differ in either discrimination or in difficulty. The \pfun{scoreIrt} function applies basic IRT to this problem. Consider 1000 randomly generated subjects with scores on 9 true/false items differing in difficulty. Selectively drop the hardest items for the 1/3 lowest subjects, and the 4 easiest items for the 1/3 top subjects (this is a crude example of what tailored testing would do). Then score these subjects: \begin{scriptsize} <>= v9 <- sim.irt(9,1000,-2.,2.,mod="normal") #dichotomous items items <- v9$items test <- irt.fa(items) total <- rowSums(items) ord <- order(total) items <- items[ord,] #now delete some of the data - note that they are ordered by score items[1:333,5:9] <- NA items[334:666,3:7] <- NA items[667:1000,1:4] <- NA scores <- scoreIrt(test,items) unitweighted <- scoreIrt(items=items,keys=rep(1,9)) scores.df <- data.frame(true=v9$theta[ord],scores,unitweighted) colnames(scores.df) <- c("True theta","irt theta","total","fit","rasch","total","fit") @ \end{scriptsize} These results are seen in Figure~\ref{fig:score.irt.pdf}. \begin{figure}[htbp] \begin{center} \caption{IRT based scoring and total test scores for 1000 simulated subjects. True theta values are reported and then the IRT and total scoring systems. } <>= pairs.panels(scores.df,pch='.',gap=0) title('Comparing true theta for IRT, Rasch and classically based scoring',line=3) @ \label{fig:score.irt.pdf} \end{center} \end{figure} \subsubsection{1 versus 2 parameter IRT scoring} In Item Response Theory, items can be assumed to be equally discriminating but to differ in their difficulty (the Rasch model) or to vary in their discriminability. Two functions (\pfun{scoreIrt.1pl} and \pfun{scoreIrt.2pl}) are meant to find multiple IRT based scales using the Rasch model or the 2 parameter model. Both allow for negatively keyed as well as positively keyed items. Consider the \pfun{bfi} data set with scoring keys key.list and items listed as an item.list. (This is the same as the key.list, but with the negative signs removed.) \begin{scriptsize} <>= keys.list <- list(agree=c("-A1","A2","A3","A4","A5"), conscientious=c("C1","C2","C3","-C4","-C5"), extraversion=c("-E1","-E2","E3","E4","E5"), neuroticism=c("N1","N2","N3","N4","N5"), openness = c("O1","-O2","O3","O4","-O5")) item.list <- list(agree=c("A1","A2","A3","A4","A5"), conscientious=c("C1","C2","C3","C4","C5"), extraversion=c("E1","E2","E3","E4","E5"), neuroticism=c("N1","N2","N3","N4","N5"), openness = c("O1","O2","O3","O4","O5")) bfi.1pl <- scoreIrt.1pl(keys.list,bfi) #the one parameter solution bfi.2pl <- scoreIrt.2pl(item.list,bfi) #the two parameter solution bfi.ctt <- scoreFast(keys.list,bfi) # fast scoring function @ \end{scriptsize} We can compare these three ways of doing the analysis using the \pfun{cor2} function which correlates two separate data frames. All three models produce vey simillar results for the case of almost complete data. It is when we have massively missing completely at random data (MMCAR) that the results show the superiority of the irt scoring. \begin{scriptsize} <>= #compare the solutions using the cor2 function cor2(bfi.1pl,bfi.ctt) cor2(bfi.2pl,bfi.ctt) cor2(bfi.2pl,bfi.1pl) @ \end{scriptsize} \section{Multilevel modeling} Correlations between individuals who belong to different natural groups (based upon e.g., ethnicity, age, gender, college major, or country) reflect an unknown mixture of the pooled correlation within each group as well as the correlation of the means of these groups. These two correlations are independent and do not allow inferences from one level (the group) to the other level (the individual). When examining data at two levels (e.g., the individual and by some grouping variable), it is useful to find basic descriptive statistics (means, sds, ns per group, within group correlations) as well as between group statistics (over all descriptive statistics, and overall between group correlations). Of particular use is the ability to decompose a matrix of correlations at the individual level into correlations within group and correlations between groups. \subsection{Decomposing data into within and between level correlations using \pfun{statsBy}} There are at least two very powerful packages (\Rpkg{nlme} and \Rpkg{multilevel}) which allow for complex analysis of hierarchical (multilevel) data structures. \pfun{statsBy} is a much simpler function to give some of the basic descriptive statistics for two level models. This follows the decomposition of an observed correlation into the pooled correlation within groups (rwg) and the weighted correlation of the means between groups which is discussed by \cite{pedhazur:97} and by \cite{bliese:09} in the multilevel package. \begin{equation} r_{xy} = \eta_{x_{wg}} * \eta_{y_{wg}} * r_{xy_{wg}} + \eta_{x_{bg}} * \eta_{y_{bg}} * r_{xy_{bg} } \end{equation} where $r_{xy} $ is the normal correlation which may be decomposed into a within group and between group correlations $r_{xy_{wg}}$ and $r_{xy_{bg}} $ and $\eta$ (eta) is the correlation of the data with the within group values, or the group means. \subsection{Generating and displaying multilevel data} \pfun{withinBetween} is an example data set of the mixture of within and between group correlations. The within group correlations between 9 variables are set to be 1, 0, and -1 while those between groups are also set to be 1, 0, -1. These two sets of correlations are crossed such that V1, V4, and V7 have within group correlations of 1, as do V2, V5 and V8, and V3, V6 and V9. V1 has a within group correlation of 0 with V2, V5, and V8, and a -1 within group correlation with V3, V6 and V9. V1, V2, and V3 share a between group correlation of 1, as do V4, V5 and V6, and V7, V8 and V9. The first group has a 0 between group correlation with the second and a -1 with the third group. See the help file for \pfun{withinBetween} to display these data. \pfun{sim.multilevel} will generate simulated data with a multilevel structure. The \pfun{statsBy.boot} function will randomize the grouping variable ntrials times and find the statsBy output. This can take a long time and will produce a great deal of output. This output can then be summarized for relevant variables using the \pfun{statsBy.boot.summary} function specifying the variable of interest. Consider the case of the relationship between various tests of ability when the data are grouped by level of education (statsBy(sat.act)) or when affect data are analyzed within and between an affect manipulation (statsBy(affect) ). \ \subsection{Factor analysis by groups} Confirmatory factor analysis comparing the structures in multiple groups can be done in the \Rpkg{lavaan} package. However, for exploratory analyses of the structure within each of multiple groups, the \pfun{faBy} function may be used in combination with the \pfun{statsBy} function. First run pfun{statsBy} with the correlation option set to TRUE, and then run \pfun{faBy} on the resulting output. \begin{scriptsize} \begin{Schunk} \begin{Sinput} sb <- statsBy(bfi[c(1:25,27)], group="education",cors=TRUE) faBy(sb,nfactors=5) #find the 5 factor solution for each education level \end{Sinput} \end{Schunk} \end{scriptsize} \subsection{Multilevel reliability} The \pfun{mlr} and \pfun{multilevelReliablity} functions follow the advice of \cite{shrout:12a} for estimating multievel reliablilty. A detailed discussion of this procedure is given in \cite{rw:paid:17} which is available at \url{https://personality-project.org/revelle/publications/rw.paid.17.final.pdf}. \section{Set Correlation and Multiple Regression from the correlation matrix} An important generalization of multiple regression and multiple correlation is \iemph{set correlation} developed by \cite{cohen:set} and discussed by \cite{cohen:03}. Set correlation is a multivariate generalization of multiple regression and estimates the amount of variance shared between two sets of variables. Set correlation also allows for examining the relationship between two sets when controlling for a third set. This is implemented in the \pfun{setCor} function. Set correlation is $$R^{2} = 1 - \prod_{i=1}^n(1-\lambda_{i})$$ where $\lambda_{i}$ is the ith eigen value of the eigen value decomposition of the matrix $$R = R_{xx}^{-1}R_{xy}R_{xx}^{-1}R_{xy}^{-1}.$$ Unfortunately, there are several cases where set correlation will give results that are much too high. This will happen if some variables from the first set are highly related to those in the second set, even though most are not. In this case, although the set correlation can be very high, the degree of relationship between the sets is not as high. In this case, an alternative statistic, based upon the average canonical correlation might be more appropriate. \pfun{setCor} has the additional feature that it will calculate multiple and partial correlations from the correlation or covariance matrix rather than the original data. Consider the correlations of the 6 variables in the \pfun{sat.act} data set. First do the normal multiple regression, and then compare it with the results using \pfun{setCor}. Two things to notice. \pfun{setCor} works on the \emph{correlation} or \emph{covariance} or \emph{raw data} matrix, and thus if using the correlation matrix, will report standardized or raw $\hat{\beta}$ weights. Secondly, it is possible to do several multiple regressions simultaneously. If the number of observations is specified, or if the analysis is done on raw data, statistical tests of significance are applied. For this example, the analysis is done on the correlation matrix rather than the raw data. \begin{scriptsize} <>= C <- cov(sat.act,use="pairwise") model1 <- lm(ACT~ gender + education + age, data=sat.act) summary(model1) @ Compare this with the output from \pfun{setCor}. <>= #compare with setCor setCor(gender + education + age ~ ACT + SATV + SATQ, data = C, n.obs=700) @ \end{scriptsize} Note that the \pfun{setCor} analysis also reports the amount of shared variance between the predictor set and the criterion (dependent) set. This set correlation is symmetric. That is, the $R^{2}$ is the same independent of the direction of the relationship. \section{Simulation functions} It is particularly helpful, when trying to understand psychometric concepts, to be able to generate sample data sets that meet certain specifications. By knowing ``truth" it is possible to see how well various algorithms can capture it. Several of the \pfun{sim} functions create artificial data sets with known structures. A number of functions in the psych package will generate simulated data. These functions include \pfun{sim} for a factor simplex, and \pfun{sim.simplex} for a data simplex, \pfun{sim.circ} for a circumplex structure, \pfun{sim.congeneric} for a one factor factor congeneric model, \pfun{sim.dichot} to simulate dichotomous items, \pfun{sim.hierarchical} to create a hierarchical factor model, \pfun{sim.item} is a more general item simulation, \pfun{sim.minor} to simulate major and minor factors, \pfun{sim.omega} to test various examples of omega, \pfun{sim.parallel} to compare the efficiency of various ways of determining the number of factors, \pfun{sim.rasch} to create simulated rasch data, \pfun{sim.irt} to create general 1 to 4 parameter IRT data by calling \pfun{sim.npl} 1 to 4 parameter logistic IRT or \pfun{sim.npn} 1 to 4 paramater normal IRT, \pfun{sim.structural} a general simulation of structural models, and \pfun{sim.anova} for ANOVA and lm simulations, and \pfun{sim.vss}. Some of these functions are separately documented and are listed here for ease of the help function. See each function for more detailed help. \begin{description} \item [\pfun{sim}] The default version is to generate a four factor simplex structure over three occasions, although more general models are possible. \item [\pfun{sim.simple}] Create major and minor factors. The default is for 12 variables with 3 major factors and 6 minor factors. \item [\pfun{sim.structure}] To combine a measurement and structural model into one data matrix. Useful for understanding structural equation models. \item [\pfun{sim.hierarchical}] To create data with a hierarchical (bifactor) structure. \item [\pfun{sim.congeneric}] To create congeneric items/tests for demonstrating classical test theory. This is just a special case of sim.structure. \item [\pfun{sim.circ}] To create data with a circumplex structure. \item [\pfun{sim.item}]To create items that either have a simple structure or a circumplex structure. \item [\pfun{sim.dichot}] Create dichotomous item data with a simple or circumplex structure. \item[\pfun{sim.rasch}] Simulate a 1 parameter logistic (Rasch) model. \item[\pfun{sim.irt}] Simulate a 2 parameter logistic (2PL) or 2 parameter Normal model. Will also do 3 and 4 PL and PN models. \item[\pfun{sim.multilevel}] Simulate data with different within group and between group correlational structures. \end{description} Some of these functions are described in more detail in the companion vignette: \href{"psych_for_sem.pdf"}{psych for sem}. The default values for \pfun{sim.structure} is to generate a 4 factor, 12 variable data set with a simplex structure between the factors. Two data structures that are particular challenges to exploratory factor analysis are the simplex structure and the presence of minor factors. Simplex structures \pfun{sim.simplex} will typically occur in developmental or learning contexts and have a correlation structure of r between adjacent variables and $r^n$ for variables n apart. Although just one latent variable (r) needs to be estimated, the structure will have nvar-1 factors. Many simulations of factor structures assume that except for the major factors, all residuals are normally distributed around 0. An alternative, and perhaps more realistic situation, is that the there are a few major (big) factors and many minor (small) factors. The challenge is thus to identify the major factors. \pfun{sim.minor} generates such structures. The structures generated can be thought of as having a a major factor structure with some small correlated residuals. Although coefficient $\omega_h$ is a very useful indicator of the general factor saturation of a unifactorial test (one with perhaps several sub factors), it has problems with the case of multiple, independent factors. In this situation, one of the factors is labelled as ``general'' and the omega estimate is too large. This situation may be explored using the \pfun{sim.omega} function. The four irt simulations, \pfun{sim.rasch}, \pfun{sim.irt}, \pfun{sim.npl} and \pfun{sim.npn}, simulate dichotomous items following the Item Response model. \pfun{sim.irt} just calls either \pfun{sim.npl} (for logistic models) or \pfun{sim.npn} (for normal models) depending upon the specification of the model. The logistic model is \begin{equation} P(x | \theta_i, \delta_j, \gamma_j, \zeta_j )= \gamma_j + \frac{\zeta_j - \gamma_j}{1+e^{\alpha_j(\delta_j - \theta_i}}. \end{equation} where $\gamma$ is the lower asymptote or guessing parameter, $\zeta$ is the upper asymptote (normally 1), $\alpha_j$ is item discrimination and $\delta_j$ is item difficulty. For the 1 Paramater Logistic (Rasch) model, gamma=0, zeta=1, alpha=1 and item difficulty is the only free parameter to specify. (Graphics of these may be seen in the demonstrations for the logistic function.) The normal model (\pfun{irt.npn} calculates the probability using \fun{pnorm} instead of the logistic function used in \pfun{irt.npl}, but the meaning of the parameters are otherwise the same. With the a = $\alpha$ parameter = 1.702 in the logiistic model the two models are practically identical. \section{Graphical Displays} Many of the functions in the \Rpkg{psych} package include graphic output and examples have been shown in the previous figures. After running \pfun{fa}, \pfun{iclust}, \pfun{omega}, \pfun{irt.fa}, plotting the resulting object is done by the \pfun{plot.psych} function as well as specific diagram functions. e.g., (but not shown) \begin{scriptsize} \begin{Schunk} \begin{Sinput} f3 <- fa(Thurstone,3) plot(f3) fa.diagram(f3) c <- iclust(Thurstone) plot(c) #a pretty boring plot iclust.diagram(c) #a better diagram c3 <- iclust(Thurstone,3) plot(c3) #a more interesting plot data(bfi) e.irt <- irt.fa(bfi[11:15]) plot(e.irt) ot <- omega(Thurstone) plot(ot) omega.diagram(ot) \end{Sinput} \end{Schunk} \end{scriptsize} The ability to show path diagrams to represent factor analytic and structural models is discussed in somewhat more detail in the accompanying vignette, \href{"psych_for_sem.pdf"}{psych for sem}. Basic routines to draw path diagrams are included in the \pfun{dia.rect} and accompanying functions. These are used by the \pfun{fa.diagram}, \pfun{structure.diagram} and \pfun{iclust.diagram} functions. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= xlim=c(0,10) ylim=c(0,10) plot(NA,xlim=xlim,ylim=ylim,main="Demonstration of dia functions",axes=FALSE,xlab="",ylab="") ul <- dia.rect(1,9,labels="upper left",xlim=xlim,ylim=ylim) ll <- dia.rect(1,3,labels="lower left",xlim=xlim,ylim=ylim) lr <- dia.ellipse(9,3,"lower right",xlim=xlim,ylim=ylim,e.size=.09) ur <- dia.ellipse(7,9,"upper right",xlim=xlim,ylim=ylim,e.size=.1) ml <- dia.ellipse(3,6,"middle left",xlim=xlim,ylim=ylim,e.size=.1) mr <- dia.ellipse(7,6,"middle right",xlim=xlim,ylim=ylim,e.size=.08) bl <- dia.ellipse(1,1,"bottom left",xlim=xlim,ylim=ylim,e.size=.08) br <- dia.rect(9,1,"bottom right",xlim=xlim,ylim=ylim) dia.arrow(from=lr,to=ul,labels="right to left") dia.arrow(from=ul,to=ur,labels="left to right") dia.curved.arrow(from=lr,to=ll$right,labels ="right to left") dia.curved.arrow(to=ur,from=ul$right,labels ="left to right") dia.curve(ll$top,ul$bottom,"double",-1) #for rectangles, specify where to point dia.curved.arrow(mr,ur,"up") #but for ellipses, just point to it. dia.curve(ml,mr,"across") dia.curved.arrow(ur,lr,"top down") dia.curved.arrow(br$top,lr$bottom,"up") dia.curved.arrow(bl,br,"left to right") dia.arrow(bl$top,ll$bottom) dia.curved.arrow(ml,ll$top,scale=-1) dia.curved.arrow(mr,lr$top) @ \end{scriptsize} \caption{The basic graphic capabilities of the dia functions are shown in this figure.} \label{fig:dia} \end{center} \end{figure} \section{Converting output to APA style tables using \LaTeX} Although for most purposes, using the \Rpkg{Sweave} or \Rpkg{KnitR} packages produces clean output, some prefer output pre formatted for APA style tables. This can be done using the \Rpkg{xtable} package for almost anything, but there are a few simple functions in \Rpkg{psych} for the most common tables. \pfun{fa2latex} will convert a factor analysis or components analysis output to a \LaTeX table, \pfun{cor2latex} will take a correlation matrix and show the lower (or upper diagonal), \pfun{irt2latex} converts the item statistics from the \pfun{irt.fa} function to more convenient \LaTeX output, and finally, \pfun{df2latex} converts a generic data frame to \LaTeX. An example of converting the output from \pfun{fa} to \LaTeX appears in Table~\ref{falatex}. % fa2latex % f3 % Called in the psych package fa2latex % Called in the psych package f3 \begin{scriptsize} \begin{table}[htpb] \caption{fa2latex} \begin{center} \begin{tabular} {l r r r r r r } \multicolumn{ 6 }{l}{ A factor analysis table from the psych package in R } \cr \hline Variable & MR1 & MR2 & MR3 & h2 & u2 & com \cr \hline Sentences & 0.91 & -0.04 & 0.04 & 0.82 & 0.18 & 1.01 \cr Vocabulary & 0.89 & 0.06 & -0.03 & 0.84 & 0.16 & 1.01 \cr Sent.Completion & 0.83 & 0.04 & 0.00 & 0.73 & 0.27 & 1.00 \cr First.Letters & 0.00 & 0.86 & 0.00 & 0.73 & 0.27 & 1.00 \cr 4.Letter.Words & -0.01 & 0.74 & 0.10 & 0.63 & 0.37 & 1.04 \cr Suffixes & 0.18 & 0.63 & -0.08 & 0.50 & 0.50 & 1.20 \cr Letter.Series & 0.03 & -0.01 & 0.84 & 0.72 & 0.28 & 1.00 \cr Pedigrees & 0.37 & -0.05 & 0.47 & 0.50 & 0.50 & 1.93 \cr Letter.Group & -0.06 & 0.21 & 0.64 & 0.53 & 0.47 & 1.23 \cr \hline \cr SS loadings & 2.64 & 1.86 & 1.5 & \cr\cr \hline \cr MR1 & 1.00 & 0.59 & 0.54 \cr MR2 & 0.59 & 1.00 & 0.52 \cr MR3 & 0.54 & 0.52 & 1.00 \cr \hline \end{tabular} \end{center} \label{falatex} \end{table} \end{scriptsize} \newpage \section{Miscellaneous functions} A number of functions have been developed for some very specific problems that don't fit into any other category. The following is an incomplete list. Look at the \iemph{Index} for \Rpkg{psych} for a list of all of the functions. \begin{description} \item [\pfun{block.random}] Creates a block randomized structure for n independent variables. Useful for teaching block randomization for experimental design. \item [\pfun{df2latex}] is useful for taking tabular output (such as a correlation matrix or that of \pfun{describe} and converting it to a \LaTeX{} table. May be used when Sweave is not convenient. \item [\pfun{cor2latex}] Will format a correlation matrix in APA style in a \LaTeX{} table. See also \pfun{fa2latex} and \pfun{irt2latex}. \item [\pfun{cosinor}] One of several functions for doing \iemph{circular statistics}. This is important when studying mood effects over the day which show a diurnal pattern. See also \pfun{circadian.mean}, \pfun{circadian.cor} and \pfun{circadian.linear.cor} for finding circular means, circular correlations, and correlations of circular with linear data. \item[\pfun{fisherz}] Convert a correlation to the corresponding Fisher z score. \item [\pfun{geometric.mean}] also \pfun{harmonic.mean} find the appropriate mean for working with different kinds of data. \item [\pfun{ICC}] and \pfun{cohen.kappa} are typically used to find the reliability for raters. \item [\pfun{headtail}] combines the \fun{head} and \fun{tail} functions to show the first and last lines of a data set or output. \item [\pfun{topBottom}] Same as headtail. Combines the \fun{head} and \fun{tail} functions to show the first and last lines of a data set or output, but does not add ellipsis between. \item [\pfun{mardia}] calculates univariate or multivariate (Mardia's test) skew and kurtosis for a vector, matrix, or data.frame \item [\pfun{p.rep}] finds the probability of replication for an F, t, or r and estimate effect size. \item [\pfun{partial.r}] partials a y set of variables out of an x set and finds the resulting partial correlations. (See also \pfun{setCor}.) \item [\pfun{rangeCorrection}] will correct correlations for restriction of range. \item [\pfun{reverse.code}] will reverse code specified items. Done more conveniently in most \Rpkg{psych} functions, but supplied here as a helper function when using other packages. \item [\pfun{superMatrix}] Takes two or more matrices, e.g., A and B, and combines them into a ``Super matrix'' with A on the top left, B on the lower right, and 0s for the other two quadrants. A useful trick when forming complex keys, or when forming example problems. \end{description} \section{Data sets} A number of data sets for demonstrating psychometric techniques are included in the \Rpkg{psych} package. These include six data sets showing a hierarchical factor structure (five cognitive examples, \pfun{Thurstone}, \pfun{Thurstone.33}, \pfun{Holzinger}, \pfun{Bechtoldt.1}, \pfun{Bechtoldt.2}, and one from health psychology \pfun{Reise}). One of these (\pfun{Thurstone}) is used as an example in the \Rpkg{sem} package as well as \cite{mcdonald:tt}. The original data are from \cite{thurstone:41} and reanalyzed by \cite{bechtoldt:61}. Personality item data representing five personality factors on 25 items (\pfun{bfi}) or 13 personality inventory scores (\pfun{epi.bfi}), and 14 multiple choice iq items (\pfun{iqitems}). The \pfun{vegetables} example has paired comparison preferences for 9 vegetables. This is an example of Thurstonian scaling used by \cite{guilford:54} and \cite{nunnally:67}. Other data sets include \pfun{cubits}, \pfun{peas}, and \pfun{heights} from Galton. \begin{description} \item[Thurstone] Holzinger-Swineford (1937) introduced the bifactor model of a general factor and uncorrelated group factors. The Holzinger correlation matrix is a 14 * 14 matrix from their paper. The Thurstone correlation matrix is a 9 * 9 matrix of correlations of ability items. The Reise data set is 16 * 16 correlation matrix of mental health items. The Bechtholdt data sets are both 17 x 17 correlation matrices of ability tests. \item [bfi] 25 personality self report items taken from the International Personality Item Pool (ipip.ori.org) were included as part of the Synthetic Aperture Personality Assessment (\iemph{SAPA}) web based personality assessment project. The data from 2800 subjects are included here as a demonstration set for scale construction, factor analysis and Item Response Theory analyses. \item [sat.act] Self reported scores on the SAT Verbal, SAT Quantitative and ACT were collected as part of the Synthetic Aperture Personality Assessment (\iemph{SAPA}) web based personality assessment project. Age, gender, and education are also reported. The data from 700 subjects are included here as a demonstration set for correlation and analysis. \item [epi.bfi] A small data set of 5 scales from the Eysenck Personality Inventory, 5 from a Big 5 inventory, a Beck Depression Inventory, and State and Trait Anxiety measures. Used for demonstrations of correlations, regressions, graphic displays. \item[epiR] The EPI was given twice to 474 participants. This is a useful data set for exploring test-retest reliability, \item[sai, msqR] 20 anxiety items and 75 mood items were given at least twice to 3032 participants. These are useful for understanding reliability structures. \item [iq] 14 multiple choice ability items were included as part of the Synthetic Aperture Personality Assessment (\iemph{SAPA}) web based personality assessment project. The data from 1000 subjects are included here as a demonstration set for scoring multiple choice inventories and doing basic item statistics. \item [galton] Two of the earliest examples of the correlation coefficient were Francis Galton's data sets on the relationship between mid parent and child height and the similarity of parent generation peas with child peas. \pfun{galton} is the data set for the Galton height. \pfun{peas} is the data set Francis Galton used to ntroduce the correlation coefficient with an analysis of the similarities of the parent and child generation of 700 sweet peas. \item[Dwyer] \cite{dwyer:37} introduced a method for \emph{factor extension} (see \pfun{fa.extension} that finds loadings on factors from an original data set for additional (extended) variables. This data set includes his example. \item [miscellaneous] \pfun{cities} is a matrix of airline distances between 11 US cities and may be used for demonstrating multiple dimensional scaling. \pfun{vegetables} is a classic data set for demonstrating Thurstonian scaling and is the preference matrix of 9 vegetables from \cite{guilford:54}. Used by \cite{guilford:54,nunnally:67,nunnally:bernstein:94}, this data set allows for examples of basic scaling techniques. \end{description} \section{Development version and a users guide} The most recent development version is available as a source file at the repository maintained at \href{ href="https://personality-project.org/r"}{\url{https://personality-project.org/r}}. That version will have removed the most recently discovered bugs (but perhaps introduced other, yet to be discovered ones). To download that version, go to the repository %\href{"https://personality-project.org/r/src/contrib/}{ \url{https://personality-project.org/r/src/contrib/} and wander around. For a Mac and PC this version can be installed directly using the ``other repository" option in the package installer. \begin{Schunk} \begin{Sinput} > install.packages("psych", repos="https://personality-project.org/r", type="source") \end{Sinput} \end{Schunk} Although the individual help pages for the \Rpkg{psych} package are available as part of \R{} and may be accessed directly (e.g. ?psych) , the full manual for the \pfun{psych} package is also available as a pdf at \url{https://personality-project.org/r/psych_manual.pdf} %psych\_manual.pdf. News and a history of changes are available in the NEWS and CHANGES files in the source files. To view the most recent news, \begin{Schunk} \begin{Sinput} > news(Version > "1.8.4", package="psych") \end{Sinput} \end{Schunk} \section{Psychometric Theory} The \Rpkg{psych} package has been developed to help psychologists do basic research. Many of the functions were developed to supplement a book (\url{https://personality-project.org/r/book} An introduction to Psychometric Theory with Applications in \R{} \citep{revelle:intro} More information about the use of some of the functions may be found in the book . For more extensive discussion of the use of \Rpkg{psych} in particular and \R{} in general, consult \url{https://personality-project.org/r/r.guide.html} A short guide to R. \section{SessionInfo} This document was prepared using the following settings. \begin{tiny} <>= sessionInfo() @ \end{tiny} \newpage %\bibliography{/Volumes/WR/Documents/Active/book/all} \begin{thebibliography}{} \bibitem[\protect\astroncite{Bechtoldt}{1961}]{bechtoldt:61} Bechtoldt, H. 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(2011). \newblock Methodological advances in differential psychology. \newblock In Chamorro-Premuzic, T., Furnham, A., and von Stumm, S., editors, {\em Handbook of Individual Differences}, chapter~2, pages 39--73. Wiley-Blackwell. \bibitem[\protect\astroncite{Revelle and Condon}{2018}]{rc:reliability} Revelle, W. and Condon, D.~M. (2018). \newblock Reliability. \newblock In Irwing, P., Booth, T., and Hughes, D., editors, {\em Wiley-Blackwell Handbook of Psychometric Testing}. Wiley-Blackwell. \bibitem[\protect\astroncite{Revelle and Rocklin}{1979}]{revelle:vss} Revelle, W. and Rocklin, T. (1979). \newblock {Very Simple Structure} - alternative procedure for estimating the optimal number of interpretable factors. \newblock {\em Multivariate Behavioral Research}, 14(4):403--414. \bibitem[\protect\astroncite{Revelle et~al.}{2010}]{rwr:sapa} Revelle, W., Wilt, J., and Rosenthal, A. (2010). \newblock Individual differences in cognition: New methods for examining the personality-cognition link. \newblock In Gruszka, A., Matthews, G., and Szymura, B., editors, {\em Handbook of Individual Differences in Cognition: Attention, Memory and Executive Control}, chapter~2, pages 27--49. Springer, New York, N.Y. \bibitem[\protect\astroncite{Revelle and Wilt}{2017}]{rw:paid:17} Revelle, W. and Wilt, J.~A. (2017). \newblock Analyzing dynamic data: a tutorial. \newblock {\em Personality and Individual Differences}, (in press). \bibitem[\protect\astroncite{Revelle and Zinbarg}{2009}]{rz:09} Revelle, W. and Zinbarg, R.~E. (2009). \newblock Coefficients alpha, beta, omega and the glb: comments on {Sijtsma}. \newblock {\em Psychometrika}, 74(1):145--154. \bibitem[\protect\astroncite{Schmid and Leiman}{1957}]{schmid:57} Schmid, J.~J. and Leiman, J.~M. (1957). \newblock The development of hierarchical factor solutions. \newblock {\em Psychometrika}, 22(1):83--90. \bibitem[\protect\astroncite{Shrout and Lane}{2012}]{shrout:12a} Shrout, P. and Lane, S.~P. (2012). \newblock Psychometrics. \newblock In {\em Handbook of research methods for studying daily life}. Guilford Press. \bibitem[\protect\astroncite{Shrout and Fleiss}{1979}]{shrout:79} Shrout, P.~E. and Fleiss, J.~L. (1979). \newblock Intraclass correlations: Uses in assessing rater reliability. \newblock {\em Psychological Bulletin}, 86(2):420--428. \bibitem[\protect\astroncite{Sneath and Sokal}{1973}]{sneath:73} Sneath, P. H.~A. and Sokal, R.~R. (1973). \newblock {\em Numerical taxonomy: the principles and practice of numerical classification}. \newblock A Series of books in biology. W. H. Freeman, San Francisco. \bibitem[\protect\astroncite{Sokal and Sneath}{1963}]{sokal:63} Sokal, R.~R. and Sneath, P. H.~A. (1963). \newblock {\em Principles of numerical taxonomy}. \newblock A Series of books in biology. W. H. Freeman, San Francisco. \bibitem[\protect\astroncite{Spearman}{1904}]{spearman:rho} Spearman, C. (1904). \newblock The proof and measurement of association between two things. \newblock {\em The American Journal of Psychology}, 15(1):72--101. \bibitem[\protect\astroncite{Thorburn}{1918}]{thornburn:1918} Thorburn, W.~M. (1918). \newblock The myth of {Occam's} razor. \newblock {\em Mind}, 27:345--353. \bibitem[\protect\astroncite{Thurstone and Thurstone}{1941}]{thurstone:41} Thurstone, L.~L. and Thurstone, T.~G. (1941). \newblock {\em Factorial studies of intelligence}. \newblock The University of Chicago press, Chicago, Ill. \bibitem[\protect\astroncite{Tryon}{1935}]{tryon:35} Tryon, R.~C. (1935). \newblock A theory of psychological components--an alternative to "mathematical factors.". \newblock {\em Psychological Review}, 42(5):425--454. \bibitem[\protect\astroncite{Tryon}{1939}]{tryon:39} Tryon, R.~C. (1939). \newblock {\em Cluster analysis}. \newblock Edwards Brothers, Ann Arbor, Michigan. \bibitem[\protect\astroncite{Velicer}{1976}]{velicer:76} Velicer, W. (1976). \newblock Determining the number of components from the matrix of partial correlations. \newblock {\em Psychometrika}, 41(3):321--327. \bibitem[\protect\astroncite{Zinbarg et~al.}{2005}]{zinbarg:pm:05} Zinbarg, R.~E., Revelle, W., Yovel, I., and Li, W. (2005). \newblock Cronbach's {$\alpha$}, {Revelle's} {$\beta$}, and {McDonald's} {$\omega_H$}: Their relations with each other and two alternative conceptualizations of reliability. \newblock {\em Psychometrika}, 70(1):123--133. \bibitem[\protect\astroncite{Zinbarg et~al.}{2006}]{zinbarg:apm:06} Zinbarg, R.~E., Yovel, I., Revelle, W., and McDonald, R.~P. (2006). \newblock Estimating generalizability to a latent variable common to all of a scale's indicators: A comparison of estimators for {$\omega_h$}. \newblock {\em Applied Psychological Measurement}, 30(2):121--144. \end{thebibliography} \printindex \end{document} psychTools/vignettes/omega.Rnw0000644000176200001440000015450713726243502016231 0ustar liggesusers% \VignetteIndexEntry{How to find Omega} % \VignettePackage{psych} % \VignetteKeywords{multivariate} % \VignetteKeyword{models} % \VignetteKeyword{Hplot} %\VignetteDepends{psych} %\documentclass[doc]{apa} \documentclass[11pt]{article} %\documentclass[11pt]{amsart} \usepackage{geometry} % See geometry.pdf to learn the layout options. There are lots. \geometry{letterpaper} % ... or a4paper or a5paper or ... %\geometry{landscape} % Activate for for rotated page geometry \usepackage[parfill]{parskip} % Activate to begin paragraphs with an empty line rather than an indent \usepackage{graphicx} \usepackage{amssymb} \usepackage{epstopdf} \usepackage{mathptmx} \usepackage{helvet} \usepackage{courier} \usepackage{epstopdf} \usepackage{makeidx} % allows index generation \usepackage[authoryear,round]{natbib} \usepackage{gensymb} \usepackage{longtable} %\usepackage{geometry} \usepackage{amssymb} \usepackage{amsmath} %\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \usepackage{Sweave} %\usepackage{/Volumes/'Macintosh HD'/Library/Frameworks/R.framework/Versions/2.13/Resources/share/texmf/tex/latex/Sweave} %\usepackage[ae]{Rd} %\usepackage[usenames]{color} %\usepackage{setspace} \bibstyle{apacite} \bibliographystyle{apa} %this one plus author year seems to work? %\usepackage{hyperref} \usepackage[colorlinks=true,citecolor=blue]{hyperref} %this makes reference links hyperlinks in pdf! \DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \usepackage{multicol} % used for the two-column index \usepackage[bottom]{footmisc}% places footnotes at page bottom \let\proglang=\textsf \newcommand{\R}{\proglang{R}} %\newcommand{\pkg}[1]{{\normalfont\fontseries{b}\selectfont #1}} \newcommand{\Rfunction}[1]{{\texttt{#1}}} \newcommand{\fun}[1]{{\texttt{#1}\index{#1}\index{R function!#1}}} \newcommand{\pfun}[1]{{\texttt{#1}\index{#1}\index{R function!#1}\index{R function!psych package!#1}}}\newcommand{\Rc}[1]{{\texttt{#1}}} %R command same as Robject \newcommand{\Robject}[1]{{\texttt{#1}}} \newcommand{\Rpkg}[1]{{\textit{#1}\index{#1}\index{R package!#1}}} %different from pkg - which is better? \newcommand{\iemph}[1]{{\emph{#1}\index{#1}}} \newcommand{\wrc}[1]{\marginpar{\textcolor{blue}{#1}}} %bill's comments \newcommand{\wra}[1]{\textcolor{blue}{#1}} %bill's comments \newcommand{\ve}[1]{{\textbf{#1}}} %trying to get a vector command \usepackage{fancyvrb} %this allows fancy boxes \fvset{fontfamily=courier} \DefineVerbatimEnvironment{Routput}{Verbatim} %{fontsize=\scriptsize, xleftmargin=0.6cm} {fontseries=b,fontsize=\scriptsize, xleftmargin=0.1cm} \DefineVerbatimEnvironment{Binput}{Verbatim} {fontseries=b, fontsize=\scriptsize,frame=single, label=\fbox{lavaan model syntax}, framesep=2mm} %\DefineShortVerb{\!} %%% generates error! %change the definition of Sinput from Sweave \DefineVerbatimEnvironment{Sinput}{Verbatim}{fontseries=b, fontsize=\scriptsize, frame=single, label=\fbox{R code},xleftmargin=0pt, framesep=1mm} \DefineVerbatimEnvironment{Rinput}{Verbatim} %{fontsize=\scriptsize, frame=single, label=\fbox{R code}, framesep=1mm} {fontseries=b, fontsize=\scriptsize, frame=single, label=\fbox{R code},xleftmargin=0pt, framesep=1mm} \DefineVerbatimEnvironment{Link}{Verbatim} {fontseries=b, fontsize=\small, formatcom=\color{darkgreen}, xleftmargin=1.0cm} \DefineVerbatimEnvironment{Toutput}{Verbatim} {fontseries=b,fontsize=\tiny, xleftmargin=0.1cm} \DefineVerbatimEnvironment{rinput}{Verbatim} {fontseries=b, fontsize=\tiny, frame=single, label=\fbox{R code}, framesep=1mm} \newcommand{\citeti}[1]{\begin{tiny}\citep{#1}\end{tiny}} \newcommand{\light}[1]{\textcolor{gray}{#1}} \newcommand{\vect}[1]{\boldsymbol{#1}} \let\vec\vect \makeindex % used for the subject index \title{Using \R{} and the \Rpkg{psych} package to find $\omega$} \author{William Revelle\\Department of Psychology\\Northwestern University} %\affiliation{Northwestern University} %\acknowledgements{Written to accompany the psych package. Comments should be directed to William Revelle \\ \url{revelle@northwestern.edu}} %\date{} % Activate to display a given date or no date \begin{document} \maketitle \tableofcontents \newpage \section{Overview of this and related documents} To do basic and advanced personality and psychological research using \R{} is not as complicated as some think. This is one of a set of ``How To'' to do various things using \R{} \citep{R}, particularly using the \Rpkg{psych} \citep{psych} package. The current list of How To's includes: \begin{enumerate} \item An \href{http://personality-project.org/r/psych/intro.pdf}{introduction} (vignette) of the \Rpkg{psych} package \item An \href{http://personality-project.org/r/psych/overview.pdf}{overview} (vignette) of the \Rpkg{psych} package \item \href{http://personality-project.org/r/psych/HowTo/getting_started.pdf}{Installing} \R{} and some useful packages \item Using \R{} and the \Rpkg{psych} package to find \href{http://personality-project.org/r/psych/HowTo/omega.pdf}{$omega_h$} and $\omega_t$ (this document).. \item Using \R{} and the \Rpkg{psych} for \href{http://personality-project.org/r/psych/HowTo/factor.pdf}{factor analysis} and principal components analysis. \item Using the \pfun{scoreItems} function to find \href{http://personality-project.org/r/psych/HowTo/scoring.pdf}{scale scores and scale statistics}. \item Using \pfun{mediate} and \pfun{setCor} to do \href{http://personality-project.org/r/psych/HowTo/mediation.pdf}{mediation, moderation and regression analysis} \end{enumerate} \subsection{$omega_h$ as an estimate of the general factor saturation of a test} Cronbach's coefficient $alpha$ \citep{cronbach:51} is pehaps the most used (and most misused) estimate of the internal consistency of a test. $\alpha$ may be found in the \Rpkg{psych} package using the \pfun{alpha} function. However, two alternative estimates of reliability that take into account the hierarchical structure of the inventory are McDonald's $\omega_h$ and $\omega_t$ \citep{mcdonald:tt}. These may be found in \R{} in one step using one of two functions in the \Rpkg{psych} package: the \pfun{omega} function for an exploratory analysis (See Figure~\ref{fig:omega.9}) or \pfun{omegaSem} for a confirmatory analysis using the \Rpkg{sem} package solution based upon the exploratory solution from \pfun{omega}. This guide explains how to do it for the non or novice \R{} user. These set of instructions are adapted from three different sets of notes that the interested reader might find helpful: A set of slides developed for a \href{http://personality-project.org/r/aps/aps-short.pdf}{ two hour short course} in \R{} given for several years to the Association of Psychological Science as well as a \href{http://personality-project.org/r/}{short guide }to \R{} for psychologists and the \href{http://cran.r-project.org/web/packages/psych/vignettes/overview.pdf}{vignette} for the \Rpkg{psych} package. McDonald has proposed coefficient omega (hierarchical) ($\omega_h$) as an estimate of the general factor saturation of a test. \cite{zinbarg:pm:05} and \cite{rz:09} compare compare McDonald's $\omega_h$ to Cronbach's $\alpha$ and Revelle's $\beta$. They conclude that $\omega_h$ is the best estimate. (See also \cite{zinbarg:apm:06} and \cite{rz:09} as well as \cite{rc:reliability,rc:pa:19}. By following these simple guides, you soon will be able to do such things as find $\omega_{h}$ by issuing just three lines of code: \begin{Rinput} library(psych) my.data <- read.file() omega(my.data) \end{Rinput} The resulting output will be both graphical and textual. This guide helps the naive \R{} user to issue those three lines. Be careful, for once you start using \R, you will want to do more. One way to find $\omega_h$ is to do a factor analysis of the original data set, rotate the factors obliquely, factor that correlation matrix, do a Schmid-Leiman (\pfun{schmid}) transformation to find general factor loadings, and then find $\omega_h$. This is done using the \pfun{omega} function in the \Rpkg{psych} package in \R{}. This requires installing and using both \R{} as well as the \Rpkg{psych} package \citep{psych}. \subsubsection{But what about $\alpha$?} Several statistics were developed in the 1930s-1950s as short cut estimates of reliability \citep{rc:pa:19}. The approaches that consider just one test are collectively known as internal consistency procedures but also borrow from the concepts of domain sampling. Some of these techniques, e.g., \cite{cronbach:51,guttman:45,kuder:37} were developed before advances in computational speed made it trivial to find the factor structure of tests, and were based upon test and item variances. These procedures ($\alpha$, $\lambda_3$, KR20) were essentially short cuts for estimating reliability. To just find Guttman's $\lambda_3$ \citep{guttman:45} which is also known as \emph{coefficient} $\alpha$ \citep{cronbach:51}, you can use the \pfun{alpha} function or the \pfun{scoreItems} function. See the tutorial on how to use the \pfun{scoreItems} function to find \href{http://personality-project.org/r/psych/HowTo/scoring.pdf}{scale scores and scale statistics}. But, with modern computers, we can find \emph{model based} estimates that consider the factor structure of the items. $\omega_h$ and $\omega_t$ are two such model based estimates and are easy to find in \R{}. ~\ <>= library(psych) #make the psych package active library(psychTools) #make psychTools active om <- omega(Thurstone) #do the analysis om #show it @ <>= png('Thurstone.png') omega.diagram(om) dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{Thurstone.png} \caption{$\omega_h$ is a reliability estimate of the general factor of a set of variables. It is based upon the correlation of lower order factors. It may be found in \R{} by using the \pfun{omega} function which is part of the \Rpkg{psych} package. The figure shows a solution for the \pfun{Thurstone} 9 variable data set. Compare this to the solution using the \pfun{omegaDirect} function from \cite{waller:17} (Figure~\ref{fig:direct})} \label{fig:omega.9} \end{center} \end{figure} \newpage To use \R{} obviously requires installing \R{} on your computer. This is very easy to do (see section~\ref{install}) and needs to be done once. (The following sections are elaborated in the \href{https://personality-project.org/r/psych/HowTo/getting_started.pdf}{``getting startedHow To" } . If you need more help in installing \R{} see the longer version.) The power of \R{} is in the supplemental \emph{packages}. There are at least 16,000 packages that have been contributed to the \R{} project. To do any of the analyses discussed in these ``How To's", you will need to install the package \Rpkg{psych} \citep{psych}. To do factor analyses or principal component analyses you will also need the \Rpkg{GPArotation} \citep{GPA} package. With these two packages, you will be be able to find $\omega_{h}$ using Exploratory Factor Analysis. If you want to find to estimate $\omega_h$ using Confirmatory Factor Analysis, you will also need to add the \Rpkg{lavaan} \citep{lavaan} package. To use \Rpkg{psych} to create simulated data sets, you also need the \Rpkg{mnormt} \citep{mnormt} package. For a more complete installation of a number of psychometric packages, you can install and activate a package (\Rpkg{ctv}) that installs a large set of psychometrically relevant packages. As is true for \R{}, you will need to install packages just once. \subsection{Install R for the first time} \begin{enumerate} \item Download from R Cran (\url{http://cran.r-project.org/}) \item Install R (current version is 4.0.2) \item Start \R{}. Note that the \R{} prompt $>$ starts off every line. This is \R{}'s way of indicating that it wants input. In addition, note that almost all commands start and finish with parentheses. \item Add useful packages (just need to do this once) (see section~\ref{installing}) \begin{enumerate} \begin{Rinput} install.packages("psych",dependencies=TRUE) #the minimum requirement or install.packages(c("psych","GPArotation"),dependencies=TRUE) #required for factor analysis \end{Rinput} \item or if you want to do CFA \begin{Rinput} install.packages(c("psych","lavaan"), dependencies=TRUE) \end{Rinput} \item or if you want to install the psychometric task views \begin{Rinput} install.packages("ctv") #this downloads the task view package library(ctv) #this activates the ctv package install.views("Psychometrics") #among others \end{Rinput} \end{enumerate} \item Take a 5 minute break while the packages are loaded. \item Activate the package(s) you want to use (e.g., \Rpkg{psych}) \begin{Rinput} library(psych) #Only need to make psych active once a session \end{Rinput} \Rpkg{psych} will automatically activate the other packages it needs, as long as they are installed. Note that \Rpkg{psych} is updated roughly quarterly, the current version is 2.0.8 Patches and improvements to \Rpkg{psych} (the bleeding edge version) are available from the repository at the personality-project web server and may be installed from there: ~\ \begin{Rinput} install.packages("psych", repos = "https://personality-project.org/r", type="source") \end{Rinput} %\item library(sem) \#will be used for a few examples \item Use \R{} \end{enumerate} \subsubsection{Install R } \label{install} Go to the \href{http://cran.r-project.org}{Comprehensive R Archive Network (CRAN)} at \url{http://cran.r-project.org}: %(Figure~\ref{fig:cran}) %\begin{figure}[htbp] %\begin{center} %%\includegraphics[width=14cm]{../../../tutorials/R_Short_APS.15/rcran3.png} %\includegraphics[width=20cm]{../../../images/CRAN.png} %\caption{The basic \href{http://cran.r-project.org}{CRAN} window allows you choose your operating system. Comprehensive R Archive Network (CRAN) is found at \href{http://cran.r-project.org}{http://cran.r-project.org}:} %\label{fig:cran} %\end{center} %\end{figure} Choose your operating system and then download and install the appropriate version %For a PC: %(Figure~\ref{fig:pc}) %\begin{figure}[htbp] %\begin{center} %%\includegraphics[width=14cm]{../../../tutorials/R_Short_APS.15/cranpc1.png} %\includegraphics[width=19cm]{../../../images/CRAN_pc.pdf} % %\caption{On a PC you want to choose the base system} %\label{fig:pc} %\end{center} %\end{figure} Download and install the appropriate version -- Mac, PC or Unix/Linux %\begin{figure}[htbp] %\begin{center} %%\includegraphics[width=14cm]{../../../tutorials/R_Short_APS.15/cran-pc15.png} %\includegraphics[width=19cm]{../../../images/CRAN_pc_16.png} %\caption{Download the Windows version} %\label{default} %\end{center} %\end{figure} %Starting R on a PC. Once you have installed \R{} you probably, and particularly if you have a PC, will want to download and install the \href{https://www.rstudio.com} {R Studio} program. It is a very nice interface for PCs and Macs that combines four windows into one screen. %\begin{figure}[htbp] %\begin{center} %\includegraphics[width=14cm]{../../../images/RStudio01.png} %\caption{Using R Studio on a PC. } %\label{fig:pcstartup} %\end{center} %\end{figure} % %When using a PC, RStudio is very helpful. (Many like it for Macs as well). % % %\begin{figure}[htbp] %\begin{center} %\includegraphics[width=14cm]{../../../images/RStudio01.png} %\caption{Using R Studio on a PC. } %\label{fig:pcRstudio} %\end{center} %\end{figure} % %\clearpage % %%For a Mac: download and install the appropriate version -- Mac (Figure~\ref{fig:mac}) %\begin{figure}[htbp] %\begin{center} %%\includegraphics[width=14cm]{../../../tutorials/R_Short_APS/cran-mac.png} %\includegraphics[width=19cm]{../../../images/cran_mac.png} %\caption{For the Mac, you want to choose the latest version which includes the GUI as well as the 32 and 64 bit versions.} %\label{fig:mac} %\end{center} %\end{figure} % %\newpage %Start up R and get ready to play (Mac version). %\begin{scriptsize} %\begin{Schunk} %\begin{Soutput} %R version 3.3.0 (2016-05-03) -- "Supposedly Educational" %Copyright (C) 2016 The R Foundation for Statistical Computing %Platform: x86_64-apple-darwin13.4.0 (64-bit) % %R is free software and comes with ABSOLUTELY NO WARRANTY. %You are welcome to redistribute it under certain conditions. %Type 'license()' or 'licence()' for distribution details. % % Natural language support but running in an English locale % %R is a collaborative project with many contributors. %Type 'contributors()' for more information and %'citation()' on how to cite R or R packages in publications. % %Type 'demo()' for some demos, 'help()' for on-line help, or %'help.start()' for an HTML browser interface to help. %Type 'q()' to quit R. % %[R.app GUI 1.68 (7202) x86_64-apple-darwin13.4.0] % %[Workspace restored from /Users/revelle/.RData] %[History restored from /Users/revelle/.Rapp.history] % %> %\end{Soutput} %\end{Schunk} %\end{scriptsize} \subsubsection{Install relevant packages} \label{installing} Once \R{} is installed on your machine, you still need to install a few relevant ``packages''. Packages are what make \R{} so powerful, for they are special sets of functions that are designed for one particular application. In the case of the \Rpkg{psych} package, this is an application for doing the kind of basic data analysis and psychometric analysis that psychologists and many others find particularly useful. \Rpkg{psych} may be thought of a ``Swiss Army Knife" for psychological statistics. While not the best tool for a particular job, it is a useful tool for many jobs. You may either install the minimum set of packages necessary to do the analysis using an Exploratory Factor Analysis (EFA) approach (recommended) or a few more packages to do both an EFA and a CFA approach. It is also possible to add many psychometrically relevant packages all at once by using the ``task views'' approach. A particularly powerful package is the \Rpkg{lavaan} \citep{lavaan} package for doing structural equation modeling. Another useful one is the \Rpkg{sem} pacakge \citep{sem}. \paragraph{Install the minimum set} This may be done by typing into the console or using menu options (e.g., the Package Installer underneath the Packages and Data menu). \begin{Rinput} install.packages(c("psych", "psychTools"), dependencies = TRUE) \end{Rinput} % %\begin{figure}[htbp] %\begin{center} %\includegraphics[width=14cm]{../../../images/RStudio02.PNG} %\caption{Installing packages using R studio on a PC. Use the install menu option.} %\label{fig:installPC} %\end{center} %\end{figure} \paragraph{Install a few more packages } If you want some more functionality for some of the more advanced statistical procedures (e.g., \pfun{omegaSem}) you will need to install a few more packages (e.g., \Rpkg{lavaan}. \begin{Rinput} install.packages(c("psych","GPArotation","lavaan"),dependencies=TRUE) \end{Rinput} \paragraph{Install a ``task view" to get lots of packages} If you know that there are a number of packages that you want to use, it is possible they are listed as a ``task view". For instance, about 50 packages will be installed at once if you install the ``psychometrics'' task view. You can Install all the psychometric packages from the ``psychometrics'' task view by first installing a package (``ctv") that in turn installs many different task views. To see the list of possible task views, go to \url{https://cran.r-project.org/web/views/}. ~\ \begin{Rinput} install.packages("ctv") } #this downloads the task view package library(ctv) #this activates the ctv package install.views("Psychometrics") #one of the many Taskviews \end{Rinput} Take a 5 minute break because you will be installing about 50 packages. \paragraph{For the more adventurous users} The \Rpkg{psych} pacakge is under (sporadic) development with a new release issued to CRAN roughly every 4-6 months. The experimental, development version (prerelease) is always available at the Personality-Project web site and may be installed for Macs or PCs directly: ~\ \begin{Rinput} install.packages("psych", repos= "https://personality-project.org/r", type ="source") \end{Rinput} This development version will have fixed any bugs reported since the last release and will have various new features that are being tested before release to CRAN. After installation, it is necessary to restart \R{} to make the new version active. \paragraph{Make the \Rpkg{psych} package active.} You are almost ready. But first, to use most of the following examples you need to make the \Rpkg{psych} and \Rpkg{psychTools} packages active. You only need to do this once per session. ~\ \begin{Rinput} library(psych) #to do the analyses described here library(psychTools) #for some useful additions such as read.file \end{Rinput} %(If you want to automate this last step, you can create a special command to be run every time you start \R{}. % %\begin{Rinput} %.First <- function() {library(psych)} %\end{Rinput} %Do this when you first start \R. Then quit with the save option. Then restart \R. You will now automatically have loaded the \Rpkg{psych} package every time you start \R{}.) % % \section{Reading in the data for analysis} \subsection{Find a file and read from it} There are of course many ways to enter data into \R. Reading from a local file using \pfun{read.file} is perhaps the most preferred. This will read in most of the standard file types (.csv, .sav, .txt, etc). \pfun{read.file} combines the \fun{file.choose} and \fun{read.table} functions: ~\ \begin{Rinput} my.data <- read.file() #note the open and closing parentheses \end{Rinput} \pfun{read.file} opens a search window on your system just like any open file command does. \pfun{read.file} assumes that the first row of your table has labels for each column. If this is not true, specify names=FALSE, e.g., ~\ \begin{Rinput} my.data <- read.file(names = FALSE) \end{Rinput} If you want to read a remote file, specify the file name and then \pfun{read.file} ~\ \begin{Rinput} datafilename <- "http://personality-project.org/r/datasets/finkel.sav" new.data <- read.file(datafilename) #the data has labels \end{Rinput} \subsection{Or: copy the data from another program using the copy and paste commands of your operating system} However, many users will enter their data in a text editor or spreadsheet program and then want to copy and paste into \R{}. This may be done by using one of the \pfun{read.clipboard} set of functions . \begin{description} \item [\pfun{read.clipboard}] is the base function for reading data from the clipboard. \item [\pfun{read.clipboard.csv}] for reading text that is comma delimited. \item [\pfun{read.clipboard.tab}] for reading text that is tab delimited (e.g., copied directly from an Excel file). \item [\pfun{read.clipboard.lower}] for reading input of a lower triangular matrix with or without a diagonal. The resulting object is a square matrix. \item [\pfun{read.clipboard.upper}] for reading input of an upper triangular matrix. \item[\pfun{read.clipboard.fwf}] for reading in fixed width fields (some very old data sets) \end{description} For example, given a data set copied to the clipboard from a spreadsheet, just enter the command ~\ \begin{Rinput} my.data <- read.clipboard() \end{Rinput} This will work if every data field has a value and even missing data are given some values (e.g., NA or -999). If the data were entered in a spreadsheet and the missing values were just empty cells, then the data should be read in as a tab delimited or by using the \pfun{read.clipboard.tab} function. ~\ \begin{Rinput} my.data <- read.clipboard(sep="\t") #define the tab option, or my.tab.data <- read.clipboard.tab() #just use the alternative function \end{Rinput} For the case of data in fixed width fields (some old data sets tend to have this format), copy to the clipboard and then specify the width of each field (in the example below, the first variable is 5 columns, the second is 2 columns, the next 5 are 1 column the last 4 are 3 columns). ~\ \begin{Rinput} my.data <- read.clipboard.fwf(widths=c(5,2,rep(1,5),rep(3,4)) \end{Rinput} \subsection{Or: import from an SPSS or SAS file} To read data from an SPSS, SAS, or Systat file, you can probably just use the \pfun{read.file} function. \pfun{read.file} examines the suffix of the data file and if it is .sav (from SPSS) or .xpt (from SAS) will attempt to read given various default options. However, if that does not work, use the \Rpkg{foreign} package. This should come with Base \R{} but still need to be loaded using the \Rfunction{library} command. \fun{read.spss} reads a file stored by the SPSS save or export commands. \begin{verbatim}read.spss(file, use.value.labels = TRUE, to.data.frame = FALSE, max.value.labels = Inf, trim.factor.names = FALSE, trim_values = TRUE, reencode = NA, use.missings = to.data.frame) \end{verbatim} The \Rfunction{read.spss} function has many parameters that need to be set. In the example, I have used the parameters that I think are most useful. \begin{description} \item [file] Character string: the name of the file or URL to read. \item [use.value.labels] Convert variables with value labels into R factors with those levels? \item [to.data.frame] return a data frame? Defaults to FALSE, probably should be TRUE in most cases. \item [max.value.labels] Only variables with value labels and at most this many unique values will be converted to factors if use.value.labels $= TRUE$. \item [trim.factor.names] Logical: trim trailing spaces from factor levels? \item [trim\_values] logical: should values and value labels have trailing spaces ignored when matching for use.value.labels $= TRUE $? \item [use.missings] logical: should information on user-defined missing values be used to set the corresponding values to NA? \end{description} The following is an example of reading from a remote SPSS file and then describing the data set to make sure that it looks ok (with thanks to Eli Finkel). ~\ \begin{Rinput} datafilename <- "http://personality-project.org/r/datasets/finkel.sav" eli <-read.file(datafilename) describe(eli,skew=FALSE) \end{Rinput} \begin{Routput} var n mean sd median trimmed mad min max range se USER* 1 69 35.00 20.06 35 35.00 25.20 1 69 68 2.42 HAPPY 2 69 5.71 1.04 6 5.82 0.00 2 7 5 0.13 SOULMATE 3 69 5.09 1.80 5 5.32 1.48 1 7 6 0.22 ENJOYDEX 4 68 6.47 1.01 7 6.70 0.00 2 7 5 0.12 UPSET 5 69 0.41 0.49 0 0.39 0.00 0 1 1 0.06 \end{Routput} \section{Some simple descriptive statistics before you start} Although you probably want to jump right in and find $\omega$, you should first make sure that your data are reasonable. Use the \pfun{describe} function to get some basic descriptive statistics. This next example takes advantage of a built in data set. ~\ \begin{Sinput} my.data <- sat.act #built in example -- replace with your data describe(my.data) \end{Sinput} \begin{Soutput} var n mean sd median trimmed mad min max range skew kurtosis se gender 1 700 1.65 0.48 2 1.68 0.00 1 2 1 -0.61 -1.62 0.02 education 2 700 3.16 1.43 3 3.31 1.48 0 5 5 -0.68 -0.07 0.05 age 3 700 25.59 9.50 22 23.86 5.93 13 65 52 1.64 2.42 0.36 ACT 4 700 28.55 4.82 29 28.84 4.45 3 36 33 -0.66 0.53 0.18 SATV 5 700 612.23 112.90 620 619.45 118.61 200 800 600 -0.64 0.33 4.27 SATQ 6 687 610.22 115.64 620 617.25 118.61 200 800 600 -0.59 -0.02 4.41 \end{Soutput} There are, of course, all kinds of things you could do with your data at this point, but read about them in the \href{http://cran.r-project.org/web/packages/psych/vignettes/intro.pdf}{introductory vignette} and \href{http://cran.r-project.org/web/packages/psychTools/vignettes/overview.pdf}{more advanced vignette} for the \Rpkg{psych} package, \section{Using the \pfun{omega} function to find $\omega$} Two alternative estimates of reliability that take into account the hierarchical structure of the inventory are McDonald's $\omega_h$ and $\omega_t$ \citep{mcdonald:tt,rz:09}. These may be found using the \pfun{omega} function for an exploratory analysis (See Figure~\ref{fig:omega.9}) or \pfun{omegaSem} for a confirmatory analysis using the \Rpkg{sem} based upon the exploratory solution from \pfun{omega}. \subsection{Background on the $\omega$ statistics} \cite{mcdonald:tt} has proposed coefficient omega (hierarchical) ($\omega_h$) as an estimate of the general factor saturation of a test. \href{http://personality-project.org/revelle/publications/zinbarg.revelle.pmet.05.pdf}{\cite{zinbarg:pm:05}} compare McDonald's $\omega_h$ to Cronbach's $\alpha$ and Revelle's $\beta$. They conclude that $\omega_h$ is the best estimate. (See also \cite{zinbarg:apm:06} and \cite{rz:09} ). One way to find $\omega_h$ is to do a factor analysis of the original data set, rotate the factors obliquely, factor that correlation matrix, do a Schmid-Leiman (\pfun{schmid}) transformation to find general factor loadings, and then find $\omega_h$. $\omega_h$ differs slightly as a function of how the factors are estimated. Three options are available, the default will do a minimum residual factor analysis, fm=``pa" does a principal axes factor analysis (\pfun{factor.pa}), and fm=``mle" provides a maximum likelihood solution. For ability items, it is typically the case that all items will have positive loadings on the general factor. However, for non-cognitive items it is frequently the case that some items are to be scored positively, and some negatively. Although probably better to specify which directions the items are to be scored by specifying a key vector, if flip =TRUE (the default), items will be reversed so that they have positive loadings on the general factor. The keys are reported so that scores can be found using the \pfun{score.items} function. Arbitrarily reversing items this way can overestimate the general factor. (See the example with a simulated circumplex). The \pfun{omega} function uses exploratory factor analysis to estimate the $\omega_h$ coefficient. It is important to remember that ``A recommendation that should be heeded, regardless of the method chosen to estimate $\omega_h$, is to always examine the pattern of the estimated general factor loadings prior to estimating $\omega_h$. Such an examination constitutes an informal test of the assumption that there is a latent variable common to all of the scale's indicators that can be conducted even in the context of EFA. If the loadings were salient for only a relatively small subset of the indicators, this would suggest that there is no true general factor underlying the covariance matrix. Just such an informal assumption test would have afforded a great deal of protection against the possibility of misinterpreting the misleading $\omega_h$ estimates occasionally produced in the simulations reported here." \citep[][p 137]{zinbarg:apm:06}. Although $\omega_h$ is uniquely defined only for cases where 3 or more subfactors are extracted, it is sometimes desired to have a two factor solution. By default this is done by forcing the \pfun{schmid} extraction to treat the two subfactors as having equal loadings. There are three possible options for this condition: setting the general factor loadings between the two lower order factors to be ``equal" which will be the $\sqrt{r_{ab}}$ where $r_{ab}$ is the oblique correlation between the factors) or to ``first" or ``second" in which case the general factor is equated with either the first or second group factor. A message is issued suggesting that the model is not really well defined. This solution discussed in Zinbarg et al., 2007. To do this in omega, add the option=``first" or option=``second" to the call. Although obviously not meaningful for a 1 factor solution, it is of course possible to find the sum of the loadings on the first (and only) factor, square them, and compare them to the overall matrix variance. This is done, with appropriate complaints. In addition to $\omega_h$, another of McDonald's coefficients is $\omega_t$. This is an estimate of the total reliability of a test. McDonald's $\omega_t$, which is similar to Guttman's $\lambda_6$, (see \pfun{guttman}) uses the estimates of uniqueness $u^2$ from factor analysis to find $e_j^2$. This is based on a decomposition of the variance of a test score, $V_x$ into four parts: that due to a general factor, $\vec{g}$, that due to a set of group factors, $\vec{f}$, (factors common to some but not all of the items), specific factors, $\vec{s}$ unique to each item, and $\vec{e}$, random error. (Because specific variance can not be distinguished from random error unless the test is given at least twice, some combine these both into error). Letting $\vec{x} = \vec{cg} + \vec{Af} + \vec {Ds} + \vec{e} $ then the communality of item$_j$, based upon general as well as group factors, $h_j^2 = c_j^2 + \sum{f_{ij}^2}$ and the unique variance for the item $u_j^2 = \sigma_j^2 (1-h_j^2)$ may be used to estimate the test reliability. That is, if $h_j^2$ is the communality of item$_j$, based upon general as well as group factors, then for standardized items, $e_j^2 = 1 - h_j^2$ and $$ \omega_t = \frac{\vec{1}\vec{cc'}\vec{1} + \vec{1}\vec{AA'}\vec{1}'}{V_x} = 1 - \frac{\sum(1-h_j^2)}{V_x} = 1 - \frac{\sum u^2}{V_x} $$ Because $h_j^2 \geq r_{smc}^2$, $\omega_t \geq \lambda_6$. It is important to distinguish here between the two $\omega$ coefficients of McDonald, 1978 and Equation 6.20a of McDonald, 1999, $\omega_t$ and $\omega_h$. While the former is based upon the sum of squared loadings on all the factors, the latter is based upon the sum of the squared loadings on the general factor. $$\omega_h = \frac{ \vec{1}\vec{cc'}\vec{1}}{V_x}$$ Another estimate reported is the omega for an infinite length test with a structure similar to the observed test. This is found by $$\omega_{\inf} = \frac{ \vec{1}\vec{cc'}\vec{1}}{\vec{1}\vec{cc'}\vec{1} + \vec{1}\vec{AA'}\vec{1}'}$$ It can be shown In the case of simulated variables, that the amount of variance attributable to a general factor ($\omega_h$) is quite large, and the reliability of the set of items is somewhat greater than that estimated by $\alpha$ or $\lambda_6$. \subsection{Yet another alternative: Coefficient $\beta$} $\beta$, an alternative to $\omega_h$, is defined as the worst split half reliability \citep{revelle:iclust}. It can be estimated by using \pfun{iclust} (Item Cluster analysis: a hierarchical clustering algorithm). For a very complimentary review of why the iclust algorithm is useful in scale construction, see \cite{cooksey:06}. For a discussion of how use \pfun{iclust} see the \href{http://cran.r-project.org/web/packages/psychTools/vignettes/factor.pdf}{factor analysis vignette}. \subsection{Using the \pfun{omega} function} This is \R{}. Just call it. For the next example, we find $\omega$ for a data set from Thurstone. To find it for your data, replace Thurstone with my.data. ~\ <>== omega(Thurstone) @ %\begin{Routput} % %Omega %Call: omega(m = Thurstone) %Alpha: 0.89 %G.6: 0.91 %Omega Hierarchical: 0.74 %Omega H asymptotic: 0.79 %Omega Total 0.93 % %Schmid Leiman Factor loadings greater than 0.2 % g F1* F2* F3* h2 u2 p2 %Sentences 0.71 0.57 0.82 0.18 0.61 %Vocabulary 0.73 0.55 0.84 0.16 0.63 %Sent.Completion 0.68 0.52 0.73 0.27 0.63 %First.Letters 0.65 0.56 0.73 0.27 0.57 %4.Letter.Words 0.62 0.49 0.63 0.37 0.61 %Suffixes 0.56 0.41 0.50 0.50 0.63 %Letter.Series 0.59 0.61 0.72 0.28 0.48 %Pedigrees 0.58 0.23 0.34 0.50 0.50 0.66 %Letter.Group 0.54 0.46 0.53 0.47 0.56 % %With eigenvalues of: % g F1* F2* F3* %3.58 0.96 0.74 0.71 % %general/max 3.71 max/min = 1.35 %mean percent general = 0.6 with sd = 0.05 and cv of 0.09 % %The degrees of freedom are 12 and the fit is 0.01 % %The root mean square of the residuals is 0 %The df corrected root mean square of the residuals is 0.01 % %Compare this with the adequacy of just a general factor and no group factors %The degrees of freedom for just the general factor are 27 and the fit is 1.48 % %The root mean square of the residuals is 0.1 %The df corrected root mean square of the residuals is 0.16 % %Measures of factor score adequacy % g F1* F2* F3* %Correlation of scores with factors 0.86 0.73 0.72 0.75 %Multiple R square of scores with factors 0.74 0.54 0.52 0.56 %Minimum correlation of factor score estimates 0.49 0.08 0.03 0.11 %> % \end{Routput} \subsection{Find three measures of reliability: $\omega_h$, $\alpha$, and $\omega_t$} In a review of various measures of reliability, \cite{rc:pa:19} suggest that one should routinely report 3 estimates of internal consistency ($\omega_h$, $\alpha$, and $\omega_t$). As an example, they use 10 items to measure anxiety taken from the state anxiety data set (\pfun{sai} in the \Rpkg{psychTools} package. First examine the descriptive statistics and then find and summarize the omega for these data. By inspection of the correlation matrix, it seems as if there are two group factors (tension and calmness) as well as an overall general factor of anxiety. We use a two factor solution to better represent the results (Figure~\ref{fig.anxiety}). ~\ <>= anxiety <- sai[c("anxious", "jittery", "nervous" ,"tense", "upset","at.ease" , "calm" , "confident", "content","relaxed")] describe(anxiety) lowerCor(anxiety) om <- omega(anxiety,2) #specify a two factor solution summary(om) #summarize the output @ <>== png('anxiety.png') omega.diagram(om, main="Omega analysis of two factors of anxiety") dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{anxiety} \caption{An \pfun{omega} solution for 10 anxiety items with two group factors. See \cite{rc:pa:19} for more measures of reliability for this data set.} \label{fig.anxiety} \end{center} \end{figure} \subsection{Estimating $\omega_h$ using a direct Schmid-Leiman transformation} The \pfun{omegaDirect} function uses Niels Waller's algorithm for finding a g factor directly without extracting a higher order model \citep{waller:17}. This has the advantage that it will work cleanly for data with just 2 group factors. Unfortunately, it will produce non-zero estimates for omega even if there is no general factor. ~\ <>= om <- omegaDirect(Thurstone) om @ <>== png('direct.png') omega.diagram(om, main="Direct Schmid Leihman solution") dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{direct} \caption{The Direct Schmid Leiman solution is taken from an algorithm by \cite{waller:17}. Compare this solution to Figure~\ref{fig:omega.9}. } \label{fig:direct} \end{center} \end{figure} \subsection{Estimating $\omega_h$ using Confirmatory Factor Analysis} The \pfun{omegaSem} function will do an exploratory analysis and then take the highest loading items on each factor and do a confirmatory factor analysis using the \Rpkg{lavaan} package. These results can produce slightly different estimates of $\omega_h$, primarily because cross loadings are modeled as part of the general factor. We use a classic data set from Holzinger and Swineford, some of the tests of which are included in the \Rpkg{lavaan} package. This analysis allows us to examine the hierarchical structure of these ability tests. The data are taken from the \pfun{holzinger.swineford} data set in the \Rpkg{psychTools} package. ~\ <>= om <- omega(holzinger.swineford[8:31],4) #the exploratory solution omegaSem(holzinger.swineford[8:31],4) #the confirmatory solution @ %\begin{Routput} %Call: omegaSem(m = r9, n.obs = 500) %Omega %Call: omega(m = m, nfactors = nfactors, fm = fm, key = key, flip = flip, % digits = digits, title = title, sl = sl, labels = labels, % plot = plot, n.obs = n.obs, rotate = rotate, Phi = Phi, option = option) %Alpha: 0.75 %G.6: 0.74 %Omega Hierarchical: 0.66 %Omega H asymptotic: 0.84 %Omega Total 0.78 % %Schmid Leiman Factor loadings greater than 0.2 % g F1* F2* F3* h2 u2 p2 %V1 0.70 0.53 0.47 0.93 %V2 0.70 0.52 0.48 0.94 %V3 0.54 0.32 0.68 0.91 %V4 0.53 0.46 0.50 0.50 0.57 %V5 0.44 0.44 0.39 0.61 0.50 %V6 0.40 0.32 0.26 0.74 0.59 %V7 0.31 0.31 0.21 0.79 0.48 %V8 0.34 0.44 0.30 0.70 0.37 %V9 0.24 0.36 0.19 0.81 0.32 % %With eigenvalues of: % g F1* F2* F3* %2.18 0.52 0.08 0.44 % %general/max 4.21 max/min = 6.17 %mean percent general = 0.62 with sd = 0.24 and cv of 0.39 % %The degrees of freedom are 12 and the fit is 0.03 %The number of observations was 500 with Chi Square = 14.23 with prob < 0.29 %The root mean square of the residuals is 0.01 %The df corrected root mean square of the residuals is 0.03 %RMSEA index = 0.02 and the 90 % confidence intervals are NA 0.052 %BIC = -60.35 % %Compare this with the adequacy of just a general factor and no group factors %The degrees of freedom for just the general factor are 27 and the fit is 0.21 %The number of observations was 500 with Chi Square = 103.64 with prob < 6.4e-11 %The root mean square of the residuals is 0.05 %The df corrected root mean square of the residuals is 0.08 % %RMSEA index = 0.076 and the 90 % confidence intervals are 0.06 0.091 %BIC = -64.15 % %Measures of factor score adequacy % g F1* F2* F3* %Correlation of scores with factors 0.86 0.63 0.25 0.59 %Multiple R square of scores with factors 0.74 0.39 0.06 0.35 %Minimum correlation of factor score estimates 0.48 -0.21 -0.88 -0.30 % % Omega Hierarchical from a confirmatory model using sem = 0.68 % Omega Total from a confirmatory model using sem = 0.78 %With loadings of % g F1* F2* F3* h2 u2 %V1 0.73 0.54 0.46 %V2 0.68 0.29 0.54 0.46 %V3 0.51 0.22 0.31 0.69 %V4 0.54 0.47 0.51 0.49 %V5 0.45 0.42 0.38 0.62 %V6 0.39 0.31 0.25 0.75 %V7 0.34 0.34 0.23 0.77 %V8 0.36 0.39 0.28 0.72 %V9 0.26 0.33 0.18 0.82 % %With eigenvalues of: % g F1* F2* F3* %2.21 0.49 0.14 0.38 %\end{Routput} <>= @ \section{Simulating a hierarchical/higher order structure} There are several simulation functions in the \Rpkg{psych} package for creating structures with a general factor. One, \pfun{sim.hierarchical} creates lower level factors which are all correlated with a general factor. The default simulation has the parameters discussed by \cite{jensen:weng}. Another way to simulate a hierarchical structure is to simulate a bifactor model directly using the \pfun{sim.structure} function. The \cite{jensen:weng} model: <>= jen <- sim.hierarchical() #use the default values om <- omega(jen) om @ \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= png('jensen.png' ) omega.diagram(om) dev.off() @ \end{scriptsize} \includegraphics{jensen} \caption{An example of a hierarchical model from Jensen.} \label{fig:outlier} \end{center} \end{figure} %\begin{Routput} %jen <- sim.hierarchical() #use the default values %> om <- omega(jen) %> om %Omega %Call: omega(m = jen) %Alpha: 0.76 %G.6: 0.76 %Omega Hierarchical: 0.69 %Omega H asymptotic: 0.86 %Omega Total 0.8 % %Schmid Leiman Factor loadings greater than 0.2 % g F1* F2* F3* h2 u2 p2 %V1 0.72 0.35 0.64 0.36 0.81 %V2 0.63 0.31 0.49 0.51 0.81 %V3 0.54 0.26 0.36 0.64 0.81 %V4 0.56 0.42 0.49 0.51 0.64 %V5 0.48 0.36 0.36 0.64 0.64 %V6 0.40 0.30 0.25 0.75 0.64 %V7 0.42 0.43 0.36 0.64 0.49 %V8 0.35 0.36 0.25 0.75 0.49 %V9 0.28 0.29 0.16 0.84 0.49 % %With eigenvalues of: % g F1* F2* F3* %2.29 0.28 0.40 0.39 % %general/max 5.78 max/min = 1.4 %mean percent general = 0.65 with sd = 0.14 and cv of 0.21 %Explained Common Variance of the general factor = 0.68 % %The degrees of freedom are 12 and the fit is 0 % %The root mean square of the residuals is 0 %The df corrected root mean square of the residuals is 0 % %Compare this with the adequacy of just a general factor and no group factors %The degrees of freedom for just the general factor are 27 and the fit is 0.18 % %The root mean square of the residuals is 0.06 %The df corrected root mean square of the residuals is 0.07 % %Measures of factor score adequacy % g F1* F2* F3* %Correlation of scores with factors 0.85 0.46 0.57 0.57 %Multiple R square of scores with factors 0.73 0.21 0.32 0.32 %Minimum correlation of factor score estimates 0.46 -0.57 -0.35 -0.35 % % Total, General and Subset omega for each subset % g F1* F2* F3* %Omega total for total scores and subscales 0.80 0.74 0.63 0.50 %Omega general for total scores and subscales 0.69 0.60 0.40 0.25 %Omega group for total scores and subscales 0.11 0.14 0.23 0.26 %> %\end{Routput} \subsubsection{Simulate a bifactor model} Simulate a bifactor model and then compare two ways of finding the solution (normal omega and directOmega). We compare the solutions using the \pfun{fa.congruence} function. \begin{Rinput} fx <- matrix(c(.7,.6,.5,.7,.6,.5,.8,.7,.6, .6,.6,.6,rep(0,9),c(.6,.5,.6),rep(0,9),.6,.6,.6),ncol=4) simx <-sim.structure(fx) lowerMat(simx$model) om <- omega(simx$model) dsl <- omegaDirect(simx$model) summary(om) summary(dsl) fa.congruence(list(om,dsl,fx)) \end{Rinput} <>== fx <- matrix(c(.7,.6,.5,.7,.6,.5,.8,.7,.6, .6,.6,.6,rep(0,9),c(.6,.5,.6),rep(0,9),.6,.6,.6),ncol=4) simx <-sim.structure(fx) om <- omega(simx$model) dsl <- omegaDirect(simx$model) @ \begin{scriptsize} <>= lowerMat(simx$model) summary(om) summary(dsl) fa.congruence(list(om,dsl,fx)) @ \end{scriptsize} %\begin{Routput} %summary(om) %Omega %Alpha: 0.9 %G.6: 0.93 %Omega Hierarchical: 0.74 %Omega H asymptotic: 0.78 %Omega Total 0.95 % %With eigenvalues of: % g F1* F2* F3* %3.67 1.08 1.08 0.97 %The degrees of freedom for the model is 12 and the fit was 0 % %The root mean square of the residuals is 0 %The df corrected root mean square of the residuals is 0 %Explained Common Variance of the general factor = 0.54 % % Total, General and Subset omega for each subset % g F1* F2* F3* %Omega total for total scores and subscales 0.95 0.95 0.89 0.87 %Omega general for total scores and subscales 0.74 0.55 0.45 0.46 %Omega group for total scores and subscales 0.21 0.40 0.44 0.41 %> summary(dsl) %Call: omegaDirect(m = simx$model) %Omega H direct: 0.71 % %With eigenvalues of: % g F1* F2* F3* %3.53 1.22 1.06 0.99 %The degrees of freedom for the model is 12 and the fit was 0 % %The root mean square of the residuals is 0 %The df corrected root mean square of the residuals is 0 % % Total, General and Subset omega for each subset % g F1* F2* F3* %Omega total for total scores and subscales 0.95 0.95 0.89 0.87 %Omega general for total scores and subscales 0.71 0.50 0.45 0.45 %Omega group for total scores and subscales 0.22 0.45 0.43 0.42 %> fa.congruence(list(om,dsl,fx)) % g F1* F2* F3* h2 g F1* F2* F3* %g 1.00 0.64 0.55 0.54 1.00 1.00 0.67 0.56 0.57 1.00 0.54 0.54 0.63 %F1* 0.64 1.00 0.00 0.00 0.65 0.62 1.00 0.02 0.04 0.64 0.00 0.00 1.00 %F2* 0.55 0.00 1.00 0.00 0.55 0.56 0.02 1.00 0.00 0.55 1.00 0.00 0.00 %F3* 0.54 0.00 0.00 1.00 0.52 0.55 0.03 0.00 1.00 0.54 0.00 1.00 0.00 %h2 1.00 0.65 0.55 0.52 1.00 1.00 0.68 0.57 0.55 1.00 0.55 0.52 0.64 %g 1.00 0.62 0.56 0.55 1.00 1.00 0.65 0.58 0.57 1.00 0.56 0.55 0.62 %F1* 0.67 1.00 0.02 0.03 0.68 0.65 1.00 0.04 0.07 0.67 0.02 0.03 1.00 %F2* 0.56 0.02 1.00 0.00 0.57 0.58 0.04 1.00 0.00 0.56 1.00 0.00 0.02 %F3* 0.57 0.04 0.00 1.00 0.55 0.57 0.07 0.00 1.00 0.57 0.00 1.00 0.03 % 1.00 0.64 0.55 0.54 1.00 1.00 0.67 0.56 0.57 1.00 0.54 0.54 0.63 % 0.54 0.00 1.00 0.00 0.55 0.56 0.02 1.00 0.00 0.54 1.00 0.00 0.00 % 0.54 0.00 0.00 1.00 0.52 0.55 0.03 0.00 1.00 0.54 0.00 1.00 0.00 % 0.63 1.00 0.00 0.00 0.64 0.62 1.00 0.02 0.03 0.63 0.00 0.00 1.00 %> %\end{Routput} \section{Summary} In the modern era of computation, there is little justification for continuing with procedures that were developed as \href{https://personality-project.org/revelle/publications/cup.18.final.pdf}{short-cuts 80 years ago} \citep{reh:20}, To find $\omega_h$, $\alpha$, and $\omega_t$ is very easy using the open source statistical system (\R{}) as well as the \pfun{omega} functions in the \Rpkg{psych} package. \section{System Info} When running any \R{} package, it is useful to find out the session information to see if you have the most recent releases. \begin{scriptsize} <>= sessionInfo() @ \end{scriptsize} \newpage \begin{thebibliography}{} \bibitem[\protect\astroncite{Azzalini and Genz}{2016}]{mnormt} Azzalini, A. and Genz, A. (2016). \newblock {\em The {R} package \texttt{mnormt}: The multivariate normal and $t$ distributions (version 1.5-5)}. \bibitem[\protect\astroncite{Bernaards and Jennrich}{2005}]{GPA} Bernaards, C. and Jennrich, R. (2005). \newblock {Gradient projection algorithms and software for arbitrary rotation criteria in factor analysis}. \newblock {\em Educational and Psychological Measurement}, 65(5):676--696. \bibitem[\protect\astroncite{Cooksey and Soutar}{2006}]{cooksey:06} Cooksey, R. and Soutar, G. (2006). \newblock Coefficient beta and hierarchical item clustering - an analytical procedure for establishing and displaying the dimensionality and homogeneity of summated scales. \newblock {\em Organizational Research Methods}, 9:78--98. \bibitem[\protect\astroncite{Cronbach}{1951}]{cronbach:51} Cronbach, L.~J. (1951). \newblock Coefficient alpha and the internal structure of tests. \newblock {\em Psychometrika}, 16:297--334. \bibitem[\protect\astroncite{Fox et~al.}{2013}]{sem} Fox, J., Nie, Z., and Byrnes, J. (2013). \newblock {\em sem: Structural Equation Models}. \newblock R package version 3.1-3. \bibitem[\protect\astroncite{Guttman}{1945}]{guttman:45} Guttman, L. (1945). \newblock A basis for analyzing test-retest reliability. \newblock {\em Psychometrika}, 10(4):255--282. \bibitem[\protect\astroncite{Jensen and Weng}{1994}]{jensen:weng} Jensen, A.~R. and Weng, L.-J. (1994). \newblock What is a good g? \newblock {\em Intelligence}, 18(3):231--258. \bibitem[\protect\astroncite{Kuder and Richardson}{1937}]{kuder:37} Kuder, G. and Richardson, M. (1937). \newblock The theory of the estimation of test reliability. \newblock {\em Psychometrika}, 2(3):151--160. \bibitem[\protect\astroncite{McDonald}{1999}]{mcdonald:tt} McDonald, R.~P. (1999). \newblock {\em Test theory: {A} unified treatment}. \newblock L. Erlbaum Associates, Mahwah, N.J. \bibitem[\protect\astroncite{{R Core Team}}{2020}]{R} {R Core Team} (2020). \newblock {\em R: A Language and Environment for Statistical Computing}. \newblock R Foundation for Statistical Computing, Vienna, Austria. \bibitem[\protect\astroncite{Revelle}{1979}]{revelle:iclust} Revelle, W. (1979). \newblock Hierarchical cluster-analysis and the internal structure of tests. \newblock {\em Multivariate Behavioral Research}, 14(1):57--74. \bibitem[\protect\astroncite{Revelle}{2020}]{psych} Revelle, W. (2020). \newblock {\em psych: Procedures for Personality and Psychological Research}. \newblock Northwestern University, Evanston, https://CRAN.r-project.org/package=psych. \newblock R package version 2.0.8. \bibitem[\protect\astroncite{Revelle and Condon}{2018}]{rc:reliability} Revelle, W. and Condon, D.~M. (2018). \newblock Reliability. \newblock In Irwing, P., Booth, T., and Hughes, D.~J., editors, {\em The {Wiley Handbook of Psychometric Testing:} A Multidisciplinary Reference on Survey, Scale and Test Development}. John Wily \& Sons, London. \bibitem[\protect\astroncite{Revelle and Condon}{2019}]{rc:pa:19} Revelle, W. and Condon, D.~M. (2019). \newblock Reliability from $\alpha$ to $\omega$: A tutorial. \newblock {\em Psychological Assessment} 31 (12) p 1395-1411. \bibitem[\protect\astroncite{Revelle et al.}{2020}]{reh:20} Revelle, W. and Elleman, L.G. and Hall, A. (2020). \newblock Statistical analyses and computer programming in personality. \newblock In Corr, P.J. editor, {\em The {Cambridge University Press Handbook of Personality}}. {Cambridge University Press}. \bibitem[\protect\astroncite{Revelle and Zinbarg}{2009}]{rz:09} Revelle, W. and Zinbarg, R.~E. (2009). \newblock Coefficients alpha, beta, omega and the glb: comments on {Sijtsma}. \newblock {\em Psychometrika}, 74(1):145--154. \bibitem[\protect\astroncite{Rosseel}{2012}]{lavaan} Rosseel, Y. (2012). \newblock {lavaan}: An {R} package for structural equation modeling. \newblock {\em Journal of Statistical Software}, 48(2):1--36. \bibitem[\protect\astroncite{Waller}{2017}]{waller:17} Waller, N.~G. (2017). \newblock Direct {Schmid-Leiman} transformations and rank-deficient loadings matrices. \newblock {\em Psychometrika.} \bibitem[\protect\astroncite{Zinbarg et~al.}{2005}]{zinbarg:pm:05} Zinbarg, R.~E., Revelle, W., Yovel, I., and Li, W. (2005). \newblock Cronbach's {$\alpha$}, {Revelle's} {$\beta$}, and {McDonald's} {$\omega_H$}: Their relations with each other and two alternative conceptualizations of reliability. \newblock {\em Psychometrika}, 70(1):123--133. \bibitem[\protect\astroncite{Zinbarg et~al.}{2006}]{zinbarg:apm:06} Zinbarg, R.~E., Yovel, I., Revelle, W., and McDonald, R.~P. (2006). \newblock Estimating generalizability to a latent variable common to all of a scale's indicators: A comparison of estimators for {$\omega_h$}. \newblock {\em Applied Psychological Measurement}, 30(2):121--144. \end{thebibliography} \end{document} psychTools/vignettes/mediation.rnw0000644000176200001440000023014714003577154017147 0ustar liggesusers% \VignetteIndexEntry{Overview of the psych package} % \VignettePackage{psych} % \VignetteKeywords{multivariate} % \VignetteKeyword{models} % \VignetteKeyword{Hplot} %\VignetteDepends{psych} %\documentclass[doc]{apa} \documentclass[11pt]{article} %\documentclass[11pt]{amsart} \usepackage{geometry} % See geometry.pdf to learn the layout options. There are lots. \geometry{letterpaper} % ... or a4paper or a5paper or ... %\geometry{landscape} % Activate for for rotated page geometry \usepackage[parfill]{parskip} % Activate to begin paragraphs with an empty line rather than an indent \usepackage{graphicx} \usepackage{amssymb} \usepackage{epstopdf} \usepackage{mathptmx} \usepackage{helvet} \usepackage{courier} \usepackage{epstopdf} \usepackage{makeidx} % allows index generation \usepackage[authoryear,round]{natbib} \usepackage{gensymb} \usepackage{longtable} %\usepackage{geometry} \usepackage{amssymb} \usepackage{amsmath} %\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \usepackage{Sweave} %\usepackage{/Volumes/'Macintosh HD'/Library/Frameworks/R.framework/Versions/2.13/Resources/share/texmf/tex/latex/Sweave} %\usepackage[ae]{Rd} %\usepackage[usenames]{color} %\usepackage{setspace} \bibstyle{apacite} \bibliographystyle{apa} %this one plus author year seems to work? %\usepackage{hyperref} \usepackage[colorlinks=true,citecolor=blue]{hyperref} %this makes reference links hyperlinks in pdf! \DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \usepackage{multicol} % used for the two-column index \usepackage[bottom]{footmisc}% places footnotes at page bottom \let\proglang=\textsf \newcommand{\R}{\proglang{R}} %\newcommand{\pkg}[1]{{\normalfont\fontseries{b}\selectfont #1}} \newcommand{\Rfunction}[1]{{\texttt{#1}}} \newcommand{\fun}[1]{{\texttt{#1}\index{#1}\index{R function!#1}}} \newcommand{\pfun}[1]{{\texttt{#1}\index{#1}\index{R function!#1}\index{R function!psych package!#1}}}\newcommand{\Rc}[1]{{\texttt{#1}}} %R command same as Robject \newcommand{\Robject}[1]{{\texttt{#1}}} \newcommand{\Rpkg}[1]{{\textit{#1}\index{#1}\index{R package!#1}}} %different from pkg - which is better? \newcommand{\iemph}[1]{{\emph{#1}\index{#1}}} \newcommand{\wrc}[1]{\marginpar{\textcolor{blue}{#1}}} %bill's comments \newcommand{\wra}[1]{\textcolor{blue}{#1}} %bill's comments \newcommand{\ve}[1]{{\textbf{#1}}} %trying to get a vector command \usepackage{fancyvrb} %this allows fancy boxes \newcommand{\vect}[1]{\boldsymbol{#1}} \let\vec\vect \fvset{fontfamily=courier} \DefineVerbatimEnvironment{Routput}{Verbatim} %{fontsize=\scriptsize, xleftmargin=0.6cm} {fontseries=b,fontsize=\scriptsize, xleftmargin=0.1cm} \DefineVerbatimEnvironment{Soutput}{Verbatim} %{fontsize=\scriptsize, xleftmargin=0.6cm} {fontseries=b,fontsize=\scriptsize, xleftmargin=0.1cm} \DefineVerbatimEnvironment{Binput}{Verbatim} {fontseries=b, fontsize=\scriptsize,frame=single, label=\fbox{lavaan model syntax}, framesep=2mm} %\DefineShortVerb{\!} %%% generates error! \DefineVerbatimEnvironment{Rinput}{Verbatim} %{fontsize=\scriptsize, frame=single, label=\fbox{R code}, framesep=1mm} {fontseries=b, fontsize=\scriptsize, frame=single, label=\fbox{R code},xleftmargin=0pt, framesep=1mm} \DefineVerbatimEnvironment{Sinput}{Verbatim} %{fontsize=\scriptsize, frame=single, label=\fbox{R code}, framesep=1mm} {fontseries=b, fontsize=\scriptsize, frame=single, label=\fbox{R code},xleftmargin=0pt, framesep=1mm} \DefineVerbatimEnvironment{Link}{Verbatim} {fontseries=b, fontsize=\small, formatcom=\color{darkgreen}, xleftmargin=1.0cm} \DefineVerbatimEnvironment{Toutput}{Verbatim} {fontseries=b,fontsize=\tiny, xleftmargin=0.1cm} \DefineVerbatimEnvironment{rinput}{Verbatim} {fontseries=b, fontsize=\tiny, frame=single, label=\fbox{R code}, framesep=1mm} \newcommand{\citeti}[1]{\begin{tiny}\citep{#1}\end{tiny}} \newcommand{\light}[1]{\textcolor{gray}{#1}} %\newcommand{\vect}[1]{\boldsymbol{#1}} %\let\vec\vect \makeindex % used for the subject index \title{How to use the psych package for mediation/moderation/regression analysis} \author{William Revelle} %the following works only with apaclass \begin{document} \maketitle %\bibliography{all} \tableofcontents \newpage \section{Overview of this and related documents} To do basic and advanced personality and psychological research using \R{} is not as complicated as some think. This is one of a set of ``How To'' to do various things using \R{} \citep{R}, particularly using the \Rpkg{psych} \citep{psych} package. The current list of How To's includes: \begin{enumerate} \item An \href{http://personality-project.org/r/psych/intro.pdf}{introduction} (vignette) of the \Rpkg{psych} package \item An \href{http://personality-project.org/r/psych/overview.pdf}{overview} (vignette) of the \Rpkg{psych} package \item \href{http://personality-project.org/r/psych/HowTo/getting_started.pdf}{Installing} \R{} and some useful packages \item Using \R{} and the \Rpkg{psych} package to find \href{http://personality-project.org/r/psych/HowTo/omega.pdf}{$omega_h$} and $\omega_t$. \item Using \R{} and the \Rpkg{psych} for \href{http://personality-project.org/r/psych/HowTo/factor.pdf}{factor analysis} and principal components analysis. \item Using the \pfun{scoreItems} function to find \href{http://personality-project.org/r/psych/HowTo/scoring.pdf}{scale scores and scale statistics}. \item Using \pfun{mediate} and \pfun{setCor} to do \href{http://personality-project.org/r/psych/HowTo/mediation.pdf}{mediation, moderation and regression analysis} (this document) \end{enumerate} \subsection{Jump starting the \Rpkg{psych} package--a guide for the impatient} You have installed \Rpkg{psych} and you want to use it without reading much more. What should you do? \begin{enumerate} \item Activate the \Rpkg{psych} and \Rpkg{psychTools} packages. <>== library(psych) library(psychTools) @ \item Input your data. If your file name ends in .sav, .text, .txt, .csv, .xpt, .rds, .Rds, .rda, or .RDATA, then just read it in directly using \pfun{read.file}. Or you can go to your friendly text editor or data manipulation program (e.g., Excel) and copy the data to the clipboard. Include a first line that has the variable labels. Paste it into \Rpkg{psych} using the \pfun{read.clipboard.tab} command: \begin{Rinput} myData <- read.file() #this will open a search window on your machine # and read or load the file. #or #first copy your file to your clipboard and then myData <- read.clipboard.tab() #if you have an excel file \end{Rinput} \item Make sure that what you just read is right. Describe it and perhaps look at the first and last few lines. If you want to ``view" the first and last few lines using a spreadsheet like viewer, use \pfun{quickView}. \begin{Rinput} describe(myData) headTail(myData) #or quickView(myData) \end{Rinput} \item Look at the patterns in the data. If you have fewer than about 10 variables, look at the SPLOM (Scatter Plot Matrix) of the data using \pfun{pairs.panels}. \begin{Rinput} pairs.panels(myData) \end{Rinput} \item Find the correlations of all of your data. \begin{itemize} \item Descriptively (just the values) \begin{Rinput} lowerCor(myData) \end{Rinput} \item Graphically \begin{Rinput} corPlot(myData) #does not show the values of the correlations corPlot(myData, numbers=TRUE) #show the numbers,\ #scales the character size by "significance" corPlot(myData, numbers=TRUE,scale=FALSE) #show the numbers, # all characters the same size corPlot(lowerCor(myData), numbers =TRUE) #print the correlations # and show them graphically \end{Rinput} \end{itemize} \end{enumerate} \subsection{For the not impatient} The following pages are meant to lead you through the use of the \pfun{setCor} and \pfun{mediate} functions. The assumption is that you have already made \Rpkg{psych} active and want some example code. \section{Multiple regression and mediation} Mediation and moderation are merely different uses of the linear model $\hat{\vec{Y}}= \mu + \beta_{y.x} \vec{X } $ and are implemented in \Rpkg{psych} with two functions: \pfun{setCor} and \pfun{mediate}. Given a set of predictor variables, $\vec{X}$ and a set of criteria variables, $\vec{Y}$, multiple regression solves the equation $\hat{\vec{Y}} = \mu + \beta_{y.x} \vec{X } $ by finding $\beta_{y.x} = \vec{C_{xx}}^{-1} C_{yx} $ where $\vec{C_{xx}}$ is the covariances of the $\vec{X}$ variables and $\vec{C_{yx}}$ is the covariances of predictors and the criteria. Although typically done using the raw data, clearly this can also be done by using the covariance or correlation matrices. \pfun{setCor} was developed to handle the correlation matrix solution but has been generalized to the case of raw data. In the later case, it assumes a Missing Completely at Random (MCAR) structure, and thus uses all the data and finds pair.wise complete correlations. For complete data sets, the results are identical to using \pfun{lm}. By default, \pfun{setCor} uses standardized variables, but to compare with \pfun{lm}, it can use unstandardized variables. \section{Regression using \pfun{setCor}} Although typically done from a raw data matrix (using the \fun{lm} function), it is sometimes useful to do the regression from a correlation or covariance matrix. \pfun{setCor} was developed for this purpose. From a correlation/covariance matrix, it will do normal regression as well as regression on partialled correlation matrices. With the raw data, it will also do moderated regression (centered or non-centered). In particular, for the raw data, it will work with missing data. An interesting option, if using categorical or dichotomous data is first find the appropriate polychoric, tetrachoric, or poly-serial correlations using \pfun{mixedCor} and then use the resulting correlation matrix for analysis. The resulting correlations and multiple correlations will not match those of the \pfun{lm} analysis. \subsection{Comparison with \pfun{lm} on complete data} We use the \pfun{attitude} data set for our first example. \subsubsection{It is important to know your data by describing it first} <>== psych::describe(attitude) @ \subsubsection{Now do the regressions} <>== #do not standardize mod1 <- setCor(rating ~ complaints + privileges, data=attitude,std=FALSE) mod1 @ Compare this solution with the results of the \pfun{lm} function. <>== summary(lm(rating ~ complaints + privileges, data=attitude)) @ The graphic for the standardized regression is shown in (Figure~\ref{fig:attitude}). <>== png('attitude.png') # standardize by default mod2 <- setCor(rating ~ complaints + privileges, data=attitude) mod2 setCor.diagram(mod2, main="A simple regression model") dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{attitude.png} \caption{A simple multiple regression using the attitude data set (standardized solution is shown).} \label{fig:attitude} \end{center} \end{figure} \subsection{From a correlation matrix} Perhaps most usefully, \pfun{setCor} will find the beta weights between a set of X variables, and a set of Y variables. Consider seven variables in the \pfun{atttitude} data set. We first find the correlation matrix (normally, this could just be supplied by the user). Then we find the regressions from the correlation matrix. Compare this regression to the (standardized) solution shown above. By specifying the number of observations (n.obs), we are able to apply various inferential tests. <>== R <- lowerCor(attitude) setCor(rating ~ complaints + privileges, data=R, n.obs =30) @ Compare this solution (from the correlation matrix) with the \emph{standardized} solution for the raw data. \pfun{setCor} does several things: \begin{itemize} \item Finds the regression weights (betas) between the predictor variables and each of the criterion variables. \item If the number of subjects is specified, or if the raw data are used, it also compares each of these betas to its standard error, finds a $t$ statistic, and reports the probability of the $t > 0$. \item It reports the Multiple R and $R^2$ based upon these beta weights. In addition, following the tradition of the robust beauty of the improper linear models \citep{dawes:79} it also reports the unit weighted multiple correlations. \item The canonical correlations between the two sets \citep{hotelling:36} is reported. \item Cohen's set correlation \citep{cohen:82} as well as the unweighted correlation between the two sets of variables are reported. \end{itemize} \subsection{The Hotelling example} <>== #the second Kelley data from Hotelling kelley <- structure(list(speed = c(1, 0.4248, 0.042, 0.0215, 0.0573), power = c(0.4248, 1, 0.1487, 0.2489, 0.2843), words = c(0.042, 0.1487, 1, 0.6693, 0.4662), symbols = c(0.0215, 0.2489, 0.6693, 1, 0.6915), meaningless = c(0.0573, 0.2843, 0.4662, 0.6915, 1)), .Names = c("speed", "power", "words", "symbols", "meaningless"), class = "data.frame", row.names = c("speed", "power", "words", "symbols", "meaningless")) #first show the correlations lowerMat(kelley) #now find and draw the regression sc <- setCor(power + speed ~ words + symbols + meaningless,data=kelley) #formula mode sc #show it @ %First show the correlation matrix. %\begin{Routput} % %lowerMat(kelley) % speed power words symbl mnngl %speed 1.00 %power 0.42 1.00 %words 0.04 0.15 1.00 %symbols 0.02 0.25 0.67 1.00 %meaningless 0.06 0.28 0.47 0.69 1.00 %\end{Routput} % %Now, use the \pfun{setCor} function. % %\begin{Routput} %Call: setCor(y = power + speed ~ words + symbols + meaningless, data = kelley) % %Multiple Regression from matrix input % % DV = power % slope VIF %words -0.03 1.81 %symbols 0.12 2.72 %meaningless 0.22 1.92 % % Multiple Regression % R R2 Ruw R2uw %power 0.29 0.09 0.26 0.07 % % DV = speed % slope VIF %words 0.05 1.81 %symbols -0.07 2.72 %meaningless 0.08 1.92 % % Multiple Regression % R R2 Ruw R2uw %speed 0.07 0.01 0.05 0 % %Various estimates of between set correlations %Squared Canonical Correlations %[1] 0.0946 0.0035 % % Average squared canonical correlation = 0.05 % Cohen's Set Correlation R2 = 0.1 %Unweighted correlation between the two sets = 0.18 % %\end{Routput} A plot of the regression model is shown as well (Figure~\ref{fig:hotelling}). <>== png('hotelling.png') setCor.diagram(sc, main="The Kelley data set") dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{hotelling.png} \caption{The relationship between three predictors and two criteria from \pfun{setCor}. The data are from the Kelley data set reported by \cite{hotelling:36}.} \label{fig:hotelling} \end{center} \end{figure} %\subsection{From the raw data} % % % %If the data are available, \pfun{setCor} will find the regressions between variables in an X set and those in a Y set. The first analysis (Figure~\ref{fig:2pred}) is perhaps the more typical (one criterion, two predictors), while the second example is more complicated, with three predictors of 3 dependent variables (Figure~\ref{fig:3x3}). % % %<>== %mod2 <- setCor(ACT ~ SATV + SATQ, data=sat.act) %mod2 %@ % %<>== %png('mod2.png') %setCor.diagram(mod2, main="Regressions for sat.act data") %dev.off() %@ % %\begin{Rinput} % %# a typical use of setCor %mod2 <- setCor(ACT ~ SATV + SATQ, data=sat.act) %mod2 %\end{Rinput} %\begin{Routput} %Call: setCor(y = ACT ~ SATV + SATQ, data = sat.act) % %Multiple Regression from raw data % % DV = ACT % slope se t p VIF %SATV 0.31 0.04 8.09 2.7e-15 1.72 %SATQ 0.39 0.04 10.08 0.0e+00 1.72 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %ACT 0.63 0.4 0.63 0.4 0.4 0.03 234.26 2 697 0 %\end{Routput} %\begin{figure}[htbp] %\begin{center} %\includegraphics{mod2.png} %\caption{The relationship between two predictors and one criterion from \pfun{setCor}. The data are from the \pfun{sat.act} data set } %\label{fig:2pred} %\end{center} %\end{figure} % But, we can also do multiple predictors \emph{and} multiple criteria in the same call: <>== png('satact.png') mod3 <- setCor(SATV + SATQ + ACT ~ gender + education + age, data = sat.act) dev.off() @ %<>== %png('satact.png') %setCor.diagram(mod3, main="Three predictors, 3 criteria") %dev.off() @ %\begin{Rinput} %mod3 <- setCor(SATV + SATQ + ACT ~ gender + education + age, data = sat.act) % %\end{Rinput} % %\begin{Routput} %Multiple Regression from raw data % % DV = SATV % slope se t p VIF %gender -0.03 0.04 -0.79 0.430 1.01 %education 0.10 0.05 2.29 0.022 1.45 %age -0.10 0.05 -2.21 0.028 1.44 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %SATV 0.1 0.01 0.05 0 0.01 0.01 2.26 3 696 0.0808 % % DV = SATQ % slope se t p VIF %gender -0.18 0.04 -4.71 3.0e-06 1.01 %education 0.10 0.04 2.25 2.5e-02 1.45 %age -0.09 0.04 -2.08 3.8e-02 1.44 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %SATQ 0.19 0.04 0.11 0.01 0.03 0.01 8.63 3 696 1.24e-05 % % DV = ACT % slope se t p VIF %gender -0.05 0.04 -1.28 0.2000 1.01 %education 0.14 0.05 3.14 0.0017 1.45 %age 0.03 0.04 0.71 0.4800 1.44 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %ACT 0.16 0.03 0.15 0.02 0.02 0.01 6.49 3 696 0.000248 % %Various estimates of between set correlations %Squared Canonical Correlations %[1] 0.050 0.033 0.008 %Chisq of canonical correlations %[1] 35.8 23.1 5.6 % % Average squared canonical correlation = 0.03 % Cohen's Set Correlation R2 = 0.09 % Shrunken Set Correlation R2 = 0.08 % F and df of Cohen's Set Correlation 7.26 9 1681.86 %Unweighted correlation between the two sets = 0.01 % %\end{Routput} % \begin{figure}[htbp] \begin{center} \includegraphics{satact.png} \caption{The relationship between three predictors and three criteria from \pfun{setCor}. The data are from the \pfun{sat.act} data set.} \label{fig:3x3} \end{center} \end{figure} \subsection{Moderated multiple regression} If we have the raw data, we can also find interactions (known as moderated multiple regression). This is done by zero centering the data \citep{cohen:03} and then multiplying the two terms of the interaction. As an option, we can not zero center the data \citep{hayes:13} which results in different ``main effects" but the same interaction term. To show the equivalence of the interaction terms, we also must not standardize the results. <>== mod <- setCor(SATQ ~ SATV*gender + ACT, data=sat.act, std =FALSE, plot=FALSE) mod mod0 <- setCor(SATQ ~ SATV*gender + ACT, data=sat.act, zero=FALSE, std=FALSE,plot=FALSE) mod0 @ <>== png('moderation.png') setCor.diagram(mod, main="not zero centered") dev.off() @ <>== png('moderation0.png') setCor.diagram(mod0, main="zero centered") dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{moderation.png} \caption{Showing a moderated multiple regression using \pfun{setCor}. The data are from the \pfun{sat.act} data set.} \label{fig:mod} \end{center} \end{figure} %\begin{Routput} %Call: setCor(y = SATQ ~ SATV * gender + ACT, data = sat.act, std = FALSE) % %Multiple Regression from raw data % % DV = SATQ % slope se t p VIF %SATV 0.47 0.03 14.47 0.0e+00 1.46 %gender -35.08 6.40 -5.48 6.0e-08 1.00 %ACT 7.72 0.77 10.05 0.0e+00 1.47 %SATV*gender -0.03 0.06 -0.47 6.4e-01 1.01 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %SATQ 0.72 0.51 70.33 4946.43 0.51 0.03 183.23 4 695 0 % % %Call: setCor(y = SATQ ~ SATV * gender + ACT, data = sat.act, std = FALSE, % zero = FALSE) % %Multiple Regression from raw data % % DV = SATQ % slope se t p VIF %SATV 0.52 0.10 5.20 2.7e-07 13.52 %gender -18.71 35.31 -0.53 6.0e-01 30.44 %ACT 7.72 0.77 10.05 0.0e+00 1.47 %SATV*gender -0.03 0.06 -0.47 6.4e-01 41.50 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %SATQ 0.72 0.51 40.02 1601.68 0.51 0.03 183.23 4 695 0 % %\end{Routput} \begin{figure}[htbp] \begin{center} \includegraphics[width=7cm]{moderation.png} \includegraphics[width=7cm]{moderation0.png} \caption{The difference between 0 and not 0 centering \pfun{setCor}. The data are from the \pfun{sat.act} data set. In both cases, the data are not standarized.} \label{default} \end{center} \end{figure} \subsection{Plotting the interactions} To visualize the effect of zero (mean) centering, it is useful to plot the various elements that go into the linear model. \pfun{setCor} returns the product terms as well as the original data. I combine the two datasets to make it clearer. Note that the correlations of the centered SATQ, SATV, and gender with the uncentered are 1.0, but that the correlations with the product terms depend upon centering versus not. I drop the ACT variables from the figure for clarity (Figure~\ref{fig:splom}). <>== both <- cbind(mod$data[-1],mod0$data[-1]) png('splom.png') pairs.panels(both[,-c(4,9)],gap=0) #show the mean centered data dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{splom.png} \caption{The effect of not mean centering versus mean centering on the product terms. The first four variables were not zero centered, the second four were. } \label{fig:splom} \end{center} \end{figure} \subsection{Comparisons to \fun{lm}} The \pfun{setCor} function duplicates the functionality of the \fun{lm} function for complete data, although \fun{lm} does not zero center and \pfun{setCor} will (by default). In addition, \pfun{setCor} finds correlations based upon pair.wise deletion of missing data, while \fun{lm} does case.wise deletion. We compare the \fun{lm} and \pfun{setCor} results for complete data by setting the \texttt{ use = "complete"} option. <>== lm(SATQ ~ SATV*gender + ACT, data=sat.act) mod <- setCor(SATQ ~ SATV*gender + ACT, data=(sat.act), zero=FALSE, std=FALSE,use="complete") print(mod,digits=5) @ % lm(SATQ ~ SATV*gender + ACT, data=sat.act) %Call: %lm(formula = SATQ ~ SATV * gender + ACT, data = sat.act) % %Coefficients: %(Intercept) SATV gender ACT SATV:gender % 138.52395 0.50280 -22.24995 7.71702 -0.01984 % %> mod <- setCor(SATQ ~ SATV*gender + ACT, data=(sat.act), zero=FALSE, std=FALSE,use="complete") % %print(mod,digits=5) %Call: setCor(y = SATQ ~ SATV * gender + ACT, data = (sat.act), use = "complete", % std = FALSE, zero = FALSE) % %Multiple Regression from raw data % % DV = SATQ % slope se t p VIF %SATV 0.50280 0.09936 5.06050 5.3589e-07 13.43994 %gender -22.24995 35.25783 -0.63106 5.2821e-01 30.29663 %ACT 7.71702 0.76977 10.02511 0.0000e+00 1.46678 %SATV*gender -0.01984 0.05652 -0.35105 7.2566e-01 41.25607 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %SATQ 0.71414 0.51 39.93879 1595.107 0.50718 0.02621 180.8401 4 695 0 %\end{Routput} \section{Mediation using the \pfun{mediate} function} Mediation analysis is just linear regression reorganized slightly to show the direct effects of an X variable upon Y, partialling out the effect of a ``mediator" (Figure~\ref{fig:mediation}). Although the statistical ``significance" of the (c) path and the (c') path are both available from standard regression, the mediation effect (ab) is best found by boot strapping the regression model and displaying the empirical confidence intervals. \begin{figure}[htbp] \begin{center} \begin{picture}(200,200) \put(10,50){\framebox(20,20){$X_{1}$}} \put(85,123){\framebox(20,20){$M_{1}$}} \put(160,50){\framebox(20,20){$Y_{1}$}} \put(30,70){\vector(1,1){54}} \put(105,123){\vector(1,-1){54}} \put(30,60){\vector(1,0){130}} \put(50,98){a} \put(134,98){b} \put(95,65){c} \put(78,51){c'= c - ab} \end{picture} \caption{The classic mediation model. The Direct Path from X -> Y (c) is said to be mediated by the indirect path (a) to the mediator (X -> M) and (b) from the mediator to Y (M -> Y). The mediation effect is (ab). } \label{fig:mediation} \end{center} \end{figure} A number of papers discuss how to test for the effect of mediation and there are some very popular `macros' for SPSS and SAS to do so \citep{hayes:13,preacher:04,preacher:07,preacher:15}. A useful discussion of mediation and moderation with sample data sets is found in \cite{hayes:13}. More recently, the \Rpkg{processR} package \citep{processR} has been released with these data sets. Although these data used to be be available from \href{"http://www.afhayes.com/public/hayes2018data.zip"}{http://www.afhayes.com/public/hayes2018data.zip} this now longer seems to be case.\footnote{The Hayes data sets (2018) do not correspond exactly with those from the 2013 book. Those data files were at \href{"http://www.afhayes.com/public/hayes2013data.zip"}{http://www.afhayes.com/public/hayes2013data.zip}.}. I use these for comparisons with the results in \cite{hayes:13}. Four of these data sets are now included in the \Rpkg{psych} package with the kind permission of their authors: \pfun{Garcia} is from \cite{garcia:10}, and \pfun{Tal\_Or} is from \cite{talor:10}, The \pfun{Pollack} correlation matrix is taken from an article by \cite{pollack}. The \pfun{globalWarm} data set is the \pfun{glbwarm} data set in the \Rpkg{processR} package and added to \Rpkg{psychTools} with the kind permission of the original author, Erik Nisbet. To find the confidence intervals of the effect of mediation (the reduction between the c and c' paths, where c' = c - ab), I bootstrap the results by randomly sampling from the data with replacement (n.iter = 5000) times. For these examples, the data files \pfun{Garcia} \citep{garcia:10} and \pfun{Tal\_Or} \citep{talor:10} are included in the \pfun{psych} package. The \pfun{estrss} data set and \pfun{glbwarm} were originally downloaded from the \cite{hayes:13} data seta and stored in a local directory on my computer. They are now available from the \Rpkg{processR} package \cite{processR}. The syntax is that $ y \sim x + (m) $ where m is the mediating variable. By default the output is to two decimals, as is the graphic output. This can be increased by returning the output to an object and then printing that object with the desired number of decimals. \subsection{Simple mediation} The first example \citep[mod.4.5]{hayes:13} is taken from \citep{talor:10} and examines the mediating effect of ``Presumed Media Influence'' (pmi) on the intention to act (reaction) based upon the importance of a message (import). The data are in the \pfun{Tal\_Or} data set in \Rpkg{psych} (with the kind permission of Nurit Tal-Or, Jonanathan Cohen, Yariv Tasfati, and Albert Gunther). In the \cite{hayes:13} book, this is the \pfun{pmi} data set. <>== data(Tal.Or) psych::describe(Tal_Or) #descriptive statistics mod4.4 <- mediate(reaction ~ cond + (pmi), data =Tal_Or) mod4.4 #print(mod4.4, digits = 4) # in order to get the precision of the Hayes (2013) p 99 example @ %\begin{Routput} %data(Tal_Or) %describe(Tal_Or) #descriptive statistics % vars n mean sd median trimmed mad min max range skew kurtosis se %cond 1 123 0.47 0.50 0.00 0.46 0.00 0 1 1 0.11 -2.00 0.05 %pmi 2 123 5.60 1.32 6.00 5.78 1.48 1 7 6 -1.17 1.30 0.12 %import 3 123 4.20 1.74 4.00 4.26 1.48 1 7 6 -0.26 -0.89 0.16 %reaction 4 123 3.48 1.55 3.25 3.44 1.85 1 7 6 0.21 -0.90 0.14 %gender 5 123 1.65 0.48 2.00 1.69 0.00 1 2 1 -0.62 -1.62 0.04 %age 6 123 24.63 5.80 24.00 23.76 1.48 18 61 43 4.71 24.76 0.52 % % % mod4.4 <- mediate(reaction ~ cond + (pmi), data =Tal_Or) %> mod4.4 %Mediation/Moderation Analysis %Call: mediate(y = reaction ~ cond + (pmi), data = Tal_Or) % %The DV (Y) was reaction . The IV (X) was cond . The mediating variable(s) = pmi . % %Total effect(c) of cond on reaction = 0.5 S.E. = 0.28 t = 1.79 df= 120 with p = 0.077 %Direct effect (c') of cond on reaction removing pmi = 0.25 S.E. = 0.26 % t = 0.99 df= 120 with p = 0.32 %Indirect effect (ab) of cond on reaction through pmi = 0.24 %Mean bootstrapped indirect effect = 0.24 with standard error = 0.13 Lower CI = 0 Upper CI = 0.52 %R = 0.45 R2 = 0.21 F = 15.56 on 2 and 120 DF p-value: 9.83e-07 % % To see the longer output, specify short = FALSE in the print statement or ask for the summary % % Full output % Total effect estimates (c) % reaction se t df Prob %cond 0.5 0.28 1.79 120 0.0766 % %Direct effect estimates (c') % reaction se t df Prob %cond 0.25 0.26 0.99 120 3.22e-01 %pmi 0.51 0.10 5.22 120 7.66e-07 % %R = 0.45 R2 = 0.21 F = 15.56 on 2 and 120 DF p-value: 9.83e-07 % % 'a' effect estimates % pmi se t df Prob %cond 0.48 0.24 2.02 121 0.0454 % % 'b' effect estimates % reaction se t df Prob %pmi 0.51 0.1 5.22 120 7.66e-07 % % 'ab' effect estimates % reaction boot sd lower upper %cond 0.24 0.24 0.13 0 0.52 % %\end{Routput} <>== png('mediate99.png') mediate.diagram(mod4.4) dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{mediate99.png} \caption{A simple mediation model \citep[p 99] {hayes:13} with data derived from \cite{talor:10}. The effect of a salience manipulation (cond) on the intention to buy a product (reaction) is mediated through the presumed media influence (pmi).} \label{default} \end{center} \end{figure} A second example from \citep{hayes:13} is an example of moderated mediated effect. The data are from \citep{garcia:10} and report on the effect of protest on reactions to a case of sexual discrimination. <>== data(GSBE) #alias to Garcia data set #compare two models (bootstrapping n.iter set to 50 for speed # 1) mean center the variables prior to taking product terms mod1 <- mediate(respappr ~ prot2 * sexism +(sexism),data=Garcia,n.iter=50 ,main="Moderated mediation (mean centered)") # 2) do not mean center mod2 <- mediate(respappr ~ prot2 * sexism +(sexism),data=Garcia,zero=FALSE, n.iter=50, main="Moderated mediation (not centered") summary(mod1) summary(mod2) @ %A second example of simple mediation from \cite[p 118-121]{hayes:13} is the effect of economic stress. The original data are from a study by \cite{pollack:12} and are available from the \cite{hayes:13} website. Is the effect of economic stress (estress) on subsequent disengagement from entreprenuerial activities (withdraw) mediated through depressed affect (affect)? % %\begin{Rinput} %estress <- read.file() #read the external file %describe(estress) %mod4.5 <- mediate(withdraw ~ estress + (affect), data =estress) %mod4.5 #normal printing is to 2 decimals %# and show the graphic to 2 decimals %#print(mod4.5, digits=4) #print to four decimals to confirm output with Hayes %#mediate.diagram(mod4.5,digits=3) to show the graphic to 3 decimals % %\end{Rinput} % %\begin{Routput} %estress <- read.file() %re-encoding from CP1252 %Data from the SPSS sav file" % /Users/WR/Box Sync/pmc_folder/tutorials/HowTo/mediation/hayes2018data/estress/estress.sav has been loaded. %> describe(estress) % vars n mean sd median trimmed mad min max range skew kurtosis se %tenure 1 262 5.93 6.58 4.00 4.73 5.07 0.00 33 33.00 1.64 2.67 0.41 %estress 2 262 4.62 1.42 4.50 4.66 1.48 1.00 7 6.00 -0.27 -0.49 0.09 %affect 3 262 1.60 0.72 1.33 1.46 0.49 1.00 5 4.00 1.97 4.57 0.04 %withdraw 4 262 2.32 1.25 2.00 2.19 1.48 1.00 7 6.00 0.70 -0.17 0.08 %sex 5 262 0.62 0.49 1.00 0.65 0.00 0.00 1 1.00 -0.48 -1.77 0.03 %age 6 262 43.79 10.36 44.00 43.78 11.86 23.00 71 48.00 -0.01 -0.82 0.64 %ese 7 262 5.61 0.94 5.73 5.67 1.08 2.53 7 4.47 -0.55 -0.13 0.06 % %\end{Routput} %\begin{Toutput} %mod4.5 % %Mediation/Moderation Analysis %Call: mediate(y = withdraw ~ estress + (affect), data = estress) % %The DV (Y) was withdraw . The IV (X) was estress . The mediating variable(s) = affect . % %Total effect(c) of estress on withdraw = 0.06 S.E. = 0.05 t = 1.04 df= 259 with p = 0.3 %Direct effect (c') of estress on withdraw removing affect = -0.08 S.E. = 0.05 t = -1.47 df= 259 with p = 0.14 %Indirect effect (ab) of estress on withdraw through affect = 0.13 %Mean bootstrapped indirect effect = 0.13 with standard error = 0.03 Lower CI = 0.07 Upper CI = 0.2 %R = 0.42 R2 = 0.18 F = 28.49 on 2 and 259 DF p-value: 6.53e-12 %\end{Toutput} %\begin{Routput} % Full output % Total effect estimates (c) % withdraw se t df Prob %estress 0.06 0.05 1.04 259 0.302 % %Direct effect estimates (c') % withdraw se t df Prob %estress -0.08 0.05 -1.47 259 1.44e-01 %affect 0.77 0.10 7.46 259 1.29e-12 % %R = 0.42 R2 = 0.18 F = 28.49 on 2 and 259 DF p-value: 6.53e-12 % % 'a' effect estimates % affect se t df Prob %estress 0.17 0.03 5.83 260 1.63e-08 % % 'b' effect estimates % withdraw se t df Prob %affect 0.77 0.1 7.46 259 1.29e-12 % % 'ab' effect estimates % withdraw boot sd lower upper %estress 0.13 0.13 0.03 0.07 0.2 %\end{Routput} % %\begin{figure}[htbp] %\begin{center} % %\includegraphics{mediate118.pdf} %\caption{A simple mediation model \citep[p 118] {hayes:13}. The data are from \cite{pollack:12} taken from the \cite{hayes:13} website. Is the effect of economic stress (estress) on subsequent disengagement from entreprenuerial activities (withdraw) mediated through depressed affect (affect)?} %\label{default} %\end{center} %\end{figure} \subsection{Multiple mediators} It is trivial to show the effect of multiple mediators. I do this by adding the second (or third) mediator into our equation. I use the \fun{Tal\_Or} data set \citep{talor:10} again. I show the graphical representation in Figure~\ref{fig:2m}. <>== mod5.4 <- mediate(reaction ~ cond + (import) + (pmi), data = Tal_Or) print(mod5.4, digits=4) #to compare with Hayes @ <>== png('mediate131.png') mediate.diagram(mod5.4, digits=3, main="Hayes example 5.3") dev.off() @ %\begin{Toutput} %Call: mediate(y = reaction ~ cond + (import) + (pmi), data = Tal_Or) % %The DV (Y) was reaction . The IV (X) was cond . The mediating variable(s) = import pmi . Variable(s) partialled out were % %Total Direct effect(c) of cond on reaction = 0.4957 S.E. = 0.2775 t direct = 1.786 with probability = 0.07661 %Direct effect (c') of cond on reaction removing import pmi = 0.1034 S.E. = 0.2391 t direct = 0.4324 with probability = 0.6662 %Indirect effect (ab) of cond on reaction through import pmi = 0.3923 %Mean bootstrapped indirect effect = 0.3964 with standard error = 0.1658 Lower CI = 0.0895 Upper CI = 0.7317 %R2 of model = 0.3251 % To see the longer output, specify short = FALSE in the print statement % % %\end{Toutput} %\begin{Routput} % Full output % % Total effect estimates (c) % reaction se t Prob %cond 0.4957 0.2775 1.786 0.076608 % %Direct effect estimates (c') % reaction se t Prob %cond 0.1034 0.2391 0.4324 6.6622e-01 %import 0.3244 0.0707 4.5857 1.1267e-05 %pmi 0.3965 0.0930 4.2645 4.0383e-05 % % 'a' effect estimates % cond se t Prob %import 0.6268 0.3098 2.0234 0.045235 %pmi 0.4765 0.2357 2.0218 0.045401 % % 'b' effect estimates % reaction se t Prob %import 0.3244 0.0707 4.5857 1.1267e-05 %pmi 0.3965 0.0930 4.2645 4.0383e-05 % % 'ab' effect estimates % reaction boot sd lower upper %cond 0.3923 0.3965 0.1645 0.0896 0.7392 %> \end{Routput} % \begin{figure}[htbp] \begin{center} \includegraphics{mediate131.png} \caption{A mediation model with two mediators \citep[p 131] {hayes:13}. The data are data derived from \cite{talor:10}. The effect of a salience manipulation (cond) on the intention to buy a product (reaction) is mediated through the presumed media influence (pmi) and importance of the message (import).} \label{fig:2m} \end{center} \end{figure} \subsection{Serial mediators} The example from \cite{hayes:13} for two mediators, where one effects the second, is a bit more complicated and currently can be done by combining two separate analyses. The first is just model 5.4, the second is the effect of cond on pmi mediated by import. Combining the two results leads to the output found on \cite[page 153]{hayes:13}. <>== png('mediate131.png') mediate.diagram(mod5.4, digits=3, main="Hayes example 5.3") dev.off() @ <>== #model 5.4 + mod5.7 is the two chained mediator model mod5.7 <- mediate(pmi ~ cond + (import) , data = Tal_Or) summary(mod5.7, digits=4) @ % %\begin{Routput} %Call: mediate(y = pmi ~ cond + (import), data = Tal_Or) % % Total effect estimates (c) % pmi se t df Prob %cond 0.4765 0.2357 2.0218 120 0.045419 % %Direct effect estimates (c') % pmi se t df Prob %cond 0.3536 0.2325 1.5207 120 0.1309600 %import 0.1961 0.0671 2.9228 120 0.0041467 % %R = 0.3114 R2 = 0.097 F = 6.4428 on 2 and 120 DF p-value: 0.0021989 % % 'a' effect estimates % import se t df Prob %cond 0.6268 0.3098 2.0234 121 0.045235 % % 'b' effect estimates % pmi se t df Prob %import 0.1961 0.0671 2.9228 120 0.0041467 % % 'ab' effect estimates % pmi boot sd lower upper %cond 0.1229 0.1226 0.0825 -0.0017 0.3152 %> % %\end{Routput} % \subsection{Single mediators, multiple covariates} The \fun{Pollack} data set \citep{pollack:12} is used as an example of multiple covariates (included in \Rpkg{psychTools} as a correlation matrix). The raw data are available from the \Rpkg{processR} package as \pfun{estress}. Confidence in executive decision making (``Entrepeneurial self-effiicacy), gender (sex), and length of time in business (tenure) are used as covariates. There are two ways of doing this: enter them as predictors of the criterion or to partial them out. The first approach estimates their effects, the second just removes them. <>== lowerMat(Pollack) mod6.2 <- mediate(withdrawal ~ economic.stress + self.efficacy + sex + tenure + (depression), data=Pollack, n.obs=262) summary(mod6.2) @ <>== png('mediate177.png') mediate.diagram(mod6.2, digits=3, main = "Simple mediation, 3 covariates") dev.off() @ The graphical output (Figure~\ref{fig:3cov}) looks a bit more complicated than the figure in \cite[p 177]{hayes:13} because I are showing the covariates as causal paths. %\begin{Toutput} %Call: mediate(y = withdraw ~ estress + ese + sex + tenure + (affect), % data = estress) % %The DV (Y) was withdraw . The IV (X) was estress ese sex tenure . The mediating variable(s) = affect . % %Total effect(c) of estress on withdraw = 0.02 S.E. = 0.05 t = 0.35 df= 256 with p = 0.72 %Direct effect (c') of estress on withdraw removing affect = -0.09 S.E. = 0.05 t = -1.78 df= 256 with p = 0.077 %Indirect effect (ab) of estress on withdraw through affect = 0.11 %Mean bootstrapped indirect effect = 0.11 with standard error = 0.03 Lower CI = 0.06 Upper CI = 0.17 % %Total effect(c) of ese on withdraw = -0.32 S.E. = 0.08 t = -3.98 df= 256 with p = 9e-05 %Direct effect (c') of ese on NA removing affect = -0.21 S.E. = 0.08 t = -2.78 df= 256 with p = 0.0059 %Indirect effect (ab) of ese on withdraw through affect = -0.11 %Mean bootstrapped indirect effect = 0.11 with standard error = 0.03 Lower CI = -0.19 Upper CI = -0.03 % %Total effect(c) of sex on withdraw = 0.14 S.E. = 0.16 t = 0.88 df= 256 with p = 0.38 %Direct effect (c') of sex on NA removing affect = 0.13 S.E. = 0.14 t = 0.88 df= 256 with p = 0.38 %Indirect effect (ab) of sex on withdraw through affect = 0.01 %Mean bootstrapped indirect effect = 0.11 with standard error = 0.03 Lower CI = -0.09 Upper CI = 0.15 % %Total effect(c) of tenure on withdraw = -0.01 S.E. = 0.01 t = -0.85 df= 256 with p = 0.4 %Direct effect (c') of tenure on NA removing affect = 0 S.E. = 0.01 t = -0.19 df= 256 with p = 0.85 %Indirect effect (ab) of tenure on withdraw through affect = -0.01 %Mean bootstrapped indirect effect = 0.11 with standard error = 0.03 Lower CI = -0.02 Upper CI = 0 %R = 0.45 R2 = 0.21 F = 13.28 on 5 and 256 DF p-value: 1.63e-11 %\end{Toutput} % %\begin{Routput} % Full output % % % Total effect estimates (c) % withdraw se t df Prob %estress 0.02 0.05 0.35 256 7.24e-01 %ese -0.32 0.08 -3.98 256 9.02e-05 %sex 0.14 0.16 0.88 256 3.78e-01 %tenure -0.01 0.01 -0.85 256 3.97e-01 % %Direct effect estimates (c') % withdraw se t df Prob %estress -0.09 0.05 -1.78 256 0.07710 %ese -0.21 0.08 -2.78 256 0.00589 %sex 0.13 0.14 0.88 256 0.37800 %tenure 0.00 0.01 -0.19 256 0.84600 % %R = 0.45 R2 = 0.21 F = 13.28 on 5 and 256 DF p-value: 1.63e-11 % % 'a' effect estimates % affect se t df Prob %estress 0.16 0.03 5.36 257 1.84e-07 %ese -0.15 0.04 -3.49 257 5.70e-04 %sex 0.01 0.09 0.17 257 8.63e-01 %tenure -0.01 0.01 -1.72 257 8.61e-02 % % 'b' effect estimates % withdraw se t df Prob %affect 0.71 0.1 6.74 256 1.03e-10 % % 'ab' effect estimates % withdraw boot sd lower upper %estress 0.11 0.11 0.03 0.06 0.17 %ese -0.11 -0.11 0.04 -0.19 -0.03 %sex 0.01 0.02 0.06 -0.09 0.15 %tenure -0.01 -0.01 0.00 -0.02 0.00 %> %\end{Routput} \begin{figure}[htbp] \begin{center} \includegraphics{mediate177.png} \caption{A mediation model with three covariates \citep[p 177] {hayes:13}. Compare this to the solution in which they are partialled out. (Figure~\ref{fig:mod6.2a}).} \label{fig:3cov} \end{center} \end{figure} \subsection{Single predictor, single criterion, multiple covariates} An alternative way to display the previous results is to remove the three covariates from the mediation model. I do this by partialling out the covariates. This is represented in the \pfun{mediate} code by a negative sign. (Figure~\ref{fig:mod6.2a}) <>== mod6.2a <- mediate(withdrawal ~ economic.stress -self.efficacy - sex - tenure + (depression), data=Pollack, n.obs=262) summary(mod6.2a) @ %\begin{Rinput} %mod6.2a <- mediate(withdraw ~ estress - ese - sex - tenure + (affect), data=estress) %mod6.2a #give the output %\end{Rinput} %\begin{Toutput} % %The DV (Y) was withdraw . The IV (X) was estress . The mediating variable(s) = affect . Variable(s) partialled out were ese sex tenure % %Total effect(c) of estress on withdraw = 0.02 S.E. = 0.05 t = 0.36 df= 256 with p = 0.72 %Direct effect (c') of estress on withdraw removing affect = -0.09 S.E. = 0.05 t = -1.77 df= 256 with p = 0.078 %Indirect effect (ab) of estress on withdraw through affect = 0.11 %Mean bootstrapped indirect effect = 0.11 with standard error = 0.03 Lower CI = 0.06 Upper CI = 0.17 %R = 0.39 R2 = 0.15 F = 22.8 on 2 and 256 DF p-value: 7.71e-10 % %\end{Toutput} %\begin{Routput} % Full output %Total effect estimates (c) % withdraw se t df Prob %estress 0.02 0.05 0.36 256 0.722 % %Direct effect estimates (c') % withdraw se t df Prob %estress -0.09 0.05 -1.77 256 7.78e-02 %affect 0.71 0.11 6.72 256 1.14e-10 % %R = 0.39 R2 = 0.15 F = 22.8 on 2 and 256 DF p-value: 7.71e-10 % % 'a' effect estimates % affect se t df Prob %estress 0.16 0.03 5.39 257 1.58e-07 % % 'b' effect estimates % withdraw se t df Prob %affect 0.71 0.11 6.72 256 1.14e-10 % % 'ab' effect estimates % withdraw boot sd lower upper %estress 0.11 0.11 0.03 0.06 0.17 % % %\end{Routput} % <>== png('mod62partial.png') mediate.diagram(mod6.2a, digits=3, main = "Simple mediation, 3 covariates (partialled out)") dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{mod62partial.png} \caption{Show the mediation model from Figure~\ref{fig:3cov} with the covariates (ese, sex, tenure) removed.} \label{fig:mod6.2a} \end{center} \end{figure} \subsection{Multiple predictors, single criterion} It is straightforward to use multiple predictors see \cite[p196]{hayes:13} and in fact I did so in the previous example where the predictors were treated as \emph{covariates}. \pfun{mediate} also allows for multiple criteria. \section{Mediation and moderation} We already saw how to do moderation in the discussion of \pfun{setCor}. Combining the concepts of mediation with moderation is done in \pfun{mediate}. That is, I can find the linear model of product terms as they are associated with dependent variables and regressed on the mediating variables. The \fun{Garcia} data set \citep{garcia:10} can be used for an example of moderation. (This was taken from \citep{hayes:13} but is used with kind permission of Donna M. Garcia, Michael T. Schmitt, Nyla R. Branscombe, and Naomi Ellemers.) Just as \pfun{setCor} and \fun{lm} will find the interaction term by forming a product, so will \pfun{mediate}. Notice that by default, \pfun{setCor} reports zero centered and standardized regressions, \pfun{mediate} reports zero centered but not standardized regressions, and some of the examples from \cite{hayes:13} do not zero center the data. Thus, I specify zero=FALSE to get the \cite{hayes:13} results. It is important to note that the \fun{protest} data set discussed here is from the 2013 examples and not the more recent 2018 examples available from \href{http://afhayes.com}{afhayes.com}. The 2013 data have a dichotomous protest variable, while the 2018 data set has three levels for the protest variable. The \pfun{Garcia} data set is composed of the 2018 data set with the addition of a dichotomous variable (prot2) to match the 2013 exampes. We consider how the interaction of sexism with protest affects the mediation effect of sexism \citep[p 362]{hayes:13}, I contrast the \fun{lm}, \pfun{setCor} and \pfun{mediate} approaches. For reasons to be discussed in the next section, I do not zero center the variables. The graphic output is in Figure~\ref{fig:modmed} and the output is below. For comparison purposes, I show the results from the \fun{lm} as well as \pfun{setCor} and \pfun{mediate}. <>== summary(lm(respappr ~ prot2 * sexism,data = Garcia)) #show the lm results for comparison #show the setCor analysis setCor(respappr ~ prot2* sexism ,data=Garcia,zero=FALSE,main="Moderation",std=FALSE) #then show the mediate results modgarcia <-mediate(respappr ~ prot2 * sexism +(sexism),data=Garcia,zero=FALSE,main="Moderated mediation") summary(modgarcia) @ <>== png('moderatedmediation.png') mediate.diagram(modgarcia, main= "An example of moderated mediation") dev.off() @ %lm(formula = respappr ~ prot2 * sexism, data = Garcia) % %Residuals: % Min 1Q Median 3Q Max %-3.4984 -0.7540 0.0801 0.8301 3.1853 % %Coefficients: % Estimate Std. Error t value Pr(>|t|) %(Intercept) 6.5667 1.2095 5.429 2.83e-07 *** %prot2 -2.6866 1.4515 -1.851 0.06654 . %sexism -0.5290 0.2359 -2.243 0.02668 * %prot2:sexism 0.8100 0.2819 2.873 0.00478 ** %--- %Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1 % %Residual standard error: 1.144 on 125 degrees of freedom %Multiple R-squared: 0.2962, Adjusted R-squared: 0.2793 %F-statistic: 17.53 on 3 and 125 DF, p-value: 1.456e-09 % %setCor(respappr ~ prot2* sexism ,data=Garcia,zero=FALSE,main="Moderation",std=FALSE) %Call: setCor(y = respappr ~ prot2 * sexism, data = Garcia, std = FALSE, % main = "Moderation", zero = FALSE) % %Multiple Regression from raw data % % DV = respappr % slope se t p VIF %prot2 -2.69 1.45 -1.85 0.0670 44.99 %sexism -0.53 0.24 -2.24 0.0270 3.34 %prot2*sexism 0.81 0.28 2.87 0.0048 48.14 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %respappr 0.54 0.3 0.65 0.43 0.28 0.06 17.53 3 125 1.46e-09 % %> summary( mediate(respappr ~ prot2 * sexism +(sexism),data=Garcia,zero=FALSE,main="Moderated mediation")) %Call: mediate(y = respappr ~ prot2 * sexism + (sexism), data = Garcia, % zero = FALSE, main = "Moderated mediation") % % Total effect estimates (c) % respappr se t df Prob %prot2 0.00 0.84 0.00 125 0.9960 %prot2*sexism 0.28 0.16 1.79 125 0.0756 % %Direct effect estimates (c') % respappr se t df Prob %prot2 -2.69 1.45 -1.85 125 0.06650 %prot2*sexism 0.81 0.28 2.87 125 0.00478 % %R = 0.54 R2 = 0.3 F = 17.53 on 3 and 125 DF p-value: 1.46e-09 % % 'a' effect estimates % sexism se t df Prob %prot2 -5.07 0.31 -16.33 126 6.81e-33 %prot2*sexism 1.00 0.06 17.15 126 9.41e-35 % % 'b' effect estimates % respappr se t df Prob %sexism -0.53 0.24 -2.24 125 0.0267 % % 'ab' effect estimates % respappr boot sd lower upper %prot2 2.68 2.65 1.60 -0.69 5.60 %prot2*sexism -0.53 -0.52 0.32 -1.11 0.14 %\end{Routput} \begin{figure}[htbp] \begin{center} \includegraphics{moderatedmediation.png} \caption{Moderated mediation from \citep[p 362]{hayes:13}. The data are from \cite{garcia:10}.} \label{fig:modmed} \end{center} \end{figure} \subsection{To center or not to center, that is the question} We have discussed the difference between zero centering and not zero centering. Although \cite{hayes:13} seems to prefer not centering, some of his examples are in fact centered. So, when we examine Table 8.2 and try to replicate the regression, we need to zero center the data. With the global warming data from \cite{hayes:13}, the default (uncentered) regression does not reproduce his Table, but zero centering does. To this in \fun{lm} requires two steps, but we can do this in \pfun{setCor} with the zero=TRUE or zero=FALSE option. <>== lm(govact ~ age * negemot + posemot + ideology + sex, data=globalWarm) # but zero center and try again glbwarmc <-data.frame(scale(globalWarm,scale=FALSE)) lm(govact ~ age * negemot + posemot + ideology + sex, data=glbwarmc) mod.glb <- setCor(govact ~ age * negemot + posemot + ideology + sex, data=globalWarm,zero=FALSE,std=FALSE) print(mod.glb,digits=6) mod.glb0 <- setCor(govact ~ age * negemot + posemot + ideology + sex, data=globalWarm,std=FALSE) print(mod.glb0,digits=6) @ %\begin{Routput} %> lm(govact ~ age * negemot + posemot + ideology + sex, data=glbwarm) %Call: %lm(formula = govact ~ age * negemot + posemot + ideology + sex, % data = glbwarm) %Coefficients: %(Intercept) age negemot posemot ideology sex age:negemot % 5.173849 -0.023879 0.119583 -0.021419 -0.211515 -0.011191 0.006331 %> # but zero center and try again %> glbwarmc <-data.frame(scale(glbwarm,scale=FALSE)) %> lm(govact ~ age * negemot + posemot + ideology + sex, data=glbwarmc) % %Call: %lm(formula = govact ~ age * negemot + posemot + ideology + sex, % data = glbwarmc) % %Coefficients: %(Intercept) age negemot posemot ideology sex age:negemot % 0.008979 -0.001354 0.433184 -0.021419 -0.211515 -0.011191 0.006331 % %> mod.glb <- setCor(govact ~ age * negemot + posemot + ideology + sex, data=glbwarm,zero=FALSE,std=FALSE) %> print(mod.glb,digits=6) %Call: setCor(y = govact ~ age * negemot + posemot + ideology + sex, % data = glbwarm, std = FALSE, zero = FALSE) % %Multiple Regression from raw data % % DV = govact % slope se t p VIF %age -0.023879 0.005980 -3.992944 7.12038e-05 6.949401 %negemot 0.119583 0.082535 1.448881 1.47759e-01 11.594520 %posemot -0.021419 0.027904 -0.767597 4.42951e-01 1.028663 %ideology -0.211515 0.026833 -7.882678 1.02141e-14 1.198910 %sex -0.011191 0.076003 -0.147240 8.82979e-01 1.052907 %age*negemot 0.006331 0.001543 4.103542 4.48155e-05 16.455422 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %govact 0.633093 0.400806 0.571703 0.326844 0.396357 0.026299 90.07983 6 808 0 % %> mod.glb0 <- setCor(govact ~ age * negemot + posemot + ideology + sex, data=glbwarm,std=FALSE) %> print(mod.glb0,digits=6) %Call: setCor(y = govact ~ age * negemot + posemot + ideology + sex, % data = glbwarm, std = FALSE) % %Multiple Regression from raw data % % DV = govact % slope se t p VIF %age -0.001354 0.002348 -0.576864 5.64192e-01 1.071058 %negemot 0.433184 0.026243 16.506679 0.00000e+00 1.172207 %posemot -0.021419 0.027904 -0.767597 4.42951e-01 1.028663 %ideology -0.211515 0.026833 -7.882678 1.02141e-14 1.198910 %sex -0.011191 0.076003 -0.147240 8.82979e-01 1.052907 %age*negemot 0.006331 0.001543 4.103542 4.48155e-05 1.014744 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %govact 0.633093 0.400806 0.34298 0.117635 0.396357 0.026299 90.07983 6 808 0 % %\end{Routput} % So, when we do the mediated moderation model, we need to use the zero centered option to match the \cite{hayes:13} results from Figure 8.5. <>== #by default, mediate zero centers before finding the products mod.glb <- mediate(govact ~ age * negemot + posemot + ideology + sex + (age), data=globalWarm,zero=TRUE) summary(mod.glb,digits=4) @ Compare this output to that of Table 8.2 and Figure 8.5 (p 258 - 259). %\begin{Routput} %Call: mediate(y = govact ~ age * negemot + posemot + ideology + sex + % (age), data = glbwarm, zero = TRUE) % % Total effect estimates (c) % govact se t df Prob %negemot 0.4328 0.0262 16.5043 808 5.9317e-53 %posemot -0.0220 0.0279 -0.7890 808 4.3036e-01 %ideology -0.2145 0.0263 -8.1510 808 1.3712e-15 %sex -0.0173 0.0752 -0.2304 808 8.1783e-01 %age*negemot 0.0063 0.0015 4.1025 808 4.5004e-05 % %Direct effect estimates (c') % govact se t df Prob %negemot 0.4332 0.0262 16.5067 808 5.7578e-53 %posemot -0.0214 0.0279 -0.7676 808 4.4295e-01 %ideology -0.2115 0.0268 -7.8827 808 1.0360e-14 %sex -0.0112 0.0760 -0.1472 808 8.8298e-01 %age*negemot 0.0063 0.0015 4.1035 808 4.4816e-05 % %R = 0.6331 R2 = 0.4008 F = 90.0798 on 6 and 808 DF p-value: 1.8246e-86 % % 'a' effect estimates % age se t df Prob %negemot 0.2757 0.3929 0.7017 809 4.8305e-01 %posemot 0.4232 0.4176 1.0135 809 3.1112e-01 %ideology 2.2079 0.3943 5.6002 809 2.9334e-08 %sex 4.5345 1.1269 4.0238 809 6.2643e-05 %age*negemot 0.0031 0.0231 0.1346 809 8.9294e-01 % % 'b' effect estimates % govact se t df Prob %age -0.0014 0.0023 -0.5769 808 0.56419 % % 'ab' effect estimates % govact boot sd lower upper %negemot -0.0004 -0.0004 0.0012 -0.0033 0.0016 %posemot -0.0006 -0.0005 0.0014 -0.0038 0.0021 %ideology -0.0030 -0.0029 0.0051 -0.0136 0.0070 %sex -0.0061 -0.0057 0.0106 -0.0273 0.0150 %age*negemot 0.0000 0.0000 0.0001 -0.0002 0.0002 % \end{Routput} % \subsection{Another example of moderated medation} The \pfun{Garcia} data set (\pfun{protest} in \cite{hayes:13}) is another example of a moderated analysis. We can use either \pfun{setCor} or \pfun{mediate} to examine this data set. The defaults for these two differ, in that \pfun{setCor} assumes we want to zero center \emph{and} standardize, while \pfun{mediate} defaults to not standardizing but also defaults to zero (mean) centering. Note that in the next examples we specify we do not want to standardize nor to mean center. <>== psych::describe(Garcia) lm(liking ~ prot2* sexism + respappr, data=Garcia) setCor(liking ~ prot2* sexism + respappr, data = Garcia, zero=FALSE,std=FALSE) mod7.4 <- mediate(liking ~ prot2 * sexism +respappr, data = Garcia, zero=FALSE) summary(mod7.4) @ %\begin{Routput} % describe(Garcia) % vars n mean sd median trimmed mad min max range skew kurtosis se %protest 1 129 1.03 0.82 1.00 1.04 1.48 0.00 2 2.00 -0.06 -1.52 0.07 %sexism 2 129 5.12 0.78 5.12 5.10 0.74 2.87 7 4.13 0.12 -0.32 0.07 %anger 3 129 2.12 1.66 1.00 1.84 0.00 1.00 7 6.00 1.29 0.26 0.15 %liking 4 129 5.64 1.05 5.83 5.73 0.99 1.00 7 6.00 -1.15 2.48 0.09 %respappr 5 129 4.87 1.35 5.25 4.98 1.11 1.50 7 5.50 -0.75 -0.18 0.12 %prot2 6 129 0.68 0.47 1.00 0.72 0.00 0.00 1 1.00 -0.77 -1.41 0.04 % % % %Call: %lm(formula = liking ~ prot2 * sexism + respappr, data = Garcia) % %Coefficients: % (Intercept) prot2 sexism respappr prot2:sexism % 5.3471 -2.8075 -0.2824 0.3593 0.5426 % %> setCor(liking ~ prot2* sexism + respappr, data = Garcia, zero=FALSE,std=FALSE) %Call: setCor(y = liking ~ prot2 * sexism + respappr, data = Garcia, % std = FALSE, zero = FALSE) % %Multiple Regression from raw data % % DV = liking % slope se t p VIF %prot2 -2.81 1.16 -2.42 1.7e-02 46.22 %sexism -0.28 0.19 -1.49 1.4e-01 3.47 %respappr 0.36 0.07 5.09 1.3e-06 1.42 %prot2*sexism 0.54 0.23 2.36 2.0e-02 51.32 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %liking 0.53 0.28 0.39 0.15 0.26 0.06 12.26 4 124 1.99e-08 %> mod7.4m <- mediate(liking ~ protest * sexism, data = protest, zero=FALSE) %> mod7.4m %Call: mediate(y = liking ~ prot2 * sexism + respappr, data = Garcia, % zero = FALSE) % %The DV (Y) was liking . The IV (X) was prot2 sexism respappr prot2*sexism . The mediating variable(s) = . % DV = liking % slope se t p %prot2 -2.81 1.16 -2.42 1.7e-02 %sexism -0.28 0.19 -1.49 1.4e-01 %respappr 0.36 0.07 5.09 1.3e-06 %prot2*sexism 0.54 0.23 2.36 2.0e-02 % %With R2 = 0.28 %R = 0.53 R2 = 0.28 F = 12.26 on 4 and 124 DF p-value: 1.99e-08 %\end{Routput} <>== png('moderation.png') mediate.diagram(mod7.4, main= "Another example of moderated mediation") dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{moderation.png} \caption{A simple moderated regression analysis of the \fun{protest} data set. The data were not zero centered. This shows the strength of the three regressions. Figure~\ref{fig:garcia} shows the actual data and the three regression lines. } \label{fig:moderation} \end{center} \end{figure} \subsection{Graphic Displays of Interactions} In order to graphically display interactions, particularly if one of the variable is categorical, we can plot separate regression lines for each value of the categorical variable. We do this for the \pfun{Garcia} data set to show the interaction of protest with sexism. (see Figure~\ref{fig:garcia}). This is just an example of how to use Core-R to do graphics and is not a feature of \Rpkg{psych}. <>== png('garciainteraction.png') plot(respappr ~ sexism, pch = 23- protest, bg = c("black","red", "blue")[protest], data=Garcia, main = "Response to sexism varies as type of protest") by(Garcia,Garcia$protest, function(x) abline(lm(respappr ~ sexism, data =x),lty=c("solid","dashed","dotted")[x$protest+1])) text(6.5,3.5,"No protest") text(3,3.9,"Individual") text(3,5.2,"Collective") dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{garciainteraction.png} \caption{Showing the interaction between type of protest and sexism from the \fun{Garcia} data set. The strength of the regression effects is shown in Fig~\ref{fig:moderation}.} \label{fig:garcia} \end{center} \end{figure} \section{Partial Correlations} Although not strickly speaking part of mediation or moderation, the use of \emph{partial correlations} can be addressed here. \subsection{Partial some variables from the rest of the variables} Given a set of X variables and a set of Y variables, we can control for an additional set of Z variables when we find the correlations between X and Y. This is effectively what happens when we want to add covariates into a model. We see this when we compare the regression model for government action as a function of the iteraction of ideology and age with some covariates, or when we partial them out first. <>== #first, the more complicated model mod.glb <- setCor(govact ~ age * negemot + posemot + ideology + sex, data=globalWarm,std=FALSE) print(mod.glb,digits=3) # compare this to the partialled model mod.glb.partialled <- setCor(govact ~ age * negemot - posemot - ideology - sex,data = globalWarm) @ % %\begin{Routput} % print(mod.glb,digits=3) %Call: setCor(y = govact ~ age * negemot + posemot + ideology + sex, % data = glbwarm, std = FALSE) % %Multiple Regression from raw data % % DV = govact % slope se t p VIF %age -0.001 0.002 -0.577 5.64e-01 1.071 %negemot 0.433 0.026 16.507 0.00e+00 1.172 %posemot -0.021 0.028 -0.768 4.43e-01 1.029 %ideology -0.212 0.027 -7.883 1.02e-14 1.199 %sex -0.011 0.076 -0.147 8.83e-01 1.053 %age*negemot 0.006 0.002 4.104 4.48e-05 1.015 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %govact 0.633 0.401 0.343 0.118 0.396 0.026 90.08 6 808 0 %> %mod.glb.partialled <- setCor(govact ~ age * negemot - posemot - ideology - sex, %+ data=glbwarm,std=FALSE) % %mod.glb.partialled %Call: setCor(y = govact ~ age * negemot - posemot - ideology - sex, % data = glbwarm) % %Multiple Regression from raw data % % DV = govact % slope se t p VIF %age -0.02 0.03 -0.54 0.59000 1.00 %negemot 0.49 0.03 16.19 0.00000 1.01 %age*negemot 0.11 0.03 3.75 0.00019 1.01 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %govact 0.52 0.27 0.33 0.11 0.27 0.03 100.5 3 811 0 %> %\end{Routput} Note how the beta weights for the age, negemot and interaction terms are identical. \subsection{Partial everything from everything} Sometimes we want to examine just the independent effects of all our variables. That is to say, we want to partial all the variables from all the other variables. I do this with the \pfun{partial.r} function. To show the results, I compare the partialed rs to the original rs. I show the lower off diagonal matrix using \pfun{lowerMat}. Then to compare the partial matrix to the original matrix, I form the square matrix where the lower off diagonal is the original matrix and the upper off diagonal is the partial matrix. <>== upper <-partial.r(globalWarm) lowerMat(upper) #show it lower <- lowerCor(globalWarm) lowup <- lowerUpper(lower,upper) @ %\begin{Routput} %upper <-partial.r(glbwarm) %> lowerMat(upper) #show it % govct posmt negmt idlgy age sex prtyd %govact 1.00 %posemot -0.03 1.00 %negemot 0.50 0.13 1.00 %ideology -0.19 0.00 -0.07 1.00 %age -0.02 0.04 0.03 0.14 1.00 %sex 0.00 0.08 -0.07 0.04 0.14 1.00 %partyid -0.08 -0.01 -0.09 0.53 0.03 0.02 1.00 %> lower <- lowerCor(glbwarm) % govct posmt negmt idlgy age sex prtyd %govact 1.00 %posemot 0.04 1.00 %negemot 0.58 0.13 1.00 %ideology -0.42 -0.03 -0.35 1.00 %age -0.10 0.04 -0.06 0.21 1.00 %sex -0.10 0.07 -0.12 0.13 0.17 1.00 %partyid -0.36 -0.04 -0.32 0.62 0.15 0.11 1.00 % %\end{Routput} <>== png('partials.png') psych::corPlot(lowup,numbers = TRUE) dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics[width=9cm]{partials.png} \caption{Correlations (below diagonal) and partial correlations (above the diagonal) } \label{default} \end{center} \end{figure} \section{Related packages} \pfun{mediate} and \pfun{setCor} are just two functions in the \Rpkg{psych} package. There are several additional packages available in \R{} to do mediation. The \Rpkg{mediation} package \citep{mediation} seems the most powerful, in that is tailor made for mediation. \Rpkg{MBESS} \citep{MBESS} has a mediation function. Steven Short has a nice tutorial on mediation analysis available for download \href{http://docs.wixstatic.com/ugd/bb3887\_73181065d7c744c4a0925844302cf813.pdf}{that discusses how to use R for mediation.} And, of course, the \Rpkg{lavaan} package \citep{lavaan} is the recommended package to do SEM and path models. \newpage \section{Development version and a users guide} The \Rpkg{psych} package is available from the CRAN repository. However, the most recent development version of the \Rpkg{psych} package is available as a source file at the repository maintained at \href{ href="http://personality-project.org/r"}{\url{http://personality-project.org/r}}. That version will have removed the most recently discovered bugs (but perhaps introduced other, yet to be discovered ones). To install this development version, either for PCs or Macs, \begin{Rinput} install.packages("psych", repos = "http://personality-project.org/r", type = "source") \end{Rinput} After doing this, it is important to restart \R{} to get the new package. Although the individual help pages for the \Rpkg{psych} package are available as part of \R{} and may be accessed directly (e.g. ?psych) , the full manual for the \pfun{psych} package is also available as a pdf at \url{http://personality-project.org/r/psych_manual.pdf} %psych\_manual.pdf. News and a history of changes are available in the NEWS and CHANGES files in the source files. To view the most recent news, \begin{Schunk} \begin{Sinput} > news(Version >-= "2.0.12",package="psych") \end{Sinput} \end{Schunk} \section{Psychometric Theory} The \Rpkg{psych} package has been developed to help psychologists do basic research. Many of the functions were developed to supplement a book (\url{http://personality-project.org/r/book} An introduction to Psychometric Theory with Applications in \R{} \citep{revelle:intro} More information about the use of some of the functions may be found in the book . For more extensive discussion of the use of \Rpkg{psych} in particular and \R{} in general, consult \url{http://personality-project.org/r/r.guide.html} A short guide to R. \section{SessionInfo} This document was prepared using the following settings. \begin{tiny} <>== sessionInfo() @ \end{tiny} \ \newpage %\bibliography{../../../all} \begin{thebibliography}{} \bibitem[\protect\astroncite{Cohen}{1982}]{cohen:82} Cohen, J. (1982). \newblock Set correlation as a general mulitivariate data-analytic method. \newblock {\em Multivariate Behavioral Research}, 17(3):301--341. \bibitem[\protect\astroncite{Cohen et~al.}{2003}]{cohen:03} Cohen, J., Cohen, P., West, S.~G., and Aiken, L.~S. (2003). \newblock {\em Applied multiple regression/correlation analysis for the behavioral sciences}. \newblock L. Erlbaum Associates, Mahwah, N.J., 3rd ed edition. \bibitem[\protect\astroncite{Dawes}{1979}]{dawes:79} Dawes, R.~M. (1979). \newblock The robust beauty of improper linear models in decision making. \newblock {\em American Psychologist}, 34(7):571--582. \bibitem[\protect\astroncite{Garcia et~al.}{2010}]{garcia:10} Garcia, D.~M., Schmitt, M.~T., Branscombe, N.~R., and Ellemers, N. (2010). \newblock Women's reactions to ingroup members who protest discriminatory treatment: The importance of beliefs about inequality and response appropriateness. \newblock {\em European Journal of Social Psychology}, 40(5):733--745. \bibitem[\protect\astroncite{Hayes}{2013}]{hayes:13} Hayes, A.~F. (2013). \newblock {\em Introduction to mediation, moderation, and conditional process analysis: A regression-based approach}. \newblock Guilford Press, New York. \bibitem[\protect\astroncite{Hotelling}{1936}]{hotelling:36} Hotelling, H. (1936). \newblock Relations between two sets of variates. \newblock {\em Biometrika}, 28(3/4):321--377. \bibitem[\protect\astroncite{Kelley}{2017}]{MBESS} Kelley, K. (2017). \newblock {\em {MBESS: The MBESS R} Package}. \newblock R package version 4.4.1. \bibitem[\protect\astroncite{Pollack et~al.}{2012}]{pollack:12} Pollack, J.~M., Vanepps, E.~M., and Hayes, A.~F. (2012). \newblock The moderating role of social ties on entrepreneurs' depressed affect and withdrawal intentions in response to economic stress. \newblock {\em Journal of Organizational Behavior}, 33(6):789--810. \bibitem[\protect\astroncite{Preacher}{2015}]{preacher:15} Preacher, K.~J. (2015). \newblock Advances in mediation analysis: A survey and synthesis of new developments. \newblock {\em Annual Review of Psychology}, 66:825--852. \bibitem[\protect\astroncite{Preacher and Hayes}{2004}]{preacher:04} Preacher, K.~J. and Hayes, A.~F. (2004). \newblock {SPSS and SAS} procedures for estimating indirect effects in simple mediation models. \newblock {\em Behavior Research Methods, Instruments, \& Computers}, 36(4):717--731. \bibitem[\protect\astroncite{Preacher et~al.}{2007}]{preacher:07} Preacher, K.~J., Rucker, D.~D., and Hayes, A.~F. (2007). \newblock Addressing moderated mediation hypotheses: Theory, methods, and prescriptions. \newblock {\em Multivariate behavioral research}, 42(1):185--227. \bibitem[\protect\astroncite{{R Core Team}}{2020}]{R} {R Core Team} (2020). \newblock {\em R: A Language and Environment for Statistical Computing}. \newblock R Foundation for Statistical Computing, Vienna, Austria. \bibitem[\protect\astroncite{Revelle}{2020}]{psych} Revelle, W. (2020). \newblock {\em \href{https://cran.r-project.org/web/packages/psych/index.html}{psych}: Procedures for Personality and Psychological Research}. \newblock Northwestern University, Evanston, https://CRAN.r-project.org/package=psych. \newblock R package version 2.0.9. \bibitem[\protect\astroncite{Revelle}{prep}]{revelle:intro} Revelle, W. ({in prep}). \newblock {\em An introduction to psychometric theory with applications in {R}}. \newblock Springer. \bibitem[\protect\astroncite{Rosseel}{2012}]{lavaan} Rosseel, Y. (2012). \newblock {lavaan}: An {R} package for structural equation modeling. \newblock {\em Journal of Statistical Software}, 48(2):1--36. \bibitem[\protect\astroncite{Tal-Or et~al.}{2010}]{talor:10} Tal-Or, N., Cohen, J., Tsfati, Y., and Gunther, A.~C. (2010). \newblock Testing causal direction in the influence of presumed media influence. \newblock {\em Communication Research}, 37(6):801--824. \bibitem[\protect\astroncite{Tingley et~al.}{2014}]{mediation} Tingley, D., Yamamoto, T., Hirose, K., Keele, L., and Imai, K. (2014). \newblock {mediation}: {R} package for causal mediation analysis. \newblock {\em Journal of Statistical Software}, 59(5):1--38. \end{thebibliography} \begin{thebibliography}{} \bibitem[\protect\astroncite{Cohen}{1982}]{cohen:82} Cohen, J. (1982). \newblock Set correlation as a general mulitivariate data-analytic method. \newblock {\em Multivariate Behavioral Research}, 17(3):301--341. \bibitem[\protect\astroncite{Cohen et~al.}{2003}]{cohen:03} Cohen, J., Cohen, P., West, S.~G., and Aiken, L.~S. (2003). \newblock {\em Applied multiple regression/correlation analysis for the behavioral sciences}. \newblock L. Erlbaum Associates, Mahwah, N.J., 3rd ed edition. \bibitem[\protect\astroncite{Dawes}{1979}]{dawes:79} Dawes, R.~M. (1979). \newblock The robust beauty of improper linear models in decision making. \newblock {\em American Psychologist}, 34(7):571--582. \bibitem[\protect\astroncite{Garcia et~al.}{2010}]{garcia:10} Garcia, D.~M., Schmitt, M.~T., Branscombe, N.~R., and Ellemers, N. (2010). \newblock Women's reactions to ingroup members who protest discriminatory treatment: The importance of beliefs about inequality and response appropriateness. \newblock {\em European Journal of Social Psychology}, 40(5):733--745. \bibitem[\protect\astroncite{Hayes}{2013}]{hayes:13} Hayes, A.~F. (2013). \newblock {\em Introduction to mediation, moderation, and conditional process analysis: A regression-based approach}. \newblock Guilford Press, New York. \bibitem[\protect\astroncite{Hotelling}{1936}]{hotelling:36} Hotelling, H. (1936). \newblock Relations between two sets of variates. \newblock {\em Biometrika}, 28(3/4):321--377. \bibitem[\protect\astroncite{Kelley}{2017}]{MBESS} Kelley, K. (2017). \newblock {\em {MBESS: The MBESS R} Package}. \newblock R package version 4.4.1. \bibitem[\protect\astroncite{Pollack et~al.}{2012}]{pollack:12} Pollack, J.~M., Vanepps, E.~M., and Hayes, A.~F. (2012). \newblock The moderating role of social ties on entrepreneurs' depressed affect and withdrawal intentions in response to economic stress. \newblock {\em Journal of Organizational Behavior}, 33(6):789--810. \bibitem[\protect\astroncite{Preacher}{2015}]{preacher:15} Preacher, K.~J. (2015). \newblock Advances in mediation analysis: A survey and synthesis of new developments. \newblock {\em Annual Review of Psychology}, 66:825--852. \bibitem[\protect\astroncite{Preacher and Hayes}{2004}]{preacher:04} Preacher, K.~J. and Hayes, A.~F. (2004). \newblock {SPSS and SAS} procedures for estimating indirect effects in simple mediation models. \newblock {\em Behavior Research Methods, Instruments, \& Computers}, 36(4):717--731. \bibitem[\protect\astroncite{Preacher et~al.}{2007}]{preacher:07} Preacher, K.~J., Rucker, D.~D., and Hayes, A.~F. (2007). \newblock Addressing moderated mediation hypotheses: Theory, methods, and prescriptions. \newblock {\em Multivariate behavioral research}, 42(1):185--227. \bibitem[\protect\astroncite{{R Core Team}}{2020}]{R} {R Core Team} (2020). \newblock {\em R: A Language and Environment for Statistical Computing}. \newblock R Foundation for Statistical Computing, Vienna, Austria. \bibitem[\protect\astroncite{{Moon}}{2020}]{processR} {Keon-Woong Moon} (2020). \newblock {\em processR: Implementation of the 'PROCESS' Macro}. \newblock https://CRAN.R-project.org/package=processR \bibitem[\protect\astroncite{Revelle}{2020}]{psych} Revelle, W. (2020). \newblock {\em \href{https://cran.r-project.org/web/packages/psych/index.html}{psych}: Procedures for Personality and Psychological Research}. \newblock Northwestern University, Evanston, https://CRAN.r-project.org/package=psych. \newblock R package version 2.0.12. \bibitem[\protect\astroncite{Revelle}{prep}]{revelle:intro} Revelle, W. ({in prep}). \newblock {\em An introduction to psychometric theory with applications in {R}}. \newblock Springer. \bibitem[\protect\astroncite{Rosseel}{2012}]{lavaan} Rosseel, Y. (2012). \newblock {lavaan}: An {R} package for structural equation modeling. \newblock {\em Journal of Statistical Software}, 48(2):1--36. \bibitem[\protect\astroncite{Tal-Or et~al.}{2010}]{talor:10} Tal-Or, N., Cohen, J., Tsfati, Y., and Gunther, A.~C. (2010). \newblock Testing causal direction in the influence of presumed media influence. \newblock {\em Communication Research}, 37(6):801--824. \bibitem[\protect\astroncite{Tingley et~al.}{2014}]{mediation} Tingley, D., Yamamoto, T., Hirose, K., Keele, L., and Imai, K. (2014). \newblock {mediation}: {R} package for causal mediation analysis. \newblock {\em Journal of Statistical Software}, 59(5):1--38. \end{thebibliography} %\printindex \end{document} psychTools/vignettes/factor.Rnw0000644000176200001440000045246513720521015016413 0ustar liggesusers% \VignetteIndexEntry{Using the psych package for factor analysis} % \VignettePackage{psych} % \VignetteKeywords{multivariate} % \VignetteKeyword{models} % \VignetteKeyword{Hplot} %\VignetteDepends{psych} %\documentclass[doc]{apa} \documentclass[11pt]{article} %\documentclass[11pt]{amsart} \usepackage{geometry} % See geometry.pdf to learn the layout options. There are lots. \geometry{letterpaper} % ... or a4paper or a5paper or ... %\geometry{landscape} % Activate for for rotated page geometry \usepackage[parfill]{parskip} % Activate to begin paragraphs with an empty line rather than an indent \usepackage{graphicx} \usepackage{amssymb} \usepackage{epstopdf} \usepackage{mathptmx} \usepackage{helvet} \usepackage{courier} \usepackage{epstopdf} \usepackage{makeidx} % allows index generation \usepackage[authoryear,round]{natbib} \usepackage{gensymb} \usepackage{longtable} %\usepackage{geometry} \usepackage{amssymb} \usepackage{amsmath} %\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \usepackage{Sweave} %\usepackage{/Volumes/'Macintosh HD'/Library/Frameworks/R.framework/Versions/2.13/Resources/share/texmf/tex/latex/Sweave} %\usepackage[ae]{Rd} %\usepackage[usenames]{color} %\usepackage{setspace} \bibstyle{apacite} \bibliographystyle{apa} %this one plus author year seems to work? %\usepackage{hyperref} \usepackage[colorlinks=true,citecolor=blue]{hyperref} %this makes reference links hyperlinks in pdf! \DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \usepackage{multicol} % used for the two-column index \usepackage[bottom]{footmisc}% places footnotes at page bottom \let\proglang=\textsf \newcommand{\R}{\proglang{R}} %\newcommand{\pkg}[1]{{\normalfont\fontseries{b}\selectfont #1}} \newcommand{\Rfunction}[1]{{\texttt{#1}}} \newcommand{\fun}[1]{{\texttt{#1}\index{#1}\index{R function!#1}}} \newcommand{\pfun}[1]{{\texttt{#1}\index{#1}\index{R function!#1}\index{R function!psych package!#1}}}\newcommand{\Rc}[1]{{\texttt{#1}}} %R command same as Robject \newcommand{\Robject}[1]{{\texttt{#1}}} \newcommand{\Rpkg}[1]{{\textit{#1}\index{#1}\index{R package!#1}}} %different from pkg - which is better? \newcommand{\iemph}[1]{{\emph{#1}\index{#1}}} \newcommand{\wrc}[1]{\marginpar{\textcolor{blue}{#1}}} %bill's comments \newcommand{\wra}[1]{\textcolor{blue}{#1}} %bill's comments \newcommand{\ve}[1]{{\textbf{#1}}} %trying to get a vector command \usepackage{fancyvrb} %this allows fancy boxes \fvset{fontfamily=courier} \DefineVerbatimEnvironment{Routput}{Verbatim} %{fontsize=\scriptsize, xleftmargin=0.6cm} {fontseries=b,fontsize=\scriptsize, xleftmargin=0.1cm} \DefineVerbatimEnvironment{Binput}{Verbatim} {fontseries=b, fontsize=\scriptsize,frame=single, label=\fbox{lavaan model syntax}, framesep=2mm} %\DefineShortVerb{\!} %%% generates error! \DefineVerbatimEnvironment{Rinput}{Verbatim} %{fontsize=\scriptsize, frame=single, label=\fbox{R code}, framesep=1mm} {fontseries=b, fontsize=\scriptsize, frame=single, label=\fbox{R code},xleftmargin=0pt, framesep=1mm} \DefineVerbatimEnvironment{Link}{Verbatim} {fontseries=b, fontsize=\small, formatcom=\color{darkgreen}, xleftmargin=1.0cm} \DefineVerbatimEnvironment{Toutput}{Verbatim} {fontseries=b,fontsize=\tiny, xleftmargin=0.1cm} \DefineVerbatimEnvironment{rinput}{Verbatim} {fontseries=b, fontsize=\tiny, frame=single, label=\fbox{R code}, framesep=1mm} \newcommand{\citeti}[1]{\begin{tiny}\citep{#1}\end{tiny}} \newcommand{\light}[1]{\textcolor{gray}{#1}} \newcommand{\vect}[1]{\boldsymbol{#1}} \let\vec\vect \makeindex % used for the subject index \title{How To: Use the psych package for Factor Analysis and data reduction} \author{William Revelle\\Department of Psychology\\Northwestern University} %\affiliation{Northwestern University} %\acknowledgements{Written to accompany the psych package. Comments should be directed to William Revelle \\ \url{revelle@northwestern.edu}} %\date{} % Activate to display a given date or no date \begin{document} \SweaveOpts{concordance=TRUE} \maketitle \tableofcontents \newpage \section{Overview of this and related documents} To do basic and advanced personality and psychological research using \R{} is not as complicated as some think. This is one of a set of ``How To'' to do various things using \R{} \citep{R}, particularly using the \Rpkg{psych} \citep{psych} package. The current list of How To's includes: \begin{enumerate} \item \href{http://personality-project.org/r/psych/HowTo/getting_started.pdf}{Installing} \R{} and some useful packages \item Using \R{} and the \Rpkg{psych} package to find \href{http://personality-project.org/r/psych/HowTo/omega.pdf}{$omega_h$} and $\omega_t$. \item Using \R{} and the \Rpkg{psych} for \href{http://personality-project.org/r/psych/HowTo/factor.pdf}{factor analysis} and principal components analysis. (This document). \item Using the \pfun{score.items} function to find \href{http://personality-project.org/r/psych/HowTo/scoring.pdf}{scale scores and scale statistics}. \item An \href{http://personality-project.org/r/psych/overview.pdf}{overview} (vignette) of the \Rpkg{psych} package Several functions are meant to do multiple regressions, either from the raw data or from a variance/covariance matrix, or a correlation matrix. This is discussed in more detail in \item How to do mediation and moderation analysis using \pfun{mediate} and \pfun{setCor} is discuseded in the \href{https://personality-project.org/r/psych/HowTo/mediation.pdf}{mediation, moderation and regression analysis} tutorial. \end{enumerate} \subsection{Jump starting the \Rpkg{psych} package--a guide for the impatient} You have installed \Rpkg{psych} (section \ref{sect:starting}) and you want to use it without reading much more. What should you do? \begin{enumerate} \item Activate the \Rpkg{psych} package: \begin{Rinput} library(psych) library(psychTools) \end{Rinput} \item Input your data (section \ref{sect:read}). Go to your friendly text editor or data manipulation program (e.g., Excel) and copy the data to the clipboard. Include a first line that has the variable labels. Paste it into \Rpkg{psych} using the \pfun{read.clipboard.tab} command: \begin{Rinput} myData <- read.clipboard.tab() \end{Rnput} \item Make sure that what you just read is right. Describe it (section~\ref{sect:describe}) and perhaps look at the first and last few lines: \begin{Rinput} describe(myData) headTail(myData) \end{Rinput} \item Look at the patterns in the data. If you have fewer than about 10 variables, look at the SPLOM (Scatter Plot Matrix) of the data using \pfun{pairs.panels} (section~\ref{sect:pairs}). \begin{Rinput} pairs.panels(myData) \end{Rinput} %\item Note that you have some weird subjects, probably due to data entry errors. Either edit the data by hand (use the \fun{edit} command) or just \pfun{scrub} the data (section \ref{sect:scrub}). %\begin{scriptsize} %\begin{Schunk} %\begin{Sinput} %cleaned <- scrub(myData, max=9) #e.g., change anything great than 9 to NA %\end{Sinput} %\end{Schunk} %\end{scriptsize} %\item Graph the data with error bars for each variable (section \ref{sect:errorbars}). %\begin{scriptsize} %\begin{Schunk} %\begin{Sinput} %error.bars(myData) %\end{Sinput} %\end{Schunk} %\end{scriptsize} \item Find the correlations of all of your data. \begin{itemize} \item Descriptively (just the values) (section \ref{sect:lowerCor}) \begin{Rinput} lowerCor(myData) \end{Rinput} \item Graphically (section \ref{sect:corplot}) \begin{Rinput} corPlot(r) \end{Rinput} \end{itemize} % %\item Inferentially (the values, the ns, and the p values) (section \ref{sect:corr.test}) %\begin{scriptsize} %\begin{Schunk} %\begin{Sinput} %corr.test(myData) % %\end{Sinput} %\end{Schunk} %\end{scriptsize} %\end{itemize} \item Test for the number of factors in your data using parallel analysis (\pfun{fa.parallel}, section \ref{sect:fa.parallel}) or Very Simple Structure (\pfun{vss}, \ref{sect:vss}) . \begin{Rinput} fa.parallel(myData) vss(myData) \end{Rinput} \item Factor analyze (see section \ref{sect:fa}) the data with a specified number of factors (the default is 1), the default method is minimum residual, the default rotation for more than one factor is oblimin. There are many more possibilities (see sections \ref{sect:minres}-\ref{sect:wls}). Compare the solution to a hierarchical cluster analysis using the ICLUST algorithm \citep{revelle:iclust} (see section \ref{sect:iclust}). Also consider a hierarchical factor solution to find coefficient $\omega$ (see \ref{sect:omega}). Yet another option is to do a series of factor analyses in what is known as the ``bass akward" procedure \citep{goldberg:06} which considers the correlation between factors at multiple levels of analysis (see \ref{sect:bassAckward}). \begin{Rinput} fa(myData) iclust(myData) omega(myData) bassAckward(myData) \end{Rinput} \item Some people like to find coefficient $\alpha$ as an estimate of reliability. This may be done for a single scale using the \pfun{alpha} function (see \ref{sect:alpha}). Perhaps more useful is the ability to create several scales as unweighted averages of specified items using the \pfun{scoreIems} function (see \ref{sect:score}) and to find various estimates of internal consistency for these scales, find their intercorrelations, and find scores for all the subjects. \begin{Rinput} alpha(myData) #score all of the items as part of one scale. myKeys <- make.keys(nvar=20,list(first = c(1,-3,5,-7,8:10),second=c(2,4,-6,11:15,-16))) my.scores <- scoreItems(myKeys,myData) #form several scales my.scores #show the highlights of the results \end{Rinput} \end{enumerate} At this point you have had a chance to see the highlights of the \Rpkg{psych} package and to do some basic (and advanced) data analysis. You might find reading the entire \href{http://personality-project.org/r/psych/overview.pdf}{overview} vignette helpful to get a broader understanding of what can be done in \R{} using the \Rpkg{psych}. Remember that the help command (?) is available for every function. Try running the examples for each help page. \newpage \section{Overview of this and related documents} The \Rpkg{psych} package \citep{psych} has been developed at Northwestern University since 2005 to include functions most useful for personality, psychometric, and psychological research. The package is also meant to supplement a text on psychometric theory \citep{revelle:intro}, a draft of which is available at \url{http://personality-project.org/r/book/}. Some of the functions (e.g., \pfun{read.clipboard}, \pfun{describe}, \pfun{pairs.panels}, \pfun{scatter.hist}, \pfun{error.bars}, \pfun{multi.hist}, \pfun{bi.bars}) are useful for basic data entry and descriptive analyses. Psychometric applications emphasize techniques for dimension reduction including factor analysis, cluster analysis, and principal components analysis. The \pfun{fa} function includes five methods of \iemph{factor analysis} (\iemph{minimum residual}, \iemph{principal axis}, \iemph{weighted least squares}, \iemph{generalized least squares} and \iemph{maximum likelihood} factor analysis). Determining the number of factors or components to extract may be done by using the Very Simple Structure \citep{revelle:vss} (\pfun{vss}), Minimum Average Partial correlation \citep{velicer:76} (\pfun{MAP}) or parallel analysis (\pfun{fa.parallel}) criteria. Item Response Theory (IRT) models for dichotomous or polytomous items may be found by factoring \pfun{tetrachoric} or \pfun{polychoric} correlation matrices and expressing the resulting parameters in terms of location and discrimination using \pfun{irt.fa}. Bifactor and hierarchical factor structures may be estimated by using Schmid Leiman transformations \citep{schmid:57} (\pfun{schmid}) to transform a hierarchical factor structure into a \iemph{bifactor} solution \citep{holzinger:37}. Scale construction can be done using the Item Cluster Analysis \citep{revelle:iclust} (\pfun{iclust}) function to determine the structure and to calculate reliability coefficients $\alpha$ \citep{cronbach:51}(\pfun{alpha}, \pfun{scoreItems}, \pfun{score.multiple.choice}), $\beta$ \citep{revelle:iclust,rz:09} (\pfun{iclust}) and McDonald's $\omega_h$ and $\omega_t$ \citep{mcdonald:tt} (\pfun{omega}). Guttman's six estimates of internal consistency reliability (\cite{guttman:45}, as well as additional estimates \citep{rz:09} are in the \pfun{guttman} function. The six measures of Intraclass correlation coefficients (\pfun{ICC}) discussed by \cite{shrout:79} are also available. Graphical displays include Scatter Plot Matrix (SPLOM) plots using \pfun{pairs.panels}, correlation ``heat maps'' (\pfun{cor.plot}) factor, cluster, and structural diagrams using \pfun{fa.diagram}, \pfun{iclust.diagram}, \pfun{structure.diagram}, as well as item response characteristics and item and test information characteristic curves \pfun{plot.irt} and \pfun{plot.poly}. %This vignette is meant to give an overview of the \Rpkg{psych} package. That is, it is meant to give a summary of the main functions in the \Rpkg{psych} package with examples of how they are used for data description, dimension reduction, and scale construction. The extended user manual at \url{psych_manual.pdf} includes examples of graphic output and more extensive demonstrations than are found in the help menus. (Also available at \url{http://personality-project.org/r/psych_manual.pdf}). The vignette, psych for sem, at \url{psych_for_sem.pdf}, discusses how to use psych as a front end to the \Rpkg{sem} package of John Fox \citep{sem}. (The vignette is also available at \href{"http://personality-project.org/r/book/psych_for_sem.pdf"}{\url{http://personality-project.org/r/book/psych_for_sem.pdf}}). % %For a step by step tutorial in the use of the psych package and the base functions in R for basic personality research, see the guide for using \R{} for personality research at \url{http://personalitytheory.org/r/r.short.html}. For an \iemph{introduction to psychometric theory with applications in \R{}}, see the draft chapters at \url{http://personality-project.org/r/book}). % % % \section{Getting started} \label{sect:starting} Some of the functions described in this overview require other packages. Particularly useful for rotating the results of factor analyses (from e.g., \pfun{fa} or \pfun {principal}) or hierarchical factor models using \pfun{omega} or \pfun{schmid}, is the \Rpkg{GPArotation} package. These and other useful packages may be installed by first installing and then using the task views (\Rpkg{ctv}) package to install the ``Psychometrics" task view, but doing it this way is not necessary. % %\begin{Schunk} %\begin{Sinput} %install.packages("ctv") %library(ctv) %task.views("Psychometrics") %\end{Sinput} %\end{Schunk} % %The ``Psychometrics'' task view will install a large number of useful packages. To install the bare minimum for the examples in this vignette, it is necessary to install just 3 packages: % %\begin{Schunk} %\begin{Sinput} %install.packages(list(c("GPArotation","mvtnorm","MASS") %\end{Sinput} %\end{Schunk} % % %Because of the difficulty of installing the package \Rpkg{Rgraphviz}, alternative graphics have been developed and are available as \iemph{diagram} functions. If \Rpkg{Rgraphviz} is available, some functions will take advantage of it. An alternative is to use ``dot'' output of commands for any external graphics package that uses the dot language. % \section{Basic data analysis} A number of \Rpkg{psych} functions facilitate the entry of data and finding basic descriptive statistics. Remember, to run any of the \Rpkg{psych} functions, it is necessary to make the package active by using the \fun{library} command: \begin{Rinput} library(psych) library(psychTools) \end{Rinput} The other packages, once installed, will be called automatically by \Rpkg{psych}. It is possible to automatically load \Rpkg{psych} and other functions by creating and then saving a ``.First" function: e.g., \begin{Rinput} .First <- function(x) {library(psych)} \end{Rinput} \subsection{Data input from a local or remote file} \label{sect:read} Find and read standard files using \pfun{read.file}. This will open a search window for your operating system which you can use to find the file. If the file has a suffix of .text, .txt, .TXT, .csv, ,dat, .data, .sav, .xpt, .XPT, .r, .R, .rds, .Rds, .rda, .Rda, .rdata, Rdata, or .RData, then the file will be opened and the data will be read in (or loaded in the case of Rda files) \begin{scriptsize} \begin{Schunk} \begin{Sinput} myData <- read.file() # find the appropriate file using your normal operating system \end{Sinput} \end{Schunk} \end{scriptsize} Alternatively, if you have a file name for a remote file, you can read it using \pfun{read.file} as well. \begin{scriptsize} \begin{Schunk} \begin{Sinput} myData <- read.file(fn) # where file name is the the remote address of the file \end{Sinput} \end{Schunk} \end{scriptsize} \subsection{Data input from the clipboard} There are of course many ways to enter data into \R. Reading from a local file using \fun{read.file} is perhaps the most preferred. However, many users will enter their data in a text editor or spreadsheet program and then want to copy and paste into \R{}. This may be done by using \fun{read.table} and specifying the input file as ``clipboard" (PCs) or ``pipe(pbpaste)" (Macs). Alternatively, the \pfun{read.clipboard} set of functions are perhaps more user friendly: \begin{description} \item [\pfun{read.clipboard}] is the base function for reading data from the clipboard. \item [\pfun{read.clipboard.csv}] for reading text that is comma delimited. \item [\pfun{read.clipboard.tab}] for reading text that is tab delimited (e.g., copied directly from an Excel file). \item [\pfun{read.clipboard.lower}] for reading input of a lower triangular matrix with or without a diagonal. The resulting object is a square matrix. \item [\pfun{read.clipboard.upper}] for reading input of an upper triangular matrix. \item[\pfun{read.clipboard.fwf}] for reading in fixed width fields (some very old data sets) \end{description} For example, given a data set copied to the clipboard from a spreadsheet, just enter the command \begin{Rinput} my.data <- read.clipboard() \end{Rinput} This will work if every data field has a value and even missing data are given some values (e.g., NA or -999). If the data were entered in a spreadsheet and the missing values were just empty cells, then the data should be read in as a tab delimited or by using the \pfun{read.clipboard.tab} function. \begin{Rinput} my.data <- read.clipboard(sep="\t") #define the tab option, or my.tab.data <- read.clipboard.tab() #just use the alternative function \end{Rinput} For the case of data in fixed width fields (some old data sets tend to have this format), copy to the clipboard and then specify the width of each field (in the example below, the first variable is 5 columns, the second is 2 columns, the next 5 are 1 column the last 4 are 3 columns). \begin{Rinput} my.data <- read.clipboard.fwf(widths=c(5,2,rep(1,5),rep(3,4)) \end{Rinput} \subsection{Basic descriptive statistics} \label{sect:describe} Once the data are read in, then \pfun{describe} will provide basic descriptive statistics arranged in a data frame format. Consider the data set \pfun{sat.act} which includes data from 700 web based participants on 3 demographic variables and 3 ability measures. \begin{description} \item[\pfun{describe}] reports means, standard deviations, medians, min, max, range, skew, kurtosis and standard errors for integer or real data. Non-numeric data, although the statistics are meaningless, will be treated as if numeric (based upon the categorical coding of the data), and will be flagged with an *. \end{description} It is very important to describe your data before you continue on doing more complicated multivariate statistics. The problem of outliers and bad data can not be overemphasized. \begin{scriptsize} <>= options(width=160) library(psych) library(psychTools) data(sat.act) describe(sat.act) #basic descriptive statistics @ \end{scriptsize} %These data may then be analyzed by groups defined in a logical statement or by some other variable. E.g., break down the descriptive data for males or females. These descriptive data can also be seen graphically using the \pfun{error.bars.by} function (Figure~\ref{fig:error.bars}). By setting skew=FALSE and ranges=FALSE, the output is limited to the most basic statistics. % %\begin{scriptsize} %<>= % #basic descriptive statistics by a grouping variable. %describeBy(sat.act,sat.act$gender,skew=FALSE,ranges=FALSE) %@ %\end{scriptsize} % % %The output from the \pfun{describeBy} function can be forced into a matrix form for easy analysis by other programs. In addition, describeBy can group by several grouping variables at the same time. % %\begin{scriptsize} %<>= %sa.mat <- describeBy(sat.act,list(sat.act$gender,sat.act$education), % skew=FALSE,ranges=FALSE,mat=TRUE) %headTail(sa.mat) %@ %\end{scriptsize} %\subsubsection{Basic data cleaning using \pfun{scrub}} %\label{sect:scrub} %If, after describing the data it is apparent that there were data entry errors that need to be globally replaced with NA, or only certain ranges of data will be analyzed, the data can be ``cleaned" using the \pfun{scrub} function. % %Consider a data set of 10 rows of 12 columns with values from 1 - 120. All values of columns 3 - 5 that are less than 30, 40, or 50 respectively, or greater than 70 in any of the three columns will be replaced with NA. In addition, any value exactly equal to 45 will be set to NA. (max and isvalue are set to one value here, but they could be a different value for every column). % %\begin{scriptsize} %<>= %x <- matrix(1:120,ncol=10,byrow=TRUE) %colnames(x) <- paste('V',1:10,sep='') %new.x <- scrub(x,3:5,min=c(30,40,50),max=70,isvalue=45,newvalue=NA) %new.x %@ %\end{scriptsize} %Note that the number of subjects for those columns has decreased, and the minimums have gone up but the maximums down. Data cleaning and examination for outliers should be a routine part of any data analysis. % %\subsubsection{Recoding categorical variables into dummy coded variables} %Sometimes categorical variables (e.g., college major, occupation, ethnicity) are to be analyzed using correlation or regression. To do this, one can form ``dummy codes'' which are merely binary variables for each category. This may be done using \pfun{dummy.code}. Subsequent analyses using these dummy coded variables may be using \pfun{biserial} or point biserial (regular Pearson r) to show effect sizes and may be plotted in e.g., \pfun{spider} plots. \subsection{Simple descriptive graphics} Graphic descriptions of data are very helpful both for understanding the data as well as communicating important results. Scatter Plot Matrices (SPLOMS) using the \pfun{pairs.panels} function are useful ways to look for strange effects involving outliers and non-linearities. \pfun{error.bars.by} will show group means with 95\% confidence boundaries. \subsubsection{Scatter Plot Matrices} Scatter Plot Matrices (SPLOMS) are very useful for describing the data. The \pfun{pairs.panels} function, adapted from the help menu for the \fun{pairs} function produces xy scatter plots of each pair of variables below the diagonal, shows the histogram of each variable on the diagonal, and shows the \iemph{lowess} locally fit regression line as well. An ellipse around the mean with the axis length reflecting one standard deviation of the x and y variables is also drawn. The x axis in each scatter plot represents the column variable, the y axis the row variable (Figure~\ref{fig:pairs.panels}). When plotting many subjects, it is both faster and cleaner to set the plot character (pch) to be '.'. (See Figure~\ref{fig:pairs.panels} for an example.) \begin{description} \label{sect:pairs} \item[\pfun{pairs.panels} ] will show the pairwise scatter plots of all the variables as well as histograms, locally smoothed regressions, and the Pearson correlation. When plotting many data points (as in the case of the sat.act data, it is possible to specify that the plot character is a period to get a somewhat cleaner graphic. \end{description} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= png( 'pairspanels.png' ) pairs.panels(sat.act,pch='.') dev.off() @ \end{scriptsize} \includegraphics{pairspanels} \caption{Using the \pfun{pairs.panels} function to graphically show relationships. The x axis in each scatter plot represents the column variable, the y axis the row variable. Note the extreme outlier for the ACT. The plot character was set to a period (pch='.') in order to make a cleaner graph. } \label{fig:pairs.panels} \end{center} \end{figure} %Another example of \pfun{pairs.panels} is to show differences between experimental groups. Consider the data in the \pfun{affect} data set. The scores reflect post test scores on positive and negative affect and energetic and tense arousal. The colors show the results for four movie conditions: depressing, frightening movie, neutral, and a comedy. % %\begin{figure}[htbp] %\begin{center} %\begin{scriptsize} %<>= %png('affect.png') %pairs.panels(affect[14:17],bg=c("red","black","white","blue")[affect$Film],pch=21, % main="Affect varies by movies ") %dev.off() %@ %\end{scriptsize} %\includegraphics{affect} %\caption{Using the \pfun{pairs.panels} function to graphically show relationships. The x axis in each scatter plot represents the column variable, the y axis the row variable. The coloring represent four different movie conditions. } %\label{fig:pairs.panels2} %\end{center} %\end{figure} % %\subsubsection{Means and error bars} %\label{sect:errorbars} %Additional descriptive graphics include the ability to draw \iemph{error bars} on sets of data, as well as to draw error bars in both the x and y directions for paired data. These are the functions % %\begin{description} %\item [\pfun{error.bars}] show the 95 \% confidence intervals for each variable in a data frame or matrix. These errors are based upon normal theory and the standard errors of the mean. Alternative options include +/- one standard deviation or 1 standard error. If the data are repeated measures, the error bars will be reflect the between variable correlations. %\item [\pfun{error.bars.by}] does the same, but grouping the data by some condition. %\item [\pfun{error.crosses}] draw the confidence intervals for an x set and a y set of the same size. %\end{description} % %The use of the \pfun{error.bars.by} function allows for graphic comparisons of different groups (see Figure~\ref{fig:error.bars}). Five personality measures are shown as a function of high versus low scores on a ``lie" scale. People with higher lie scores tend to report being more agreeable, conscientious and less neurotic than people with lower lie scores. The error bars are based upon normal theory and thus are symmetric rather than reflect any skewing in the data. % %\begin{figure}[htbp] %\begin{center} %\begin{scriptsize} %<>= %data(epi.bfi) %error.bars.by(epi.bfi[,6:10],epi.bfi$epilie<4) %@ %\end{scriptsize} %\caption{Using the \pfun{error.bars.by} function shows that self reported personality scales on the Big Five Inventory vary as a function of the Lie scale on the EPI. } %\label{fig:error.bars} %\end{center} %\end{figure} % %Although not recommended, it is possible to use the \pfun{error.bars} function to draw bar graphs with associated error bars. (This kind of`\iemph{dynamite plot} (Figure~\ref{fig:dynamite}) can be very misleading in that the scale is arbitrary. Go to a discussion of the problems in presenting data this way at \url{http://emdbolker.wikidot.com/blog:dynamite}. In the example shown, note that the graph starts at 0, although is out of the range. This is a function of using bars, which always are assumed to start at zero. Consider other ways of showing your data. % %\begin{figure}[htbp] %\begin{center} %\begin{scriptsize} %<>= %error.bars.by(sat.act[5:6],sat.act$gender,bars=TRUE, % labels=c("Male","Female"),ylab="SAT score",xlab="") %@ %\end{scriptsize} %\caption{A ``Dynamite plot" of SAT scores as a function of gender is one way of misleading the reader. By using a bar graph, the range of scores is ignored. Bar graphs start from 0. } %\label{fig:dynamite} %\end{center} %\end{figure} % % %\subsubsection{Two dimensional displays of means and errors} %Yet another way to display data for different conditions is to use the \pfun{errorCrosses} function. For instance, the effect of various movies on both ``Energetic Arousal'' and ``Tense Arousal'' can be seen in one graph and compared to the same movie manipulations on ``Positive Affect'' and ``Negative Affect''. Note how Energetic Arousal is increased by three of the movie manipulations, but that Positive Affect increases following the Happy movie only. % % %\begin{figure}[htbp] %\begin{center} %\begin{scriptsize} %<>= %op <- par(mfrow=c(1,2)) % data(affect) %colors <- c("black","red","white","blue") % films <- c("Sad","Horror","Neutral","Happy") %affect.stats <- errorCircles("EA2","TA2",data=affect,group="Film",labels=films,xlab="Energetic Arousal",ylab="Tense Arousal",ylim=c(10,22),xlim=c(8,20),pch=16,cex=2,col=colors, % main =' Movies effect on arousal') % errorCircles("PA2","NA2",data=affect.stats,labels=films,xlab="Positive Affect",ylab="Negative Affect",pch=16,cex=2,col=colors, % main ="Movies effect on affect") %op <- par(mfrow=c(1,1)) %@ %\end{scriptsize} %\caption{The use of the \pfun{errorCircles} function allows for two dimensional displays of means and error bars. The first call to \pfun{errorCircles} finds descriptive statistics for the \iemph{affect} data.frame based upon the grouping variable of Film. These data are returned and then used by the second call which examines the effect of the same grouping variable upon different measures. The size of the circles represent the relative sample sizes for each group. The data are from the PMC lab and reported in \cite{smillie:jpsp}.} %\label{fig:errorCircles} %\end{center} %\end{figure} % %\clearpage %\subsubsection{Back to back histograms} %The \pfun{bi.bars} function summarize the characteristics of two groups (e.g., males and females) on a second variable (e.g., age) by drawing back to back histograms (see Figure~\ref{fig:bibars}). %\begin{figure}[htbp] %\begin{center} %\begin{scriptsize} %<>= %data(bfi) %with(bfi,{bi.bars(age,gender,ylab="Age",main="Age by males and females")}) %@ %\end{scriptsize} %\caption{A bar plot of the age distribution for males and females shows the use of \pfun{bi.bars}. The data are males and females from 2800 cases collected using the \iemph{SAPA} procedure and are available as part of the \pfun{bfi} data set. } %\label{fig:bibars} %\end{center} %\end{figure} % %\clearpage \subsubsection{Correlational structure} \label{sect:lowerCor} There are many ways to display correlations. Tabular displays are probably the most common. The output from the \fun{cor} function in core R is a rectangular matrix. \pfun{lowerMat} will round this to (2) digits and then display as a lower off diagonal matrix. \pfun{lowerCor} calls \fun{cor} with \emph{use=`pairwise', method=`pearson'} as default values and returns (invisibly) the full correlation matrix and displays the lower off diagonal matrix. \begin{scriptsize} <>= lowerCor(sat.act) @ \end{scriptsize} When comparing results from two different groups, it is convenient to display them as one matrix, with the results from one group below the diagonal, and the other group above the diagonal. Use \pfun{lowerUpper} to do this: \begin{scriptsize} <>= female <- subset(sat.act,sat.act$gender==2) male <- subset(sat.act,sat.act$gender==1) lower <- lowerCor(male[-1]) upper <- lowerCor(female[-1]) both <- lowerUpper(lower,upper) round(both,2) @ \end{scriptsize} It is also possible to compare two matrices by taking their differences and displaying one (below the diagonal) and the difference of the second from the first above the diagonal: \begin{scriptsize} <>= diffs <- lowerUpper(lower,upper,diff=TRUE) round(diffs,2) @ \end{scriptsize} \subsubsection{Heatmap displays of correlational structure} \label{sect:corplot} Perhaps a better way to see the structure in a correlation matrix is to display a \emph{heat map} of the correlations. This is just a matrix color coded to represent the magnitude of the correlation. This is useful when considering the number of factors in a data set. Consider the \pfun{Thurstone} data set which has a clear 3 factor solution (Figure~\ref{fig:cor.plot}) or a simulated data set of 24 variables with a circumplex structure (Figure~\ref{fig:cor.plot.circ}). The color coding represents a ``heat map'' of the correlations, with darker shades of red representing stronger negative and darker shades of blue stronger positive correlations. As an option, the value of the correlation can be shown. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= png('corplot.png') cor.plot(Thurstone,numbers=TRUE,main="9 cognitive variables from Thurstone") dev.off() @ \end{scriptsize} \includegraphics{corplot.png} \caption{The structure of correlation matrix can be seen more clearly if the variables are grouped by factor and then the correlations are shown by color. By using the 'numbers' option, the values are displayed as well. } \label{fig:cor.plot} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= png('circplot.png') circ <- sim.circ(24) r.circ <- cor(circ) cor.plot(r.circ,main='24 variables in a circumplex') dev.off() @ \end{scriptsize} \includegraphics{circplot.png} \caption{Using the cor.plot function to show the correlations in a circumplex. Correlations are highest near the diagonal, diminish to zero further from the diagonal, and the increase again towards the corners of the matrix. Circumplex structures are common in the study of affect.} \label{fig:cor.plot.circ} \end{center} \end{figure} %Yet another way to show structure is to use ``spider'' plots. Particularly if variables are ordered in some meaningful way (e.g., in a circumplex), a spider plot will show this structure easily. This is just a plot of the magnitude of the correlation as a radial line, with length ranging from 0 (for a correlation of -1) to 1 (for a correlation of 1). (See Figure~\ref{fig:cor.plot.spider}). % %\begin{figure}[htbp] %\begin{center} %\begin{scriptsize} %<>= %png('spider.png') %op<- par(mfrow=c(2,2)) %spider(y=c(1,6,12,18),x=1:24,data=r.circ,fill=TRUE,main="Spider plot of 24 circumplex variables") %op <- par(mfrow=c(1,1)) %dev.off() %@ %\end{scriptsize} %\includegraphics{spider.png} %\caption{A spider plot can show circumplex structure very clearly. Circumplex structures are common in the study of affect.} %\label{fig:cor.plot.spider} %\end{center} %\end{figure} % %\subsection{Testing correlations} %\label{sect:corr.test} %Correlations are wonderful descriptive statistics of the data but some people like to test whether these correlations differ from zero, or differ from each other. The \fun{cor.test} function (in the \Rpkg{stats} package) will test the significance of a single correlation, and the \fun{rcorr} function in the \Rpkg{Hmisc} package will do this for many correlations. In the \Rpkg{psych} package, the \pfun{corr.test} function reports the correlation (Pearson, Spearman, or Kendall) between all variables in either one or two data frames or matrices, as well as the number of observations for each case, and the (two-tailed) probability for each correlation. Unfortunately, these probability values have not been corrected for multiple comparisons and so should be taken with a great deal of salt. Thus, in \pfun{corr.test} and \pfun{corr.p} the raw probabilities are reported below the diagonal and the probabilities adjusted for multiple comparisons using (by default) the Holm correction are reported above the diagonal (Table~\ref{tab:corr.test}). (See the \fun{p.adjust} function for a discussion of \cite{holm:79} and other corrections.) % %\begin{table}[htpb] %\caption{The \pfun{corr.test} function reports correlations, cell sizes, and raw and adjusted probability values. \pfun{corr.p} reports the probability values for a correlation matrix. By default, the adjustment used is that of \cite{holm:79}.} %\begin{scriptsize} %<>= %corr.test(sat.act) %@ %\end{scriptsize} %\label{tab:corr.test} %\end{table}% % % %Testing the difference between any two correlations can be done using the \pfun{r.test} function. The function actually does four different tests (based upon an article by \cite{steiger:80b}, depending upon the input: % %1) For a sample size n, find the t and p value for a single correlation as well as the confidence interval. %\begin{scriptsize} %<>= %r.test(50,.3) %@ %\end{scriptsize} % %2) For sample sizes of n and n2 (n2 = n if not specified) find the z of the difference between the z transformed correlations divided by the standard error of the difference of two z scores. %\begin{scriptsize} %<>= %r.test(30,.4,.6) %@ %\end{scriptsize} % % %3) For sample size n, and correlations ra= r12, rb= r23 and r13 specified, test for the difference of two dependent correlations (Steiger case A). %\begin{scriptsize} %<>= %r.test(103,.4,.5,.1) %@ %\end{scriptsize} % %4) For sample size n, test for the difference between two dependent correlations involving different variables. (Steiger case B). %\begin{scriptsize} %<>= %r.test(103,.5,.6,.7,.5,.5,.8) #steiger Case B %@ %\end{scriptsize} % % %To test whether a matrix of correlations differs from what would be expected if the population correlations were all zero, the function \pfun{cortest} follows \cite{steiger:80b} who pointed out that the sum of the squared elements of a correlation matrix, or the Fisher z score equivalents, is distributed as chi square under the null hypothesis that the values are zero (i.e., elements of the identity matrix). This is particularly useful for examining whether correlations in a single matrix differ from zero or for comparing two matrices. Although obvious, \pfun{cortest} can be used to test whether the \pfun{sat.act} data matrix produces non-zero correlations (it does). This is a much more appropriate test when testing whether a residual matrix differs from zero. % %\begin{scriptsize} %<>= %cortest(sat.act) %@ %\end{scriptsize} % \subsection{Polychoric, tetrachoric, polyserial, and biserial correlations} The Pearson correlation of dichotomous data is also known as the $\phi$ coefficient. If the data, e.g., ability items, are thought to represent an underlying continuous although latent variable, the $\phi$ will underestimate the value of the Pearson applied to these latent variables. One solution to this problem is to use the \pfun{tetrachoric} correlation which is based upon the assumption of a bivariate normal distribution that has been cut at certain points. The \pfun{draw.tetra} function demonstrates the process (Figure~\ref{fig:tetra}). A simple generalization of this to the case of the multiple cuts is the \pfun{polychoric} correlation. % %\begin{figure}[htbp] %\begin{center} %\begin{scriptsize} %<>= %draw.tetra() %@ %\end{scriptsize} %\caption{The tetrachoric correlation estimates what a Pearson correlation would be given a two by two table of observed values assumed to be sampled from a bivariate normal distribution. The $\phi$ correlation is just a Pearson r performed on the observed values.} %\label{fig:tetra} %\end{center} %\end{figure} Other estimated correlations based upon the assumption of bivariate normality with cut points include the \pfun{biserial} and \pfun{polyserial} correlation. If the data are a mix of continuous, polytomous and dichotomous variables, the \pfun{mixed.cor} function will calculate the appropriate mixture of Pearson, polychoric, tetrachoric, biserial, and polyserial correlations. The correlation matrix resulting from a number of tetrachoric or polychoric correlation matrix sometimes will not be positive semi-definite. This will also happen if the correlation matrix is formed by using pair-wise deletion of cases. The \pfun{cor.smooth} function will adjust the smallest eigen values of the correlation matrix to make them positive, rescale all of them to sum to the number of variables, and produce a ``smoothed'' correlation matrix. An example of this problem is a data set of \pfun{burt} which probably had a typo in the original correlation matrix. Smoothing the matrix corrects this problem. %\subsection{Multiple regression from data or correlation matrices} % %The typical application of the \fun{lm} function is to do a linear model of one Y variable as a function of multiple X variables. Because \fun{lm} is designed to analyze complex interactions, it requires raw data as input. It is, however, sometimes convenient to do \iemph{multiple regression} from a correlation or covariance matrix. The \pfun{setCor} function will do this, taking a set of y variables predicted from a set of x variables, perhaps with a set of z covariates removed from both x and y. Consider the \iemph{Thurstone} correlation matrix and find the multiple correlation of the last five variables as a function of the first 4. % %\begin{scriptsize} %<>= %setCor(y = 5:9,x=1:4,data=Thurstone) %@ %\end{scriptsize} % %By specifying the number of subjects in correlation matrix, appropriate estimates of standard errors, t-values, and probabilities are also found. The next example finds the regressions with variables 1 and 2 used as covariates. The $\hat{\beta}$ weights for variables 3 and 4 do not change, but the multiple correlation is much less. It also shows how to find the residual correlations between variables 5-9 with variables 1-4 removed. % %\begin{scriptsize} %<>= %sc <- setCor(y = 5:9,x=3:4,data=Thurstone,z=1:2) %round(sc$residual,2) %@ %\end{scriptsize} \section{Item and scale analysis} The main functions in the \Rpkg{psych} package are for analyzing the structure of items and of scales and for finding various estimates of scale reliability. These may be considered as problems of dimension reduction (e.g., factor analysis, cluster analysis, principal components analysis) and of forming and estimating the reliability of the resulting composite scales. \subsection{Dimension reduction through factor analysis and cluster analysis} \label{sect:fa} Parsimony of description has been a goal of science since at least the famous dictum commonly attributed to William of Ockham to not multiply entities beyond necessity\footnote{Although probably neither original with Ockham nor directly stated by him \citep{thornburn:1918}, Ockham's razor remains a fundamental principal of science.}. The goal for parsimony is seen in psychometrics as an attempt either to describe (components) or to explain (factors) the relationships between many observed variables in terms of a more limited set of components or latent factors. The typical data matrix represents multiple items or scales usually thought to reflect fewer underlying constructs\footnote{\cite{cattell:fa78} as well as \cite{maccallum:07} argue that the data are the result of many more factors than observed variables, but are willing to estimate the major underlying factors.}. At the most simple, a set of items can be be thought to represent a random sample from one underlying domain or perhaps a small set of domains. The question for the psychometrician is how many domains are represented and how well does each item represent the domains. Solutions to this problem are examples of \iemph{factor analysis} (\iemph{FA}), \iemph{principal components analysis} (\iemph{PCA}), and \iemph{cluster analysis} (\emph{CA}). All of these procedures aim to reduce the complexity of the observed data. In the case of FA, the goal is to identify fewer underlying constructs to explain the observed data. In the case of PCA, the goal can be mere data reduction, but the interpretation of components is frequently done in terms similar to those used when describing the latent variables estimated by FA. Cluster analytic techniques, although usually used to partition the subject space rather than the variable space, can also be used to group variables to reduce the complexity of the data by forming fewer and more homogeneous sets of tests or items. At the data level the data reduction problem may be solved as a \iemph{Singular Value Decomposition} of the original matrix, although the more typical solution is to find either the \iemph{principal components} or \iemph{factors} of the covariance or correlation matrices. Given the pattern of regression weights from the variables to the components or from the factors to the variables, it is then possible to find (for components) individual \index{component scores} \emph{component} or \iemph{cluster scores} or estimate (for factors) \iemph{factor scores}. Several of the functions in \Rpkg{psych} address the problem of data reduction. \begin{description} \item[\pfun{fa}] incorporates five alternative algorithms: \iemph{minres factor analysis}, \iemph{principal axis factor analysis}, \iemph{weighted least squares factor analysis}, \iemph{generalized least squares factor analysis} and \iemph{maximum likelihood factor analysis}. That is, it includes the functionality of three other functions that will be eventually phased out. \item[\pfun(bassAckward)] will do multiple factor and principal components analyses and then show the relationship between factor levels by finding the interfactor correlations. \item[\pfun{fa.extend}] will extend the factor solution for an X set of variables into a Y set (perhaps of criterion variables). %\item [\pfun{factor.minres}] Minimum residual factor analysis is a least squares, iterative solution to the factor problem. minres attempts to minimize the residual (off-diagonal) correlation matrix. It produces solutions similar to maximum likelihood solutions, but will work even if the matrix is singular. % %\item [\pfun{factor.pa}] Principal Axis factor analysis is a least squares, iterative solution to the factor problem. PA will work for cases where maximum likelihood techniques (\fun{factanal}) will not work. The original communality estimates are either the squared multiple correlations (\pfun{smc}) for each item or 1. % %\item [\pfun{factor.wls}] Weighted least squares factor analysis is a least squares, iterative solution to the factor problem. It minimizes the (weighted) squared residual matrix. The weights are based upon the independent contribution of each variable. % \item [\pfun{principal}] Principal Components Analysis reports the largest n eigen vectors rescaled by the square root of their eigen values. \item [\pfun{factor.congruence}] The congruence between two factors is the cosine of the angle between them. This is just the cross products of the loadings divided by the sum of the squared loadings. This differs from the correlation coefficient in that the mean loading is not subtracted before taking the products. \pfun{factor.congruence} will find the cosines between two (or more) sets of factor loadings. \item [\pfun{vss}] Very Simple Structure \cite{revelle:vss} applies a goodness of fit test to determine the optimal number of factors to extract. It can be thought of as a quasi-confirmatory model, in that it fits the very simple structure (all except the biggest c loadings per item are set to zero where c is the level of complexity of the item) of a factor pattern matrix to the original correlation matrix. For items where the model is usually of complexity one, this is equivalent to making all except the largest loading for each item 0. This is typically the solution that the user wants to interpret. The analysis includes the \pfun{MAP} criterion of \cite{velicer:76} and a $\chi^2$ estimate. \item [\pfun{fa.parallel}] The parallel factors technique compares the observed eigen values of a correlation matrix with those from random data. \item [\pfun{fa.plot}] will plot the loadings from a factor, principal components, or cluster analysis (just a call to plot will suffice). If there are more than two factors, then a SPLOM of the loadings is generated. \item[\pfun{nfactors}] A number of different tests for the number of factors problem are run. \item[\pfun{fa.diagram}] replaces \pfun{fa.graph} and will draw a path diagram representing the factor structure. It does not require Rgraphviz and thus is probably preferred. \item[\pfun{fa.graph}] requires \fun{Rgraphviz} and will draw a graphic representation of the factor structure. If factors are correlated, this will be represented as well. \item[\pfun{iclust} ] is meant to do item cluster analysis using a hierarchical clustering algorithm specifically asking questions about the reliability of the clusters \citep{revelle:iclust}. Clusters are formed until either coefficient $\alpha$ \cite{cronbach:51} or $\beta$ \cite{revelle:iclust} fail to increase. \end{description} \subsubsection{Minimum Residual Factor Analysis} \label{sect:minres} The factor model is an approximation of a correlation matrix by a matrix of lower rank. That is, can the correlation matrix, $\vec{_nR_n}$ be approximated by the product of a factor matrix, $\vec{_nF_k}$ and its transpose plus a diagonal matrix of uniqueness. \begin{equation} R = FF' + U^2 \end{equation} The maximum likelihood solution to this equation is found by \fun{factanal} in the \Rpkg{stats} package. Five alternatives are provided in \Rpkg{psych}, all of them are included in the \pfun{fa} function and are called by specifying the factor method (e.g., fm=``minres", fm=``pa", fm=``"wls", fm="gls" and fm="ml"). In the discussion of the other algorithms, the calls shown are to the \pfun{fa} function specifying the appropriate method. \pfun{factor.minres} attempts to minimize the off diagonal residual correlation matrix by adjusting the eigen values of the original correlation matrix. This is similar to what is done in \fun{factanal}, but uses an ordinary least squares instead of a maximum likelihood fit function. The solutions tend to be more similar to the MLE solutions than are the \pfun{factor.pa} solutions. \iemph{min.res} is the default for the \pfun{fa} function. A classic data set, collected by \cite{thurstone:41} and then reanalyzed by \cite{bechtoldt:61} and discussed by \cite{mcdonald:tt}, is a set of 9 cognitive variables with a clear bi-factor structure \cite{holzinger:37}. The minimum residual solution was transformed into an oblique solution using the default option on rotate which uses an oblimin transformation (Table~\ref{tab:factor.minres}). Alternative rotations and transformations include ``none", ``varimax", ``quartimax", ``bentlerT", and ``geominT" (all of which are orthogonal rotations). as well as ``promax", ``oblimin", ``simplimax", ``bentlerQ, and``geominQ" and ``cluster" which are possible oblique transformations of the solution. The default is to do a oblimin transformation, although prior versions defaulted to varimax. The measures of factor adequacy reflect the multiple correlations of the factors with the best fitting linear regression estimates of the factor scores \citep{grice:01}. \begin{table}[htpb] \caption{Three correlated factors from the Thurstone 9 variable problem. By default, the solution is transformed obliquely using oblimin. The extraction method is (by default) minimum residual.} \begin{scriptsize} \begin{center} <>= f3t <- fa(Thurstone,3,n.obs=213) f3t @ \end{center} \end{scriptsize} \label{tab:factor.minres} \end{table}% \subsubsection{Principal Axis Factor Analysis} An alternative, least squares algorithm, \pfun{factor.pa}, does a Principal Axis factor analysis by iteratively doing an eigen value decomposition of the correlation matrix with the diagonal replaced by the values estimated by the factors of the previous iteration. This OLS solution is not as sensitive to improper matrices as is the maximum likelihood method, and will sometimes produce more interpretable results. It seems as if the SAS example for PA uses only one iteration. Setting the max.iter parameter to 1 produces the SAS solution. The solutions from the \pfun{fa}, the \pfun{factor.minres} and \pfun{factor.pa} as well as the \pfun{principal} functions can be rotated or transformed with a number of options. Some of these call the \Rpkg{GPArotation} package. Orthogonal rotations are \fun{varimax} and \fun{quartimax}. Oblique transformations include \fun{oblimin}, \fun{quartimin} and then two targeted rotation functions \pfun{Promax} and \pfun{target.rot}. The latter of these will transform a loadings matrix towards an arbitrary target matrix. The default is to transform towards an independent cluster solution. Using the Thurstone data set, three factors were requested and then transformed into an independent clusters solution using \pfun{target.rot} (Table~\ref{tab:Thurstone}). \begin{table}[htpb] \caption{The 9 variable problem from Thurstone is a classic example of factoring where there is a higher order factor, g, that accounts for the correlation between the factors. The extraction method was principal axis. The transformation was a targeted transformation to a simple cluster solution.} \begin{center} \begin{scriptsize} <>= f3 <- fa(Thurstone,3,n.obs = 213,fm="pa") f3o <- target.rot(f3) f3o @ \end{scriptsize} \end{center} \label{tab:Thurstone} \end{table} \subsubsection{Weighted Least Squares Factor Analysis} \label{sect:wls} Similar to the minres approach of minimizing the squared residuals, factor method ``wls" weights the squared residuals by their uniquenesses. This tends to produce slightly smaller overall residuals. In the example of weighted least squares, the output is shown by using the \pfun{print} function with the cut option set to 0. That is, all loadings are shown (Table~\ref{tab:Thurstone.wls}). \begin{table}[htpb] \caption{The 9 variable problem from Thurstone is a classic example of factoring where there is a higher order factor, g, that accounts for the correlation between the factors. The factors were extracted using a weighted least squares algorithm. All loadings are shown by using the cut=0 option in the \pfun{print.psych} function.} \begin{scriptsize} <>= f3w <- fa(Thurstone,3,n.obs = 213,fm="wls") print(f3w,cut=0,digits=3) @ \end{scriptsize} \label{tab:Thurstone.wls} \end{table} The unweighted least squares solution may be shown graphically using the \pfun{fa.plot} function which is called by the generic \fun{plot} function (Figure~\ref{fig:thurstone}. Factors were transformed obliquely using a oblimin. These solutions may be shown as item by factor plots (Figure~\ref{fig:thurstone} or by a structure diagram (Figure~\ref{fig:thurstone.diagram}. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= plot(f3t) @ \end{scriptsize} \caption{A graphic representation of the 3 oblique factors from the Thurstone data using \pfun{plot}. Factors were transformed to an oblique solution using the oblimin function from the GPArotation package.} \label{fig:thurstone} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= fa.diagram(f3t) @ \end{scriptsize} \caption{A graphic representation of the 3 oblique factors from the Thurstone data using \pfun{fa.diagram}. Factors were transformed to an oblique solution using oblimin.} \label{fig:thurstone.diagram} \end{center} \end{figure} A comparison of these three approaches suggests that the minres solution is more similar to a maximum likelihood solution and fits slightly better than the pa or wls solutions. Comparisons with SPSS suggest that the pa solution matches the SPSS OLS solution, but that the minres solution is slightly better. At least in one test data set, the weighted least squares solutions, although fitting equally well, had slightly different structure loadings. Note that the rotations used by SPSS will sometimes use the ``Kaiser Normalization''. By default, the rotations used in psych do not normalize, but this can be specified as an option in \pfun{fa}. \subsubsection{Principal Components analysis (PCA)} An alternative to factor analysis, which is unfortunately frequently confused with \iemph{factor analysis}, is \iemph{principal components analysis}. Although the goals of \iemph{PCA} and \iemph{FA} are similar, PCA is a descriptive model of the data, while FA is a structural model. Psychologists typically use PCA in a manner similar to factor analysis and thus the \pfun{principal} function produces output that is perhaps more understandable than that produced by \fun{princomp} in the \Rpkg{stats} package. Table~\ref{tab:pca} shows a PCA of the Thurstone 9 variable problem rotated using the \pfun{Promax} function. Note how the loadings from the factor model are similar but smaller than the principal component loadings. This is because the PCA model attempts to account for the entire variance of the correlation matrix, while FA accounts for just the \iemph{common variance}. This distinction becomes most important for small correlation matrices. Also note how the goodness of fit statistics, based upon the residual off diagonal elements, is much worse than the \pfun{fa} solution. \begin{table}[htpb] \caption{The Thurstone problem can also be analyzed using Principal Components Analysis. Compare this to Table~\ref{tab:Thurstone}. The loadings are higher for the PCA because the model accounts for the unique as well as the common variance.The fit of the off diagonal elements, however, is much worse than the \pfun{fa} results.} \begin{center} \begin{scriptsize} <>= p3p <-principal(Thurstone,3,n.obs = 213,rotate="Promax") p3p @ \end{scriptsize} \end{center} \label{tab:pca} \end{table} \subsubsection{Hierarchical and bi-factor solutions} \label{sect:omega} For a long time structural analysis of the ability domain have considered the problem of factors that are themselves correlated. These correlations may themselves be factored to produce a higher order, general factor. An alternative \citep{holzinger:37,jensen:weng} is to consider the general factor affecting each item, and then to have group factors account for the residual variance. Exploratory factor solutions to produce a hierarchical or a bifactor solution are found using the \pfun{omega} function. This technique has more recently been applied to the personality domain to consider such things as the structure of neuroticism (treated as a general factor, with lower order factors of anxiety, depression, and aggression). Consider the 9 Thurstone variables analyzed in the prior factor analyses. The correlations between the factors (as shown in Figure~\ref{fig:thurstone.diagram} can themselves be factored. This results in a higher order factor model (Figure~\ref{fig:omega}). An an alternative solution is to take this higher order model and then solve for the general factor loadings as well as the loadings on the residualized lower order factors using the \iemph{Schmid-Leiman} procedure. (Figure ~\ref{fig:omega.2}). Yet another solution is to use structural equation modeling to directly solve for the general and group factors. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= om.h <- omega(Thurstone,n.obs=213,sl=FALSE) op <- par(mfrow=c(1,1)) @ \end{scriptsize} \caption{A higher order factor solution to the Thurstone 9 variable problem} \label{fig:omega} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= om <- omega(Thurstone,n.obs=213) @ \end{scriptsize} \caption{A bifactor factor solution to the Thurstone 9 variable problem} \label{fig:omega.2} \end{center} \end{figure} Yet another approach to the bifactor structure is do use the \pfun{bifactor} rotation function in either \Rpkg{psych} or in \Rpkg{GPArotation}. This does the rotation discussed in \cite{jennrich:11}. \subsubsection{Item Cluster Analysis: iclust} \label{sect:iclust} An alternative to factor or components analysis is \iemph{cluster analysis}. The goal of cluster analysis is the same as factor or components analysis (reduce the complexity of the data and attempt to identify homogeneous subgroupings). Mainly used for clustering people or objects (e.g., projectile points if an anthropologist, DNA if a biologist, galaxies if an astronomer), clustering may be used for clustering items or tests as well. Introduced to psychologists by \cite{tryon:39} in the 1930's, the cluster analytic literature exploded in the 1970s and 1980s \citep{blashfield:80,blashfield:88,everitt:74,hartigan:75}. Much of the research is in taxonmetric applications in biology \citep{sneath:73,sokal:63} and marketing \citep{cooksey:06} where clustering remains very popular. It is also used for taxonomic work in forming clusters of people in family \citep{henry:05} and clinical psychology \citep{martinent:07,mun:08}. Interestingly enough it has has had limited applications to psychometrics. This is unfortunate, for as has been pointed out by e.g. \citep{tryon:35,loevinger:53}, the theory of factors, while mathematically compelling, offers little that the geneticist or behaviorist or perhaps even non-specialist finds compelling. \cite{cooksey:06} reviews why the \pfun{iclust} algorithm is particularly appropriate for scale construction in marketing. \emph{Hierarchical cluster analysis} \index{hierarchical cluster analysis} forms clusters that are nested within clusters. The resulting \iemph{tree diagram} (also known somewhat pretentiously as a \iemph{rooted dendritic structure}) shows the nesting structure. Although there are many hierarchical clustering algorithms in \R{} (e.g., \fun{agnes}, \fun{hclust}, and \pfun{iclust}), the one most applicable to the problems of scale construction is \pfun{iclust} \citep{revelle:iclust}. \begin{enumerate} \item Find the proximity (e.g. correlation) matrix, \item Identify the most similar pair of items \item Combine this most similar pair of items to form a new variable (cluster), \item Find the similarity of this cluster to all other items and clusters, \item Repeat steps 2 and 3 until some criterion is reached (e.g., typicallly, if only one cluster remains or in \pfun{iclust} if there is a failure to increase reliability coefficients $\alpha$ or $\beta$). \item Purify the solution by reassigning items to the most similar cluster center. \end{enumerate} \pfun{iclust} forms clusters of items using a hierarchical clustering algorithm until one of two measures of internal consistency fails to increase \citep{revelle:iclust}. The number of clusters may be specified a priori, or found empirically. The resulting statistics include the average split half reliability, $\alpha$ \citep{cronbach:51}, as well as the worst split half reliability, $\beta$ \citep{revelle:iclust}, which is an estimate of the general factor saturation of the resulting scale (Figure~\ref{fig:iclust}). Cluster loadings (corresponding to the structure matrix of factor analysis) are reported when printing (Table~\ref{tab:iclust}). The pattern matrix is available as an object in the results. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= data(bfi) ic <- iclust(bfi[1:25]) @ \end{scriptsize} \caption{Using the \pfun{iclust} function to find the cluster structure of 25 personality items (the three demographic variables were excluded from this analysis). When analyzing many variables, the tree structure may be seen more clearly if the graphic output is saved as a pdf and then enlarged using a pdf viewer.} \label{fig:iclust} \end{center} \end{figure} \begin{table}[htpb] \caption{The summary statistics from an iclust analysis shows three large clusters and smaller cluster.} \begin{center} \begin{scriptsize} <>= summary(ic) #show the results @ \end{scriptsize} \end{center} \label{tab:iclust} \end{table}% The previous analysis (Figure~\ref{fig:iclust}) was done using the Pearson correlation. A somewhat cleaner structure is obtained when using the \pfun{polychoric} function to find polychoric correlations (Figure~\ref{fig:iclust.poly}). Note that the first time finding the polychoric correlations some time, but the next three analyses were done using that correlation matrix (r.poly\$rho). When using the console for input, \pfun{polychoric} will report on its progress while working using \pfun{progressBar}. \begin{table}[htpb] \caption{The \pfun{polychoric} and the \pfun{tetrachoric} functions can take a long time to finish and report their progress by a series of dots as they work. The dots are suppressed when creating a Sweave document.} \begin{center} \begin{tiny} <>= data(bfi) r.poly <- polychoric(bfi[1:25]) #the ... indicate the progress of the function @ \end{tiny} \end{center} \label{tab:bad} \end{table}% \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= ic.poly <- iclust(r.poly$rho,title="ICLUST using polychoric correlations") iclust.diagram(ic.poly) @ \end{scriptsize} \caption{ICLUST of the BFI data set using polychoric correlations. Compare this solution to the previous one (Figure~\ref{fig:iclust}) which was done using Pearson correlations. } \label{fig:iclust.poly} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= ic.poly <- iclust(r.poly$rho,5,title="ICLUST using polychoric correlations for nclusters=5") iclust.diagram(ic.poly) @ \end{scriptsize} \caption{ICLUST of the BFI data set using polychoric correlations with the solution set to 5 clusters. Compare this solution to the previous one (Figure~\ref{fig:iclust.poly}) which was done without specifying the number of clusters and to the next one (Figure~\ref{fig:iclust.3}) which was done by changing the beta criterion. } \label{fig:iclust.5} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= ic.poly <- iclust(r.poly$rho,beta.size=3,title="ICLUST beta.size=3") @ \end{scriptsize} \caption{ICLUST of the BFI data set using polychoric correlations with the beta criterion set to 3. Compare this solution to the previous three (Figure~\ref{fig:iclust},~\ref{fig:iclust.poly}, \ref{fig:iclust.5}).} \label{fig:iclust.3} \end{center} \end{figure} \begin{table}[htpb] \caption{The output from \pfun{iclust}includes the loadings of each item on each cluster. These are equivalent to factor structure loadings. By specifying the value of cut, small loadings are suppressed. The default is for cut=0.su } \begin{center} \begin{scriptsize} <>= print(ic,cut=.3) @ \end{scriptsize} \end{center} \label{tab:iclust} \end{table}% A comparison of these four cluster solutions suggests both a problem and an advantage of clustering techniques. The problem is that the solutions differ. The advantage is that the structure of the items may be seen more clearly when examining the clusters rather than a simple factor solution. \subsection{Confidence intervals using bootstrapping techniques} Exploratory factoring techniques are sometimes criticized because of the lack of statistical information on the solutions. Overall estimates of goodness of fit including $\chi^{2}$ and RMSEA are found in the \pfun{fa} and \pfun{omega} functions. Confidence intervals for the factor loadings may be found by doing multiple bootstrapped iterations of the original analysis. This is done by setting the n.iter parameter to the desired number of iterations. This can be done for factoring of Pearson correlation matrices as well as polychoric/tetrachoric matrices (See Table~\ref{tab:bootstrap}). Although the example value for the number of iterations is set to 20, more conventional analyses might use 1000 bootstraps. This will take much longer. \begin{table}[htpb] \caption{An example of bootstrapped confidence intervals on 10 items from the Big 5 inventory. The number of bootstrapped samples was set to 20. More conventional bootstrapping would use 100 or 1000 replications. } \begin{tiny} \begin{center} <>= fa(bfi[1:10],2,n.iter=20) @ \end{center} \end{tiny} \label{tab:bootstrap} \end{table}% \subsection{Comparing factor/component/cluster solutions} Cluster analysis, factor analysis, and principal components analysis all produce structure matrices (matrices of correlations between the dimensions and the variables) that can in turn be compared in terms of the \iemph{congruence coefficient} which is just the cosine of the angle between the dimensions $$c_{f_{i}f_{j}} = \frac{\sum_{k=1}^{n}{f_{ik}f_{jk}}} {\sum{f_{ik}^{2}}\sum{f_{jk}^{2}}}.$$ Consider the case of a four factor solution and four cluster solution to the Big Five problem. \begin{scriptsize} <>= f4 <- fa(bfi[1:25],4,fm="pa") factor.congruence(f4,ic) @ \end{scriptsize} A more complete comparison of oblique factor solutions (both minres and principal axis), bifactor and component solutions to the Thurstone data set is done using the \pfun{factor.congruence} function. (See table~\ref{tab:congruence}). \begin{table}[htpb] \caption{Congruence coefficients for oblique factor, bifactor and component solutions for the Thurstone problem.} \begin{scriptsize} <>= factor.congruence(list(f3t,f3o,om,p3p)) @ \end{scriptsize} \label{tab:congruence} \end{table}% \subsubsection{Factor correlations} Factor congruences may be found between any two sets of factor loadings. If given the same data set/correlation matrix, factor correlations may be found using \pfun{faCor} which finds the correlations between the factors. This procedure is also used in the \pfun{bassAckward} function which compares multiple solutions with a different number of factors. Consider the correlation of three versus five factors of the \pfun{bfi} data set. \begin{table}[htpb] \caption{Factor correlations and factor congruences between ``minres" factor analysis and ``pca" principal components using ``oblimin" rotation for both.} \begin{center} \begin{scriptsize} <>= faCor(Thurstone,c(3,3),fm=c("minres","pca"), rotate=c("oblimin","oblimin")) @ \end{scriptsize} \end{center} \label{tab:faCor} \end{table} \subsection{Determining the number of dimensions to extract.} How many dimensions to use to represent a correlation matrix is an unsolved problem in psychometrics. There are many solutions to this problem, none of which is uniformly the best. Henry Kaiser once said that ``a solution to the number-of factors problem in factor analysis is easy, that he used to make up one every morning before breakfast. But the problem, of course is to find \emph{the} solution, or at least a solution that others will regard quite highly not as the best" \cite{horn:79}. Techniques most commonly used include 1) Extracting factors until the chi square of the residual matrix is not significant. 2) Extracting factors until the change in chi square from factor n to factor n+1 is not significant. 3) Extracting factors until the eigen values of the real data are less than the corresponding eigen values of a random data set of the same size (parallel analysis) \pfun{fa.parallel} \citep{horn:65}. 4) Plotting the magnitude of the successive eigen values and applying the scree test (a sudden drop in eigen values analogous to the change in slope seen when scrambling up the talus slope of a mountain and approaching the rock face \citep{cattell:scree}. 5) Extracting factors as long as they are interpretable. 6) Using the Very Structure Criterion (\pfun{vss}) \citep{revelle:vss}. 7) Using Wayne Velicer's Minimum Average Partial (\pfun{MAP}) criterion \citep{velicer:76}. 8) Extracting principal components until the eigen value < 1. Each of the procedures has its advantages and disadvantages. Using either the chi square test or the change in square test is, of course, sensitive to the number of subjects and leads to the nonsensical condition that if one wants to find many factors, one simply runs more subjects. Parallel analysis is partially sensitive to sample size in that for large samples the eigen values of random factors will be very small. The scree test is quite appealing but can lead to differences of interpretation as to when the scree``breaks". Extracting interpretable factors means that the number of factors reflects the investigators creativity more than the data. vss, while very simple to understand, will not work very well if the data are very factorially complex. (Simulations suggests it will work fine if the complexities of some of the items are no more than 2). The eigen value of 1 rule, although the default for many programs, seems to be a rough way of dividing the number of variables by 3 and is probably the worst of all criteria. An additional problem in determining the number of factors is what is considered a factor. Many treatments of factor analysis assume that the residual correlation matrix after the factors of interest are extracted is composed of just random error. An alternative concept is that the matrix is formed from major factors of interest but that there are also numerous minor factors of no substantive interest but that account for some of the shared covariance between variables. The presence of such minor factors can lead one to extract too many factors and to reject solutions on statistical grounds of misfit that are actually very good fits to the data. This problem is partially addressed later in the discussion of simulating complex structures using \pfun{sim.structure} and of small extraneous factors using the \pfun{sim.minor} function. \subsubsection{Very Simple Structure} \label{sect:vss} The \pfun{vss} function compares the fit of a number of factor analyses with the loading matrix ``simplified" by deleting all except the c greatest loadings per item, where c is a measure of factor complexity \cite{revelle:vss}. Included in \pfun{vss} is the MAP criterion (Minimum Absolute Partial correlation) of \cite{velicer:76}. Using the Very Simple Structure criterion for the bfi data suggests that 4 factors are optimal (Figure~\ref{fig:vss}). However, the MAP criterion suggests that 5 is optimal. \begin{figure}[htbp] \begin{center} <>= vss <- vss(bfi[1:25],title="Very Simple Structure of a Big 5 inventory") @ \caption{The Very Simple Structure criterion for the number of factors compares solutions for various levels of item complexity and various numbers of factors. For the Big 5 Inventory, the complexity 1 and 2 solutions both achieve their maxima at four factors. This is in contrast to parallel analysis which suggests 6 and the MAP criterion which suggests 5. } \label{fig:vss} \end{center} \end{figure} \begin{scriptsize} <>= vss @ \end{scriptsize} \subsubsection{Parallel Analysis} \label{sect:fa.parallel} An alternative way to determine the number of factors is to compare the solution to random data with the same properties as the real data set. If the input is a data matrix, the comparison includes random samples from the real data, as well as normally distributed random data with the same number of subjects and variables. For the BFI data, parallel analysis suggests that 6 factors might be most appropriate (Figure~\ref{fig:parallel}). It is interesting to compare \pfun{fa.parallel} with the \fun{paran} from the \Rpkg{paran} package. This latter uses smcs to estimate communalities. Simulations of known structures with a particular number of major factors but with the presence of trivial, minor (but not zero) factors, show that using smcs will tend to lead to too many factors. \begin{figure}[htbp] \begin{scriptsize} \begin{center} <>= fa.parallel(bfi[1:25],main="Parallel Analysis of a Big 5 inventory") @ \caption{Parallel analysis compares factor and principal components solutions to the real data as well as resampled data. Although vss suggests 4 factors, MAP 5, parallel analysis suggests 6. One more demonstration of Kaiser's dictum.} \label{fig:parallel} \end{center} \end{scriptsize} \end{figure} A more tedious problem in terms of computation is to do parallel analysis of \iemph{polychoric} correlation matrices. This is done by \pfun{fa.parallel.poly} or \pfun{fa.parallel} with the cor option="poly". By default the number of replications is 20. This is appropriate when choosing the number of factors from dicthotomous or polytomous data matrices. \subsection{Factor extension} Sometimes we are interested in the relationship of the factors in one space with the variables in a different space. One solution is to find factors in both spaces separately and then find the structural relationships between them. This is the technique of structural equation modeling in packages such as \Rpkg{sem} or \Rpkg{lavaan}. An alternative is to use the concept of \iemph{factor extension} developed by \citep{dwyer:37}. Consider the case of 16 variables created to represent one two dimensional space. If factors are found from eight of these variables, they may then be extended to the additional eight variables (See Figure~\ref{fig:fa.extension}). \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= v16 <- sim.item(16) s <- c(1,3,5,7,9,11,13,15) f2 <- fa(v16[,s],2) fe <- fa.extension(cor(v16)[s,-s],f2) fa.diagram(f2,fe=fe) @ \end{scriptsize} \caption{Factor extension applies factors from one set (those on the left) to another set of variables (those on the right). \pfun{fa.extension} is particularly useful when one wants to define the factors with one set of variables and then apply those factors to another set. \pfun{fa.diagram} is used to show the structure. } \label{fig:fa.extension} \end{center} \end{figure} Factor extension may also be used to see the validity of a certain factor solution compared to a set of criterion variables. Consider the case of 5 factors from the 25 items of the \pfun{bfi} data set and how they predict gender, age, and education (See Figure~\ref{fig:fa:extend}). \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= fe <- fa.extend(bfi,5,ov=1:25,ev=26:28) extension.diagram(fe) @ \end{scriptsize} \caption{Factor extension applies factors from one set (those on the left) to another set of variables (those on the right). \pfun{fa.extend} is particularly useful when one wants to define the factors with one set of variables and then apply those factors to another set. \pfun{diagram} is used to show the structure. } \label{fig:fa.extend} \end{center} \end{figure} Another way to examine the overlap between two sets is the use of \iemph{set correlation} found by \pfun{setCor} (discussed later). \subsection{Comparing multiple solutions} A procedure dubbed ``bass Ackward" by Lew Goldberg \citep{goldberg:06} compares solutions at multiple levels of complexity. Here we show a 2, 3, 4 and 5 dimensional solution to the \pfun{bfi} data set. (Figure~\ref{fig:bass.ack}). This is done by finding the factor correlations between solutions (see \pfun{faCor}) and then organizing them sequentially. The factor correlations for two solutions from the same correlation matrix, $\vec{R}$ , $\vec{F_1} $ and $\vec{F_2}$ are found by using the two weights matrices, $\vec{W_1}$ and $\vec{W_2}$ (for finding factor scores for the first and second model) and then finding the factor covariances, $C = \vec{W_1' R W_2} $ which may then be converted to factor correlations by dividing by the square root of the diagonal of $\vec{C}$. By default \pfun{bassAckward} uses the correlation preserving weights discussed by \cite{tenBerge.99}, although other options (e.g. regression weights) may also be used. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= ba5 <- bassAckward(bfi[1:25], nfactors =c(2,3,4,5),plot=FALSE) baf <- bassAckward.diagram(ba5) @ \end{scriptsize} \caption{\pfun{bassAckward} compares solutions at multiple levels by successive factoring and the finding the factor correlations across levels. Compare the three factor solution to the five factor solution. The dimensions of social approach, withdrawal, and constraint seen at the three factor level become the more traditional CANOE factors at the five factor level. } \label{fig:bass.ack} \end{center} \end{figure} And we show the items associated with this solution by using \pfun{fa.lookup} (Table~\ref{tab:bfi}) \begin{table}[htpb] \caption{bfi items sorted in the order of the five factors from \pfun{bassAckward}} \begin{center} \begin{scriptsize} <>= # fa.lookup(baf$bass.ack[[5]],dictionary=bfi.dictionary[2]) @ \end{scriptsize} \end{center} \label{tab:bfi} \end{table} \section{Classical Test Theory and Reliability} Surprisingly, 107 years after \cite{spearman:rho} introduced the concept of reliability to psychologists, there are still multiple approaches for measuring it. Although very popular, Cronbach's $\alpha$ \citep{cronbach:51} underestimates the reliability of a test and over estimates the first factor saturation \citep{rz:09}. $\alpha$ \citep{cronbach:51} is the same as Guttman's $\lambda3$ \citep{guttman:45} and may be found by $$ \lambda_3 = \frac{n}{n-1}\Bigl(1 - \frac{tr(\vec{V})_x}{V_x}\Bigr) = \frac{n}{n-1} \frac{V_x - tr(\vec{V}_x)}{V_x} = \alpha $$ Perhaps because it is so easy to calculate and is available in most commercial programs, alpha is without doubt the most frequently reported measure of internal consistency reliability. Alpha is the mean of all possible spit half reliabilities (corrected for test length). For a unifactorial test, it is a reasonable estimate of the first factor saturation, although if the test has any microstructure (i.e., if it is ``lumpy") coefficients $\beta$ \citep{revelle:iclust} (see \pfun{iclust}) and $\omega_h$ (see \pfun{omega}) are more appropriate estimates of the general factor saturation. $\omega_t$is a better estimate of the reliability of the total test. Guttman's $\lambda _6$ (G6) considers the amount of variance in each item that can be accounted for the linear regression of all of the other items (the squared multiple correlation or smc), or more precisely, the variance of the errors, $e_j^2$, and is $$ \lambda_6 = 1 - \frac{\sum e_j^2}{V_x} = 1 - \frac{\sum(1-r_{smc}^2)}{V_x}. $$ The squared multiple correlation is a lower bound for the item communality and as the number of items increases, becomes a better estimate. G6 is also sensitive to lumpiness in the test and should not be taken as a measure of unifactorial structure. For lumpy tests, it will be greater than alpha. For tests with equal item loadings, alpha > G6, but if the loadings are unequal or if there is a general factor, G6 > alpha. G6 estimates item reliability by the squared multiple correlation of the other items in a scale. A modification of G6, G6*, takes as an estimate of an item reliability the smc with all the items in an inventory, including those not keyed for a particular scale. This will lead to a better estimate of the reliable variance of a particular item. Alpha, G6 and G6* are positive functions of the number of items in a test as well as the average intercorrelation of the items in the test. When calculated from the item variances and total test variance, as is done here, raw alpha is sensitive to differences in the item variances. Standardized alpha is based upon the correlations rather than the covariances. More complete reliability analyses of a single scale can be done using the \pfun{omega} function which finds $\omega_h$ and $\omega_t$ based upon a hierarchical factor analysis. Alternative functions \pfun{scoreItems} and \pfun{cluster.cor} will also score multiple scales and report more useful statistics. ``Standardized" alpha is calculated from the inter-item correlations and will differ from raw alpha. Functions for examining the reliability of a single scale or a set of scales include: \begin{description} \item [alpha] Internal consistency measures of reliability range from $\omega_h$ to $\alpha$ to $\omega_t$. The \pfun{alpha} function reports two estimates: Cronbach's coefficient $\alpha$ and Guttman's $\lambda_6$. Also reported are item - whole correlations, $\alpha$ if an item is omitted, and item means and standard deviations. \item [guttman] Eight alternative estimates of test reliability include the six discussed by \cite{guttman:45}, four discussed by ten Berge and Zergers (1978) ($\mu_0 \dots \mu_3$) as well as $\beta$ \citep[the worst split half,][]{revelle:iclust}, the glb (greatest lowest bound) discussed by Bentler and Woodward (1980), and $\omega_h$ and$\omega_t$ (\citep{mcdonald:tt,zinbarg:pm:05}. \item [omega] Calculate McDonald's omega estimates of general and total factor saturation. (\cite{rz:09} compare these coefficients with real and artificial data sets.) \item [cluster.cor] Given a n x c cluster definition matrix of -1s, 0s, and 1s (the keys) , and a n x n correlation matrix, find the correlations of the composite clusters. \item [scoreItems] Given a matrix or data.frame of k keys for m items (-1, 0, 1), and a matrix or data.frame of items scores for m items and n people, find the sum scores or average scores for each person and each scale. If the input is a square matrix, then it is assumed that correlations or covariances were used, and the raw scores are not available. In addition, report Cronbach's alpha, coefficient G6*, the average r, the scale intercorrelations, and the item by scale correlations (both raw and corrected for item overlap and scale reliability). Replace missing values with the item median or mean if desired. Will adjust scores for reverse scored items. \item [score.multiple.choice] Ability tests are typically multiple choice with one right answer. score.multiple.choice takes a scoring key and a data matrix (or data.frame) and finds total or average number right for each participant. Basic test statistics (alpha, average r, item means, item-whole correlations) are also reported. \end{description} \subsection{Reliability of a single scale} \label{sect:alpha} A conventional (but non-optimal) estimate of the internal consistency reliability of a test is coefficient $\alpha$ \citep{cronbach:51}. Alternative estimates are Guttman's $\lambda_6$, Revelle's $\beta$, McDonald's $\omega_h$ and $\omega_t$. Consider a simulated data set, representing 9 items with a hierarchical structure and the following correlation matrix. Then using the \pfun{alpha} function, the $\alpha$ and $\lambda_6$ estimates of reliability may be found for all 9 items, as well as the if one item is dropped at a time. \begin{scriptsize} <>= set.seed(17) r9 <- sim.hierarchical(n=500,raw=TRUE)$observed round(cor(r9),2) alpha(r9) @ \end{scriptsize} Some scales have items that need to be reversed before being scored. Rather than reversing the items in the raw data, it is more convenient to just specify which items need to be reversed scored. This may be done in \pfun{alpha} by specifying a \iemph{keys} vector of 1s and -1s. (This concept of keys vector is more useful when scoring multiple scale inventories, see below.) As an example, consider scoring the 7 attitude items in the attitude data set. Assume a conceptual mistake in that item 2 is to be scored (incorrectly) negatively. \begin{scriptsize} <>= keys <- c(1,-1,1,1,1,1,1) alpha(attitude,keys) @ \end{scriptsize} Note how the reliability of the 7 item scales with an incorrectly reversed item is very poor, but if the item 2 is dropped then the reliability is improved substantially. This suggests that item 2 was incorrectly scored. Doing the analysis again with item 2 positively scored produces much more favorable results. \begin{scriptsize} <>= keys <- c(1,1,1,1,1,1,1) alpha(attitude,keys) @ \end{scriptsize} It is useful when considering items for a potential scale to examine the item distribution. This is done in \pfun{scoreItems} as well as in \pfun{alpha}. \begin{scriptsize} <>= items <- sim.congeneric(N=500,short=FALSE,low=-2,high=2,categorical=TRUE) #500 responses to 4 discrete items alpha(items$observed) #item response analysis of congeneric measures @ \end{scriptsize} \subsection{Using \pfun{omega} to find the reliability of a single scale} Two alternative estimates of reliability that take into account the hierarchical structure of the inventory are McDonald's $\omega_h$ and $\omega_t$. These may be found using the \pfun{omega} function for an exploratory analysis (See Figure~\ref{fig:omega.9}) or \pfun{omegaSem} for a confirmatory analysis using the \Rpkg{sem} based upon the exploratory solution from \pfun{omega}. McDonald has proposed coefficient omega (hierarchical) ($\omega_h$) as an estimate of the general factor saturation of a test. \cite{zinbarg:pm:05} \url{http://personality-project.org/revelle/publications/zinbarg.revelle.pmet.05.pdf} compare McDonald's $\omega_h$ to Cronbach's $\alpha$ and Revelle's $\beta$. They conclude that $\omega_h$ is the best estimate. (See also \cite{zinbarg:apm:06} and \cite{rz:09} \url{http://personality-project.org/revelle/publications/revelle.zinbarg.08.pdf} ). One way to find $\omega_h$ is to do a factor analysis of the original data set, rotate the factors obliquely, factor that correlation matrix, do a Schmid-Leiman (\pfun{schmid}) transformation to find general factor loadings, and then find $\omega_h$. $\omega_h$ differs slightly as a function of how the factors are estimated. Four options are available, the default will do a minimum residual factor analysis, fm=``pa" does a principal axes factor analysis (\pfun{factor.pa}), fm=``mle" uses the factanal function, and fm=``pc" does a principal components analysis (\pfun{principal}). For ability items, it is typically the case that all items will have positive loadings on the general factor. However, for non-cognitive items it is frequently the case that some items are to be scored positively, and some negatively. Although probably better to specify which directions the items are to be scored by specifying a key vector, if flip =TRUE (the default), items will be reversed so that they have positive loadings on the general factor. The keys are reported so that scores can be found using the \pfun{scoreItems} function. Arbitrarily reversing items this way can overestimate the general factor. (See the example with a simulated circumplex). $\beta$, an alternative to $\omega$, is defined as the worst split half reliability. It can be estimated by using \pfun{iclust} (Item Cluster analysis: a hierarchical clustering algorithm). For a very complimentary review of why the iclust algorithm is useful in scale construction, see \cite{cooksey:06}. The \pfun{omega} function uses exploratory factor analysis to estimate the $\omega_h$ coefficient. It is important to remember that ``A recommendation that should be heeded, regardless of the method chosen to estimate $\omega_h$, is to always examine the pattern of the estimated general factor loadings prior to estimating $\omega_h$. Such an examination constitutes an informal test of the assumption that there is a latent variable common to all of the scale's indicators that can be conducted even in the context of EFA. If the loadings were salient for only a relatively small subset of the indicators, this would suggest that there is no true general factor underlying the covariance matrix. Just such an informal assumption test would have afforded a great deal of protection against the possibility of misinterpreting the misleading $\omega_h$ estimates occasionally produced in the simulations reported here." \citep[][p 137]{zinbarg:apm:06}. Although $\omega_h$ is uniquely defined only for cases where 3 or more subfactors are extracted, it is sometimes desired to have a two factor solution. By default this is done by forcing the \pfun{schmid} extraction to treat the two subfactors as having equal loadings. There are three possible options for this condition: setting the general factor loadings between the two lower order factors to be ``equal" which will be the $\sqrt{r_{ab}}$ where $r_{ab}$ is the oblique correlation between the factors) or to ``first" or ``second" in which case the general factor is equated with either the first or second group factor. A message is issued suggesting that the model is not really well defined. This solution discussed in Zinbarg et al., 2007. To do this in omega, add the option=``first" or option=``second" to the call. Although obviously not meaningful for a 1 factor solution, it is of course possible to find the sum of the loadings on the first (and only) factor, square them, and compare them to the overall matrix variance. This is done, with appropriate complaints. In addition to $\omega_h$, another of McDonald's coefficients is $\omega_t$. This is an estimate of the total reliability of a test. McDonald's $\omega_t$, which is similar to Guttman's $\lambda_6$, (see \pfun{guttman}) uses the estimates of uniqueness $u^2$ from factor analysis to find $e_j^2$. This is based on a decomposition of the variance of a test score, $V_x$ into four parts: that due to a general factor, $\vec{g}$, that due to a set of group factors, $\vec{f}$, (factors common to some but not all of the items), specific factors, $\vec{s}$ unique to each item, and $\vec{e}$, random error. (Because specific variance can not be distinguished from random error unless the test is given at least twice, some combine these both into error). Letting $\vec{x} = \vec{cg} + \vec{Af} + \vec {Ds} + \vec{e} $ then the communality of item$_j$, based upon general as well as group factors, $h_j^2 = c_j^2 + \sum{f_{ij}^2}$ and the unique variance for the item $u_j^2 = \sigma_j^2 (1-h_j^2)$ may be used to estimate the test reliability. That is, if $h_j^2$ is the communality of item$_j$, based upon general as well as group factors, then for standardized items, $e_j^2 = 1 - h_j^2$ and $$ \omega_t = \frac{\vec{1}\vec{cc'}\vec{1} + \vec{1}\vec{AA'}\vec{1}'}{V_x} = 1 - \frac{\sum(1-h_j^2)}{V_x} = 1 - \frac{\sum u^2}{V_x} $$ Because $h_j^2 \geq r_{smc}^2$, $\omega_t \geq \lambda_6$. It is important to distinguish here between the two $\omega$ coefficients of McDonald, 1978 and Equation 6.20a of McDonald, 1999, $\omega_t$ and $\omega_h$. While the former is based upon the sum of squared loadings on all the factors, the latter is based upon the sum of the squared loadings on the general factor. $$\omega_h = \frac{ \vec{1}\vec{cc'}\vec{1}}{V_x}$$ Another estimate reported is the omega for an infinite length test with a structure similar to the observed test. This is found by $$\omega_{\inf} = \frac{ \vec{1}\vec{cc'}\vec{1}}{\vec{1}\vec{cc'}\vec{1} + \vec{1}\vec{AA'}\vec{1}'}$$ \begin{figure}[htbp] \begin{center} <>= om.9 <- omega(r9,title="9 simulated variables") @ \caption{A bifactor solution for 9 simulated variables with a hierarchical structure. } \label{fig:omega.9} \end{center} \end{figure} In the case of these simulated 9 variables, the amount of variance attributable to a general factor ($\omega_h$) is quite large, and the reliability of the set of 9 items is somewhat greater than that estimated by $\alpha$ or $\lambda_6$. \begin{scriptsize} <>= om.9 @ \end{scriptsize} \subsection{Estimating $\omega_h$ using Confirmatory Factor Analysis} The \pfun{omegaSem} function will do an exploratory analysis and then take the highest loading items on each factor and do a confirmatory factor analysis using the \Rpkg{sem} package. These results can produce slightly different estimates of $\omega_h$, primarily because cross loadings are modeled as part of the general factor. \begin{scriptsize} <>= omegaSem(r9,n.obs=500) @ \end{scriptsize} \subsubsection{Other estimates of reliability} Other estimates of reliability are found by the \pfun{splitHalf} function. These are described in more detail in \cite{rz:09}. They include the 6 estimates from Guttman, four from TenBerge, and an estimate of the greatest lower bound. \begin{scriptsize} <>= splitHalf(r9) @ \end{scriptsize} \subsection{Reliability and correlations of multiple scales within an inventory} \label{sect:score} A typical research question in personality involves an inventory of multiple items purporting to measure multiple constructs. For example, the data set \pfun{bfi} includes 25 items thought to measure five dimensions of personality (Extraversion, Emotional Stability, Conscientiousness, Agreeableness, and Openness). The data may either be the raw data or a correlation matrix (\pfun{scoreItems}) or just a correlation matrix of the items ( \pfun{cluster.cor} and \pfun{cluster.loadings}). When finding reliabilities for multiple scales, item reliabilities can be estimated using the squared multiple correlation of an item with all other items, not just those that are keyed for a particular scale. This leads to an estimate of G6*. \subsubsection{Scoring from raw data} To score these five scales from the 25 items, use the \pfun{scoreItems} function with the helper function \pfun{make.keys}. Logically, scales are merely the weighted composites of a set of items. The weights used are -1, 0, and 1. 0 implies do not use that item in the scale, 1 implies a positive weight (add the item to the total score), -1 a negative weight (subtract the item from the total score, i.e., reverse score the item). Reverse scoring an item is equivalent to subtracting the item from the maximum + minimum possible value for that item. The minima and maxima can be estimated from all the items, or can be specified by the user. There are two different ways that scale scores tend to be reported. Social psychologists and educational psychologists tend to report the scale score as the \emph{average item score} while many personality psychologists tend to report the \emph{total item score}. The default option for \pfun{scoreItems} is to report item averages (which thus allows interpretation in the same metric as the items) but totals can be found as well. Personality researchers should be encouraged to report scores based upon item means and avoid using the total score although some reviewers are adamant about the following the tradition of total scores. The printed output includes coefficients $\alpha$ and G6*, the average correlation of the items within the scale (corrected for item overlap and scale relliability), as well as the correlations between the scales (below the diagonal, the correlations above the diagonal are corrected for attenuation. As is the case for most of the \Rpkg{psych} functions, additional information is returned as part of the object. First, create keys matrix using the \pfun{make.keys} function. (The keys matrix could also be prepared externally using a spreadsheet and then copying it into \R{}). Although not normally necessary, show the keys to understand what is happening. Note that the number of items to specify in the \pfun{make.keys} function is the total number of items in the inventory. That is, if scoring just 5 items from a 25 item inventory, \pfun{make.keys} should be told that there are 25 items. \pfun{make.keys} just changes a list of items on each scale to make up a scoring matrix. Because the \pfun{bfi} data set has 25 items as well as 3 demographic items, the number of variables is specified as 28. \begin{scriptsize} <>= keys <- make.keys(nvars=28,list(Agree=c(-1,2:5),Conscientious=c(6:8,-9,-10), Extraversion=c(-11,-12,13:15),Neuroticism=c(16:20), Openness = c(21,-22,23,24,-25)), item.labels=colnames(bfi)) keys @ \end{scriptsize} The use of multiple key matrices for different inventories is facilitated by using the \pfun{superMatrix} function to combine two or more matrices. This allows convenient scoring of large data sets combining multiple inventories with keys based upon each individual inventory. Pretend for the moment that the big 5 items were made up of two inventories, one consisting of the first 10 items, the second the last 18 items. (15 personality items + 3 demographic items.) Then the following code would work: \begin{scriptsize} <>= keys.1<- make.keys(10,list(Agree=c(-1,2:5),Conscientious=c(6:8,-9,-10))) keys.2 <- make.keys(15,list(Extraversion=c(-1,-2,3:5),Neuroticism=c(6:10), Openness = c(11,-12,13,14,-15))) keys.25 <- superMatrix(list(keys.1,keys.2)) @ \end{scriptsize} The resulting keys matrix is identical to that found above except that it does not include the extra 3 demographic items. This is useful when scoring the raw items because the response frequencies for each category are reported, and for the demographic data, This use of making multiple key matrices and then combining them into one super matrix of keys is particularly useful when combining demographic information with items to be scores. A set of demographic keys can be made and then these can be combined with the keys for the particular scales. Now use these keys in combination with the raw data to score the items, calculate basic reliability and intercorrelations, and find the item-by scale correlations for each item and each scale. By default, missing data are replaced by the median for that variable. \begin{scriptsize} <>= scores <- scoreItems(keys,bfi) scores @ \end{scriptsize} To see the additional information (the raw correlations, the individual scores, etc.), they may be specified by name. Then, to visualize the correlations between the raw scores, use the \pfun{pairs.panels} function on the scores values of scores. (See figure~\ref{fig:scores} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= png('scores.png') pairs.panels(scores$scores,pch='.',jiggle=TRUE) dev.off() @ \end{scriptsize} \includegraphics{scores} \caption{A graphic analysis of the Big Five scales found by using the scoreItems function. The pair.wise plot allows us to see that some participants have reached the ceiling of the scale for these 5 items scales. Using the pch='.' option in pairs.panels is recommended when plotting many cases. The data points were ``jittered'' by setting jiggle=TRUE. Jiggling this way shows the density more clearly. To save space, the figure was done as a png. For a clearer figure, save as a pdf.} \label{fig:scores} \end{center} \end{figure} \subsubsection{Forming scales from a correlation matrix} There are some situations when the raw data are not available, but the correlation matrix between the items is available. In this case, it is not possible to find individual scores, but it is possible to find the reliability and intercorrelations of the scales. This may be done using the \pfun{cluster.cor} function or the \pfun{scoreItems} function. The use of a keys matrix is the same as in the raw data case. Consider the same \pfun{bfi} data set, but first find the correlations, and then use \pfun{cluster.cor}. \begin{scriptsize} <>= r.bfi <- cor(bfi,use="pairwise") scales <- cluster.cor(keys,r.bfi) summary(scales) @ \end{scriptsize} To find the correlations of the items with each of the scales (the ``structure" matrix) or the correlations of the items controlling for the other scales (the ``pattern" matrix), use the \pfun{cluster.loadings} function. To do both at once (e.g., the correlations of the scales as well as the item by scale correlations), it is also possible to just use \pfun{scoreItems}. \subsection{Scoring Multiple Choice Items} Some items (typically associated with ability tests) are not themselves mini-scales ranging from low to high levels of expression of the item of interest, but are rather multiple choice where one response is the correct response. Two analyses are useful for this kind of item: examining the response patterns to all the alternatives (looking for good or bad distractors) and scoring the items as correct or incorrect. Both of these operations may be done using the \pfun{score.multiple.choice} function. Consider the 16 example items taken from an online ability test at the Personality Project: \url{http://test.personality-project.org}. This is part of the Synthetic Aperture Personality Assessment (\iemph{SAPA}) study discussed in \cite{rcw:methods,rwr:sapa}. \begin{scriptsize} <>= data(iqitems) iq.keys <- c(4,4,4, 6,6,3,4,4, 5,2,2,4, 3,2,6,7) score.multiple.choice(iq.keys,iqitems) #just convert the items to true or false iq.tf <- score.multiple.choice(iq.keys,iqitems,score=FALSE) describe(iq.tf) #compare to previous results @ \end{scriptsize} Once the items have been scored as true or false (assigned scores of 1 or 0), they made then be scored into multiple scales using the normal \pfun{scoreItems} function. \subsection{Item analysis} Basic item analysis starts with describing the data (\pfun{describe}, finding the number of dimensions using factor analysis (\pfun{fa}) and cluster analysis \pfun{iclust} perhaps using the Very Simple Structure criterion (\pfun{vss}), or perhaps parallel analysis \pfun{fa.parallel}. Item whole correlations may then be found for scales scored on one dimension (\pfun{alpha} or many scales simultaneously (\pfun{scoreItems}). Scales can be modified by changing the keys matrix (i.e., dropping particular items, changing the scale on which an item is to be scored). This analysis can be done on the normal Pearson correlation matrix or by using polychoric correlations. Validities of the scales can be found using multiple correlation of the raw data or based upon correlation matrices using the \pfun{setCor} function. However, more powerful item analysis tools are now available by using Item Response Theory approaches. Although the \pfun{response.frequencies} output from \pfun{score.multiple.choice} is useful to examine in terms of the probability of various alternatives being endorsed, it is even better to examine the pattern of these responses as a function of the underlying latent trait or just the total score. This may be done by using \pfun{irt.responses} (Figure~\ref{fig:irt.response}). \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= data(iqitems) iq.keys <- c(4,4,4, 6,6,3,4,4, 5,2,2,4, 3,2,6,7) scores <- score.multiple.choice(iq.keys,iqitems,score=TRUE,short=FALSE) #note that for speed we can just do this on simple item counts rather than IRT based scores. op <- par(mfrow=c(2,2)) #set this to see the output for multiple items irt.responses(scores$scores,iqitems[1:4],breaks=11) @ \end{scriptsize} \caption{ The pattern of responses to multiple choice ability items can show that some items have poor distractors. This may be done by using the the \pfun{irt.responses} function. A good distractor is one that is negatively related to ability.} \label{fig:irt.response} \end{center} \end{figure} \section{Item Response Theory analysis} The use of Item Response Theory has become is said to be the ``new psychometrics". The emphasis is upon item properties, particularly those of item difficulty or location and item discrimination. These two parameters are easily found from classic techniques when using factor analyses of correlation matrices formed by \pfun{polychoric} or \pfun{tetrachoric} correlations. The \pfun{irt.fa} function does this and then graphically displays item discrimination and item location as well as item and test information (see Figure~\ref{fig:irt}). \subsection{Factor analysis and Item Response Theory} If the correlations of all of the items reflect one underlying latent variable, then factor analysis of the matrix of tetrachoric correlations should allow for the identification of the regression slopes ($\alpha$) of the items on the latent variable. These regressions are, of course just the factor loadings. Item difficulty, $\delta_j$ and item discrimination, $\alpha_j$ may be found from factor analysis of the tetrachoric correlations where $\lambda_j$ is just the factor loading on the first factor and $\tau_j$ is the normal threshold reported by the \pfun{tetrachoric} function. \begin{equation} \delta_j = \frac{D\tau}{\sqrt{1-\lambda_j^2}}, \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\;\; \alpha_j = \frac{\lambda_j}{\sqrt{1-\lambda_j^2}} \label{eq:irt:diff} \end{equation} where D is a scaling factor used when converting to the parameterization of \iemph{logistic} model and is 1.702 in that case and 1 in the case of the normal ogive model. Thus, in the case of the normal model, factor loadings ($\lambda_j$) and item thresholds ($\tau$) are just \begin{equation*} \lambda_j = \frac{\alpha_j}{\sqrt{1+\alpha_j^2}}, \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\;\;\tau_j = \frac{\delta_j}{\sqrt{1+\alpha_j^2}}. \end{equation*} Consider 9 dichotomous items representing one factor but differing in their levels of difficulty \begin{scriptsize} <>= set.seed(17) d9 <- sim.irt(9,1000,-2.,2.,mod="normal") #dichotomous items test <- irt.fa(d9$items) test @ \end{scriptsize} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= op <- par(mfrow=c(3,1)) plot(test,type="ICC") plot(test,type="IIC") plot(test,type="test") op <- par(mfrow=c(1,1)) @ \end{scriptsize} \caption{A graphic analysis of 9 dichotomous (simulated) items. The top panel shows the probability of item endorsement as the value of the latent trait increases. Items differ in their location (difficulty) and discrimination (slope). The middle panel shows the information in each item as a function of latent trait level. An item is most informative when the probability of endorsement is 50\%. The lower panel shows the total test information. These items form a test that is most informative (most accurate) at the middle range of the latent trait.} \label{fig:irt} \end{center} \end{figure} Similar analyses can be done for polytomous items such as those of the bfi extraversion scale: \begin{scriptsize} <>= data(bfi) e.irt <- irt.fa(bfi[11:15]) e.irt @ \end{scriptsize} The item information functions show that not all of items are equally good (Figure~\ref{fig:e.irt}): \begin{figure}[htbp] \begin{center} <>= e.info <- plot(e.irt,type="IIC") @ \caption{A graphic analysis of 5 extraversion items from the bfi. The curves represent the amount of information in the item as a function of the latent score for an individual. That is, each item is maximally discriminating at a different part of the latent continuum. Print e.info to see the average information for each item.} \label{fig:e.irt} \end{center} \end{figure} These procedures can be generalized to more than one factor by specifying the number of factors in \pfun{irt.fa}. The plots can be limited to those items with discriminations greater than some value of cut. An invisible object is returned when plotting the output from \pfun{irt.fa} that includes the average information for each item that has loadings greater than cut. \begin{scriptsize} <>= print(e.info,sort=TRUE) @ \end{scriptsize} More extensive IRT packages include the \Rpkg{ltm} and \Rpkg{eRm} and should be used for serious Item Response Theory analysis. \subsection{Speeding up analyses} Finding tetrachoric or polychoric correlations is very time consuming. Thus, to speed up the process of analysis, the original correlation matrix is saved as part of the output of both \pfun{irt.fa} and \pfun{omega}. Subsequent analyses may be done by using this correlation matrix. This is done by doing the analysis not on the original data, but rather on the output of the previous analysis. For example, taking the output from the 16 ability items from the \iemph{SAPA} project when scored for True/False using \pfun{score.multiple.choice} we can first do a simple IRT analysis of one factor (Figure~\ref{fig:iq.irt}) and then use that correlation matrix to do an \pfun{omega} analysis to show the sub-structure of the ability items . \begin{figure}[htbp] \begin{tiny} \begin{center} <>= iq.irt <- irt.fa(iq.tf) @ \end{center} \end{tiny} \caption{A graphic analysis of 16 ability items sampled from the \iemph{SAPA} project. The curves represent the amount of information in the item as a function of the latent score for an individual. That is, each item is maximally discriminating at a different part of the latent continuum. Print iq.irt to see the average information for each item. Partly because this is a power test (it is given on the web) and partly because the items have not been carefully chosen, the items are not very discriminating at the high end of the ability dimension.} \label{fig:iq.irt} \end{figure} \begin{scriptsize} <>= iq.irt @ \end{scriptsize} \begin{figure}[htbp] \begin{center} <>= om <- omega(iq.irt$rho,4) @ \caption{An Omega analysis of 16 ability items sampled from the SAPA project. The items represent a general factor as well as four lower level factors. The analysis is done using the tetrachoric correlations found in the previous \pfun{irt.fa} analysis. The four matrix items have some serious problems, which may be seen later when examine the item response functions.} \label{fig:iq.irt} \end{center} \end{figure} \subsection{IRT based scoring} The primary advantage of IRT analyses is examining the item properties (both difficulty and discrimination). With complete data, the scores based upon simple total scores and based upon IRT are practically identical (this may be seen in the examples for \pfun{scoreIrt}). However, when working with data such as those found in the Synthetic Aperture Personality Assessment (\iemph{SAPA}) project, it is advantageous to use IRT based scoring. \iemph{SAPA} data might have 2-3 items/person sampled from scales with 10-20 items. Simply finding the average of the three (classical test theory) fails to consider that the items might differ in either discrimination or in difficulty. The \pfun{scoreIrt} function applies basic IRT to this problem. Consider 1000 randomly generated subjects with scores on 9 true/false items differing in difficulty. Selectively drop the hardest items for the 1/3 lowest subjects, and the 4 easiest items for the 1/3 top subjects (this is a crude example of what tailored testing would do). Then score these subjects: \begin{scriptsize} <>= v9 <- sim.irt(9,1000,-2.,2.,mod="normal") #dichotomous items items <- v9$items test <- irt.fa(items) total <- rowSums(items) ord <- order(total) items <- items[ord,] #now delete some of the data - note that they are ordered by score items[1:333,5:9] <- NA items[334:666,3:7] <- NA items[667:1000,1:4] <- NA scores <- scoreIrt(test,items) unitweighted <- scoreIrt(items=items,keys=rep(1,9)) scores.df <- data.frame(true=v9$theta[ord],scores,unitweighted) colnames(scores.df) <- c("True theta","irt theta","total","fit","rasch","total","fit") @ \end{scriptsize} These results are seen in Figure~\ref{fig:scoreIrt.pdf}. \begin{figure}[htbp] \begin{center} \caption{IRT based scoring and total test scores for 1000 simulated subjects. True theta values are reported and then the IRT and total scoring systems. } <>= pairs.panels(scores.df,pch='.',gap=0) title('Comparing true theta for IRT, Rasch and classically based scoring',line=3) @ \label{fig:scoreIrt.pdf} \end{center} \end{figure} \section{Multilevel modeling} Correlations between individuals who belong to different natural groups (based upon e.g., ethnicity, age, gender, college major, or country) reflect an unknown mixture of the pooled correlation within each group as well as the correlation of the means of these groups. These two correlations are independent and do not allow inferences from one level (the group) to the other level (the individual). When examining data at two levels (e.g., the individual and by some grouping variable), it is useful to find basic descriptive statistics (means, sds, ns per group, within group correlations) as well as between group statistics (over all descriptive statistics, and overall between group correlations). Of particular use is the ability to decompose a matrix of correlations at the individual level into correlations within group and correlations between groups. \subsection{Decomposing data into within and between level correlations using \pfun{statsBy}} There are at least two very powerful packages (\Rpkg{nlme} and \Rpkg{multilevel}) which allow for complex analysis of hierarchical (multilevel) data structures. \pfun{statsBy} is a much simpler function to give some of the basic descriptive statistics for two level models. This follows the decomposition of an observed correlation into the pooled correlation within groups (rwg) and the weighted correlation of the means between groups which is discussed by \cite{pedhazur:97} and by \cite{bliese:09} in the multilevel package. \begin{equation} r_{xy} = \eta_{x_{wg}} * \eta_{y_{wg}} * r_{xy_{wg}} + \eta_{x_{bg}} * \eta_{y_{bg}} * r_{xy_{bg} } \end{equation} where $r_{xy} $ is the normal correlation which may be decomposed into a within group and between group correlations $r_{xy_{wg}}$ and $r_{xy_{bg}} $ and $\eta$ (eta) is the correlation of the data with the within group values, or the group means. \subsection{Generating and displaying multilevel data} \pfun{withinBetween} is an example data set of the mixture of within and between group correlations. The within group correlations between 9 variables are set to be 1, 0, and -1 while those between groups are also set to be 1, 0, -1. These two sets of correlations are crossed such that V1, V4, and V7 have within group correlations of 1, as do V2, V5 and V8, and V3, V6 and V9. V1 has a within group correlation of 0 with V2, V5, and V8, and a -1 within group correlation with V3, V6 and V9. V1, V2, and V3 share a between group correlation of 1, as do V4, V5 and V6, and V7, V8 and V9. The first group has a 0 between group correlation with the second and a -1 with the third group. See the help file for \pfun{withinBetween} to display these data. \pfun{sim.multilevel} will generate simulated data with a multilevel structure. The \pfun{statsBy.boot} function will randomize the grouping variable ntrials times and find the statsBy output. This can take a long time and will produce a great deal of output. This output can then be summarized for relevant variables using the \pfun{statsBy.boot.summary} function specifying the variable of interest. Consider the case of the relationship between various tests of ability when the data are grouped by level of education (statsBy(sat.act)) or when affect data are analyzed within and between an affect manipulation (statsBy(affect) ). \section{Set Correlation and Multiple Regression from the correlation matrix} An important generalization of multiple regression and multiple correlation is \iemph{set correlation} developed by \cite{cohen:set} and discussed by \cite{cohen:03}. Set correlation is a multivariate generalization of multiple regression and estimates the amount of variance shared between two sets of variables. Set correlation also allows for examining the relationship between two sets when controlling for a third set. This is implemented in the \pfun{setCor} function. Set correlation is $$R^{2} = 1 - \prod_{i=1}^n(1-\lambda_{i})$$ where $\lambda_{i}$ is the ith eigen value of the eigen value decomposition of the matrix $$R = R_{xx}^{-1}R_{xy}R_{xx}^{-1}R_{xy}^{-1}.$$ Unfortunately, there are several cases where set correlation will give results that are much too high. This will happen if some variables from the first set are highly related to those in the second set, even though most are not. In this case, although the set correlation can be very high, the degree of relationship between the sets is not as high. In this case, an alternative statistic, based upon the average canonical correlation might be more appropriate. \pfun{setCor} has the additional feature that it will calculate multiple and partial correlations from the correlation or covariance matrix rather than the original data. Consider the correlations of the 6 variables in the \pfun{sat.act} data set. First do the normal multiple regression, and then compare it with the results using \pfun{setCor}. Two things to notice. \pfun{setCor} works on the \emph{correlation} or \emph{covariance} or \emph{raw data} matrix, and thus if using the correlation matrix, will report standardized $\hat{\beta}$ weights. Secondly, it is possible to do several multiple regressions simultaneously. If the number of observations is specified, or if the analysis is done on raw data, statistical tests of significance are applied. For this example, the analysis is done on the correlation matrix rather than the raw data. \begin{scriptsize} <>= C <- cov(sat.act,use="pairwise") model1 <- lm(ACT~ gender + education + age, data=sat.act) summary(model1) @ Compare this with the output from \pfun{setCor}. <>= #compare with mat.regress setCor(c(4:6),c(1:3),C, n.obs=700) @ \end{scriptsize} Note that the \pfun{setCor} analysis also reports the amount of shared variance between the predictor set and the criterion (dependent) set. This set correlation is symmetric. That is, the $R^{2}$ is the same independent of the direction of the relationship. For a much more detailed discussion of \pfun{setCor} see the \href{https://personality-project.org/r/psych/HowTo/mediation.pdf}{mediation, moderation and regression analysis} tutorial. \section{Simulation functions} It is particularly helpful, when trying to understand psychometric concepts, to be able to generate sample data sets that meet certain specifications. By knowing ``truth" it is possible to see how well various algorithms can capture it. Several of the \pfun{sim} functions create artificial data sets with known structures. A number of functions in the psych package will generate simulated data. These functions include \pfun{sim} for a factor simplex, and \pfun{sim.simplex} for a data simplex, \pfun{sim.circ} for a circumplex structure, \pfun{sim.congeneric} for a one factor factor congeneric model, \pfun{sim.dichot} to simulate dichotomous items, \pfun{sim.hierarchical} to create a hierarchical factor model, \pfun{sim.item} is a more general item simulation, \pfun{sim.minor} to simulate major and minor factors, \pfun{sim.omega} to test various examples of omega, \pfun{sim.parallel} to compare the efficiency of various ways of determining the number of factors, \pfun{sim.rasch} to create simulated rasch data, \pfun{sim.irt} to create general 1 to 4 parameter IRT data by calling \pfun{sim.npl} 1 to 4 parameter logistic IRT or \pfun{sim.npn} 1 to 4 paramater normal IRT, \pfun{sim.structural} a general simulation of structural models, and \pfun{sim.anova} for ANOVA and lm simulations, and \pfun{sim.vss}. Some of these functions are separately documented and are listed here for ease of the help function. See each function for more detailed help. \begin{description} \item [\pfun{sim}] The default version is to generate a four factor simplex structure over three occasions, although more general models are possible. \item [\pfun{sim.simple}] Create major and minor factors. The default is for 12 variables with 3 major factors and 6 minor factors. \item [\pfun{sim.structure}] To combine a measurement and structural model into one data matrix. Useful for understanding structural equation models. \item [\pfun{sim.hierarchical}] To create data with a hierarchical (bifactor) structure. \item [\pfun{sim.congeneric}] To create congeneric items/tests for demonstrating classical test theory. This is just a special case of sim.structure. \item [\pfun{sim.circ}] To create data with a circumplex structure. \item [\pfun{sim.item}]To create items that either have a simple structure or a circumplex structure. \item [\pfun{sim.dichot}] Create dichotomous item data with a simple or circumplex structure. \item[\pfun{sim.rasch}] Simulate a 1 parameter logistic (Rasch) model. \item[\pfun{sim.irt}] Simulate a 2 parameter logistic (2PL) or 2 parameter Normal model. Will also do 3 and 4 PL and PN models. \item[\pfun{sim.multilevel}] Simulate data with different within group and between group correlational structures. \end{description} Some of these functions are described in more detail in the companion vignette: \href{"psych_for_sem.pdf"}{psych for sem}. The default values for \pfun{sim.structure} is to generate a 4 factor, 12 variable data set with a simplex structure between the factors. Two data structures that are particular challenges to exploratory factor analysis are the simplex structure and the presence of minor factors. Simplex structures \pfun{sim.simplex} will typically occur in developmental or learning contexts and have a correlation structure of r between adjacent variables and $r^n$ for variables n apart. Although just one latent variable (r) needs to be estimated, the structure will have nvar-1 factors. Many simulations of factor structures assume that except for the major factors, all residuals are normally distributed around 0. An alternative, and perhaps more realistic situation, is that the there are a few major (big) factors and many minor (small) factors. The challenge is thus to identify the major factors. \pfun{sim.minor} generates such structures. The structures generated can be thought of as having a a major factor structure with some small correlated residuals. Although coefficient $\omega_h$ is a very useful indicator of the general factor saturation of a unifactorial test (one with perhaps several sub factors), it has problems with the case of multiple, independent factors. In this situation, one of the factors is labelled as ``general'' and the omega estimate is too large. This situation may be explored using the \pfun{sim.omega} function. The four irt simulations, \pfun{sim.rasch}, \pfun{sim.irt}, \pfun{sim.npl} and \pfun{sim.npn}, simulate dichotomous items following the Item Response model. \pfun{sim.irt} just calls either \pfun{sim.npl} (for logistic models) or \pfun{sim.npn} (for normal models) depending upon the specification of the model. The logistic model is \begin{equation} P(x | \theta_i, \delta_j, \gamma_j, \zeta_j )= \gamma_j + \frac{\zeta_j - \gamma_j}{1+e^{\alpha_j(\delta_j - \theta_i}}. \end{equation} where $\gamma$ is the lower asymptote or guessing parameter, $\zeta$ is the upper asymptote (normally 1), $\alpha_j$ is item discrimination and $\delta_j$ is item difficulty. For the 1 Paramater Logistic (Rasch) model, gamma=0, zeta=1, alpha=1 and item difficulty is the only free parameter to specify. (Graphics of these may be seen in the demonstrations for the logistic function.) The normal model (\pfun{irt.npn} calculates the probability using \fun{pnorm} instead of the logistic function used in \pfun{irt.npl}, but the meaning of the parameters are otherwise the same. With the a = $\alpha$ parameter = 1.702 in the logiistic model the two models are practically identical. \section{Graphical Displays} Many of the functions in the \Rpkg{psych} package include graphic output and examples have been shown in the previous figures. After running \pfun{fa}, \pfun{iclust}, \pfun{omega}, \pfun{irt.fa}, plotting the resulting object is done by the \pfun{plot.psych} function as well as specific diagram functions. e.g., (but not shown) \begin{scriptsize} \begin{Schunk} \begin{Sinput} f3 <- fa(Thurstone,3) plot(f3) fa.diagram(f3) c <- iclust(Thurstone) plot(c) #a pretty boring plot iclust.diagram(c) #a better diagram c3 <- iclust(Thurstone,3) plot(c3) #a more interesting plot data(bfi) e.irt <- irt.fa(bfi[11:15]) plot(e.irt) ot <- omega(Thurstone) plot(ot) omega.diagram(ot) \end{Sinput} \end{Schunk} \end{scriptsize} The ability to show path diagrams to represent factor analytic and structural models is discussed in somewhat more detail in the accompanying vignette, \href{"psych_for_sem.pdf"}{psych for sem}. Basic routines to draw path diagrams are included in the \pfun{dia.rect} and accompanying functions. These are used by the \pfun{fa.diagram}, \pfun{structure.diagram} and \pfun{iclust.diagram} functions. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= xlim=c(0,10) ylim=c(0,10) plot(NA,xlim=xlim,ylim=ylim,main="Demontration of dia functions",axes=FALSE,xlab="",ylab="") ul <- dia.rect(1,9,labels="upper left",xlim=xlim,ylim=ylim) ll <- dia.rect(1,3,labels="lower left",xlim=xlim,ylim=ylim) lr <- dia.ellipse(9,3,"lower right",xlim=xlim,ylim=ylim) ur <- dia.ellipse(9,9,"upper right",xlim=xlim,ylim=ylim) ml <- dia.ellipse(3,6,"middle left",xlim=xlim,ylim=ylim) mr <- dia.ellipse(7,6,"middle right",xlim=xlim,ylim=ylim) bl <- dia.ellipse(1,1,"bottom left",xlim=xlim,ylim=ylim) br <- dia.rect(9,1,"bottom right",xlim=xlim,ylim=ylim) dia.arrow(from=lr,to=ul,labels="right to left") dia.arrow(from=ul,to=ur,labels="left to right") dia.curved.arrow(from=lr,to=ll$right,labels ="right to left") dia.curved.arrow(to=ur,from=ul$right,labels ="left to right") dia.curve(ll$top,ul$bottom,"double") #for rectangles, specify where to point dia.curved.arrow(mr,ur,"up") #but for ellipses, just point to it. dia.curve(ml,mr,"across") dia.arrow(ur,lr,"top down") dia.curved.arrow(br$top,lr$bottom,"up") dia.curved.arrow(bl,br,"left to right") dia.arrow(bl,ll$bottom) dia.curved.arrow(ml,ll$right) dia.curved.arrow(mr,lr$top) @ \end{scriptsize} \caption{The basic graphic capabilities of the dia functions are shown in this figure.} \label{fig:dia} \end{center} \end{figure} \section{Miscellaneous functions} A number of functions have been developed for some very specific problems that don't fit into any other category. The following is an incomplete list. Look at the \iemph{Index} for \Rpkg{psych} for a list of all of the functions. \begin{description} \item [\pfun{block.random}] Creates a block randomized structure for n independent variables. Useful for teaching block randomization for experimental design. \item [\pfun{df2latex}] is useful for taking tabular output (such as a correlation matrix or that of \pfun{describe} and converting it to a \LaTeX{} table. May be used when Sweave is not convenient. \item [\pfun{cor2latex}] Will format a correlation matrix in APA style in a \LaTeX{} table. See also \pfun{fa2latex} and \pfun{irt2latex}. \item [\pfun{cosinor}] One of several functions for doing \iemph{circular statistics}. This is important when studying mood effects over the day which show a diurnal pattern. See also \pfun{circadian.mean}, \pfun{circadian.cor} and \pfun{circadian.linear.cor} for finding circular means, circular correlations, and correlations of circular with linear data. \item[\pfun{fisherz}] Convert a correlation to the corresponding Fisher z score. \item [\pfun{geometric.mean}] also \pfun{harmonic.mean} find the appropriate mean for working with different kinds of data. \item [\pfun{ICC}] and \pfun{cohen.kappa} are typically used to find the reliability for raters. \item [\pfun{headtail}] combines the \fun{head} and \fun{tail} functions to show the first and last lines of a data set or output. \item [\pfun{topBottom}] Same as headtail. Combines the \fun{head} and \fun{tail} functions to show the first and last lines of a data set or output, but does not add ellipsis between. \item [\pfun{mardia}] calculates univariate or multivariate (Mardia's test) skew and kurtosis for a vector, matrix, or data.frame \item [\pfun{p.rep}] finds the probability of replication for an F, t, or r and estimate effect size. \item [\pfun{partial.r}] partials a y set of variables out of an x set and finds the resulting partial correlations. (See also \pfun{setCor}.) \item [\pfun{rangeCorrection}] will correct correlations for restriction of range. \item [\pfun{reverse.code}] will reverse code specified items. Done more conveniently in most \Rpkg{psych} functions, but supplied here as a helper function when using other packages. \item [\pfun{superMatrix}] Takes two or more matrices, e.g., A and B, and combines them into a ``Super matrix'' with A on the top left, B on the lower right, and 0s for the other two quadrants. A useful trick when forming complex keys, or when forming example problems. \end{description} \section{Data sets} A number of data sets for demonstrating psychometric techniques are included in the \Rpkg{psych} package. These include six data sets showing a hierarchical factor structure (five cognitive examples, \pfun{Thurstone}, \pfun{Thurstone.33}, \pfun{Holzinger}, \pfun{Bechtoldt.1}, \pfun{Bechtoldt.2}, and one from health psychology \pfun{Reise}). One of these (\pfun{Thurstone}) is used as an example in the \Rpkg{sem} package as well as \cite{mcdonald:tt}. The original data are from \cite{thurstone:41} and reanalyzed by \cite{bechtoldt:61}. Personality item data representing five personality factors on 25 items (\pfun{bfi}) or 13 personality inventory scores (\pfun{epi.bfi}), and 14 multiple choice iq items (\pfun{iqitems}). The \pfun{vegetables} example has paired comparison preferences for 9 vegetables. This is an example of Thurstonian scaling used by \cite{guilford:54} and \cite{nunnally:67}. Other data sets include \pfun{cubits}, \pfun{peas}, and \pfun{heights} from Galton. \begin{description} \item[Thurstone] Holzinger-Swineford (1937) introduced the bifactor model of a general factor and uncorrelated group factors. The Holzinger correlation matrix is a 14 * 14 matrix from their paper. The Thurstone correlation matrix is a 9 * 9 matrix of correlations of ability items. The Reise data set is 16 * 16 correlation matrix of mental health items. The Bechtholdt data sets are both 17 x 17 correlation matrices of ability tests. \item [bfi] 25 personality self report items taken from the International Personality Item Pool (ipip.ori.org) were included as part of the Synthetic Aperture Personality Assessment (\iemph{SAPA}) web based personality assessment project. The data from 2800 subjects are included here as a demonstration set for scale construction, factor analysis and Item Response Theory analyses. \item [sat.act] Self reported scores on the SAT Verbal, SAT Quantitative and ACT were collected as part of the Synthetic Aperture Personality Assessment (\iemph{SAPA}) web based personality assessment project. Age, gender, and education are also reported. The data from 700 subjects are included here as a demonstration set for correlation and analysis. \item [epi.bfi] A small data set of 5 scales from the Eysenck Personality Inventory, 5 from a Big 5 inventory, a Beck Depression Inventory, and State and Trait Anxiety measures. Used for demonstrations of correlations, regressions, graphic displays. \item [iq] 14 multiple choice ability items were included as part of the Synthetic Aperture Personality Assessment (\iemph{SAPA}) web based personality assessment project. The data from 1000 subjects are included here as a demonstration set for scoring multiple choice inventories and doing basic item statistics. \item [galton] Two of the earliest examples of the correlation coefficient were Francis Galton's data sets on the relationship between mid parent and child height and the similarity of parent generation peas with child peas. \pfun{galton} is the data set for the Galton height. \pfun{peas} is the data set Francis Galton used to ntroduce the correlation coefficient with an analysis of the similarities of the parent and child generation of 700 sweet peas. \item[Dwyer] \cite{dwyer:37} introduced a method for \emph{factor extension} (see \pfun{fa.extension} that finds loadings on factors from an original data set for additional (extended) variables. This data set includes his example. \item [miscellaneous] \pfun{cities} is a matrix of airline distances between 11 US cities and may be used for demonstrating multiple dimensional scaling. \pfun{vegetables} is a classic data set for demonstrating Thurstonian scaling and is the preference matrix of 9 vegetables from \cite{guilford:54}. Used by \cite{guilford:54,nunnally:67,nunnally:bernstein:84}, this data set allows for examples of basic scaling techniques. \end{description} \section{Development version and a users guide} The most recent development version is available as a source file at the repository maintained at \href{ href="http://personality-project.org/r"}{\url{http://personality-project.org/r}}. That version will have removed the most recently discovered bugs (but perhaps introduced other, yet to be discovered ones). To download and install that version for either Macs or PCs: \begin{Rinput} install.packages("psych",repos="http://personality-project.org/r", type="source") \end{Rinput} Although the individual help pages for the \Rpkg{psych} package are available as part of \R{} and may be accessed directly (e.g. ?psych) , the full manual for the \pfun{psych} package is also available as a pdf at \url{http://personality-project.org/r/psych_manual.pdf} %psych\_manual.pdf. News and a history of changes are available in the NEWS and CHANGES files in the source files. To view the most recent news, \begin{Rinput} news(Version > "1.2.8",package="psych") \end{Rinput} \section{Psychometric Theory} The \Rpkg{psych} package has been developed to help psychologists do basic research. Many of the functions were developed to supplement a book (\url{http://personality-project.org/r/book} An introduction to Psychometric Theory with Applications in \R{} \citep{revelle:intro} More information about the use of some of the functions may be found in the book . For more extensive discussion of the use of \Rpkg{psych} in particular and \R{} in general, consult \url{http://personality-project.org/r/r.guide.html} A short guide to R. \section{SessionInfo} This document was prepared using the following settings. \begin{tiny} <>= sessionInfo() @ \end{tiny} \newpage %\bibliography{/Volumes/WR/Documents/Active/book/all} %\bibliography{../../../../all} \begin{thebibliography}{} \bibitem[\protect\astroncite{Bechtoldt}{1961}]{bechtoldt:61} Bechtoldt, H. (1961). \newblock An empirical study of the factor analysis stability hypothesis. \newblock {\em Psychometrika}, 26(4):405--432. \bibitem[\protect\astroncite{Blashfield}{1980}]{blashfield:80} Blashfield, R.~K. 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({in prep}). \newblock {\em An introduction to psychometric theory with applications in {R}}. \newblock Springer. \bibitem[\protect\astroncite{Revelle et~al.}{2011}]{rcw:methods} Revelle, W., Condon, D., and Wilt, J. (2011). \newblock Methodological advances in differential psychology. \newblock In Chamorro-Premuzic, T., Furnham, A., and von Stumm, S., editors, {\em Handbook of Individual Differences}, chapter~2, pages 39--73. Wiley-Blackwell. \bibitem[\protect\astroncite{Revelle and Rocklin}{1979}]{revelle:vss} Revelle, W. and Rocklin, T. (1979). \newblock {Very Simple Structure} - alternative procedure for estimating the optimal number of interpretable factors. \newblock {\em Multivariate Behavioral Research}, 14(4):403--414. \bibitem[\protect\astroncite{Revelle et~al.}{2010}]{rwr:sapa} Revelle, W., Wilt, J., and Rosenthal, A. (2010). \newblock Individual differences in cognition: New methods for examining the personality-cognition link. \newblock In Gruszka, A., Matthews, G., and Szymura, B., editors, {\em Handbook of Individual Differences in Cognition: Attention, Memory and Executive Control}, chapter~2, pages 27--49. Springer, New York, N.Y. \bibitem[\protect\astroncite{Revelle and Zinbarg}{2009}]{rz:09} Revelle, W. and Zinbarg, R.~E. (2009). \newblock Coefficients alpha, beta, omega and the glb: comments on {Sijtsma}. \newblock {\em Psychometrika}, 74(1):145--154. \bibitem[\protect\astroncite{Schmid and Leiman}{1957}]{schmid:57} Schmid, J.~J. and Leiman, J.~M. (1957). \newblock The development of hierarchical factor solutions. \newblock {\em Psychometrika}, 22(1):83--90. \bibitem[\protect\astroncite{Shrout and Fleiss}{1979}]{shrout:79} Shrout, P.~E. and Fleiss, J.~L. (1979). \newblock Intraclass correlations: Uses in assessing rater reliability. \newblock {\em Psychological Bulletin}, 86(2):420--428. \bibitem[\protect\astroncite{Sneath and Sokal}{1973}]{sneath:73} Sneath, P. H.~A. and Sokal, R.~R. (1973). \newblock {\em Numerical taxonomy: the principles and practice of numerical classification}. \newblock A Series of books in biology. W. H. Freeman, San Francisco. \bibitem[\protect\astroncite{Sokal and Sneath}{1963}]{sokal:63} Sokal, R.~R. and Sneath, P. H.~A. (1963). \newblock {\em Principles of numerical taxonomy}. \newblock A Series of books in biology. W. H. Freeman, San Francisco. \bibitem[\protect\astroncite{Spearman}{1904}]{spearman:rho} Spearman, C. (1904). \newblock The proof and measurement of association between two things. \newblock {\em The American Journal of Psychology}, 15(1):72--101. \bibitem[\protect\astroncite{ten Berge et~al.}{1999}]{tenBerge.99} ten Berge, J.~M., Krijnen, W.~P., Wansbeek, T., and Shapiro, A. (1999). \newblock Some new results on correlation-preserving factor scores prediction methods. \newblock {\em Linear Algebra and its Applications}, 289(1-3):311 -- 318. \bibitem[\protect\astroncite{Thorburn}{1918}]{thornburn:1918} Thorburn, W.~M. (1918). \newblock The myth of {Occam's} razor. \newblock {\em Mind}, 27:345--353. \bibitem[\protect\astroncite{Thurstone and Thurstone}{1941}]{thurstone:41} Thurstone, L.~L. and Thurstone, T.~G. (1941). \newblock {\em Factorial studies of intelligence}. \newblock The University of Chicago press, Chicago, Ill. \bibitem[\protect\astroncite{Tryon}{1935}]{tryon:35} Tryon, R.~C. (1935). \newblock A theory of psychological components--an alternative to "mathematical factors.". \newblock {\em Psychological Review}, 42(5):425--454. \bibitem[\protect\astroncite{Tryon}{1939}]{tryon:39} Tryon, R.~C. (1939). \newblock {\em Cluster analysis}. \newblock Edwards Brothers, Ann Arbor, Michigan. \bibitem[\protect\astroncite{Velicer}{1976}]{velicer:76} Velicer, W. (1976). \newblock Determining the number of components from the matrix of partial correlations. \newblock {\em Psychometrika}, 41(3):321--327. \bibitem[\protect\astroncite{Zinbarg et~al.}{2005}]{zinbarg:pm:05} Zinbarg, R.~E., Revelle, W., Yovel, I., and Li, W. (2005). \newblock Cronbach's {$\alpha$}, {Revelle's} {$\beta$}, and {McDonald's} {$\omega_H$}: Their relations with each other and two alternative conceptualizations of reliability. \newblock {\em Psychometrika}, 70(1):123--133. \bibitem[\protect\astroncite{Zinbarg et~al.}{2006}]{zinbarg:apm:06} Zinbarg, R.~E., Yovel, I., Revelle, W., and McDonald, R.~P. (2006). \newblock Estimating generalizability to a latent variable common to all of a scale's indicators: A comparison of estimators for {$\omega_h$}. \newblock {\em Applied Psychological Measurement}, 30(2):121--144. \end{thebibliography} \printindex \end{document} psychTools/R/0000755000176200001440000000000014153443073012626 5ustar liggesuserspsychTools/R/utlilites.r0000644000176200001440000001027013606736744015043 0ustar liggesusers#Various useful utility functions # list the files in a directory holding a particular file, or a particular directory "filesList" <- function(f=NULL) { if(is.null(f)) { f <- file.choose()} if(dir.exists(f)) {dir <- f } else {dir <- dirname(f)} #find a file in the directory you want files.list <- list.files(dir) message("\nFiles in the directory", dir, "\n") #although I prefer cat, CRAN seems to prefer message return(files.list) } "filesInfo" <- function(f=NULL,max=NULL) { if(is.null(f)) { f <- file.choose()} if(dir.exists(f)) {dir <- f } else {dir <- dirname(f)} files.list <- list.files(dir) if(is.null(max)) max <- length(files.list) info <- list(max) for(i in 1:max) { info[[i]] <- file.info(file.path(dir,files.list[i]))} info.df <- info[[1]] for (i in 2:max) { info.df <- rbind(info.df,info[[i]])} info.df <-cbind(file=1:max,info.df) return(info.df) } "fileScan" <- function(f=NULL,nlines=3,max=NULL,from=1,filter=NULL) { cat("\n Just the content of files will be shown (not directories)\n") if(is.null(f)) {f <- file.choose()} #find a file in the directory you want dir <- dirname(f) #the directory where the file was found files.list <- list.files(dir) dir.list <- list.dirs(dir,full.names=FALSE) files.list <- files.list[!files.list %in% dir.list] #get rid of directories if(!is.null(filter)) {select <- grep(filter,files.list,ignore.case=TRUE) #these are the ones that match filter files.list <- files.list[select]} n.files <- length(files.list) if(!is.null(max)) n.files <- max + from for (i in from:n.files) { file <- files.list[i] path <- file.path(dir,file) suffix <- file_ext(file) if(suffix %in% c("xls","xlsx","doc","sav","data","dat","rds","R","r","RDS", "XPT","xpt","Rda","rda","Rdata","RData","rdata","SYD","syd","sys","jmp","sas7bdat")) { cat("\nFile = ",i, "Name = ", file, "Was skipped") } else { # temp <- scan(path,what="raw",nlines=nlines) temp <- readLines(path,n=nlines) cat("\nFile = ",i, "Name = ", file, "\n",temp,"\n")} } return(dir) } #a work around the failure of file.choose(new=TRUE) to work in Rstudio "fileCreate" <- function(newName="new.file") { cat("Search for a file in the directory where you want to create a new file") fn <- file.choose() dir <- dirname(fn) new.path <- file.path(dir,newName) message("\nAre you sure you want to create a new file named ",new.path,"?\n") ok <- readline(prompt="Yes or No ") if(any(c("Y","y") %in% ok)) { if(!file.exists(new.path)) { file.create(new.path) return(new.path) } else {message('\nFile already exists, try a different name')} }else {message("fileCreate was cancelled")} } #Completely rewritten 1/20/18 to follow the help pages for order more closely #sort a data frame according to one or multiple columns #will only work for data.frames (not matrices) #needs to not quit if there is nothing to do dfOrder <- function(object,columns=NULL,absolute=FALSE,ascending=TRUE) { if(is.matrix(object)) {mat<- TRUE object <- as.data.frame(object)} else {mat<-FALSE} if(is.null(ncol(object))) {return(object)} else { if(is.null(columns)) columns <- 1:ncol(object) nc <- length(columns) cn <- colnames(object) if(ascending) {temp <- rep(1,nc)} else {temp <- rep(-1,nc)} if(is.character(columns)) { #treat character strings temp [strtrim(columns,1)=="-"] <- -1 if(any(temp < 0 ) ) {columns <- sub("-","",columns) } } else {temp[columns < 0] <- -1 columns <- abs(columns) } if(is.character(columns) ) { for (i in 1:length(columns)) {columns[i] <- (which(colnames(object) == columns[i])) } } columns <- colnames(object)[as.numeric(columns)] if(absolute) { temp.object<- t(t(abs(psych::char2numeric(object[columns]))) * temp) } else { temp.object<- t(t(psych::char2numeric(object[columns])) * temp)} temp.object <- data.frame(temp.object) ord <- do.call(order,temp.object) if(mat) object <- as.matrix(object) if(length(ord) > 1) { return(object[ord,]) }else {return(object)} #added length test 4/26/18 } } psychTools/R/read.clipboard.R0000644000176200001440000002646313727766077015657 0ustar liggesusers# a number of functions to read data from the clipboard for both Macs and PCs "read.clipboard" <- function(header=TRUE,...) { MAC<-Sys.info()[1]=="Darwin" #are we on a Mac using the Darwin system? if (!MAC ) {if (header) return(read.table(file("clipboard"),header=TRUE,...)) else return(read.table(file("clipboard"),...)) } else { if (header) {return(read.table(pipe("pbpaste"),header=TRUE,...))} else { return(read.table(pipe("pbpaste"),...))}} } "read.clipboard.csv" <- function(header=TRUE,sep=',',...) { #same as read.clipboard(sep=',') MAC<-Sys.info()[1]=="Darwin" #are we on a Mac using the Darwin system? if (!MAC ) {if (header) read.clipboard<-read.table(file("clipboard"),header=TRUE,sep,...) else read.clipboard<-read.table(file("clipboard"),sep=sep,...) } else { if (header) read.clipboard<- read.table(pipe("pbpaste"),header=TRUE,sep,...) else read.clipboard<- read.table(pipe("pbpaste") ,sep=sep,...)} } #corrected November 8, 2008 to work with header=FALSE "read.clipboard.tab" <- function(header=TRUE,sep='\t',...) { #same as read.clipboard(sep='\t') MAC<-Sys.info()[1]=="Darwin" #are we on a Mac using the Darwin system? if (!MAC ) {if (header) read.clipboard<-read.table(file("clipboard"),header=TRUE,sep,...) else read.clipboard<-read.table(file("clipboard"),sep=sep,...) } else { if (header) read.clipboard<- read.table(pipe("pbpaste"),header=TRUE,sep,...) else read.clipboard<- read.table(pipe("pbpaste") ,sep=sep,...)} } #corrected November 8, 2008 to work with header=FALSE #adapted from John Fox's read.moments function #modified October 31, 2010 to be able to read row names as first column #corrected September 2, 2011 to be able to read row names as first column but without the diagonal "read.clipboard.lower" <- function( diag = TRUE,names=FALSE,...) { MAC<-Sys.info()[1]=="Darwin" #are we on a Mac using the Darwin system? if (!MAC ) { con <- file("clipboard") } else { con <- pipe("pbpaste" )} xij <- scan(con,what="char") close(con) m <- length(xij) d <- if (diag |names) 1 else -1 n <- floor((sqrt(1 + 8 * m) - d)/2) if(names) {name <- xij[cumsum(1:n)] xij <- xij[-cumsum(seq(1:n))] d <- if (diag ) 1 else -1 n <- floor((sqrt(1 + 8 * (m-n)) - d)/2) } xij <- as.numeric(xij) X <- diag(n) X[upper.tri(X, diag = diag)] <- xij diagonal <- diag(X) X <- t(X) + X diag(X) <- diagonal if(!names) name <- paste("V",1:n,sep="") if(!names) name <- paste("V",1:n,sep="") if(names && !diag) {rownames(X) <- colnames(X) <- c(name,paste("V",n,sep="")) } else {rownames(X) <- colnames(X) <- name } return(X) } #fixed April 30, 2016 "read.clipboard.upper" <- function( diag = TRUE,names=FALSE,...) { MAC<-Sys.info()[1]=="Darwin" #are we on a Mac using the Darwin system? if (!MAC ) { con <- file("clipboard") } else { con <- pipe("pbpaste" )} xij <- scan(con,what="char") close(con) m <- length(xij) d <- if (diag | names) 1 else -1 n <- floor((sqrt(1 + 8 * m) - d )/2) #solve the quadratic for n if(names) { name <- xij[1:n] xij <- xij[-c(1:n)] } xij <- as.numeric(xij) X <- diag(n) X[lower.tri(X, diag = diag)] <- xij diagonal <- diag(X) X <- t(X) + X diag(X) <- diagonal if(!names) name <- paste("V",1:n,sep="") rownames(X) <- colnames(X) <- name return(X) } #added March, 2010 to read fixed width input "read.clipboard.fwf" <- function(header=FALSE,widths=rep(1,10),...) { # MAC<-Sys.info()[1]=="Darwin" #are we on a Mac using the Darwin system? if (!MAC ) {if (header) read.clipboard<-read.fwf(file("clipboard"),header=TRUE,widths=widths,...) else read.clipboard<-read.fwf(file("clipboard"),widths=widths,...) } else { if (header) read.clipboard<- read.fwf(pipe("pbpaste"),header=TRUE,widths=widths,...) else read.clipboard<- read.fwf(pipe("pbpaste"),widths=widths,...)} } #added May, 2014 to read from https files "read.https" <- function(filename,header=TRUE) { temp <- tempfile() #create a temporary file download.file(filename,destfile=temp,method="curl") #copy the https file to temp result <- read.table(temp,header=header) #now, do the normal read.table command unlink(temp) #get rid of the temporary file return(result)} #give us the result #Some useful helper functions #August, 2016 #modified Jan/April 2017 to include SAS xpt #modifed May, 2019 to not load files into the .environment, but give instructions of how to do. #modified August 2020 to be a little cleaner in code "read.file" <- function(file=NULL,header=TRUE,use.value.labels=FALSE,to.data.frame=TRUE,sep=",",quote="\"",widths=NULL,f=NULL,filetype=NULL,...) { if(missing(f) && missing(file)) f <- file.choose() if(missing(f) && !missing(file)) f <- file suffix <- file_ext(f) if(!missing(filetype)) suffix <- filetype #consider the various abbreviations possible if(suffix %in% c("txt","TXT","text","data","dat","DAT")) suffix <- "txt" if(suffix %in% c("R","r")) suffix <- "R" if (suffix %in% c("rds","RDS","Rds")) suffix <- "rds" if (suffix %in% c("Rda","rda","Rdata","RData","rdata")) suffix <- "Rda" if (suffix %in% c("SYD","syd","sys","SYS")) suffix <- "SYD" if(!missing(widths)) { result <- read.fwf(f,widths,...) message("The fixed width file ", f, "has been loaded.") } else { switch(suffix, sav = {result <- read.spss(f,use.value.labels=use.value.labels,to.data.frame=to.data.frame) message('Data from the SPSS sav file ', f ,' has been loaded.')}, csv = {result <- read.table(f,header=header,sep=sep,quote=quote,...) message('Data from the .csv file ', f ,' has been loaded.')}, tab = {result <- read.table(f,header=header,sep="\t",...) message('Data from the .tab file ', f , ' has been loaded.')}, txt = {result <- read.table(f,header=header,...) message('Data from the .txt file ', f , ' has been loaded.') }, # TXT = {result <- read.table(f,header=header,...) # message('Data from the .TXT file ', f , ' has been loaded.') }, # text = {result <- read.table(f,header=header,...) # message('Data from the .text file ', f , ' has been loaded.')}, # data = {result <- read.table(f,header=header,...) # message('Data from the .data file ', f , ' has been loaded.')}, # dat = {result <- read.table(f,header=header,...) # message('Data from the .data file ', f , ' has been loaded.')}, # DAT = {result <- read.table(f,header=header,...) # message('Data from the .data file ', f , ' has been loaded.')}, rds = {result <- try(readRDS(f,...),silent=TRUE) if(class(result) == "try-error") { result <- f message("I had problems reading this file, \ntry load('",f,"') instead. \nCaution, this might replace an object currently in your environment.")} else { message('File ',f ,' has been loaded.')}}, R = {result <- dget(f,...) message('File ',f ,' has been loaded.')}, # r = {result <- dget(f,...) # message('File ',f ,' has been loaded.')}, # Rds = {result <- readRDS(f,...) # message('File ',f ,' has been loaded.')}, # RDS = {result <- readRDS(f,...) # message('File ',f ,' has been loaded.')}, XPT = { result <- read.xport(f,...) message('File ',f ,' has been loaded.')}, xpt = { result <- read.xport(f,...) message('File ',f ,' has been loaded.')}, #the next options use load rather than read #if we return f and it has the same name as the file loaded, this wipes out the file Rda = {result <- f #not helpful if the # load(f, .GlobalEnv) # load(f) message("To load this ",suffix," file (or these files) you need to load('",f,"') \nCaution, this might replace an object currently in your environment.") }, # rda = {result <- f # load(result) # message("To load this ",suffix," file (or these files) you need to load('",f,"') \nCaution, this might replace an object currently in your environment.") }, # Rdata = {result <- f # # message("To load this file (or these files) you need to load('",f,"') \nCaution, this might replace an object currently in your environment.") }, # RData = {result <- f # # message("To load this file (or these files) you need to load('",f,"') \nCaution, this might replace an object currently in your environment.") }, # rdata = {result <- f # # message("To load this file (or these files) you need to load('",f,"') \nCaution, this might replace an object currently in your environment.") }, SYD = {result <- read.systat(f,to.data.frame=to.data.frame ) message('Data from the systat SYD file ', f ,' has been loaded.')}, # syd = {result <- read.systat(f,to.data.frame=to.data.frame ) # message('Data from the systat syd file ', f ,' has been loaded.')}, # sys = {result <- read.systat(f,to.data.frame=to.data.frame ) # message('Data from the systat sys file ', f ,' has been loaded.')}, #this section handles (or complains) about jmp and SAS files. jmp = {result <- f message('I am sorrry. To read this .jmp file, it must first be saved as either a "txt" or "csv" file. If you insist on using SAS formats, try .xpt or .XPT')}, sas7bdat = {result <- f message('I am sorry. To read this .sas7bdat file, it must first be saved as either a xpt, or XPT file in SAS, or as a "txt" or "csv" file. ?read.ssd in foreign for help.')}, {message ("I am sorry. \nI can not tell from the suffix what file type is this. Rather than try to read it, I will let you specify a better format.") } ) } return (result) } "read.file.spss" <- function(file=NULL,use.value.labels=FALSE,to.data.frame=TRUE,...) { if(missing(f) && missing(file)) f <- file.choose() if(missing(f) &&!missing(file)) f <- file result <- read.spss(f,use.value.labels=use.value.labels,to.data.frame=to.data.frame,...) message('Data from the SPSS sav file ', f ,' has been loaded.') return(result) } "read.file.csv" <- function(file=NULL,header=TRUE,f=NULL,...) { if(missing(f) && missing(file)) f <- file.choose() if(missing(f) &&!missing(file)) f <- file read.table(f,header=header,sep=",",...) } "write.file" <- function(x,file=NULL,row.names=FALSE,f=NULL,...) { if(missing(f) && missing(file)) f <- file.choose(TRUE) if(missing(f) &&!missing(file)) f <- file suffix <- file_ext(f) switch(suffix, txt = {write.table(x,f, row.names=row.names, ...)}, text = {write.table(x,f,row.names=row.names,...)}, csv = {write.table(x,f,sep=",", row.names=row.names,...) }, R = {dput(x,f,...) }, r = {dput(x,f, ...) }, rda = {save(x,file=f,...)}, Rda ={save(x,file=f,...)}, Rds = {saveRDS(x,f)}, rds = {saveRDS(x,f)}, write.table(x,f,row.names=row.names) #the default for unspecified types ) } "write.file.csv" <- function(x,file=NULL,row.names=FALSE,f=NULL,...) { if(missing(f) && missing(file)) f <- file.choose(TRUE) if(missing(f) &&!missing(file)) f <- file write.table(x,f,sep=",",row.names=row.names,...) } psychTools/R/df2latex.R0000644000176200001440000004753514064402366014501 0ustar liggesusers#modified April 6, 2015 to return the table invisibly as well so it can be embedded in a Sweave document #November 22, 2013 Modified with help from Davide Morselli to allow for "stars" #also allows for printing straight text (char=TRUE) #cor2latex was modified following Davide Morselli's suggestion to allow direct calculation of the correlations #added { and } before and after each variable name to allow siunitx to work with stars #added the absolute value in the big comparison for cor2latex and df2latex #added the ability to round numbers even though other columns are character (01/24/20) #modified May 29, 2021 to addthe ability to do long tables # "df2latex" <- function(x,digits=2,rowlabels=TRUE,apa=TRUE,short.names=TRUE, font.size ="scriptsize",big.mark=NULL, drop.na=TRUE, heading="A table from the psych package in R", caption="df2latex",label="default",char=FALSE,stars=FALSE,silent=FALSE,file=NULL,append=FALSE,cut=0,big=.0,abbrev=NULL,long=FALSE) { #first set up the table if(is.null(abbrev)) abbrev<- digits + 3 nvar <- dim(x)[2] rname<- rownames(x) tempx <- x comment <- paste("%", match.call()) if(long) { header <- paste0("\\begin{center} \\begin{",font.size,"} \\begin{longtable}") header <- c(header,"{l",rep("r",(nvar)),"}\n") header <- c(header,paste0(" \\caption{",caption,"} \\endfirsthead \\multicolumn{",nvar+1,"}{c} {{\\bfseries \\tablename\\ \\thetable{} -- continued from previous page}} \\\\ \\endhead \\hline \\multicolumn{",nvar+1,"}{|c|}{{Continued on next page}} \\\\ \\hline \\endfoot \\hline \\hline \\endlastfoot ")) #this wraps up the long table footer <- paste0("\\end{longtable} \\end{",font.size,"} \\end{center}") } else { header <- paste("\\begin{table}[htpb]", "\\caption{",caption,"} \\begin{center} \\begin{",font.size,"} \\begin{tabular}",sep="") if(stars) {if(rowlabels) { header <- c(header,"{l",rep("S",(nvar)),"}\n")} else {header <- c(header,"{",rep("S",(nvar+1)),"}\n")} } else { if(rowlabels) { header <- c(header,"{l",rep("r",(nvar)),"}\n")} else {header <- c(header,"{",rep("r",(nvar+1)),"}\n")} } if(apa) {header <- c(header, "\\multicolumn{",nvar,"}{l}{",heading,"}", '\\cr \n \\hline ') footer <- paste(" \\hline ")} else {footer <- NULL} if (stars){ footer <- paste(" \\hline \n \\multicolumn{7}{l}{\\scriptsize{\\emph{Note: }\\textsuperscript{***}$p<.001$; \\textsuperscript{**}$p<.01$; \\textsuperscript{*}$p<.05$",".}}" ,sep = "") }else{ footer <- paste(" \\hline ")} footer <- paste(footer," \\end{tabular} \\end{",font.size,"} \\end{center} \\label{",label,"} \\end{table} ",sep="" ) #end of not long } #now put the data into it if(big) all.x <- x #we need to keep the original format of the data to do the big operation if(!char) {if(!is.null(digits)) {if(is.numeric(x) ) {x <- round(x,digits=digits)} else {for(i in 1:ncol(x)) {if (is.numeric(x[,i])) x[,i] <- round(x[,i],2)} } if(cut > 0) x[abs(x) < cut] <- NA } } cname <- colnames(x) if (short.names) cname <- abbreviate(cname,minlength=abbrev) #cname <- 1:nvar names1 <- paste0("{",cname[1:(nvar-1)], "} & ") lastname <- paste0("{",cname[nvar],"}\\cr \n") if(apa) {allnames <- c("Variable & ",names1,lastname," \\hline \n")} else {if(rowlabels) {allnames <- c(" & ",names1,lastname,"\\cr \n")} else { allnames <- c(names1,lastname,"\\cr \n")}} if(!char) {if(is.null(big.mark)) { x <- format(x,drop0trailing=FALSE) if(big > 0) { for(i in 1:ncol(x)) {if (is.numeric(all.x[,i])) x[abs(all.x[,i] ) > big,i] <- paste0("\\bf{",x[abs(all.x[,i]) > big,i],"}") }} # if(is.numeric(tempx)) x[abs(tempx ) > big] <- paste0("\\bf{",x[abs(tempx) > big],"}") } } else #to keep the digits the same {x <- prettyNum(x,big.mark=",",drop0trailing=FALSE)} } else {if(big > 0) { x[!is.na(abs(as.numeric(all.x))>big) & abs(as.numeric(all.x))>big ] <- paste0("\\bf{", x[!is.na(abs(as.numeric(all.x))>big) & abs(as.numeric(all.x))>big ],"}") } } # x[!is.na(abs(as.numeric(x)) > big)]<- paste0("\\bf{", x[!is.na(abs(as.numeric(x)) > big)],"}") }} value <- apply(x,1,paste,collapse=" & ") #insert & between columns if(rowlabels) {value <- paste(sanitize.latex(rname)," & ",value)} else {value <- paste(" & ",value)} values <- paste(value, "\\cr", "\n") #add \\cr at the end of each row if(drop.na) values <- gsub("NA"," ",values,fixed=TRUE) #now put it all together if(!silent) {cat(comment,"\n") #a comment field saying where the data came from cat(header) #the header information cat(allnames) #the variable names cat(values) #the data cat(footer) #close it up with a footer } result <- c(header,allnames,values,footer) if(!is.null(file)) write.table(result,file=file,row.names=FALSE,col.names=FALSE,quote=FALSE,append=append) invisible(result) } #end df2latex cor2latex <- function (x, use = "pairwise", method="pearson", adjust="holm", stars = FALSE, digits=2, rowlabels = TRUE, lower = TRUE, apa = TRUE, short.names = TRUE, font.size = "scriptsize", heading = "A correlation table from the psych package in R.", caption = "cor2latex", label = "default",silent=FALSE,file=NULL,append=FALSE,cut=0,big=.0) { if(stars) heading <- paste(heading, "Adjust for multiple tests = ",adjust ) if (!is.na(class(x)[2]) & class(x)[2]=="corr.test") { #we already did the analysis, just report it r <- x$r p <- x$p} else { if (nrow(x) > ncol(x)) { #find the correlations x <- psych::corr.test(x, use=use,method=method,adjust=adjust) r <- x$r p <- x$p } else { #take the correlations as given r <- x p <- NULL } } r <- round(r, digits) r <- format(r, nsmall = digits,drop0trailing=FALSE) #this converts to character but keeps the right number of digits) if (lower) { r[upper.tri(r)] <- "~" } else { r[lower.tri(r)] <- "~" } if(isTRUE(stars && is.null(p))) stop("To print significance levels, x must be be either a data frame of observations or a correlation matrix created with the corr.test function of the package psych. If you are not interested in displaying signicance level set stars = FALSE") #p[upper.tri(p,diag=FALSE)] #the adjusted probability values mystars <- ifelse(p < .001, "{***}", ifelse(p < .01, "{**}", ifelse(p < .05, "{*}", ""))) mystars <- t(mystars) if(stars) { R <- matrix(paste(r,mystars,sep=""),ncol=ncol(r))} else {R <- r} diag(R) <- paste(diag(r), " ", sep="") rownames(R) <- colnames(r) colnames(R) <- colnames(r) if (lower) { R[upper.tri(R, diag = FALSE)] <- "" } else { R[lower.tri(R, diag = FALSE)] <- "" } if(stars) {char<- TRUE} else {char <- FALSE} return(df2latex(R, digits = digits, rowlabels = rowlabels, apa = apa, short.names = short.names, font.size = font.size, heading = heading, caption = caption, label = label, char=TRUE,stars = stars,silent=silent,file=file,append=append,cut=cut,big=big)) } "fa2latex" <- function(f,digits=2,rowlabels=TRUE,apa=TRUE,short.names=FALSE,cumvar=FALSE,cut=0,big=.3,alpha=.05,font.size ="scriptsize", heading="A factor analysis table from the psych package in R",caption="fa2latex",label="default",silent=FALSE,file=NULL,append=FALSE) { if(class(f)[2] == "fa.ci") { if(is.null(f$cip)) {px <- f$cis$p} else {px <- f$cip}} else {px <- NULL} #get the probabilities if we did fa.ci #if(class(f)[2] !="fa") f <- f$fa x <- unclass(f$loadings) if(!is.null(f$Phi)) {Phi <- f$Phi} else {Phi <- NULL} nfactors <- ncol(x) if(nfactors > 1) {if(is.null(Phi)) {h2 <- rowSums(x^2)} else {h2 <- diag(x %*% Phi %*% t(x)) }} else {h2 <-x^2} u2 <- 1- h2 vtotal <- sum(h2 + u2) if(cut > 0) x[abs(x) < cut] <- NA #modified May 13 following a suggestion from Daniel Zingaro if(!is.null(f$complexity)) {x <- data.frame(x,h2=h2,u2=u2,com=f$complexity) } else {x <- data.frame(x,h2=h2,u2=u2)} colnames(x)[which(colnames(x)=='h2')] <- '$h^2$' #added following a request from Alex Weiss 11/28/19 colnames(x)[which(colnames(x)=='u2')] <- '$u^2$' #first set up the table nvar <- dim(x)[2] comment <- paste("% Called in the psych package ", match.call()) header <- paste("\\begin{table}[htpb]", "\\caption{",caption,"} \\begin{center} \\begin{",font.size,"} \\begin{tabular}",sep="") header <- c(header,"{l",rep("r",nvar),"}\n") if(apa) header <- c(header, "\\multicolumn{",nvar,"}{l}{",heading,"}", '\\cr \n \\hline ') if(apa) {footer <- paste(" \\hline ")} footer <- paste(footer," \\end{tabular} \\end{",font.size,"} \\end{center} \\label{",label,"} \\end{table} ",sep="" ) #now put the data into it x <- round(x,digits=digits) cname <- colnames(x) if (short.names) cname <- 1:nvar names1 <- paste(cname[1:(nvar-1)], " & ") lastname <- paste(cname[nvar],"\\cr \n") if(apa) {allnames <- c("Variable & ",names1,lastname," \\hline \n")} else {allnames <- c(" & ",names1,lastname,"\\cr \n")} fx <- format(x,drop0trailing=FALSE) #to keep the digits the same {if(!is.null(px) && (cut == 0)) { temp <- fx[1:nfactors] temp[px < alpha] <- paste("\\bf{",temp[px < alpha],"}",sep="") fx[1:nfactors] <- temp } if(big > 0) {temp <- fx[1:nfactors] x <- x[1:nfactors] temp[!is.na(x) & (abs(x) > big)] <- paste("\\bf{",temp[!is.na(x) & (abs(x) > big)],"}",sep="") fx[1:nfactors] <- temp } value <- apply(fx,1,paste,collapse=" & ") #insert & between columns value <- gsub("NA", " ", value, fixed = TRUE) if(rowlabels) value <- {paste(sanitize.latex(names(value))," & ",value)} else {paste(" & ",value)} values <- paste(value, "\\cr", "\n") #add \\cr at the end of each row #now put it all together if(!silent) { cat(comment,"\n") #a comment field saying where the data came from cat(header) #the header information cat(allnames) #the variable names cat(values) #the factor loadings } #now find and show the variance accounted for x <- f$loadings #use the original values not the rounded ones nvar <- nrow(x) if(is.null(Phi)) {if(nfactors > 1) {vx <- colSums(x^2) } else { vx <- diag(t(x) %*% x) vx <- vx*nvar/vtotal }} else {vx <- diag(Phi %*% t(x) %*% x) vx <- vx*nvar/vtotal } #names(vx) <- colnames(x)[1:nvar] vx <- round(vx,digits) loads <- c("\\hline \\cr SS loadings &",paste(vx," & ",sep=""),"\\cr \n") if(!silent) { cat(loads)} summ <- NULL #varex <- rbind("SS loadings " = vx) if(cumvar) { provar <- round(vx/nvar,digits) summ <- c("Proportion Var &" ,paste( provar, " & ",sep=""),"\\cr \n") # cat("Proportion Var &" ,paste( provar, " & ",sep=""),"\\cr \n") if (nfactors > 1) {cumvar <- round(cumsum(vx/nvar),digits) cumfavar <- round(cumsum(vx/sum(vx)),digits=digits) summ <- c(summ, "Cumulative Var & ",paste( cumvar," & ", sep=""),"\\cr \n", "Cum. factor Var & ",paste(round(cumsum(vx/sum(vx)),digits=digits)," & ",sep=""),"\\cr \n") } if(!silent) {cat(summ) } } loads <- c(loads,summ) if(!is.null(Phi)) { summ <- c("\\cr \\hline \\cr \n") if(!silent) {cat(summ) } Phi <- round(Phi,digits) phi <- format(Phi,nsmall=digits) phi <-apply(phi,1,paste,collapse=" & ") phi <-paste(colnames(x)," &",phi) phi <- paste(phi, "\\cr", "\n") loads <- c(loads,summ,phi) if(!silent) { cat(phi)} } if(!silent) { cat(footer)} #close it up with a footer } values <- c(values,loads) result <- c(header,allnames,values,footer) if(!is.null(file)) write.table(result,file=file,row.names=FALSE,col.names=FALSE,quote=FALSE,append=append) invisible(result) } "irt2latex" <- function(f,digits=2,rowlabels=TRUE,apa=TRUE,short.names=FALSE,font.size ="scriptsize", heading="An IRT factor analysis table from R",caption="fa2latex" ,label="default",silent=FALSE,file=NULL,append=FALSE) { if(class(f)[2] != "polyinfo" ) {nf <- length(f$plot$sumInfo) } else {nf <- length(f$sumInfo) } #create nf tables for(i in (1:nf)) { if(class(f)[2] != "polyinfo" ) {x <- f$plot$sumInfo[[i]]} else {x <- f$sumInfo[[i]] } if(nf>1) { rowmax <- apply(x,1,max, na.rm=TRUE) rowmax <- which(rowmax <.001,arr.ind=TRUE) if(!is.null(rowmax)) x <- x[-rowmax,]} #first set up the table nvar <- ncol(x) comment <- paste("%", match.call()) header <- paste("\\begin{",font.size,"} \\begin{table}[htpb]", "\\caption{",caption,"} \\begin{center} \\begin{tabular}",sep="") header <- c(header,"{l",rep("r",nvar),"}\n") if(apa) header <- c(header, "\\multicolumn{",nvar,"}{l}{",heading," for factor " , i, " }", "\\cr \\hline \\cr", "\n & \\multicolumn{7}{c}{Item information at $\\theta$} \\cr \\cline{2-8} ") if(apa) {footer <- paste(" \\hline ")} footer <- paste(footer," \\end{tabular} \\end{center} \\label{",label,"} \\end{table} \\end{",font.size,"} ",sep="" ) #now put the data into it x <- round(x,digits=digits) cname <- colnames(x) if (short.names) cname <- 1:nvar names1 <- paste(cname[1:(nvar-1)], " & ") lastname <- paste(cname[nvar],"\\cr \n") if(apa) {allnames <- c("Item & ",names1,lastname," \\hline \n")} else {allnames <- c(" & ",names1,lastname,"\\cr \n")} x <- format(x,drop0trailing=FALSE) #to keep the digits the same value <- apply(x,1,paste,collapse=" & ") #insert & between columns if(rowlabels) value <- paste(sanitize.latex(names(value))," & ",value) values <- paste(value, "\\cr", "\n") #add \\cr at the end of each row #now put it all together if(class(f)[2] != "polyinfo" ) {test.info <- colSums(f$plot$sumInfo[[i]])} else {test.info <- colSums(f$sumInfo[[i]])} sem <- sqrt(1/test.info) reliab <- 1 - 1/test.info summary <- rbind(test.info,sem,reliab) summary <- round(summary,digits) summary <- format(summary,nsmall=digits) summary <- cbind(c("Test.info","SEM","Reliability"),summary) summary <- apply(summary,1,paste,collapse=" & ") summary <- paste(summary,"\\cr \n") if(!silent) { cat(comment,"\n") #a comment field saying where the data came from cat(header) #the header information cat(allnames) #the variable names cat(values) #the item information cat("\\hline \n & \\multicolumn{7}{c}{Summary statistics at $\\theta$} \\cr \\cline{2-8}") cat(summary) cat(footer) #close it up with a footer' } } result <- c(header,allnames,values,summary,footer) if(!is.null(file)) write.table(result,file=file,row.names=FALSE,col.names=FALSE,quote=FALSE,append=append) invisible(result) } #adapted from various sources, including xtable "sanitize.latex" <- function(astring) { result <- astring result <- gsub("&", "\\&", result, fixed = TRUE) result <- gsub("_", "\\_", result, fixed = TRUE) result <- gsub("%", "\\%", result, fixed = TRUE) return(result) } #added December 28, 2013 "omega2latex" <- function(f,digits=2,rowlabels=TRUE,apa=TRUE,short.names=FALSE,cumvar=FALSE,cut=.2,font.size ="scriptsize", heading="An omega analysis table from the psych package in R",caption="omega2latex",label="default",silent=FALSE,file=NULL,append=FALSE) { if(class(f)[2] == "omega" ) f$loadings <- f$schmid$sl x <- unclass(f$loadings) nfactors <- ncol(x) h2 <- rowSums(x^2) u2 <- 1- h2 vtotal <- sum(h2 + u2) #first set up the table nvar <- dim(x)[2] comment <- paste("% Called in the psych package ", match.call()) header <- paste("\\begin{",font.size,"} \\begin{table}[htpb]", "\\caption{",caption," with cut = ",cut,"\n $\\omega_h = ",round(f$omega_h,digits), "\\;\\;\\;\\alpha (\\lambda_3) = ",round(f$alpha,digits), "\\;\\;\\;\\lambda_6^* = ",round(f$G6,digits),"\\;\\;\\; \\omega_t = ",round(f$omega.tot,digits),"$ } \\begin{center} \\begin{tabular}",sep="") header <- c(header,"{l",rep("r",nvar),"}\n") if(apa) header <- c(header, "\\multicolumn{",nvar,"}{l}{",heading,"}", '\\cr \n \\hline ') if(apa) {footer <- paste(" \\hline ")} footer <- paste(footer," \\end{tabular} \\end{center} \\label{",label,"} \\end{table} \\end{",font.size,"} ",sep="" ) #now put the data into it x[abs(x) < cut] <- NA x <- round(x,digits=digits) cname <- colnames(x) if (short.names) cname <- 1:nvar names1 <- paste(cname[1:(nvar-1)], " & ") lastname <- paste(cname[nvar],"\\cr \n") if(apa) {allnames <- c("Variable & ",names1,lastname," \\hline \n")} else {allnames <- c(" & ",names1,lastname,"\\cr \n")} x <- format(x,drop0trailing=FALSE) #to keep the digits the same value <- apply(x,1,paste,collapse=" & ") #insert & between columns value <- gsub("NA", " ", value, fixed = TRUE) if(rowlabels) value <- {paste(sanitize.latex(names(value))," & ",value)} else {paste(" & ",value)} values <- paste(value, "\\cr", "\n") #add \\cr at the end of each row #now put it all together #now find and show the variance accounted for x <- f$loadings #use the original values nvar <- nrow(x) vx <- colSums(x^2)[1:(ncol(x)-3)] vx <- round(vx,digits) loads <- c("\\hline \\cr SS loadings &",paste(vx," & ",sep=""),"\\cr \n") if(!silent) { cat(comment,"\n") #a comment field saying where the data came from cat(header) #the header information cat(allnames) #the variable names cat(values) #the factor loadings cat(loads) cat(footer) #close it up with a footer } result <- c(header,allnames,values,loads,footer) if(!is.null(file)) write.table(result,file=file,row.names=FALSE,col.names=FALSE,quote=FALSE,append=append) invisible(result) } #added 1/6/14 "ICC2latex" <- function(icc,digits=2,rowlabels=TRUE,apa=TRUE,ci=TRUE, font.size ="scriptsize",big.mark=NULL, drop.na=TRUE, heading="A table from the psych package in R", caption="ICC2latex",label="default",char=FALSE,silent=FALSE,file=NULL,append=FALSE) { if((length(class(icc)) < 2 ) | (class(icc)[2] !="ICC")) icc <- psych::ICC(icc) #do the analysis in case we have not done it yet #first set up the table x <- icc$results nvar <- dim(x)[2] rname<- rownames(x) comment <- paste("%", match.call()) header <- paste("\\begin{",font.size,"} \\begin{table}[[htpb]", "\\caption{",caption,"} \\begin{tabular}",sep="") if(rowlabels) { header <- c(header,"{l",rep("r",(nvar)),"}\n")} else {header <- c(header,"{",rep("r",(nvar+1)),"}\n") } if(apa) {header <- c(header, "\\multicolumn{",5,"}{l}{",heading,"}", '\\cr \n \\hline ') footer <- paste(" \\hline \\cr \\multicolumn{ 5 }{c}{ Number of subjects = ", icc$n.obs, "Number of raters = ",icc$n.judge,"}")} else {footer <- NULL} footer <- paste(footer," \\end{tabular} \\label{",label,"} \\end{table} \\end{",font.size,"} ",sep="" ) #now put the data into it x[2:nvar] <- try(round(x[2:nvar],digits=digits)) cname <- colnames(x) if(!ci) nvar <- nvar-2 names1 <- paste(cname[1:(nvar-1)], " & ") lastname <- paste(cname[nvar],"\\cr \n") if(apa) {allnames <- c("Variable & ",names1,lastname," \\hline \n")} else {if(rowlabels) {allnames <- c(" & ",names1,lastname,"\\cr \n")} else { allnames <- c(names1,lastname,"\\cr \n")}} if(!char) {if(is.null(big.mark)) { x <- format(x[1:nvar],drop0trailing=FALSE)} else #to keep the digits the same {x <- prettyNum(x,big.mark=",",drop0trailing=FALSE)} } value <- apply(x,1,paste,collapse=" & ") #insert & between columns if(rowlabels) {value <- paste(sanitize.latex(rname)," & ",value)} else {value <- paste(" & ",value)} values <- paste(value, "\\cr", "\n") #add \\cr at the end of each row if(drop.na) values <- gsub("NA"," ",values,fixed=TRUE) #now put it all together if(!silent) { cat(comment,"\n") #a comment field saying where the data came from cat(header) #the header information cat(allnames) #the variable names cat(values) #the data cat(footer) #close it up with a 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endstream endobj 551 0 obj << /Type /XRef /Index [0 552] /Size 552 /W [1 3 1] /Root 549 0 R /Info 550 0 R /ID [<7E991076FC2327D1AE056BFB9D205EED> <7E991076FC2327D1AE056BFB9D205EED>] /Length 1326 /Filter /FlateDecode >> stream x%KlVE;W(ZJ)J[z/BKi镶 n4&F6b۰5€7&Jcb`cjssO$$X(|'xAf2 f&HՐ5Rf@!1J[~B:BfC?5Upӱq,4MX)L##8}㮣#KVMm?`6J@o '}`|“/t_ttY.u>g7t0v:v:v:vc4e >+cudAʬ]iP̶TMh+ n6K6@[d9Ƹv"Aݭ1]_rfGF;Xek5f? 9$L٬i,$wmf#l59%켶Jv 8 :&.0z); zy2uʆًTYn׹ VE-πQ0&Wuڸ%wRk LZk*DM[3"`GM%0ow-0RMeh'X+XA бlɽՑ^cod-yۧorPZ@+h1  $ڳCiKN endstream endobj startxref 456895 %%EOF psychTools/inst/doc/factor.R0000644000176200001440000003655114153443020015552 0ustar liggesusers### R code from vignette source 'factor.Rnw' ################################################### ### code chunk number 1: factor.Rnw:384-389 ################################################### options(width=160) library(psych) library(psychTools) data(sat.act) describe(sat.act) #basic descriptive statistics ################################################### ### code chunk number 2: pairspanels ################################################### png( 'pairspanels.png' ) pairs.panels(sat.act,pch='.') dev.off() ################################################### ### code chunk number 3: factor.Rnw:564-565 ################################################### lowerCor(sat.act) ################################################### ### code chunk number 4: factor.Rnw:572-578 ################################################### female <- subset(sat.act,sat.act$gender==2) male <- subset(sat.act,sat.act$gender==1) lower <- lowerCor(male[-1]) upper <- lowerCor(female[-1]) both <- lowerUpper(lower,upper) round(both,2) ################################################### ### code chunk number 5: factor.Rnw:584-586 ################################################### diffs <- lowerUpper(lower,upper,diff=TRUE) round(diffs,2) ################################################### ### code chunk number 6: corplot.png ################################################### png('corplot.png') cor.plot(Thurstone,numbers=TRUE,main="9 cognitive variables from Thurstone") dev.off() ################################################### ### code chunk number 7: circplot.png ################################################### png('circplot.png') circ <- sim.circ(24) r.circ <- cor(circ) cor.plot(r.circ,main='24 variables in a circumplex') dev.off() ################################################### ### code chunk number 8: factor.Rnw:808-810 ################################################### f3t <- fa(Thurstone,3,n.obs=213) f3t ################################################### ### code chunk number 9: factor.Rnw:830-833 ################################################### f3 <- fa(Thurstone,3,n.obs = 213,fm="pa") f3o <- target.rot(f3) f3o ################################################### ### code chunk number 10: factor.Rnw:854-856 ################################################### f3w <- fa(Thurstone,3,n.obs = 213,fm="wls") print(f3w,cut=0,digits=3) ################################################### ### code chunk number 11: factor.Rnw:868-869 ################################################### plot(f3t) ################################################### ### code chunk number 12: factor.Rnw:881-882 ################################################### fa.diagram(f3t) ################################################### ### code chunk number 13: factor.Rnw:901-903 ################################################### p3p <-principal(Thurstone,3,n.obs = 213,rotate="Promax") p3p ################################################### ### code chunk number 14: factor.Rnw:922-924 ################################################### om.h <- omega(Thurstone,n.obs=213,sl=FALSE) op <- par(mfrow=c(1,1)) ################################################### ### code chunk number 15: factor.Rnw:935-936 ################################################### om <- omega(Thurstone,n.obs=213) ################################################### ### code chunk number 16: factor.Rnw:969-971 ################################################### data(bfi) ic <- iclust(bfi[1:25]) ################################################### ### code chunk number 17: factor.Rnw:983-984 ################################################### summary(ic) #show the results ################################################### ### code chunk number 18: factor.Rnw:997-999 ################################################### data(bfi) r.poly <- polychoric(bfi[1:25]) #the ... indicate the progress of the function ################################################### ### code chunk number 19: factor.Rnw:1012-1014 ################################################### ic.poly <- iclust(r.poly$rho,title="ICLUST using polychoric correlations") iclust.diagram(ic.poly) ################################################### ### code chunk number 20: factor.Rnw:1025-1027 ################################################### ic.poly <- iclust(r.poly$rho,5,title="ICLUST using polychoric correlations for nclusters=5") iclust.diagram(ic.poly) ################################################### ### code chunk number 21: factor.Rnw:1038-1039 ################################################### ic.poly <- iclust(r.poly$rho,beta.size=3,title="ICLUST beta.size=3") ################################################### ### code chunk number 22: factor.Rnw:1051-1052 ################################################### print(ic,cut=.3) ################################################### ### code chunk number 23: factor.Rnw:1069-1071 ################################################### fa(bfi[1:10],2,n.iter=20) ################################################### ### code chunk number 24: factor.Rnw:1084-1086 ################################################### f4 <- fa(bfi[1:25],4,fm="pa") factor.congruence(f4,ic) ################################################### ### code chunk number 25: factor.Rnw:1095-1096 ################################################### factor.congruence(list(f3t,f3o,om,p3p)) ################################################### ### code chunk number 26: factor.Rnw:1111-1113 ################################################### faCor(Thurstone,c(3,3),fm=c("minres","pca"), rotate=c("oblimin","oblimin")) ################################################### ### code chunk number 27: factor.Rnw:1159-1160 ################################################### vss <- vss(bfi[1:25],title="Very Simple Structure of a Big 5 inventory") ################################################### ### code chunk number 28: factor.Rnw:1168-1169 ################################################### vss ################################################### ### code chunk number 29: factor.Rnw:1179-1180 ################################################### fa.parallel(bfi[1:25],main="Parallel Analysis of a Big 5 inventory") ################################################### ### code chunk number 30: factor.Rnw:1198-1203 ################################################### v16 <- sim.item(16) s <- c(1,3,5,7,9,11,13,15) f2 <- fa(v16[,s],2) fe <- fa.extension(cor(v16)[s,-s],f2) fa.diagram(f2,fe=fe) ################################################### ### code chunk number 31: factor.Rnw:1216-1218 ################################################### fe <- fa.extend(bfi,5,ov=1:25,ev=26:28) extension.diagram(fe) ################################################### ### code chunk number 32: factor.Rnw:1236-1238 ################################################### ba5 <- bassAckward(bfi[1:25], nfactors =c(2,3,4,5),plot=FALSE) baf <- bassAckward.diagram(ba5) ################################################### ### code chunk number 33: factor.Rnw:1252-1254 ################################################### # fa.lookup(baf$bass.ack[[5]],dictionary=bfi.dictionary[2]) ################################################### ### code chunk number 34: factor.Rnw:1307-1311 ################################################### set.seed(17) r9 <- sim.hierarchical(n=500,raw=TRUE)$observed round(cor(r9),2) alpha(r9) ################################################### ### code chunk number 35: factor.Rnw:1318-1320 ################################################### keys <- c(1,-1,1,1,1,1,1) alpha(attitude,keys) ################################################### ### code chunk number 36: factor.Rnw:1327-1329 ################################################### keys <- c(1,1,1,1,1,1,1) alpha(attitude,keys) ################################################### ### code chunk number 37: factor.Rnw:1336-1338 ################################################### items <- sim.congeneric(N=500,short=FALSE,low=-2,high=2,categorical=TRUE) #500 responses to 4 discrete items alpha(items$observed) #item response analysis of congeneric measures ################################################### ### code chunk number 38: factor.Rnw:1391-1392 ################################################### om.9 <- omega(r9,title="9 simulated variables") ################################################### ### code chunk number 39: factor.Rnw:1403-1404 ################################################### om.9 ################################################### ### code chunk number 40: factor.Rnw:1412-1413 ################################################### omegaSem(r9,n.obs=500) ################################################### ### code chunk number 41: factor.Rnw:1422-1423 ################################################### splitHalf(r9) ################################################### ### code chunk number 42: factor.Rnw:1445-1450 ################################################### keys <- make.keys(nvars=28,list(Agree=c(-1,2:5),Conscientious=c(6:8,-9,-10), Extraversion=c(-11,-12,13:15),Neuroticism=c(16:20), Openness = c(21,-22,23,24,-25)), item.labels=colnames(bfi)) keys ################################################### ### code chunk number 43: factor.Rnw:1457-1461 ################################################### keys.1<- make.keys(10,list(Agree=c(-1,2:5),Conscientious=c(6:8,-9,-10))) keys.2 <- make.keys(15,list(Extraversion=c(-1,-2,3:5),Neuroticism=c(6:10), Openness = c(11,-12,13,14,-15))) keys.25 <- superMatrix(list(keys.1,keys.2)) ################################################### ### code chunk number 44: factor.Rnw:1471-1473 ################################################### scores <- scoreItems(keys,bfi) scores ################################################### ### code chunk number 45: scores ################################################### png('scores.png') pairs.panels(scores$scores,pch='.',jiggle=TRUE) dev.off() ################################################### ### code chunk number 46: factor.Rnw:1499-1502 ################################################### r.bfi <- cor(bfi,use="pairwise") scales <- cluster.cor(keys,r.bfi) summary(scales) ################################################### ### code chunk number 47: factor.Rnw:1512-1518 ################################################### data(iqitems) iq.keys <- c(4,4,4, 6,6,3,4,4, 5,2,2,4, 3,2,6,7) score.multiple.choice(iq.keys,iqitems) #just convert the items to true or false iq.tf <- score.multiple.choice(iq.keys,iqitems,score=FALSE) describe(iq.tf) #compare to previous results ################################################### ### code chunk number 48: factor.Rnw:1536-1542 ################################################### data(iqitems) iq.keys <- c(4,4,4, 6,6,3,4,4, 5,2,2,4, 3,2,6,7) scores <- score.multiple.choice(iq.keys,iqitems,score=TRUE,short=FALSE) #note that for speed we can just do this on simple item counts rather than IRT based scores. op <- par(mfrow=c(2,2)) #set this to see the output for multiple items irt.responses(scores$scores,iqitems[1:4],breaks=11) ################################################### ### code chunk number 49: factor.Rnw:1568-1572 ################################################### set.seed(17) d9 <- sim.irt(9,1000,-2.,2.,mod="normal") #dichotomous items test <- irt.fa(d9$items) test ################################################### ### code chunk number 50: factor.Rnw:1579-1584 ################################################### op <- par(mfrow=c(3,1)) plot(test,type="ICC") plot(test,type="IIC") plot(test,type="test") op <- par(mfrow=c(1,1)) ################################################### ### code chunk number 51: factor.Rnw:1595-1598 ################################################### data(bfi) e.irt <- irt.fa(bfi[11:15]) e.irt ################################################### ### code chunk number 52: factor.Rnw:1605-1606 ################################################### e.info <- plot(e.irt,type="IIC") ################################################### ### code chunk number 53: factor.Rnw:1617-1618 ################################################### print(e.info,sort=TRUE) ################################################### ### code chunk number 54: factor.Rnw:1632-1633 ################################################### iq.irt <- irt.fa(iq.tf) ################################################### ### code chunk number 55: factor.Rnw:1643-1644 ################################################### iq.irt ################################################### ### code chunk number 56: factor.Rnw:1650-1651 ################################################### om <- omega(iq.irt$rho,4) ################################################### ### code chunk number 57: factor.Rnw:1665-1679 ################################################### v9 <- sim.irt(9,1000,-2.,2.,mod="normal") #dichotomous items items <- v9$items test <- irt.fa(items) total <- rowSums(items) ord <- order(total) items <- items[ord,] #now delete some of the data - note that they are ordered by score items[1:333,5:9] <- NA items[334:666,3:7] <- NA items[667:1000,1:4] <- NA scores <- scoreIrt(test,items) unitweighted <- scoreIrt(items=items,keys=rep(1,9)) scores.df <- data.frame(true=v9$theta[ord],scores,unitweighted) colnames(scores.df) <- c("True theta","irt theta","total","fit","rasch","total","fit") ################################################### ### code chunk number 58: factor.Rnw:1688-1690 ################################################### pairs.panels(scores.df,pch='.',gap=0) title('Comparing true theta for IRT, Rasch and classically based scoring',line=3) ################################################### ### code chunk number 59: factor.Rnw:1739-1743 ################################################### C <- cov(sat.act,use="pairwise") model1 <- lm(ACT~ gender + education + age, data=sat.act) summary(model1) ################################################### ### code chunk number 60: factor.Rnw:1746-1748 ################################################### #compare with mat.regress setCor(c(4:6),c(1:3),C, n.obs=700) ################################################### ### code chunk number 61: factor.Rnw:1833-1857 ################################################### xlim=c(0,10) ylim=c(0,10) plot(NA,xlim=xlim,ylim=ylim,main="Demontration of dia functions",axes=FALSE,xlab="",ylab="") ul <- dia.rect(1,9,labels="upper left",xlim=xlim,ylim=ylim) ll <- dia.rect(1,3,labels="lower left",xlim=xlim,ylim=ylim) lr <- dia.ellipse(9,3,"lower right",xlim=xlim,ylim=ylim) ur <- dia.ellipse(9,9,"upper right",xlim=xlim,ylim=ylim) ml <- dia.ellipse(3,6,"middle left",xlim=xlim,ylim=ylim) mr <- dia.ellipse(7,6,"middle right",xlim=xlim,ylim=ylim) bl <- dia.ellipse(1,1,"bottom left",xlim=xlim,ylim=ylim) br <- dia.rect(9,1,"bottom right",xlim=xlim,ylim=ylim) dia.arrow(from=lr,to=ul,labels="right to left") dia.arrow(from=ul,to=ur,labels="left to right") dia.curved.arrow(from=lr,to=ll$right,labels ="right to left") dia.curved.arrow(to=ur,from=ul$right,labels ="left to right") dia.curve(ll$top,ul$bottom,"double") #for rectangles, specify where to point dia.curved.arrow(mr,ur,"up") #but for ellipses, just point to it. dia.curve(ml,mr,"across") dia.arrow(ur,lr,"top down") dia.curved.arrow(br$top,lr$bottom,"up") dia.curved.arrow(bl,br,"left to right") dia.arrow(bl,ll$bottom) dia.curved.arrow(ml,ll$right) dia.curved.arrow(mr,lr$top) ################################################### ### code chunk number 62: factor.Rnw:1933-1934 ################################################### sessionInfo() psychTools/inst/doc/overview.Rnw0000644000176200001440000042511613714654447016531 0ustar liggesusers% \VignetteIndexEntry{Overview of the psych package for psychometrics} % \VignettePackage{psych} % \VignetteKeywords{multivariate} % \VignetteKeyword{models} % \VignetteKeyword{Hplot} %\VignetteDepends{psych} %\documentclass[doc]{apa} \documentclass[11pt]{article} %\documentclass[11pt]{amsart} \usepackage{geometry} % See geometry.pdf to learn the layout options. There are lots. \geometry{letterpaper} % ... or a4paper or a5paper or ... %\geometry{landscape} % Activate for for rotated page geometry \usepackage[parfill]{parskip} % Activate to begin paragraphs with an empty line rather than an indent \usepackage{graphicx} \usepackage{amssymb} \usepackage{epstopdf} \usepackage{mathptmx} \usepackage{helvet} \usepackage{courier} \usepackage{epstopdf} \usepackage{makeidx} % allows index generation \usepackage[authoryear,round]{natbib} \usepackage{gensymb} \usepackage{longtable} %\usepackage{geometry} \usepackage{amssymb} \usepackage{amsmath} %\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \usepackage{Sweave} %\usepackage{/Volumes/'Macintosh HD'/Library/Frameworks/R.framework/Versions/2.13/Resources/share/texmf/tex/latex/Sweave} %\usepackage[ae]{Rd} %\usepackage[usenames]{color} %\usepackage{setspace} \bibstyle{apacite} \bibliographystyle{apa} %this one plus author year seems to work? %\usepackage{hyperref} \usepackage[colorlinks=true,citecolor=blue]{hyperref} %this makes reference links hyperlinks in pdf! \DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \usepackage{multicol} % used for the two-column index \usepackage[bottom]{footmisc}% places footnotes at page bottom \let\proglang=\textsf \newcommand{\R}{\proglang{R}} %\newcommand{\pkg}[1]{{\normalfont\fontseries{b}\selectfont #1}} \newcommand{\Rfunction}[1]{{\texttt{#1}}} \newcommand{\fun}[1]{{\texttt{#1}\index{#1}\index{R function!#1}}} \newcommand{\pfun}[1]{{\texttt{#1}\index{#1}\index{R function!#1}\index{R function!psych package!#1}}}\newcommand{\Rc}[1]{{\texttt{#1}}} %R command same as Robject \newcommand{\Robject}[1]{{\texttt{#1}}} \newcommand{\Rpkg}[1]{{\textit{#1}\index{#1}\index{R package!#1}}} %different from pkg - which is better? \newcommand{\iemph}[1]{{\emph{#1}\index{#1}}} \newcommand{\wrc}[1]{\marginpar{\textcolor{blue}{#1}}} %bill's comments \newcommand{\wra}[1]{\textcolor{blue}{#1}} %bill's comments \newcommand{\ve}[1]{{\textbf{#1}}} %trying to get a vector command \makeindex % used for the subject index \title{An introduction to the psych package: Part II\\Scale construction and psychometrics} \author{William Revelle\\Department of Psychology\\Northwestern University} %\affiliation{Northwestern University} %\acknowledgements{Written to accompany the psych package. Comments should be directed to William Revelle \\ \url{revelle@northwestern.edu}} %\date{} % Activate to display a given date or no date \begin{document} \SweaveOpts{concordance=TRUE} \maketitle \tableofcontents \newpage \subsection{Jump starting the \Rpkg{psych} package--a guide for the impatient} You have installed \Rpkg{psych} (section \ref{sect:starting}) and you want to use it without reading much more. What should you do? \begin{enumerate} \item Activate the \Rpkg{psych} package: @ \begin{scriptsize} \begin{Schunk} \begin{Sinput} library(psych) library(psychTools) \end{Sinput} \end{Schunk} \end{scriptsize} \item Input your data (see the \href{https://personality-project.org/r/psych/intro.pdf}{Introduction to Psych} vignette section 3.1). There are two ways to do this: \begin{itemize} \item Find and read standard files using \pfun{read.file}. This will open a search window for your operating system which you can use to find the file. If the file has a suffix of .text, .txt, .csv, .data, .sav, .r, .R, .rds, .Rds, .rda, .Rda, .rdata, or .RData, then the file will be opened and the data will be read in. \begin{scriptsize} \begin{Schunk} \begin{Sinput} myData <- read.file() # find the appropriate file using your normal operating system \end{Sinput} \end{Schunk} \end{scriptsize} \item Alternatively, go to your friendly text editor or data manipulation program (e.g., Excel) and copy the data to the clipboard. Include a first line that has the variable labels. Paste it into \Rpkg{psych} using the \pfun{read.clipboard.tab} command: \begin{scriptsize} \begin{Schunk} \begin{Sinput} myData <- read.clipboard.tab() # if on the clipboard \end{Sinput} \end{Schunk} \end{scriptsize} Note that there are number of options for \pfun{read.clipboard} for reading in Excel based files, lower triangular files, etc. \end{itemize} \item Make sure that what you just read is right. Describe it (see the \href{https://personality-project.org/r/psych/intro.pdf}{Introduction to Psych} vignette section 3.3) on how to \pfun{describe} data) and perhaps look at the first and last few lines. If you have multiple groups, try \pfun{describeBy}. \begin{scriptsize} \begin{Schunk} \begin{Sinput} dim(myData) #What are the dimensions of the data? describe(myData) # or descrbeBy(myData,groups="mygroups") #for descriptive statistics by groups headTail(myData) #show the first and last n lines of a file \end{Sinput} \end{Schunk} \end{scriptsize} \item Look at the patterns in the data. If you have fewer than about 12 variables, look at the SPLOM (Scatter Plot Matrix) of the data using \pfun{pairs.panels} ( (see the \href{https://personality-project.org/r/psych/intro.pdf}{Introduction to Psych} vignette section 3.4 for a discussion of graphics)) . Then, use the \pfun{outlier} function to detect outliers. \begin{scriptsize} \begin{Schunk} \begin{Sinput} pairs.panels(myData) outlier(myData) \end{Sinput} \end{Schunk} \end{scriptsize} \item Note that you might have some weird subjects, probably due to data entry errors. Either edit the data by hand (use the \fun{edit} command) or just \pfun{scrub} the data). \begin{scriptsize} \begin{Schunk} \begin{Sinput} cleaned <- scrub(myData, max=9) #e.g., change anything great than 9 to NA \end{Sinput} \end{Schunk} \end{scriptsize} \item Graph the data with error bars for each variable ( (see the \href{https://personality-project.org/r/psych/intro.pdf}{Introduction to Psych} vignette section 3.1)). \begin{scriptsize} \begin{Schunk} \begin{Sinput} error.bars(myData) \end{Sinput} \end{Schunk} \end{scriptsize} \item Find the correlations of all of your data. \pfun{lowerCor} will by default find the pairwise correlations, round them to 2 decimals, and display the lower off diagonal matrix. \begin{itemize} \item Descriptively (just the values) (section \ref{sect:lowerCor}) \begin{scriptsize} \begin{Schunk} \begin{Sinput} r <- lowerCor(myData) #The correlation matrix, rounded to 2 decimals \end{Sinput} \end{Schunk} \end{scriptsize} \item Graphically (section \ref{sect:corplot}). Another way is to show a heat map of the correlations with the correlation values included. \begin{scriptsize} \begin{Schunk} \begin{Sinput} corPlot(r) #examine the many options for this function. \end{Sinput} \end{Schunk} \end{scriptsize} \item Inferentially (the values, the ns, and the p values) (section \ref{sect:corr.test}) \begin{scriptsize} \begin{Schunk} \begin{Sinput} corr.test(myData) \end{Sinput} \end{Schunk} \end{scriptsize} \end{itemize} \item Apply various regression models. Several functions are meant to do multiple regressions, either from the raw data or from a variance/covariance matrix, or a correlation matrix. \begin{itemize} \item \pfun{setCor} will take raw data or a correlation matrix and find (and graph the path diagram) for multiple y variables depending upon multiple x variables. \begin{scriptsize} \begin{Schunk} \begin{Sinput} myData <- sat.act colnames(myData) <- c("mod1","med1","x1","x2","y1","y2") setCor(y1 + y2 ~ x1 + x2 , data = myData) \end{Sinput} \end{Schunk} \end{scriptsize} \item \pfun{mediate} will take raw data or a correlation matrix and find (and graph the path diagram) for multiple y variables depending upon multiple x variables mediated through a mediation variable. It then tests the mediation effect using a boot strap. \begin{scriptsize} \begin{Schunk} \begin{Sinput} mediate(y1 + y2 ~ x1 + x2 + (med1) , data = myData) \end{Sinput} \end{Schunk} \end{scriptsize} \item \pfun{mediate} will take raw data and find (and graph the path diagram) a moderated multiple regression model for multiple y variables depending upon multiple x variables mediated through a mediation variable. It then tests the mediation effect using a boot strap. \begin{scriptsize} \begin{Schunk} \begin{Sinput} mediate(y1 + y2 ~ x1 + x2* mod1 +(med1), data = myData) \end{Sinput} \end{Schunk} \end{scriptsize} \end{itemize} \subsection{Psychometric functions are summarized in this vignette} Many additional functions, particularly designed for basic and advanced psychometrics are discussed more fully in this Vignette. A brief review of the functions available is included here. For basic data entry and descriptive statistics, see the Vignette Intro to Psych \url{https://personality-project.org/r}. In addition, there are helpful tutorials for \emph{Finding omega}, \emph{How to score scales and find reliability}, and for \emph{Using psych for factor analysis} at \url{https://personality-project.org/r}. \begin{itemize} \item Test for the number of factors in your data using parallel analysis (\pfun{fa.parallel}, section \ref{sect:fa.parallel}) or Very Simple Structure (\pfun{vss}, \ref{sect:vss}) . \begin{scriptsize} \begin{Schunk} \begin{Sinput} fa.parallel(myData) vss(myData) \end{Sinput} \end{Schunk} \end{scriptsize} \item Factor analyze (see section \ref{sect:fa}) the data with a specified number of factors (the default is 1), the default method is minimum residual, the default rotation for more than one factor is oblimin. There are many more possibilities (see sections \ref{sect:minres}-\ref{sect:wls}). Compare the solution to a hierarchical cluster analysis using the ICLUST algorithm \citep{revelle:iclust} (see section \ref{sect:iclust}). Also consider a hierarchical factor solution to find coefficient $\omega$ (see \ref{sect:omega}). \begin{scriptsize} \begin{Schunk} \begin{Sinput} fa(myData) iclust(myData) omega(myData) \end{Sinput} \end{Schunk} \end{scriptsize} If you prefer to do a principal components analysis you may use the \pfun{principal} function. The default is one component. \begin{scriptsize} \begin{Schunk} \begin{Sinput} principal(myData) \end{Sinput} \end{Schunk} \end{scriptsize} \item Some people like to find coefficient $\alpha$ as an estimate of reliability. This may be done for a single scale using the \pfun{alpha} function (see \ref{sect:alpha}). Perhaps more useful is the ability to create several scales as unweighted averages of specified items using the \pfun{scoreItems} function (see \ref{sect:score}) and to find various estimates of internal consistency for these scales, find their intercorrelations, and find scores for all the subjects. \begin{scriptsize} \begin{Schunk} \begin{Sinput} alpha(myData) #score all of the items as part of one scale. myKeys <- make.keys(nvar=20,list(first = c(1,-3,5,-7,8:10),second=c(2,4,-6,11:15,-16))) my.scores <- scoreItems(myKeys,myData) #form several scales my.scores #show the highlights of the results \end{Sinput} \end{Schunk} \end{scriptsize} \end{itemize} \end{enumerate} At this point you have had a chance to see the highlights of the \Rpkg{psych} package and to do some basic (and advanced) data analysis. You might find reading this entire vignette as well as the Overview Vignette to be helpful to get a broader understanding of what can be done in \R{} using the \Rpkg{psych}. Remember that the help command (?) is available for every function. Try running the examples for each help page. \newpage\newpage \section{Overview of this and related documents} The \Rpkg{psych} package \citep{psych} has been developed at Northwestern University since 2005 to include functions most useful for personality, psychometric, and psychological research. The package is also meant to supplement a text on psychometric theory \citep{revelle:intro}, a draft of which is available at \url{https://personality-project.org/r/book/}. Some of the functions (e.g., \pfun{read.file}, \pfun{read.clipboard}, \pfun{describe}, \pfun{pairs.panels}, \pfun{scatter.hist}, \pfun{error.bars}, \pfun{multi.hist}, \pfun{bi.bars}) are useful for basic data entry and descriptive analyses. Psychometric applications emphasize techniques for dimension reduction including factor analysis, cluster analysis, and principal components analysis. The \pfun{fa} function includes five methods of \iemph{factor analysis} (\iemph{minimum residual}, \iemph{principal axis}, \iemph{weighted least squares}, \iemph{generalized least squares} and \iemph{maximum likelihood} factor analysis). Principal Components Analysis (PCA) is also available through the use of the \pfun{principal} or \pfun{pca} functions. Determining the number of factors or components to extract may be done by using the Very Simple Structure \citep{revelle:vss} (\pfun{vss}), Minimum Average Partial correlation \citep{velicer:76} (\pfun{MAP}) or parallel analysis (\pfun{fa.parallel}) criteria. These and several other criteria are included in the \pfun{nfactors} function. Two parameter Item Response Theory (IRT) models for dichotomous or polytomous items may be found by factoring \pfun{tetrachoric} or \pfun{polychoric} correlation matrices and expressing the resulting parameters in terms of location and discrimination using \pfun{irt.fa}. Bifactor and hierarchical factor structures may be estimated by using Schmid Leiman transformations \citep{schmid:57} (\pfun{schmid}) to transform a hierarchical factor structure into a \iemph{bifactor} solution \citep{holzinger:37}. Higher order models can also be found using \pfun{fa.multi}. Scale construction can be done using the Item Cluster Analysis \citep{revelle:iclust} (\pfun{iclust}) function to determine the structure and to calculate reliability coefficients $\alpha$ \citep{cronbach:51}(\pfun{alpha}, \pfun{scoreItems}, \pfun{score.multiple.choice}), $\beta$ \citep{revelle:iclust,rz:09} (\pfun{iclust}) and McDonald's $\omega_h$ and $\omega_t$ \citep{mcdonald:tt} (\pfun{omega}). Guttman's six estimates of internal consistency reliability (\cite{guttman:45}, as well as additional estimates \citep{rz:09} are in the \pfun{guttman} function. The six measures of Intraclass correlation coefficients (\pfun{ICC}) discussed by \cite{shrout:79} are also available. For data with a a multilevel structure (e.g., items within subjects across time, or items within subjects across groups), the \pfun{describeBy}, \pfun{statsBy} functions will give basic descriptives by group. \pfun{StatsBy} also will find within group (or subject) correlations as well as the between group correlation. \pfun{multilevel.reliability} \pfun{mlr} will find various generalizability statistics for subjects over time and items. \pfun{mlPlot} will graph items over for each subject, \pfun{mlArrange} converts wide data frames to long data frames suitable for multilevel modeling. Graphical displays include Scatter Plot Matrix (SPLOM) plots using \pfun{pairs.panels}, correlation ``heat maps'' (\pfun{corPlot}) factor, cluster, and structural diagrams using \pfun{fa.diagram}, \pfun{iclust.diagram}, \pfun{structure.diagram} and \pfun{het.diagram}, as well as item response characteristics and item and test information characteristic curves \pfun{plot.irt} and \pfun{plot.poly}. This vignette is meant to give an overview of the \Rpkg{psych} package. That is, it is meant to give a summary of the main functions in the \Rpkg{psych} package with examples of how they are used for data description, dimension reduction, and scale construction. The extended user manual at \url{psych_manual.pdf} includes examples of graphic output and more extensive demonstrations than are found in the help menus. (Also available at \url{https://personality-project.org/r/psych_manual.pdf}). The vignette, psych for sem, at \url{psych_for_sem.pdf}, discusses how to use psych as a front end to the \Rpkg{sem} package of John Fox \citep{sem}. (The vignette is also available at \href{"https://personality-project.org/r/book/psych_for_sem.pdf"}{\url{https://personality-project.org/r/book/psych_for_sem.pdf}}). In addition, there are a growing number of ``HowTo"s at the personality project. Currently these include: \begin{enumerate} \item An \href{https://personality-project.org/r/psych/intro.pdf}{introduction} (vignette) of the \Rpkg{psych} package \item An \href{https://personality-project.org/r/psych/overview.pdf}{overview} (vignette) of the \Rpkg{psych} package \item \href{https://personality-project.org/r/psych/HowTo/getting_started.pdf}{Installing} \R{} and some useful packages \item Using \R{} and the \Rpkg{psych} package to find \href{https://personality-project.org/r/psych/HowTo/omega.pdf}{$omega_h$} and $\omega_t$. \item Using \R{} and the \Rpkg{psych} for \href{https://personality-project.org/r/psych/HowTo/factor.pdf}{factor analysis} and principal components analysis. \item Using the \pfun{scoreItems} function to find \href{https://personality-project.org/r/psych/HowTo/scoring.pdf}{scale scores and scale statistics}. \item Using \pfun{mediate} and \pfun{setCor} to do \href{https://personality-project.org/r/psych/HowTo/mediation.pdf}{mediation, moderation and regression analysis}. \end{enumerate} For a step by step tutorial in the use of the psych package and the base functions in R for basic personality research, see the guide for using \R{} for personality research at \url{https://personalitytheory.org/r/r.short.html}. For an \iemph{introduction to psychometric theory with applications in \R{}}, see the draft chapters at \url{https://personality-project.org/r/book}). \section{Getting started} \label{sect:starting} Some of the functions described in this overview require other packages. Particularly useful for rotating the results of factor analyses (from e.g., \pfun{fa}, \pfun{factor.minres}, \pfun{factor.pa}, \pfun{factor.wls}, or \pfun {principal}) or hierarchical factor models using \pfun{omega} or \pfun{schmid}, is the \Rpkg{GPArotation} package. These and other useful packages may be installed by first installing and then using the task views (\Rpkg{ctv}) package to install the ``Psychometrics" task view, but doing it this way is not necessary. \begin{Schunk} \begin{Sinput} install.packages("ctv") library(ctv) task.views("Psychometrics") \end{Sinput} \end{Schunk} The ``Psychometrics'' task view will install a large number of useful packages. To install the bare minimum for the examples in this vignette, it is necessary to install just 3 packages: \begin{Schunk} \begin{Sinput} install.packages(list(c("GPArotation","mnormt","psychTools") \end{Sinput} \end{Schunk} Because of the difficulty of installing the package \Rpkg{Rgraphviz}, alternative graphics have been developed and are available as \iemph{diagram} functions. If \Rpkg{Rgraphviz} is available, some functions will take advantage of it. An alternative is to use ``dot'' output of commands for any external graphics package that uses the dot language. \section{Basic data analysis} A number of \Rpkg{psych} functions facilitate the entry of data and finding basic descriptive statistics. These are described in more detail in the companion vignette: An introduction to the psych package: Part I which is also available from the personality-project site. \url{https://personality-project.org/r/psych/vignettes/intro.pdf}. Please consult that vignette first for information on how to read data (particularly using the \pfun{read.file} and \pfun{read.clipboard} commands), Also, the \pfun{describe} and \pfun{describeBy} functions are described in more detail in the introductory vignette. For even though you probably want to jump immediately to factor analyze your data, this is a mistake. It is very important to first describe them and look for weird responses. It is also useful to \pfun{scrub} your data when removing outliers, to graphically display them using \pfun{pairs.panesl} and \pfun{corPlot}. Basic multiple regression and moderated or mediated regressions may be done from either the raw data or from correlation matrices using \pfun{setCor}, or \pfun{mediation}. Remember, to run any of the \Rpkg{psych} functions, it is necessary to make the package active by using the \fun{library} command: \begin{Schunk} \begin{Sinput} library(psych) \end{Sinput} \end{Schunk} The other packages, once installed, will be called automatically by \Rpkg{psych}. It is possible to automatically load \Rpkg{psych} and other functions by creating and then saving a ``.First" function: e.g., \begin{Schunk} \begin{Sinput} .First <- function(x) {library(psych)} \end{Sinput} \end{Schunk} \section{Item and scale analysis} The main functions in the \Rpkg{psych} package are for analyzing the structure of items and of scales and for finding various estimates of scale reliability. These may be considered as problems of dimension reduction (e.g., factor analysis, cluster analysis, principal components analysis) and of forming and estimating the reliability of the resulting composite scales. \subsection{Dimension reduction through factor analysis and cluster analysis} \label{sect:fa} Parsimony of description has been a goal of science since at least the famous dictum commonly attributed to William of Ockham to not multiply entities beyond necessity\footnote{Although probably neither original with Ockham nor directly stated by him \citep{thornburn:1918}, Ockham's razor remains a fundamental principal of science.}. The goal for parsimony is seen in psychometrics as an attempt either to describe (components) or to explain (factors) the relationships between many observed variables in terms of a more limited set of components or latent factors. The typical data matrix represents multiple items or scales usually thought to reflect fewer underlying constructs\footnote{\cite{cattell:fa78} as well as \cite{maccallum:07} argue that the data are the result of many more factors than observed variables, but are willing to estimate the major underlying factors.}. At the most simple, a set of items can be be thought to represent a random sample from one underlying domain or perhaps a small set of domains. The question for the psychometrician is how many domains are represented and how well does each item represent the domains. Solutions to this problem are examples of \iemph{factor analysis} (\iemph{FA}), \iemph{principal components analysis} (\iemph{PCA}), and \iemph{cluster analysis} (\emph{CA}). All of these procedures aim to reduce the complexity of the observed data. In the case of FA, the goal is to identify fewer underlying constructs to explain the observed data. In the case of PCA, the goal can be mere data reduction, but the interpretation of components is frequently done in terms similar to those used when describing the latent variables estimated by FA. Cluster analytic techniques, although usually used to partition the subject space rather than the variable space, can also be used to group variables to reduce the complexity of the data by forming fewer and more homogeneous sets of tests or items. At the data level the data reduction problem may be solved as a \iemph{Singular Value Decomposition} of the original matrix, although the more typical solution is to find either the \iemph{principal components} or \iemph{factors} of the covariance or correlation matrices. Given the pattern of regression weights from the variables to the components or from the factors to the variables, it is then possible to find (for components) individual \index{component scores} \emph{component} or \iemph{cluster scores} or estimate (for factors) \iemph{factor scores}. Several of the functions in \Rpkg{psych} address the problem of data reduction. \begin{description} \item[\pfun{fa}] incorporates six alternative algorithms: \iemph{minres factor analysis}, \iemph{principal axis factor analysis}, \iemph{alpha factor analysis}, \iemph{weighted least squares factor analysis}, \iemph{generalized least squares factor analysis} and \iemph{maximum likelihood factor analysis}. That is, it includes the functionality of three other functions that are deprecated and will be eventually phased out. \begin{tiny} \item[\pfun{fa.poly} (deprecated) ] is useful when finding the factor structure of categorical items. \pfun{fa.poly} first finds the tetrachoric or polychoric correlations between the categorical variables and then proceeds to do a normal factor analysis. By setting the n.iter option to be greater than 1, it will also find confidence intervals for the factor solution. Warning. Finding polychoric correlations is very slow, so think carefully before doing so. These options are now part of the \iemph{fa} function and can be controlled by setting the cor parameter to `tet' or `poly'. \item [\pfun{factor.minres} (deprecated)] Minimum residual factor analysis is a least squares, iterative solution to the factor problem. minres attempts to minimize the residual (off-diagonal) correlation matrix. It produces solutions similar to maximum likelihood solutions, but will work even if the matrix is singular. \item [\pfun{factor.pa} (deprecated)] Principal Axis factor analysis is a least squares, iterative solution to the factor problem. PA will work for cases where maximum likelihood techniques (\fun{factanal}) will not work. The original communality estimates are either the squared multiple correlations (\pfun{smc}) for each item or 1. \item [\pfun{factor.wls} (deprecated)] Weighted least squares factor analysis is a least squares, iterative solution to the factor problem. It minimizes the (weighted) squared residual matrix. The weights are based upon the independent contribution of each variable. \end{tiny} \item [\pfun{principal}] Principal Components Analysis reports the largest n eigen vectors rescaled by the square root of their eigen values. Note that PCA is not the same as factor analysis and the two should not be confused. \item [\pfun{factor.congruence}] The congruence between two factors is the cosine of the angle between them. This is just the cross products of the loadings divided by the sum of the squared loadings. This differs from the correlation coefficient in that the mean loading is not subtracted before taking the products. \pfun{factor.congruence} will find the cosines between two (or more) sets of factor loadings. \item [\pfun{vss}] Very Simple Structure \cite{revelle:vss} applies a goodness of fit test to determine the optimal number of factors to extract. It can be thought of as a quasi-confirmatory model, in that it fits the very simple structure (all except the biggest c loadings per item are set to zero where c is the level of complexity of the item) of a factor pattern matrix to the original correlation matrix. For items where the model is usually of complexity one, this is equivalent to making all except the largest loading for each item 0. This is typically the solution that the user wants to interpret. The analysis includes the \pfun{MAP} criterion of \cite{velicer:76} and a $\chi^2$ estimate. \item [\pfun{nfactors}] combines VSS, MAP, and a number of other fit statistics. The depressing reality is that frequently these conventional fit estimates of the number of factors do not agree. \item [\pfun{fa.parallel}] The parallel factors technique compares the observed eigen values of a correlation matrix with those from random data. \item [\pfun{fa.plot}] will plot the loadings from a factor, principal components, or cluster analysis (just a call to plot will suffice). If there are more than two factors, then a SPLOM of the loadings is generated. \item[\pfun{fa.diagram}] replaces \pfun{fa.graph} and will draw a path diagram representing the factor structure. It does not require Rgraphviz and thus is probably preferred. \item[\pfun{fa.graph}] requires \fun{Rgraphviz} and will draw a graphic representation of the factor structure. If factors are correlated, this will be represented as well. \item[\pfun{iclust} ] is meant to do item cluster analysis using a hierarchical clustering algorithm specifically asking questions about the reliability of the clusters \citep{revelle:iclust}. Clusters are formed until either coefficient $\alpha$ \cite{cronbach:51} or $\beta$ \cite{revelle:iclust} fail to increase. \end{description} \subsubsection{Minimum Residual Factor Analysis} \label{sect:minres} The factor model is an approximation of a correlation matrix by a matrix of lower rank. That is, can the correlation matrix, $\vec{_nR_n}$ be approximated by the product of a factor matrix, $\vec{_nF_k}$ and its transpose plus a diagonal matrix of uniqueness. \begin{equation} R = FF' + U^2 \end{equation} The maximum likelihood solution to this equation is found by \fun{factanal} in the \Rpkg{stats} package as well as the \pfun{fa} function in \Rpkg{psych}. Seven alternatives are provided in \Rpkg{psych}, all of them are included in the \pfun{fa} function and are called by specifying the factor method (e.g., fm=``minres", fm=``pa", fm=``alpha" fm=`wls", fm=``gls", fm = ``min.rank", and fm=``ml"). In the discussion of the other algorithms, the calls shown are to the \pfun{fa} function specifying the appropriate method. \pfun{factor.minres} attempts to minimize the off diagonal residual correlation matrix by adjusting the eigen values of the original correlation matrix. This is similar to what is done in \fun{factanal}, but uses an ordinary least squares instead of a maximum likelihood fit function. The solutions tend to be more similar to the MLE solutions than are the \pfun{factor.pa} solutions. \iemph{min.res} is the default for the \pfun{fa} function. A classic data set, collected by \cite{thurstone:41} and then reanalyzed by \cite{bechtoldt:61} and discussed by \cite{mcdonald:tt}, is a set of 9 cognitive variables with a clear bi-factor structure \citep{holzinger:37}. The minimum residual solution was transformed into an oblique solution using the default option on rotate which uses an oblimin transformation (Table~\ref{tab:factor.minres}). Alternative rotations and transformations include ``none", ``varimax", ``quartimax", ``bentlerT", ``varimin'' and ``geominT" (all of which are orthogonal rotations). as well as ``promax", ``oblimin", ``simplimax", ``bentlerQ, and ``geominQ" and ``cluster" which are possible oblique transformations of the solution. The default is to do a oblimin transformation. The measures of factor adequacy reflect the multiple correlations of the factors with the best fitting linear regression estimates of the factor scores \citep{grice:01}. Note that if extracting more than one factor, and doing any oblique rotation, it is necessary to have the \Rpkg{GPArotation} installed. This is checked for in the appropriate functions. <>= if(!require('GPArotation')) {stop('GPArotation must be installed to do rotations')} @ \begin{table}[htpb] \caption{Three correlated factors from the Thurstone 9 variable problem. By default, the solution is transformed obliquely using oblimin. The extraction method is (by default) minimum residual.} \begin{scriptsize} \begin{center} <>= if(!require('GPArotation')) {stop('GPArotation must be installed to do rotations')} else { library(psych) library(psychTools) f3t <- fa(Thurstone,3,n.obs=213) f3t } @ \end{center} \end{scriptsize} \label{tab:factor.minres} \end{table}% \subsubsection{Principal Axis Factor Analysis} An alternative, least squares algorithm (included in \pfun{fa} with the fm=pa option or as a standalone function (\pfun{factor.pa}), does a Principal Axis factor analysis by iteratively doing an eigen value decomposition of the correlation matrix with the diagonal replaced by the values estimated by the factors of the previous iteration. This OLS solution is not as sensitive to improper matrices as is the maximum likelihood method, and will sometimes produce more interpretable results. It seems as if the SAS example for PA uses only one iteration. Setting the max.iter parameter to 1 produces the SAS solution. The solutions from the \pfun{fa}, the \pfun{factor.minres} and \pfun{factor.pa} as well as the \pfun{principal} functions can be rotated or transformed with a number of options. Some of these call the \Rpkg{GPArotation} package. Orthogonal rotations include \fun{varimax}, \fun{quartimax}, \pfun{varimin}, \pfun{bifactor} . Oblique transformations include \fun{oblimin}, \fun{quartimin}, \pfun{biquartimin} and then two targeted rotation functions \pfun{Promax} and \pfun{target.rot}. The latter of these will transform a loadings matrix towards an arbitrary target matrix. The default is to transform towards an independent cluster solution. Using the Thurstone data set, three factors were requested and then transformed into an independent clusters solution using \pfun{target.rot} (Table~\ref{tab:Thurstone}). \begin{table}[htpb] \caption{The 9 variable problem from Thurstone is a classic example of factoring where there is a higher order factor, g, that accounts for the correlation between the factors. The extraction method was principal axis. The transformation was a targeted transformation to a simple cluster solution.} \begin{center} \begin{scriptsize} <>= if(!require('GPArotation')) {stop('GPArotation must be installed to do rotations')} else { f3 <- fa(Thurstone,3,n.obs = 213,fm="pa") f3o <- target.rot(f3) f3o} @ \end{scriptsize} \end{center} \label{tab:Thurstone} \end{table} \subsubsection{Alpha Factor Analysis} Introduced by \cite{kaiser:65} and discussed by \cite{loehlin:17}, \emph{alpha factor analysis} factors the correlation matrix of correlations or covariances corrected for their communalities. This has the effect of making all correlations corrected for reliabiity to reflect their true, latent correlations. \emph{alpha factor analysis} was added in August, 2017 to increase the range of EFA options available. This is added more completeness rather than an endorsement of the procedure. It is worth comparing solutions from minres, alpha, and MLE, for they are not the same. \subsubsection{Weighted Least Squares Factor Analysis} \label{sect:wls} Similar to the minres approach of minimizing the squared residuals, factor method ``wls" weights the squared residuals by their uniquenesses. This tends to produce slightly smaller overall residuals. In the example of weighted least squares, the output is shown by using the \pfun{print} function with the cut option set to 0. That is, all loadings are shown (Table~\ref{tab:Thurstone.wls}). \begin{table}[htpb] \caption{The 9 variable problem from Thurstone is a classic example of factoring where there is a higher order factor, g, that accounts for the correlation between the factors. The factors were extracted using a weighted least squares algorithm. All loadings are shown by using the cut=0 option in the \pfun{print.psych} function.} \begin{scriptsize} <>= f3w <- fa(Thurstone,3,n.obs = 213,fm="wls") print(f3w,cut=0,digits=3) @ \end{scriptsize} \label{tab:Thurstone.wls} \end{table} subsection{Displaying factor solutions} The unweighted least squares solution may be shown graphically using the \pfun{fa.plot} function which is called by the generic \fun{plot} function (Figure~\ref{fig:thurstone}). Factors were transformed obliquely using a oblimin. These solutions may be shown as item by factor plots (Figure~\ref{fig:thurstone}) or by a structure diagram (Figure~\ref{fig:thurstone.diagram}). \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= plot(f3t) @ \end{scriptsize} \caption{A graphic representation of the 3 oblique factors from the Thurstone data using \pfun{plot}. Factors were transformed to an oblique solution using the oblimin function from the GPArotation package.} \label{fig:thurstone} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= fa.diagram(f3t) @ \end{scriptsize} \caption{A graphic representation of the 3 oblique factors from the Thurstone data using \pfun{fa.diagram}. Factors were transformed to an oblique solution using oblimin.} \label{fig:thurstone.diagram} \end{center} \end{figure} A comparison of these three approaches suggests that the minres solution is more similar to a maximum likelihood solution and fits slightly better than the pa or wls solutions. Comparisons with SPSS suggest that the pa solution matches the SPSS OLS solution, but that the minres solution is slightly better. At least in one test data set, the weighted least squares solutions, although fitting equally well, had slightly different structure loadings. Note that the rotations used by SPSS will sometimes use the ``Kaiser Normalization''. By default, the rotations used in psych do not normalize, but this can be specified as an option in \pfun{fa}. \subsubsection{Principal Components analysis (PCA)} An alternative to factor analysis, which is unfortunately frequently confused with \iemph{factor analysis}, is \iemph{principal components analysis}. Although the goals of \iemph{PCA} and \iemph{FA} are similar, PCA is a descriptive model of the data, while FA is a structural model. Some psychologists use PCA in a manner similar to factor analysis and thus the \pfun{principal} function produces output that is perhaps more understandable than that produced by \fun{princomp} in the \Rpkg{stats} package. Table~\ref{tab:pca} shows a PCA of the Thurstone 9 variable problem rotated using the \pfun{Promax} function. Note how the loadings from the factor model are similar but smaller than the principal component loadings. This is because the PCA model attempts to account for the entire variance of the correlation matrix, while FA accounts for just the \iemph{common variance}. This distinction becomes most important for small correlation matrices. Also note how the goodness of fit statistics, based upon the residual off diagonal elements, is much worse than the \pfun{fa} solution. \begin{table}[htpb] \caption{The Thurstone problem can also be analyzed using Principal Components Analysis. Compare this to Table~\ref{tab:Thurstone}. The loadings are higher for the PCA because the model accounts for the unique as well as the common variance.The fit of the off diagonal elements, however, is much worse than the \pfun{fa} results.} \begin{center} \begin{scriptsize} <>= p3p <-principal(Thurstone,3,n.obs = 213,rotate="Promax") p3p @ \end{scriptsize} \end{center} \label{tab:pca} \end{table} \subsubsection{Hierarchical and bi-factor solutions} \label{sect:omega} For a long time structural analysis of the ability domain have considered the problem of factors that are themselves correlated. These correlations may themselves be factored to produce a higher order, general factor. An alternative \citep{holzinger:37,jensen:weng} is to consider the general factor affecting each item, and then to have group factors account for the residual variance. Exploratory factor solutions to produce a hierarchical or a bifactor solution are found using the \pfun{omega} function. This technique has more recently been applied to the personality domain to consider such things as the structure of neuroticism (treated as a general factor, with lower order factors of anxiety, depression, and aggression). Consider the 9 Thurstone variables analyzed in the prior factor analyses. The correlations between the factors (as shown in Figure~\ref{fig:thurstone.diagram} can themselves be factored. This results in a higher order factor model (Figure~\ref{fig:omega}). An an alternative solution is to take this higher order model and then solve for the general factor loadings as well as the loadings on the residualized lower order factors using the \iemph{Schmid-Leiman} procedure. (Figure ~\ref{fig:omega.2}). Yet another solution is to use structural equation modeling to directly solve for the general and group factors. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= om.h <- omega(Thurstone,n.obs=213,sl=FALSE) op <- par(mfrow=c(1,1)) @ \end{scriptsize} \caption{A higher order factor solution to the Thurstone 9 variable problem} \label{fig:omega} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= om <- omega(Thurstone,n.obs=213) @ \end{scriptsize} \caption{A bifactor factor solution to the Thurstone 9 variable problem} \label{fig:omega.2} \end{center} \end{figure} Yet another approach to the bifactor structure is do use the \pfun{bifactor} rotation function in either \Rpkg{psych} or in \Rpkg{GPArotation}. This does the rotation discussed in \cite{jennrich:11}. \subsubsection{Item Cluster Analysis: iclust} \label{sect:iclust} An alternative to factor or components analysis is \iemph{cluster analysis}. The goal of cluster analysis is the same as factor or components analysis (reduce the complexity of the data and attempt to identify homogeneous subgroupings). Mainly used for clustering people or objects (e.g., projectile points if an anthropologist, DNA if a biologist, galaxies if an astronomer), clustering may be used for clustering items or tests as well. Introduced to psychologists by \cite{tryon:39} in the 1930's, the cluster analytic literature exploded in the 1970s and 1980s \citep{blashfield:80,blashfield:88,everitt:74,hartigan:75}. Much of the research is in taxonmetric applications in biology \citep{sneath:73,sokal:63} and marketing \citep{cooksey:06} where clustering remains very popular. It is also used for taxonomic work in forming clusters of people in family \citep{henry:05} and clinical psychology \citep{martinent:07,mun:08}. Interestingly enough it has has had limited applications to psychometrics. This is unfortunate, for as has been pointed out by e.g. \citep{tryon:35,loevinger:53}, the theory of factors, while mathematically compelling, offers little that the geneticist or behaviorist or perhaps even non-specialist finds compelling. \cite{cooksey:06} reviews why the \pfun{iclust} algorithm is particularly appropriate for scale construction in marketing. \emph{Hierarchical cluster analysis} \index{hierarchical cluster analysis} forms clusters that are nested within clusters. The resulting \iemph{tree diagram} (also known somewhat pretentiously as a \iemph{rooted dendritic structure}) shows the nesting structure. Although there are many hierarchical clustering algorithms in \R{} (e.g., \fun{agnes}, \fun{hclust}, and \pfun{iclust}), the one most applicable to the problems of scale construction is \pfun{iclust} \citep{revelle:iclust}. \begin{enumerate} \item Find the proximity (e.g. correlation) matrix, \item Identify the most similar pair of items \item Combine this most similar pair of items to form a new variable (cluster), \item Find the similarity of this cluster to all other items and clusters, \item Repeat steps 2 and 3 until some criterion is reached (e.g., typicallly, if only one cluster remains or in \pfun{iclust} if there is a failure to increase reliability coefficients $\alpha$ or $\beta$). \item Purify the solution by reassigning items to the most similar cluster center. \end{enumerate} \pfun{iclust} forms clusters of items using a hierarchical clustering algorithm until one of two measures of internal consistency fails to increase \citep{revelle:iclust}. The number of clusters may be specified a priori, or found empirically. The resulting statistics include the average split half reliability, $\alpha$ \citep{cronbach:51}, as well as the worst split half reliability, $\beta$ \citep{revelle:iclust}, which is an estimate of the general factor saturation of the resulting scale (Figure~\ref{fig:iclust}). Cluster loadings (corresponding to the structure matrix of factor analysis) are reported when printing (Table~\ref{tab:iclust}). The pattern matrix is available as an object in the results. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= data(bfi) ic <- iclust(bfi[1:25]) @ \end{scriptsize} \caption{Using the \pfun{iclust} function to find the cluster structure of 25 personality items (the three demographic variables were excluded from this analysis). When analyzing many variables, the tree structure may be seen more clearly if the graphic output is saved as a pdf and then enlarged using a pdf viewer.} \label{fig:iclust} \end{center} \end{figure} \begin{table}[htpb] \caption{The summary statistics from an iclust analysis shows three large clusters and smaller cluster.} \begin{center} \begin{scriptsize} <>= summary(ic) #show the results @ \end{scriptsize} \end{center} \label{tab:iclust} \end{table}% The previous analysis (Figure~\ref{fig:iclust}) was done using the Pearson correlation. A somewhat cleaner structure is obtained when using the \pfun{polychoric} function to find polychoric correlations (Figure~\ref{fig:iclust.poly}). Note that the first time finding the polychoric correlations some time, but the next three analyses were done using that correlation matrix (r.poly\$rho). When using the console for input, \pfun{polychoric} will report on its progress while working using \pfun{progressBar}. \begin{table}[htpb] \caption{The \pfun{polychoric} and the \pfun{tetrachoric} functions can take a long time to finish and report their progress by a series of dots as they work. The dots are suppressed when creating a Sweave document.} \begin{center} \begin{tiny} <>= data(bfi) r.poly <- polychoric(bfi[1:25],correct=0) #the ... indicate the progress of the function @ \end{tiny} \end{center} \label{tab:bad}1.7.1\end{table}% \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= ic.poly <- iclust(r.poly$rho,title="ICLUST using polychoric correlations") iclust.diagram(ic.poly) @ \end{scriptsize} \caption{ICLUST of the BFI data set using polychoric correlations. Compare this solution to the previous one (Figure~\ref{fig:iclust}) which was done using Pearson correlations. } \label{fig:iclust.poly} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= ic.poly <- iclust(r.poly$rho,5,title="ICLUST using polychoric correlations for nclusters=5") iclust.diagram(ic.poly) @ \end{scriptsize} \caption{ICLUST of the BFI data set using polychoric correlations with the solution set to 5 clusters. Compare this solution to the previous one (Figure~\ref{fig:iclust.poly}) which was done without specifying the number of clusters and to the next one (Figure~\ref{fig:iclust.3}) which was done by changing the beta criterion. } \label{fig:iclust.5} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= ic.poly <- iclust(r.poly$rho,beta.size=3,title="ICLUST beta.size=3") @ \end{scriptsize} \caption{ICLUST of the BFI data set using polychoric correlations with the beta criterion set to 3. Compare this solution to the previous three (Figure~\ref{fig:iclust},~\ref{fig:iclust.poly}, \ref{fig:iclust.5}).} \label{fig:iclust.3} \end{center} \end{figure} \begin{table}[htpb] \caption{The output from \pfun{iclust} includes the loadings of each item on each cluster. These are equivalent to factor structure loadings. By specifying the value of cut, small loadings are suppressed. The default is for cut=0.su } \begin{center} \begin{scriptsize} <>= print(ic,cut=.3) @ \end{scriptsize} \end{center} \label{tab:iclust} \end{table}% A comparison of these four cluster solutions suggests both a problem and an advantage of clustering techniques. The problem is that the solutions differ. The advantage is that the structure of the items may be seen more clearly when examining the clusters rather than a simple factor solution. \subsection{Estimates of fit} Exploratory factoring techniques are sometimes criticized because of the lack of statistical information on the solutions. There are perhaps as many fit statistics as there are psychometricians. When using Maximum Likelihood extraction, many of these various fit statistics are based upon the $\chi^{2}$ which is minimized using ML. If not using ML, these same statistics can be found, but they are no longer maximum likelihood estimates. They are, however, still useful. Overall estimates of goodness of fit including $\chi^{2}$ and RMSEA are found in the \pfun{fa} and \pfun{omega} functions. \subsection{Confidence intervals using bootstrapping techniques} Confidence intervals for the factor loadings may be found by doing multiple bootstrapped iterations of the original analysis. This is done by setting the n.iter parameter to the desired number of iterations. This can be done for factoring of Pearson correlation matrices as well as polychoric/tetrachoric matrices (See Table~\ref{tab:bootstrap}). Although the example value for the number of iterations is set to 20, more conventional analyses might use 1000 bootstraps. This will take much longer. Bootstrapped confidence intervals can also be found for the loadings of a factoring of a polychoric matrix. \pfun{fa.poly} will find the polychoric correlation matrix and if the n.iter option is greater than 1, will then randomly resample the data (case wise) to give bootstrapped samples. This will take a long time for large number of items or interations. \begin{table}[htpb] \caption{An example of bootstrapped confidence intervals on 10 items from the Big 5 inventory. The number of bootstrapped samples was set to 20. More conventional bootstrapping would use 100 or 1000 replications. } \begin{tiny} \begin{center} <>= fa(bfi[1:10],2,n.iter=20) @ \end{center} \end{tiny} \label{tab:bootstrap} \end{table}% \subsection{Comparing factor/component/cluster solutions} Cluster analysis, factor analysis, and principal components analysis all produce structure matrices (matrices of correlations between the dimensions and the variables) that can in turn be compared in terms of Burt's \iemph{congruence coefficient} (also known as Tucker's coefficient) which is just the cosine of the angle between the dimensions $$c_{f_{i}f_{j}} = \frac{\sum_{k=1}^{n}{f_{ik}f_{jk}}} {\sum{f_{ik}^{2}}\sum{f_{jk}^{2}}}.$$ Consider the case of a four factor solution and four cluster solution to the Big Five problem. \begin{scriptsize} <>= f4 <- fa(bfi[1:25],4,fm="pa") factor.congruence(f4,ic) @ \end{scriptsize} A more complete comparison of oblique factor solutions (both minres and principal axis), bifactor and component solutions to the Thurstone data set is done using the \pfun{factor.congruence} function. (See table~\ref{tab:congruence}). \begin{table}[htpb] \caption{Congruence coefficients for oblique factor, bifactor and component solutions for the Thurstone problem.} \begin{scriptsize} <>= factor.congruence(list(f3t,f3o,om,p3p)) @ \end{scriptsize} \label{tab:congruence} \end{table}% \subsection{Determining the number of dimensions to extract.} How many dimensions to use to represent a correlation matrix is an unsolved problem in psychometrics. There are many solutions to this problem, none of which is uniformly the best. Henry Kaiser once said that ``a solution to the number-of factors problem in factor analysis is easy, that he used to make up one every morning before breakfast. But the problem, of course is to find \emph{the} solution, or at least a solution that others will regard quite highly not as the best" \cite{horn:79}. Techniques most commonly used include 1) Extracting factors until the chi square of the residual matrix is not significant. 2) Extracting factors until the change in chi square from factor n to factor n+1 is not significant. 3) Extracting factors until the eigen values of the real data are less than the corresponding eigen values of a random data set of the same size (parallel analysis) \pfun{fa.parallel} \citep{horn:65}. 4) Plotting the magnitude of the successive eigen values and applying the scree test (a sudden drop in eigen values analogous to the change in slope seen when scrambling up the talus slope of a mountain and approaching the rock face \citep{cattell:scree}. 5) Extracting factors as long as they are interpretable. 6) Using the Very Structure Criterion (\pfun{vss}) \citep{revelle:vss}. 7) Using Wayne Velicer's Minimum Average Partial (\pfun{MAP}) criterion \citep{velicer:76}. 8) Extracting principal components until the eigen value < 1. Each of the procedures has its advantages and disadvantages. Using either the chi square test or the change in square test is, of course, sensitive to the number of subjects and leads to the nonsensical condition that if one wants to find many factors, one simply runs more subjects. Parallel analysis is partially sensitive to sample size in that for large samples the eigen values of random factors will all tend towards 1. The scree test is quite appealing but can lead to differences of interpretation as to when the scree ``breaks". Extracting interpretable factors means that the number of factors reflects the investigators creativity more than the data. vss, while very simple to understand, will not work very well if the data are very factorially complex. (Simulations suggests it will work fine if the complexities of some of the items are no more than 2). The eigen value of 1 rule, although the default for many programs, seems to be a rough way of dividing the number of variables by 3 and is probably the worst of all criteria. An additional problem in determining the number of factors is what is considered a factor. Many treatments of factor analysis assume that the residual correlation matrix after the factors of interest are extracted is composed of just random error. An alternative concept is that the matrix is formed from major factors of interest but that there are also numerous minor factors of no substantive interest but that account for some of the shared covariance between variables. The presence of such minor factors can lead one to extract too many factors and to reject solutions on statistical grounds of misfit that are actually very good fits to the data. This problem is partially addressed later in the discussion of simulating complex structures using \pfun{sim.structure} and of small extraneous factors using the \pfun{sim.minor} function. \subsubsection{Very Simple Structure} \label{sect:vss} The \pfun{vss} function compares the fit of a number of factor analyses with the loading matrix ``simplified" by deleting all except the c greatest loadings per item, where c is a measure of factor complexity \cite{revelle:vss}. Included in \pfun{vss} is the MAP criterion (Minimum Absolute Partial correlation) of \cite{velicer:76}. Using the Very Simple Structure criterion for the bfi data suggests that 4 factors are optimal (Figure~\ref{fig:vss}). However, the MAP criterion suggests that 5 is optimal. \begin{figure}[htbp] \begin{center} <>= vss <- vss(bfi[1:25],title="Very Simple Structure of a Big 5 inventory") @ \caption{The Very Simple Structure criterion for the number of factors compares solutions for various levels of item complexity and various numbers of factors. For the Big 5 Inventory, the complexity 1 and 2 solutions both achieve their maxima at four factors. This is in contrast to parallel analysis which suggests 6 and the MAP criterion which suggests 5. } \label{fig:vss} \end{center} \end{figure} \begin{scriptsize} <>= vss @ \end{scriptsize} \subsubsection{Parallel Analysis} \label{sect:fa.parallel} An alternative way to determine the number of factors is to compare the solution to random data with the same properties as the real data set. If the input is a data matrix, the comparison includes random samples from the real data, as well as normally distributed random data with the same number of subjects and variables. For the BFI data, parallel analysis suggests that 6 factors might be most appropriate (Figure~\ref{fig:parallel}). It is interesting to compare \pfun{fa.parallel} with the \fun{paran} from the \Rpkg{paran} package. This latter uses smcs to estimate communalities. Simulations of known structures with a particular number of major factors but with the presence of trivial, minor (but not zero) factors, show that using smcs will tend to lead to too many factors. \begin{figure}[htbp] \begin{scriptsize} \begin{center} <>= fa.parallel(bfi[1:25],main="Parallel Analysis of a Big 5 inventory") @ \caption{Parallel analysis compares factor and principal components solutions to the real data as well as resampled data. Although vss suggests 4 factors, MAP 5, parallel analysis suggests 6. One more demonstration of Kaiser's dictum.} \label{fig:parallel} \end{center} \end{scriptsize} \end{figure} Experience with problems of various sizes suggests that parallel analysis is useful for less than about 1,000 subjects, and that using the number of components greater than a random solution is more robust than using the number of factors greater than random factors. A more tedious problem in terms of computation is to do parallel analysis of \iemph{polychoric} correlation matrices. This is done by \pfun{fa.parallel.poly}. By default the number of replications is 20. This is appropriate when choosing the number of factors from dicthotomous or polytomous data matrices. \subsection{Factor extension} Sometimes we are interested in the relationship of the factors in one space with the variables in a different space. One solution is to find factors in both spaces separately and then find the structural relationships between them. This is the technique of structural equation modeling in packages such as \Rpkg{sem} or \Rpkg{lavaan}. An alternative is to use the concept of \iemph{factor extension} developed by \citep{dwyer:37}. Consider the case of 16 variables created to represent one two dimensional space. If factors are found from eight of these variables, they may then be extended to the additional eight variables (See Figure~\ref{fig:fa.extension}). \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= v16 <- sim.item(16) s <- c(1,3,5,7,9,11,13,15) f2 <- fa(v16[,s],2) fe <- fa.extension(cor(v16)[s,-s],f2) fa.diagram(f2,fe=fe) @ \end{scriptsize} \caption{Factor extension applies factors from one set (those on the left) to another set of variables (those on the right). \pfun{fa.extension} is particularly useful when one wants to define the factors with one set of variables and then apply those factors to another set. \pfun{fa.diagram} is used to show the structure. } \label{fig:fa.extension} \end{center} \end{figure} Another way to examine the overlap between two sets is the use of \iemph{set correlation} found by \pfun{setCor} (discussed later). \subsection{Exploratory Structural Equation Modeling (ESEM)} Generaizing the procedures of factor extension, we can do Exploratory Structural Equation Modeling (ESEM). Traditional Exploratory Factor Analysis (EFA) examines how latent variables can account for the correlations within a data set. All loadings and cross loadings are found and rotation is done to some approximation of simple structure. Traditional Confirmatory Factor Analysis (CFA) tests such models by fitting just a limited number of loadings and typically does not allow any (or many) cross loadings. Structural Equation Modeling then applies two such measurement models, one to a set of X variables, another to a set of Y variables, and then tries to estimate the correlation between these two sets of latent variables. (Some SEM procedures estimate all the parameters from the same model, thus making the loadings in set Y affect those in set X.) It is possible to do a similar, exploratory modeling (ESEM) by conducting two Exploratory Factor Analyses, one in set X, one in set Y, and then finding the correlations of the X factors with the Y factors, as well as the correlations of the Y variables with the X factors and the X variables with the Y factors. Consider the simulated data set of three ability variables, two motivational variables, and three outcome variables: <>= fx <-matrix(c( .9,.8,.6,rep(0,4),.6,.8,-.7),ncol=2) fy <- matrix(c(.6,.5,.4),ncol=1) rownames(fx) <- c("V","Q","A","nach","Anx") rownames(fy)<- c("gpa","Pre","MA") Phi <-matrix( c(1,0,.7,.0,1,.7,.7,.7,1),ncol=3) gre.gpa <- sim.structural(fx,Phi,fy) print(gre.gpa) @ We can fit this by using the \pfun{esem} function and then draw the solution (see Figure~\ref{fig:esem}) using the \pfun{esem.diagram} function (which is normally called automatically by \pfun{esem}. <>= esem.example <- esem(gre.gpa$model,varsX=1:5,varsY=6:8,nfX=2,nfY=1,n.obs=1000,plot=FALSE) esem.example @ \begin{figure}[htpb] \begin{center} <>= esem.diagram(esem.example) @ \caption{An example of a Exploratory Structure Equation Model.} \label{fig:esem} \end{center} \end{figure} \section{Classical Test Theory and Reliability} Surprisingly, 113 years after \cite{spearman:rho} introduced the concept of reliability to psychologists, there are still multiple approaches for measuring it. Although very popular, Cronbach's $\alpha$ \citep{cronbach:51} underestimates the reliability of a test and over estimates the first factor saturation \citep{rz:09}. $\alpha$ \citep{cronbach:51} is the same as Guttman's $\lambda3$ \citep{guttman:45} and may be found by $$ \lambda_3 = \frac{n}{n-1}\Bigl(1 - \frac{tr(\vec{V})_x}{V_x}\Bigr) = \frac{n}{n-1} \frac{V_x - tr(\vec{V}_x)}{V_x} = \alpha $$ Perhaps because it is so easy to calculate and is available in most commercial programs, alpha is without doubt the most frequently reported measure of internal consistency reliability. Alpha is the mean of all possible spit half reliabilities (corrected for test length). For a unifactorial test, it is a reasonable estimate of the first factor saturation, although if the test has any microstructure (i.e., if it is ``lumpy") coefficients $\beta$ \citep{revelle:iclust} (see \pfun{iclust}) and $\omega_h$ (see \pfun{omega}) are more appropriate estimates of the general factor saturation. $\omega_t$is a better estimate of the reliability of the total test. Guttman's $\lambda _6$ (G6) considers the amount of variance in each item that can be accounted for the linear regression of all of the other items (the squared multiple correlation or smc), or more precisely, the variance of the errors, $e_j^2$, and is $$ \lambda_6 = 1 - \frac{\sum e_j^2}{V_x} = 1 - \frac{\sum(1-r_{smc}^2)}{V_x}. $$ The squared multiple correlation is a lower bound for the item communality and as the number of items increases, becomes a better estimate. G6 is also sensitive to lumpiness in the test and should not be taken as a measure of unifactorial structure. For lumpy tests, it will be greater than alpha. For tests with equal item loadings, alpha > G6, but if the loadings are unequal or if there is a general factor, G6 > alpha. G6 estimates item reliability by the squared multiple correlation of the other items in a scale. A modification of G6, G6*, takes as an estimate of an item reliability the smc with all the items in an inventory, including those not keyed for a particular scale. This will lead to a better estimate of the reliable variance of a particular item. Alpha, G6 and G6* are positive functions of the number of items in a test as well as the average intercorrelation of the items in the test. When calculated from the item variances and total test variance, as is done here, raw alpha is sensitive to differences in the item variances. Standardized alpha is based upon the correlations rather than the covariances. More complete reliability analyses of a single scale can be done using the \pfun{omega} function which finds $\omega_h$ and $\omega_t$ based upon a hierarchical factor analysis. Alternative functions \pfun{scoreItems} and \pfun{cluster.cor} will also score multiple scales and report more useful statistics. ``Standardized" alpha is calculated from the inter-item correlations and will differ from raw alpha. Functions for examining the reliability of a single scale or a set of scales include: \begin{description} \item [alpha] Internal consistency measures of reliability range from $\omega_h$ to $\alpha$ to $\omega_t$. The \pfun{alpha} function reports two estimates: Cronbach's coefficient $\alpha$ and Guttman's $\lambda_6$. Also reported are item - whole correlations, $\alpha$ if an item is omitted, and item means and standard deviations. \item [guttman] Eight alternative estimates of test reliability include the six discussed by \cite{guttman:45}, four discussed by ten Berge and Zergers (1978) ($\mu_0 \dots \mu_3$) as well as $\beta$ \citep[the worst split half,][]{revelle:iclust}, the glb (greatest lowest bound) discussed by Bentler and Woodward (1980), and $\omega_h$ and$\omega_t$ (\citep{mcdonald:tt,zinbarg:pm:05}. \item [omega] Calculate McDonald's omega estimates of general and total factor saturation. (\cite{rz:09} compare these coefficients with real and artificial data sets.) \item [cluster.cor] Given a n x c cluster definition matrix of -1s, 0s, and 1s (the keys) , and a n x n correlation matrix, find the correlations of the composite clusters. \item [scoreItems] Given a matrix or data.frame of k keys for m items (-1, 0, 1), and a matrix or data.frame of items scores for m items and n people, find the sum scores or average scores for each person and each scale. If the input is a square matrix, then it is assumed that correlations or covariances were used, and the raw scores are not available. In addition, report Cronbach's alpha, coefficient G6*, the average r, the scale intercorrelations, and the item by scale correlations (both raw and corrected for item overlap and scale reliability). Replace missing values with the item median or mean if desired. Will adjust scores for reverse scored items. \item [score.multiple.choice] Ability tests are typically multiple choice with one right answer. score.multiple.choice takes a scoring key and a data matrix (or data.frame) and finds total or average number right for each participant. Basic test statistics (alpha, average r, item means, item-whole correlations) are also reported. \item [splitHalf] Given a set of items, consider all (if n.items < 17) or 10,000 random splits of the item into two sets. The correlation between these two split halfs is then adjusted by the Spearman-Brown prophecy formula to show the range of split half reliablities. \end{description} \subsection{Reliability of a single scale} \label{sect:alpha} A conventional (but non-optimal) estimate of the internal consistency reliability of a test is coefficient $\alpha$ \citep{cronbach:51}. Alternative estimates are Guttman's $\lambda_6$, Revelle's $\beta$, McDonald's $\omega_h$ and $\omega_t$. Consider a simulated data set, representing 9 items with a hierarchical structure and the following correlation matrix. Then using the \pfun{alpha} function, the $\alpha$ and $\lambda_6$ estimates of reliability may be found for all 9 items, as well as the if one item is dropped at a time. \begin{scriptsize} <>= set.seed(17) r9 <- sim.hierarchical(n=500,raw=TRUE)$observed round(cor(r9),2) alpha(r9) @ \end{scriptsize} Some scales have items that need to be reversed before being scored. Rather than reversing the items in the raw data, it is more convenient to just specify which items need to be reversed scored. This may be done in \pfun{alpha} by specifying a \iemph{keys} vector of 1s and -1s. (This concept of keys vector is more useful when scoring multiple scale inventories, see below.) As an example, consider scoring the 7 attitude items in the attitude data set. Assume a conceptual mistake in that items 2 and 6 (complaints and critical) are to be scored (incorrectly) negatively. \begin{scriptsize} <>= alpha(attitude,keys=c("complaints","critical")) @ \end{scriptsize} Note how the reliability of the 7 item scales with an incorrectly reversed item is very poor, but if items 2 and 6 is dropped then the reliability is improved substantially. This suggests that items 2 and 6 were incorrectly scored. Doing the analysis again with the items positively scored produces much more favorable results. \begin{scriptsize} <>= alpha(attitude) @ \end{scriptsize} It is useful when considering items for a potential scale to examine the item distribution. This is done in \pfun{scoreItems} as well as in \pfun{alpha}. \begin{scriptsize} <>= items <- sim.congeneric(N=500,short=FALSE,low=-2,high=2,categorical=TRUE) #500 responses to 4 discrete items alpha(items$observed) #item response analysis of congeneric measures @ \end{scriptsize} \subsection{Using \pfun{omega} to find the reliability of a single scale} Two alternative estimates of reliability that take into account the hierarchical structure of the inventory are McDonald's $\omega_h$ and $\omega_t$. These may be found using the \pfun{omega} function for an exploratory analysis (See Figure~\ref{fig:omega.9}) or \pfun{omegaSem} for a confirmatory analysis using the \Rpkg{lavaan} package based upon the exploratory solution from \pfun{omega}. McDonald has proposed coefficient omega (hierarchical) ($\omega_h$) as an estimate of the general factor saturation of a test. \cite{zinbarg:pm:05} \url{https://personality-project.org/revelle/publications/zinbarg.revelle.pmet.05.pdf} compare McDonald's $\omega_h$ to Cronbach's $\alpha$ and Revelle's $\beta$. They conclude that $\omega_h$ is the best estimate. (See also \cite{zinbarg:apm:06} and \cite{rz:09} \url{https://personality-project.org/revelle/publications/revelle.zinbarg.08.pdf} ). One way to find $\omega_h$ is to do a factor analysis of the original data set, rotate the factors obliquely, factor that correlation matrix, do a Schmid-Leiman (\pfun{schmid}) transformation to find general factor loadings, and then find $\omega_h$. $\omega_h$ differs slightly as a function of how the factors are estimated. Four options are available, the default will do a minimum residual factor analysis, fm=``pa" does a principal axes factor analysis (\pfun{factor.pa}), fm=``mle" uses the factanal function, and fm=``pc" does a principal components analysis (\pfun{principal}). For ability items, it is typically the case that all items will have positive loadings on the general factor. However, for non-cognitive items it is frequently the case that some items are to be scored positively, and some negatively. Although probably better to specify which directions the items are to be scored by specifying a key vector, if flip =TRUE (the default), items will be reversed so that they have positive loadings on the general factor. The keys are reported so that scores can be found using the \pfun{scoreItems} function. Arbitrarily reversing items this way can overestimate the general factor. (See the example with a simulated circumplex). $\beta$, an alternative to $\omega$, is defined as the worst split half reliability. It can be estimated by using \pfun{iclust} (Item Cluster analysis: a hierarchical clustering algorithm). For a very complimentary review of why the iclust algorithm is useful in scale construction, see \cite{cooksey:06}. The \pfun{omega} function uses exploratory factor analysis to estimate the $\omega_h$ coefficient. It is important to remember that ``A recommendation that should be heeded, regardless of the method chosen to estimate $\omega_h$, is to always examine the pattern of the estimated general factor loadings prior to estimating $\omega_h$. Such an examination constitutes an informal test of the assumption that there is a latent variable common to all of the scale's indicators that can be conducted even in the context of EFA. If the loadings were salient for only a relatively small subset of the indicators, this would suggest that there is no true general factor underlying the covariance matrix. Just such an informal assumption test would have afforded a great deal of protection against the possibility of misinterpreting the misleading $\omega_h$ estimates occasionally produced in the simulations reported here." \citep[][p 137]{zinbarg:apm:06}. Although $\omega_h$ is uniquely defined only for cases where 3 or more subfactors are extracted, it is sometimes desired to have a two factor solution. By default this is done by forcing the \pfun{schmid} extraction to treat the two subfactors as having equal loadings. There are three possible options for this condition: setting the general factor loadings between the two lower order factors to be ``equal" which will be the $\sqrt{r_{ab}}$ where $r_{ab}$ is the oblique correlation between the factors) or to ``first" or ``second" in which case the general factor is equated with either the first or second group factor. A message is issued suggesting that the model is not really well defined. This solution discussed in Zinbarg et al., 2007. To do this in omega, add the option=``first" or option=``second" to the call. Although obviously not meaningful for a 1 factor solution, it is of course possible to find the sum of the loadings on the first (and only) factor, square them, and compare them to the overall matrix variance. This is done, with appropriate complaints. In addition to $\omega_h$, another of McDonald's coefficients is $\omega_t$. This is an estimate of the total reliability of a test. McDonald's $\omega_t$, which is similar to Guttman's $\lambda_6$, (see \pfun{guttman}) uses the estimates of uniqueness $u^2$ from factor analysis to find $e_j^2$. This is based on a decomposition of the variance of a test score, $V_x$ into four parts: that due to a general factor, $\vec{g}$, that due to a set of group factors, $\vec{f}$, (factors common to some but not all of the items), specific factors, $\vec{s}$ unique to each item, and $\vec{e}$, random error. (Because specific variance can not be distinguished from random error unless the test is given at least twice, some combine these both into error). Letting $\vec{x} = \vec{cg} + \vec{Af} + \vec {Ds} + \vec{e} $ then the communality of item$_j$, based upon general as well as group factors, $h_j^2 = c_j^2 + \sum{f_{ij}^2}$ and the unique variance for the item $u_j^2 = \sigma_j^2 (1-h_j^2)$ may be used to estimate the test reliability. That is, if $h_j^2$ is the communality of item$_j$, based upon general as well as group factors, then for standardized items, $e_j^2 = 1 - h_j^2$ and $$ \omega_t = \frac{\vec{1}\vec{cc'}\vec{1} + \vec{1}\vec{AA'}\vec{1}'}{V_x} = 1 - \frac{\sum(1-h_j^2)}{V_x} = 1 - \frac{\sum u^2}{V_x} $$ Because $h_j^2 \geq r_{smc}^2$, $\omega_t \geq \lambda_6$. It is important to distinguish here between the two $\omega$ coefficients of McDonald, 1978 and Equation 6.20a of McDonald, 1999, $\omega_t$ and $\omega_h$. While the former is based upon the sum of squared loadings on all the factors, the latter is based upon the sum of the squared loadings on the general factor. $$\omega_h = \frac{ \vec{1}\vec{cc'}\vec{1}}{V_x}$$ Another estimate reported is the omega for an infinite length test with a structure similar to the observed test. This is found by $$\omega_{\inf} = \frac{ \vec{1}\vec{cc'}\vec{1}}{\vec{1}\vec{cc'}\vec{1} + \vec{1}\vec{AA'}\vec{1}'}$$ \begin{figure}[htbp] \begin{center} <>= om.9 <- omega(r9,title="9 simulated variables") @ \caption{A bifactor solution for 9 simulated variables with a hierarchical structure. } \label{fig:omega.9} \end{center} \end{figure} In the case of these simulated 9 variables, the amount of variance attributable to a general factor ($\omega_h$) is quite large, and the reliability of the set of 9 items is somewhat greater than that estimated by $\alpha$ or $\lambda_6$. \begin{scriptsize} <>= om.9 @ \end{scriptsize} \subsection{Estimating $\omega_h$ using Confirmatory Factor Analysis} The \pfun{omegaSem} function will do an exploratory analysis and then take the highest loading items on each factor and do a confirmatory factor analysis using the \Rpkg{sem} package. These results can produce slightly different estimates of $\omega_h$, primarily because cross loadings are modeled as part of the general factor. \begin{scriptsize} <>= omegaSem(r9,n.obs=500,lavaan=TRUE) @ \end{scriptsize} \subsubsection{Other estimates of reliability} Other estimates of reliability are found by the \pfun{splitHalf} and \pfun{guttman} functions. These are described in more detail in \cite{rz:09} and in \cite{rc:reliability}. They include the 6 estimates from Guttman, four from TenBerge, and an estimate of the greatest lower bound. \begin{scriptsize} <>= splitHalf(r9) @ \end{scriptsize} \subsection{Reliability and correlations of multiple scales within an inventory} \label{sect:score} A typical research question in personality involves an inventory of multiple items purporting to measure multiple constructs. For example, the data set \pfun{bfi} includes 25 items thought to measure five dimensions of personality (Extraversion, Emotional Stability, Conscientiousness, Agreeableness, and Openness). The data may either be the raw data or a correlation matrix (\pfun{scoreItems}) or just a correlation matrix of the items ( \pfun{cluster.cor} and \pfun{cluster.loadings}). When finding reliabilities for multiple scales, item reliabilities can be estimated using the squared multiple correlation of an item with all other items, not just those that are keyed for a particular scale. This leads to an estimate of G6*. \subsubsection{Scoring from raw data} To score these five scales from the 25 items, use the \pfun{scoreItems} function and a list of items to be scored on each scale (a keys.list). Items may be listed by location (convenient but dangerous), or name (probably safer). Make a keys.list by by specifying the items for each scale, preceding items to be negatively keyed with a - sign: \begin{scriptsize} <>= #the newer way is probably preferred keys.list <- list(agree=c("-A1","A2","A3","A4","A5"), conscientious=c("C1","C2","C2","-C4","-C5"), extraversion=c("-E1","-E2","E3","E4","E5"), neuroticism=c("N1","N2","N3","N4","N5"), openness = c("O1","-O2","O3","O4","-O5")) #this can also be done by location-- keys.list <- list(Agree=c(-1,2:5),Conscientious=c(6:8,-9,-10), Extraversion=c(-11,-12,13:15),Neuroticism=c(16:20), Openness = c(21,-22,23,24,-25)) #These two approaches can be mixed if desired keys.list <- list(agree=c("-A1","A2","A3","A4","A5"),conscientious=c("C1","C2","C3","-C4","-C5"), extraversion=c("-E1","-E2","E3","E4","E5"), neuroticism=c(16:20),openness = c(21,-22,23,24,-25)) keys.list @ \end{scriptsize} \begin{tiny}In the past (prior to version 1.6.9, the keys.list was then converted a keys matrix using the helper function \pfun{make.keys}. This is no longer necessary. Logically, scales are merely the weighted composites of a set of items. The weights used are -1, 0, and 1. 0 implies do not use that item in the scale, 1 implies a positive weight (add the item to the total score), -1 a negative weight (subtract the item from the total score, i.e., reverse score the item). Reverse scoring an item is equivalent to subtracting the item from the maximum + minimum possible value for that item. The minima and maxima can be estimated from all the items, or can be specified by the user. There are two different ways that scale scores tend to be reported. Social psychologists and educational psychologists tend to report the scale score as the \emph{average item score} while many personality psychologists tend to report the \emph{total item score}. The default option for \pfun{scoreItems} is to report item averages (which thus allows interpretation in the same metric as the items) but totals can be found as well. Personality researchers should be encouraged to report scores based upon item means and avoid using the total score although some reviewers are adamant about the following the tradition of total scores. The printed output includes coefficients $\alpha$ and G6*, the average correlation of the items within the scale (corrected for item ovelap and scale relliability), as well as the correlations between the scales (below the diagonal, the correlations above the diagonal are corrected for attenuation. As is the case for most of the \Rpkg{psych} functions, additional information is returned as part of the object. First, create keys matrix using the \pfun{make.keys} function. (The keys matrix could also be prepared externally using a spreadsheet and then copying it into \R{}). Although not normally necessary, show the keys to understand what is happening. There are two ways to make up the keys. You can specify the items by \emph{location} (the old way) or by \emph{name} (the newer and probably preferred way). To use the newer way you must specify the file on which you will use the keys. The example below shows how to construct keys either way. Note that the number of items to specify in the \pfun{make.keys} function is the total number of items in the inventory. This is done automatically in the new way of forming keys, but if using the older way, the number must be specified. That is, if scoring just 5 items from a 25 item inventory, \pfun{make.keys} should be told that there are 25 items. \pfun{make.keys} just changes a list of items on each scale to make up a scoring matrix. Because the \pfun{bfi} data set has 25 items as well as 3 demographic items, the number of variables is specified as 28. \end{tiny} Then, use this keys list to score the items. \begin{scriptsize} <>= scores <- scoreItems(keys.list,bfi) scores @ \end{scriptsize} To see the additional information (the raw correlations, the individual scores, etc.), they may be specified by name. Then, to visualize the correlations between the raw scores, use the \pfun{pairs.panels} function on the scores values of scores. (See figure~\ref{fig:scores} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= png('scores.png') pairs.panels(scores$scores,pch='.',jiggle=TRUE) dev.off() @ \end{scriptsize} \includegraphics{scores} \caption{A graphic analysis of the Big Five scales found by using the scoreItems function. The pair.wise plot allows us to see that some participants have reached the ceiling of the scale for these 5 items scales. Using the pch='.' option in pairs.panels is recommended when plotting many cases. The data points were ``jittered'' by setting jiggle=TRUE. Jiggling this way shows the density more clearly. To save space, the figure was done as a png. For a clearer figure, save as a pdf.} \label{fig:scores} \end{center} \end{figure} \subsubsection{Forming scales from a correlation matrix} There are some situations when the raw data are not available, but the correlation matrix between the items is available. In this case, it is not possible to find individual scores, but it is possible to find the reliability and intercorrelations of the scales. This may be done using the \pfun{cluster.cor} function or the \pfun{scoreItems} function. The use of a keys matrix is the same as in the raw data case. Consider the same \pfun{bfi} data set, but first find the correlations, and then use \pfun{scoreItems}. \begin{scriptsize} <>= r.bfi <- cor(bfi,use="pairwise") scales <- scoreItems(keys.list,r.bfi) summary(scales) @ \end{scriptsize} To find the correlations of the items with each of the scales (the ``structure" matrix) or the correlations of the items controlling for the other scales (the ``pattern" matrix), use the \pfun{cluster.loadings} function. To do both at once (e.g., the correlations of the scales as well as the item by scale correlations), it is also possible to just use \pfun{scoreItems}. \subsection{Scoring Multiple Choice Items} Some items (typically associated with ability tests) are not themselves mini-scales ranging from low to high levels of expression of the item of interest, but are rather multiple choice where one response is the correct response. Two analyses are useful for this kind of item: examining the response patterns to all the alternatives (looking for good or bad distractors) and scoring the items as correct or incorrect. Both of these operations may be done using the \pfun{score.multiple.choice} function. Consider the 16 example items taken from an online ability test at the Personality Project: \url{https://www.sapa-project.org/}. This is part of the Synthetic Aperture Personality Assessment (\iemph{SAPA}) study discussed in \cite{rcw:methods,rwr:sapa}. \begin{scriptsize} <>= data(iqitems) iq.keys <- c(4,4,4, 6,6,3,4,4, 5,2,2,4, 3,2,6,7) score.multiple.choice(iq.keys,iqitems) #just convert the items to true or false iq.tf <- score.multiple.choice(iq.keys,iqitems,score=FALSE) describe(iq.tf) #compare to previous results @ \end{scriptsize} Once the items have been scored as true or false (assigned scores of 1 or 0), they made then be scored into multiple scales using the normal \pfun{scoreItems} function. \subsection{Item analysis} Basic item analysis starts with describing the data (\pfun{describe}, finding the number of dimensions using factor analysis (\pfun{fa}) and cluster analysis \pfun{iclust} perhaps using the Very Simple Structure criterion (\pfun{vss}), or perhaps parallel analysis \pfun{fa.parallel}. Item whole correlations may then be found for scales scored on one dimension (\pfun{alpha} or many scales simultaneously (\pfun{scoreItems}). Scales can be modified by changing the keys matrix (i.e., dropping particular items, changing the scale on which an item is to be scored). This analysis can be done on the normal Pearson correlation matrix or by using polychoric correlations. Validities of the scales can be found using multiple correlation of the raw data or based upon correlation matrices using the \pfun{setCor} function. However, more powerful item analysis tools are now available by using Item Response Theory approaches. Although the \pfun{response.frequencies} output from \pfun{score.multiple.choice} is useful to examine in terms of the probability of various alternatives being endorsed, it is even better to examine the pattern of these responses as a function of the underlying latent trait or just the total score. This may be done by using \pfun{irt.responses} (Figure~\ref{fig:irt.response}). \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= data(iqitems) iq.keys <- c(4,4,4, 6,6,3,4,4, 5,2,2,4, 3,2,6,7) scores <- score.multiple.choice(iq.keys,iqitems,score=TRUE,short=FALSE) #note that for speed we can just do this on simple item counts rather than IRT based scores. op <- par(mfrow=c(2,2)) #set this to see the output for multiple items irt.responses(scores$scores,iqitems[1:4],breaks=11) @ \end{scriptsize} \caption{ The pattern of responses to multiple choice ability items can show that some items have poor distractors. This may be done by using the the \pfun{irt.responses} function. A good distractor is one that is negatively related to ability.} \label{fig:irt.response} \end{center} \end{figure} \subsubsection{Exploring the item structure of scales} The Big Five scales found above can be understood in terms of the item - whole correlations, but it is also useful to think of the endorsement frequency of the items. The \pfun{item.lookup} function will sort items by their factor loading/item-whole correlation, and then resort those above a certain threshold in terms of the item means. Item content is shown by using the dictionary developed for those items. This allows one to see the structure of each scale in terms of its endorsement range. This is a simple way of thinking of items that is also possible to do using the various IRT approaches discussed later. \begin{tiny} <>= m <- colMeans(bfi,na.rm=TRUE) item.lookup(scales$item.corrected[,1:3],m,dictionary=bfi.dictionary[1:2]) @ \end{tiny} \subsubsection{Empirical scale construction} There are some situations where one wants to identify those items that most relate to a particular criterion. Although this will capitalize on chance and the results should interpreted cautiously, it does give a feel for what is being measured. Consider the following example from the \pfun{bfi} data set. The items that best predicted gender, education, and age may be found using the \pfun{bestScales} function. This also shows the use of a dictionary that has the item content. \begin{scriptsize} <>= data(bfi) bestScales(bfi,criteria=c("gender","education","age"),cut=.1,dictionary=bfi.dictionary[,1:3]) @ \end{scriptsize} \section{Item Response Theory analysis} The use of Item Response Theory has become is said to be the ``new psychometrics". The emphasis is upon item properties, particularly those of item difficulty or location and item discrimination. These two parameters are easily found from classic techniques when using factor analyses of correlation matrices formed by \pfun{polychoric} or \pfun{tetrachoric} correlations. The \pfun{irt.fa} function does this and then graphically displays item discrimination and item location as well as item and test information (see Figure~\ref{fig:irt}). \subsection{Factor analysis and Item Response Theory} If the correlations of all of the items reflect one underlying latent variable, then factor analysis of the matrix of tetrachoric correlations should allow for the identification of the regression slopes ($\alpha$) of the items on the latent variable. These regressions are, of course just the factor loadings. Item difficulty, $\delta_j$ and item discrimination, $\alpha_j$ may be found from factor analysis of the tetrachoric correlations where $\lambda_j$ is just the factor loading on the first factor and $\tau_j$ is the normal threshold reported by the \pfun{tetrachoric} function. \begin{equation} \delta_j = \frac{D\tau}{\sqrt{1-\lambda_j^2}}, \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\;\; \alpha_j = \frac{\lambda_j}{\sqrt{1-\lambda_j^2}} \label{eq:irt:diff} \end{equation} where D is a scaling factor used when converting to the parameterization of \iemph{logistic} model and is 1.702 in that case and 1 in the case of the normal ogive model. Thus, in the case of the normal model, factor loadings ($\lambda_j$) and item thresholds ($\tau$) are just \begin{equation*} \lambda_j = \frac{\alpha_j}{\sqrt{1+\alpha_j^2}}, \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\;\;\tau_j = \frac{\delta_j}{\sqrt{1+\alpha_j^2}}. \end{equation*} Consider 9 dichotomous items representing one factor but differing in their levels of difficulty \begin{scriptsize} <>= set.seed(17) d9 <- sim.irt(9,1000,-2.0,2.0,mod="normal") #dichotomous items test <- irt.fa(d9$items,correct=0) test @ \end{scriptsize} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= op <- par(mfrow=c(3,1)) plot(test,type="ICC") plot(test,type="IIC") plot(test,type="test") op <- par(mfrow=c(1,1)) @ \end{scriptsize} \caption{A graphic analysis of 9 dichotomous (simulated) items. The top panel shows the probability of item endorsement as the value of the latent trait increases. Items differ in their location (difficulty) and discrimination (slope). The middle panel shows the information in each item as a function of latent trait level. An item is most informative when the probability of endorsement is 50\%. The lower panel shows the total test information. These items form a test that is most informative (most accurate) at the middle range of the latent trait.} \label{fig:irt} \end{center} \end{figure} Similar analyses can be done for polytomous items such as those of the bfi extraversion scale: \begin{scriptsize} <>= data(bfi) e.irt <- irt.fa(bfi[11:15]) e.irt @ \end{scriptsize} The item information functions show that not all of items are equally good (Figure~\ref{fig:e.irt}): \begin{figure}[htbp] \begin{center} <>= e.info <- plot(e.irt,type="IIC") @ \caption{A graphic analysis of 5 extraversion items from the bfi. The curves represent the amount of information in the item as a function of the latent score for an individual. That is, each item is maximally discriminating at a different part of the latent continuum. Print e.info to see the average information for each item.} \label{fig:e.irt} \end{center} \end{figure} These procedures can be generalized to more than one factor by specifying the number of factors in \pfun{irt.fa}. The plots can be limited to those items with discriminations greater than some value of cut. An invisible object is returned when plotting the output from \pfun{irt.fa} that includes the average information for each item that has loadings greater than cut. \begin{scriptsize} <>= print(e.info,sort=TRUE) @ \end{scriptsize} More extensive IRT packages include the \Rpkg{ltm} and \Rpkg{eRm} and should be used for serious Item Response Theory analysis. \subsection{Speeding up analyses} Finding tetrachoric or polychoric correlations is very time consuming. Thus, to speed up the process of analysis, the original correlation matrix is saved as part of the output of both \pfun{irt.fa} and \pfun{omega}. Subsequent analyses may be done by using this correlation matrix. This is done by doing the analysis not on the original data, but rather on the output of the previous analysis. In addition, recent releases of the \Rpkg{psych} take advantage of the \Rpkg{parallels} package and use multi-cores. The default for Macs and Unix machines is to use two cores, but this can be increased using the options command. The biggest step up in improvement is from 1 to 2 cores, but for large problems using polychoric correlations, the more cores available, the better. For example of taking the output from the 16 ability items from the \iemph{SAPA} project when scored for True/False using \pfun{score.multiple.choice} we can first do a simple IRT analysis of one factor (Figure~\ref{fig:iq.irt}) and then use that correlation matrix to do an \pfun{omega} analysis to show the sub-structure of the ability items . We can also show the total test information (merely the sum of the item information. This shows that even with just 16 items, the test is very reliable for most of the range of ability. The \pfun{fa.irt} function saves the correlation matrix and item statistics so that they can be redrawn with other options. \begin{scriptsize} \begin{Schunk} \begin{Sinput} detectCores() #how many are available options("mc.cores") #how many have been set to be used options("mc.cores"=4) #set to use 4 cores \end{Sinput} \end{Schunk} \end{scriptsize} \begin{figure}[htbp] \begin{tiny} \begin{center} <>= iq.irt <- irt.fa(ability) @ \end{center} \end{tiny} \caption{A graphic analysis of 16 ability items sampled from the \iemph{SAPA} project. The curves represent the amount of information in the item as a function of the latent score for an individual. That is, each item is maximally discriminating at a different part of the latent continuum. Print iq.irt to see the average information for each item. Partly because this is a power test (it is given on the web) and partly because the items have not been carefully chosen, the items are not very discriminating at the high end of the ability dimension. } \label{fig:iq.irt} \end{figure} \begin{figure}[htbp] \begin{tiny} \begin{center} <>= plot(iq.irt,type='test') @ \end{center} \end{tiny} \caption{A graphic analysis of 16 ability items sampled from the \iemph{SAPA} project. The total test information at all levels of difficulty may be shown by specifying the type='test' option in the plot function. } \label{fig:iq.irt.test} \end{figure} \begin{scriptsize} <>= iq.irt @ \end{scriptsize} \begin{figure}[htbp] \begin{center} <>= om <- omega(iq.irt$rho,4) @ \caption{An Omega analysis of 16 ability items sampled from the SAPA project. The items represent a general factor as well as four lower level factors. The analysis is done using the tetrachoric correlations found in the previous \pfun{irt.fa} analysis. The four matrix items have some serious problems, which may be seen later when examine the item response functions.} \label{fig:iq.irt} \end{center} \end{figure} \subsection{IRT based scoring} The primary advantage of IRT analyses is examining the item properties (both difficulty and discrimination). With complete data, the scores based upon simple total scores and based upon IRT are practically identical (this may be seen in the examples for \pfun{scoreIrt}). However, when working with data such as those found in the Synthetic Aperture Personality Assessment (\iemph{SAPA}) project, it is advantageous to use IRT based scoring. \iemph{SAPA} data might have 2-3 items/person sampled from scales with 10-20 items. Simply finding the average of the three (classical test theory) fails to consider that the items might differ in either discrimination or in difficulty. The \pfun{scoreIrt} function applies basic IRT to this problem. Consider 1000 randomly generated subjects with scores on 9 true/false items differing in difficulty. Selectively drop the hardest items for the 1/3 lowest subjects, and the 4 easiest items for the 1/3 top subjects (this is a crude example of what tailored testing would do). Then score these subjects: \begin{scriptsize} <>= v9 <- sim.irt(9,1000,-2.,2.,mod="normal") #dichotomous items items <- v9$items test <- irt.fa(items) total <- rowSums(items) ord <- order(total) items <- items[ord,] #now delete some of the data - note that they are ordered by score items[1:333,5:9] <- NA items[334:666,3:7] <- NA items[667:1000,1:4] <- NA scores <- scoreIrt(test,items) unitweighted <- scoreIrt(items=items,keys=rep(1,9)) scores.df <- data.frame(true=v9$theta[ord],scores,unitweighted) colnames(scores.df) <- c("True theta","irt theta","total","fit","rasch","total","fit") @ \end{scriptsize} These results are seen in Figure~\ref{fig:score.irt.pdf}. \begin{figure}[htbp] \begin{center} \caption{IRT based scoring and total test scores for 1000 simulated subjects. True theta values are reported and then the IRT and total scoring systems. } <>= pairs.panels(scores.df,pch='.',gap=0) title('Comparing true theta for IRT, Rasch and classically based scoring',line=3) @ \label{fig:score.irt.pdf} \end{center} \end{figure} \subsubsection{1 versus 2 parameter IRT scoring} In Item Response Theory, items can be assumed to be equally discriminating but to differ in their difficulty (the Rasch model) or to vary in their discriminability. Two functions (\pfun{scoreIrt.1pl} and \pfun{scoreIrt.2pl}) are meant to find multiple IRT based scales using the Rasch model or the 2 parameter model. Both allow for negatively keyed as well as positively keyed items. Consider the \pfun{bfi} data set with scoring keys key.list and items listed as an item.list. (This is the same as the key.list, but with the negative signs removed.) \begin{scriptsize} <>= keys.list <- list(agree=c("-A1","A2","A3","A4","A5"), conscientious=c("C1","C2","C3","-C4","-C5"), extraversion=c("-E1","-E2","E3","E4","E5"), neuroticism=c("N1","N2","N3","N4","N5"), openness = c("O1","-O2","O3","O4","-O5")) item.list <- list(agree=c("A1","A2","A3","A4","A5"), conscientious=c("C1","C2","C3","C4","C5"), extraversion=c("E1","E2","E3","E4","E5"), neuroticism=c("N1","N2","N3","N4","N5"), openness = c("O1","O2","O3","O4","O5")) bfi.1pl <- scoreIrt.1pl(keys.list,bfi) #the one parameter solution bfi.2pl <- scoreIrt.2pl(item.list,bfi) #the two parameter solution bfi.ctt <- scoreFast(keys.list,bfi) # fast scoring function @ \end{scriptsize} We can compare these three ways of doing the analysis using the \pfun{cor2} function which correlates two separate data frames. All three models produce vey simillar results for the case of almost complete data. It is when we have massively missing completely at random data (MMCAR) that the results show the superiority of the irt scoring. \begin{scriptsize} <>= #compare the solutions using the cor2 function cor2(bfi.1pl,bfi.ctt) cor2(bfi.2pl,bfi.ctt) cor2(bfi.2pl,bfi.1pl) @ \end{scriptsize} \section{Multilevel modeling} Correlations between individuals who belong to different natural groups (based upon e.g., ethnicity, age, gender, college major, or country) reflect an unknown mixture of the pooled correlation within each group as well as the correlation of the means of these groups. These two correlations are independent and do not allow inferences from one level (the group) to the other level (the individual). When examining data at two levels (e.g., the individual and by some grouping variable), it is useful to find basic descriptive statistics (means, sds, ns per group, within group correlations) as well as between group statistics (over all descriptive statistics, and overall between group correlations). Of particular use is the ability to decompose a matrix of correlations at the individual level into correlations within group and correlations between groups. \subsection{Decomposing data into within and between level correlations using \pfun{statsBy}} There are at least two very powerful packages (\Rpkg{nlme} and \Rpkg{multilevel}) which allow for complex analysis of hierarchical (multilevel) data structures. \pfun{statsBy} is a much simpler function to give some of the basic descriptive statistics for two level models. This follows the decomposition of an observed correlation into the pooled correlation within groups (rwg) and the weighted correlation of the means between groups which is discussed by \cite{pedhazur:97} and by \cite{bliese:09} in the multilevel package. \begin{equation} r_{xy} = \eta_{x_{wg}} * \eta_{y_{wg}} * r_{xy_{wg}} + \eta_{x_{bg}} * \eta_{y_{bg}} * r_{xy_{bg} } \end{equation} where $r_{xy} $ is the normal correlation which may be decomposed into a within group and between group correlations $r_{xy_{wg}}$ and $r_{xy_{bg}} $ and $\eta$ (eta) is the correlation of the data with the within group values, or the group means. \subsection{Generating and displaying multilevel data} \pfun{withinBetween} is an example data set of the mixture of within and between group correlations. The within group correlations between 9 variables are set to be 1, 0, and -1 while those between groups are also set to be 1, 0, -1. These two sets of correlations are crossed such that V1, V4, and V7 have within group correlations of 1, as do V2, V5 and V8, and V3, V6 and V9. V1 has a within group correlation of 0 with V2, V5, and V8, and a -1 within group correlation with V3, V6 and V9. V1, V2, and V3 share a between group correlation of 1, as do V4, V5 and V6, and V7, V8 and V9. The first group has a 0 between group correlation with the second and a -1 with the third group. See the help file for \pfun{withinBetween} to display these data. \pfun{sim.multilevel} will generate simulated data with a multilevel structure. The \pfun{statsBy.boot} function will randomize the grouping variable ntrials times and find the statsBy output. This can take a long time and will produce a great deal of output. This output can then be summarized for relevant variables using the \pfun{statsBy.boot.summary} function specifying the variable of interest. Consider the case of the relationship between various tests of ability when the data are grouped by level of education (statsBy(sat.act)) or when affect data are analyzed within and between an affect manipulation (statsBy(affect) ). \ \subsection{Factor analysis by groups} Confirmatory factor analysis comparing the structures in multiple groups can be done in the \Rpkg{lavaan} package. However, for exploratory analyses of the structure within each of multiple groups, the \pfun{faBy} function may be used in combination with the \pfun{statsBy} function. First run pfun{statsBy} with the correlation option set to TRUE, and then run \pfun{faBy} on the resulting output. \begin{scriptsize} \begin{Schunk} \begin{Sinput} sb <- statsBy(bfi[c(1:25,27)], group="education",cors=TRUE) faBy(sb,nfactors=5) #find the 5 factor solution for each education level \end{Sinput} \end{Schunk} \end{scriptsize} \subsection{Multilevel reliability} The \pfun{mlr} and \pfun{multilevelReliablity} functions follow the advice of \cite{shrout:12a} for estimating multievel reliablilty. A detailed discussion of this procedure is given in \cite{rw:paid:17} which is available at \url{https://personality-project.org/revelle/publications/rw.paid.17.final.pdf}. \section{Set Correlation and Multiple Regression from the correlation matrix} An important generalization of multiple regression and multiple correlation is \iemph{set correlation} developed by \cite{cohen:set} and discussed by \cite{cohen:03}. Set correlation is a multivariate generalization of multiple regression and estimates the amount of variance shared between two sets of variables. Set correlation also allows for examining the relationship between two sets when controlling for a third set. This is implemented in the \pfun{setCor} function. Set correlation is $$R^{2} = 1 - \prod_{i=1}^n(1-\lambda_{i})$$ where $\lambda_{i}$ is the ith eigen value of the eigen value decomposition of the matrix $$R = R_{xx}^{-1}R_{xy}R_{xx}^{-1}R_{xy}^{-1}.$$ Unfortunately, there are several cases where set correlation will give results that are much too high. This will happen if some variables from the first set are highly related to those in the second set, even though most are not. In this case, although the set correlation can be very high, the degree of relationship between the sets is not as high. In this case, an alternative statistic, based upon the average canonical correlation might be more appropriate. \pfun{setCor} has the additional feature that it will calculate multiple and partial correlations from the correlation or covariance matrix rather than the original data. Consider the correlations of the 6 variables in the \pfun{sat.act} data set. First do the normal multiple regression, and then compare it with the results using \pfun{setCor}. Two things to notice. \pfun{setCor} works on the \emph{correlation} or \emph{covariance} or \emph{raw data} matrix, and thus if using the correlation matrix, will report standardized or raw $\hat{\beta}$ weights. Secondly, it is possible to do several multiple regressions simultaneously. If the number of observations is specified, or if the analysis is done on raw data, statistical tests of significance are applied. For this example, the analysis is done on the correlation matrix rather than the raw data. \begin{scriptsize} <>= C <- cov(sat.act,use="pairwise") model1 <- lm(ACT~ gender + education + age, data=sat.act) summary(model1) @ Compare this with the output from \pfun{setCor}. <>= #compare with setCor setCor(gender + education + age ~ ACT + SATV + SATQ, data = C, n.obs=700) @ \end{scriptsize} Note that the \pfun{setCor} analysis also reports the amount of shared variance between the predictor set and the criterion (dependent) set. This set correlation is symmetric. That is, the $R^{2}$ is the same independent of the direction of the relationship. \section{Simulation functions} It is particularly helpful, when trying to understand psychometric concepts, to be able to generate sample data sets that meet certain specifications. By knowing ``truth" it is possible to see how well various algorithms can capture it. Several of the \pfun{sim} functions create artificial data sets with known structures. A number of functions in the psych package will generate simulated data. These functions include \pfun{sim} for a factor simplex, and \pfun{sim.simplex} for a data simplex, \pfun{sim.circ} for a circumplex structure, \pfun{sim.congeneric} for a one factor factor congeneric model, \pfun{sim.dichot} to simulate dichotomous items, \pfun{sim.hierarchical} to create a hierarchical factor model, \pfun{sim.item} is a more general item simulation, \pfun{sim.minor} to simulate major and minor factors, \pfun{sim.omega} to test various examples of omega, \pfun{sim.parallel} to compare the efficiency of various ways of determining the number of factors, \pfun{sim.rasch} to create simulated rasch data, \pfun{sim.irt} to create general 1 to 4 parameter IRT data by calling \pfun{sim.npl} 1 to 4 parameter logistic IRT or \pfun{sim.npn} 1 to 4 paramater normal IRT, \pfun{sim.structural} a general simulation of structural models, and \pfun{sim.anova} for ANOVA and lm simulations, and \pfun{sim.vss}. Some of these functions are separately documented and are listed here for ease of the help function. See each function for more detailed help. \begin{description} \item [\pfun{sim}] The default version is to generate a four factor simplex structure over three occasions, although more general models are possible. \item [\pfun{sim.simple}] Create major and minor factors. The default is for 12 variables with 3 major factors and 6 minor factors. \item [\pfun{sim.structure}] To combine a measurement and structural model into one data matrix. Useful for understanding structural equation models. \item [\pfun{sim.hierarchical}] To create data with a hierarchical (bifactor) structure. \item [\pfun{sim.congeneric}] To create congeneric items/tests for demonstrating classical test theory. This is just a special case of sim.structure. \item [\pfun{sim.circ}] To create data with a circumplex structure. \item [\pfun{sim.item}]To create items that either have a simple structure or a circumplex structure. \item [\pfun{sim.dichot}] Create dichotomous item data with a simple or circumplex structure. \item[\pfun{sim.rasch}] Simulate a 1 parameter logistic (Rasch) model. \item[\pfun{sim.irt}] Simulate a 2 parameter logistic (2PL) or 2 parameter Normal model. Will also do 3 and 4 PL and PN models. \item[\pfun{sim.multilevel}] Simulate data with different within group and between group correlational structures. \end{description} Some of these functions are described in more detail in the companion vignette: \href{"psych_for_sem.pdf"}{psych for sem}. The default values for \pfun{sim.structure} is to generate a 4 factor, 12 variable data set with a simplex structure between the factors. Two data structures that are particular challenges to exploratory factor analysis are the simplex structure and the presence of minor factors. Simplex structures \pfun{sim.simplex} will typically occur in developmental or learning contexts and have a correlation structure of r between adjacent variables and $r^n$ for variables n apart. Although just one latent variable (r) needs to be estimated, the structure will have nvar-1 factors. Many simulations of factor structures assume that except for the major factors, all residuals are normally distributed around 0. An alternative, and perhaps more realistic situation, is that the there are a few major (big) factors and many minor (small) factors. The challenge is thus to identify the major factors. \pfun{sim.minor} generates such structures. The structures generated can be thought of as having a a major factor structure with some small correlated residuals. Although coefficient $\omega_h$ is a very useful indicator of the general factor saturation of a unifactorial test (one with perhaps several sub factors), it has problems with the case of multiple, independent factors. In this situation, one of the factors is labelled as ``general'' and the omega estimate is too large. This situation may be explored using the \pfun{sim.omega} function. The four irt simulations, \pfun{sim.rasch}, \pfun{sim.irt}, \pfun{sim.npl} and \pfun{sim.npn}, simulate dichotomous items following the Item Response model. \pfun{sim.irt} just calls either \pfun{sim.npl} (for logistic models) or \pfun{sim.npn} (for normal models) depending upon the specification of the model. The logistic model is \begin{equation} P(x | \theta_i, \delta_j, \gamma_j, \zeta_j )= \gamma_j + \frac{\zeta_j - \gamma_j}{1+e^{\alpha_j(\delta_j - \theta_i}}. \end{equation} where $\gamma$ is the lower asymptote or guessing parameter, $\zeta$ is the upper asymptote (normally 1), $\alpha_j$ is item discrimination and $\delta_j$ is item difficulty. For the 1 Paramater Logistic (Rasch) model, gamma=0, zeta=1, alpha=1 and item difficulty is the only free parameter to specify. (Graphics of these may be seen in the demonstrations for the logistic function.) The normal model (\pfun{irt.npn} calculates the probability using \fun{pnorm} instead of the logistic function used in \pfun{irt.npl}, but the meaning of the parameters are otherwise the same. With the a = $\alpha$ parameter = 1.702 in the logiistic model the two models are practically identical. \section{Graphical Displays} Many of the functions in the \Rpkg{psych} package include graphic output and examples have been shown in the previous figures. After running \pfun{fa}, \pfun{iclust}, \pfun{omega}, \pfun{irt.fa}, plotting the resulting object is done by the \pfun{plot.psych} function as well as specific diagram functions. e.g., (but not shown) \begin{scriptsize} \begin{Schunk} \begin{Sinput} f3 <- fa(Thurstone,3) plot(f3) fa.diagram(f3) c <- iclust(Thurstone) plot(c) #a pretty boring plot iclust.diagram(c) #a better diagram c3 <- iclust(Thurstone,3) plot(c3) #a more interesting plot data(bfi) e.irt <- irt.fa(bfi[11:15]) plot(e.irt) ot <- omega(Thurstone) plot(ot) omega.diagram(ot) \end{Sinput} \end{Schunk} \end{scriptsize} The ability to show path diagrams to represent factor analytic and structural models is discussed in somewhat more detail in the accompanying vignette, \href{"psych_for_sem.pdf"}{psych for sem}. Basic routines to draw path diagrams are included in the \pfun{dia.rect} and accompanying functions. These are used by the \pfun{fa.diagram}, \pfun{structure.diagram} and \pfun{iclust.diagram} functions. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= xlim=c(0,10) ylim=c(0,10) plot(NA,xlim=xlim,ylim=ylim,main="Demonstration of dia functions",axes=FALSE,xlab="",ylab="") ul <- dia.rect(1,9,labels="upper left",xlim=xlim,ylim=ylim) ll <- dia.rect(1,3,labels="lower left",xlim=xlim,ylim=ylim) lr <- dia.ellipse(9,3,"lower right",xlim=xlim,ylim=ylim,e.size=.09) ur <- dia.ellipse(7,9,"upper right",xlim=xlim,ylim=ylim,e.size=.1) ml <- dia.ellipse(3,6,"middle left",xlim=xlim,ylim=ylim,e.size=.1) mr <- dia.ellipse(7,6,"middle right",xlim=xlim,ylim=ylim,e.size=.08) bl <- dia.ellipse(1,1,"bottom left",xlim=xlim,ylim=ylim,e.size=.08) br <- dia.rect(9,1,"bottom right",xlim=xlim,ylim=ylim) dia.arrow(from=lr,to=ul,labels="right to left") dia.arrow(from=ul,to=ur,labels="left to right") dia.curved.arrow(from=lr,to=ll$right,labels ="right to left") dia.curved.arrow(to=ur,from=ul$right,labels ="left to right") dia.curve(ll$top,ul$bottom,"double",-1) #for rectangles, specify where to point dia.curved.arrow(mr,ur,"up") #but for ellipses, just point to it. dia.curve(ml,mr,"across") dia.curved.arrow(ur,lr,"top down") dia.curved.arrow(br$top,lr$bottom,"up") dia.curved.arrow(bl,br,"left to right") dia.arrow(bl$top,ll$bottom) dia.curved.arrow(ml,ll$top,scale=-1) dia.curved.arrow(mr,lr$top) @ \end{scriptsize} \caption{The basic graphic capabilities of the dia functions are shown in this figure.} \label{fig:dia} \end{center} \end{figure} \section{Converting output to APA style tables using \LaTeX} Although for most purposes, using the \Rpkg{Sweave} or \Rpkg{KnitR} packages produces clean output, some prefer output pre formatted for APA style tables. This can be done using the \Rpkg{xtable} package for almost anything, but there are a few simple functions in \Rpkg{psych} for the most common tables. \pfun{fa2latex} will convert a factor analysis or components analysis output to a \LaTeX table, \pfun{cor2latex} will take a correlation matrix and show the lower (or upper diagonal), \pfun{irt2latex} converts the item statistics from the \pfun{irt.fa} function to more convenient \LaTeX output, and finally, \pfun{df2latex} converts a generic data frame to \LaTeX. An example of converting the output from \pfun{fa} to \LaTeX appears in Table~\ref{falatex}. % fa2latex % f3 % Called in the psych package fa2latex % Called in the psych package f3 \begin{scriptsize} \begin{table}[htpb] \caption{fa2latex} \begin{center} \begin{tabular} {l r r r r r r } \multicolumn{ 6 }{l}{ A factor analysis table from the psych package in R } \cr \hline Variable & MR1 & MR2 & MR3 & h2 & u2 & com \cr \hline Sentences & 0.91 & -0.04 & 0.04 & 0.82 & 0.18 & 1.01 \cr Vocabulary & 0.89 & 0.06 & -0.03 & 0.84 & 0.16 & 1.01 \cr Sent.Completion & 0.83 & 0.04 & 0.00 & 0.73 & 0.27 & 1.00 \cr First.Letters & 0.00 & 0.86 & 0.00 & 0.73 & 0.27 & 1.00 \cr 4.Letter.Words & -0.01 & 0.74 & 0.10 & 0.63 & 0.37 & 1.04 \cr Suffixes & 0.18 & 0.63 & -0.08 & 0.50 & 0.50 & 1.20 \cr Letter.Series & 0.03 & -0.01 & 0.84 & 0.72 & 0.28 & 1.00 \cr Pedigrees & 0.37 & -0.05 & 0.47 & 0.50 & 0.50 & 1.93 \cr Letter.Group & -0.06 & 0.21 & 0.64 & 0.53 & 0.47 & 1.23 \cr \hline \cr SS loadings & 2.64 & 1.86 & 1.5 & \cr\cr \hline \cr MR1 & 1.00 & 0.59 & 0.54 \cr MR2 & 0.59 & 1.00 & 0.52 \cr MR3 & 0.54 & 0.52 & 1.00 \cr \hline \end{tabular} \end{center} \label{falatex} \end{table} \end{scriptsize} \newpage \section{Miscellaneous functions} A number of functions have been developed for some very specific problems that don't fit into any other category. The following is an incomplete list. Look at the \iemph{Index} for \Rpkg{psych} for a list of all of the functions. \begin{description} \item [\pfun{block.random}] Creates a block randomized structure for n independent variables. Useful for teaching block randomization for experimental design. \item [\pfun{df2latex}] is useful for taking tabular output (such as a correlation matrix or that of \pfun{describe} and converting it to a \LaTeX{} table. May be used when Sweave is not convenient. \item [\pfun{cor2latex}] Will format a correlation matrix in APA style in a \LaTeX{} table. See also \pfun{fa2latex} and \pfun{irt2latex}. \item [\pfun{cosinor}] One of several functions for doing \iemph{circular statistics}. This is important when studying mood effects over the day which show a diurnal pattern. See also \pfun{circadian.mean}, \pfun{circadian.cor} and \pfun{circadian.linear.cor} for finding circular means, circular correlations, and correlations of circular with linear data. \item[\pfun{fisherz}] Convert a correlation to the corresponding Fisher z score. \item [\pfun{geometric.mean}] also \pfun{harmonic.mean} find the appropriate mean for working with different kinds of data. \item [\pfun{ICC}] and \pfun{cohen.kappa} are typically used to find the reliability for raters. \item [\pfun{headtail}] combines the \fun{head} and \fun{tail} functions to show the first and last lines of a data set or output. \item [\pfun{topBottom}] Same as headtail. Combines the \fun{head} and \fun{tail} functions to show the first and last lines of a data set or output, but does not add ellipsis between. \item [\pfun{mardia}] calculates univariate or multivariate (Mardia's test) skew and kurtosis for a vector, matrix, or data.frame \item [\pfun{p.rep}] finds the probability of replication for an F, t, or r and estimate effect size. \item [\pfun{partial.r}] partials a y set of variables out of an x set and finds the resulting partial correlations. (See also \pfun{setCor}.) \item [\pfun{rangeCorrection}] will correct correlations for restriction of range. \item [\pfun{reverse.code}] will reverse code specified items. Done more conveniently in most \Rpkg{psych} functions, but supplied here as a helper function when using other packages. \item [\pfun{superMatrix}] Takes two or more matrices, e.g., A and B, and combines them into a ``Super matrix'' with A on the top left, B on the lower right, and 0s for the other two quadrants. A useful trick when forming complex keys, or when forming example problems. \end{description} \section{Data sets} A number of data sets for demonstrating psychometric techniques are included in the \Rpkg{psych} package. These include six data sets showing a hierarchical factor structure (five cognitive examples, \pfun{Thurstone}, \pfun{Thurstone.33}, \pfun{Holzinger}, \pfun{Bechtoldt.1}, \pfun{Bechtoldt.2}, and one from health psychology \pfun{Reise}). One of these (\pfun{Thurstone}) is used as an example in the \Rpkg{sem} package as well as \cite{mcdonald:tt}. The original data are from \cite{thurstone:41} and reanalyzed by \cite{bechtoldt:61}. Personality item data representing five personality factors on 25 items (\pfun{bfi}) or 13 personality inventory scores (\pfun{epi.bfi}), and 14 multiple choice iq items (\pfun{iqitems}). The \pfun{vegetables} example has paired comparison preferences for 9 vegetables. This is an example of Thurstonian scaling used by \cite{guilford:54} and \cite{nunnally:67}. Other data sets include \pfun{cubits}, \pfun{peas}, and \pfun{heights} from Galton. \begin{description} \item[Thurstone] Holzinger-Swineford (1937) introduced the bifactor model of a general factor and uncorrelated group factors. The Holzinger correlation matrix is a 14 * 14 matrix from their paper. The Thurstone correlation matrix is a 9 * 9 matrix of correlations of ability items. The Reise data set is 16 * 16 correlation matrix of mental health items. The Bechtholdt data sets are both 17 x 17 correlation matrices of ability tests. \item [bfi] 25 personality self report items taken from the International Personality Item Pool (ipip.ori.org) were included as part of the Synthetic Aperture Personality Assessment (\iemph{SAPA}) web based personality assessment project. The data from 2800 subjects are included here as a demonstration set for scale construction, factor analysis and Item Response Theory analyses. \item [sat.act] Self reported scores on the SAT Verbal, SAT Quantitative and ACT were collected as part of the Synthetic Aperture Personality Assessment (\iemph{SAPA}) web based personality assessment project. Age, gender, and education are also reported. The data from 700 subjects are included here as a demonstration set for correlation and analysis. \item [epi.bfi] A small data set of 5 scales from the Eysenck Personality Inventory, 5 from a Big 5 inventory, a Beck Depression Inventory, and State and Trait Anxiety measures. Used for demonstrations of correlations, regressions, graphic displays. \item[epiR] The EPI was given twice to 474 participants. This is a useful data set for exploring test-retest reliability, \item[sai, msqR] 20 anxiety items and 75 mood items were given at least twice to 3032 participants. These are useful for understanding reliability structures. \item [iq] 14 multiple choice ability items were included as part of the Synthetic Aperture Personality Assessment (\iemph{SAPA}) web based personality assessment project. The data from 1000 subjects are included here as a demonstration set for scoring multiple choice inventories and doing basic item statistics. \item [galton] Two of the earliest examples of the correlation coefficient were Francis Galton's data sets on the relationship between mid parent and child height and the similarity of parent generation peas with child peas. \pfun{galton} is the data set for the Galton height. \pfun{peas} is the data set Francis Galton used to ntroduce the correlation coefficient with an analysis of the similarities of the parent and child generation of 700 sweet peas. \item[Dwyer] \cite{dwyer:37} introduced a method for \emph{factor extension} (see \pfun{fa.extension} that finds loadings on factors from an original data set for additional (extended) variables. This data set includes his example. \item [miscellaneous] \pfun{cities} is a matrix of airline distances between 11 US cities and may be used for demonstrating multiple dimensional scaling. \pfun{vegetables} is a classic data set for demonstrating Thurstonian scaling and is the preference matrix of 9 vegetables from \cite{guilford:54}. Used by \cite{guilford:54,nunnally:67,nunnally:bernstein:94}, this data set allows for examples of basic scaling techniques. \end{description} \section{Development version and a users guide} The most recent development version is available as a source file at the repository maintained at \href{ href="https://personality-project.org/r"}{\url{https://personality-project.org/r}}. That version will have removed the most recently discovered bugs (but perhaps introduced other, yet to be discovered ones). To download that version, go to the repository %\href{"https://personality-project.org/r/src/contrib/}{ \url{https://personality-project.org/r/src/contrib/} and wander around. For a Mac and PC this version can be installed directly using the ``other repository" option in the package installer. \begin{Schunk} \begin{Sinput} > install.packages("psych", repos="https://personality-project.org/r", type="source") \end{Sinput} \end{Schunk} Although the individual help pages for the \Rpkg{psych} package are available as part of \R{} and may be accessed directly (e.g. ?psych) , the full manual for the \pfun{psych} package is also available as a pdf at \url{https://personality-project.org/r/psych_manual.pdf} %psych\_manual.pdf. News and a history of changes are available in the NEWS and CHANGES files in the source files. To view the most recent news, \begin{Schunk} \begin{Sinput} > news(Version > "1.8.4", package="psych") \end{Sinput} \end{Schunk} \section{Psychometric Theory} The \Rpkg{psych} package has been developed to help psychologists do basic research. Many of the functions were developed to supplement a book (\url{https://personality-project.org/r/book} An introduction to Psychometric Theory with Applications in \R{} \citep{revelle:intro} More information about the use of some of the functions may be found in the book . For more extensive discussion of the use of \Rpkg{psych} in particular and \R{} in general, consult \url{https://personality-project.org/r/r.guide.html} A short guide to R. \section{SessionInfo} This document was prepared using the following settings. \begin{tiny} <>= sessionInfo() @ \end{tiny} \newpage %\bibliography{/Volumes/WR/Documents/Active/book/all} \begin{thebibliography}{} \bibitem[\protect\astroncite{Bechtoldt}{1961}]{bechtoldt:61} Bechtoldt, H. 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(1935). \newblock A theory of psychological components--an alternative to "mathematical factors.". \newblock {\em Psychological Review}, 42(5):425--454. \bibitem[\protect\astroncite{Tryon}{1939}]{tryon:39} Tryon, R.~C. (1939). \newblock {\em Cluster analysis}. \newblock Edwards Brothers, Ann Arbor, Michigan. \bibitem[\protect\astroncite{Velicer}{1976}]{velicer:76} Velicer, W. (1976). \newblock Determining the number of components from the matrix of partial correlations. \newblock {\em Psychometrika}, 41(3):321--327. \bibitem[\protect\astroncite{Zinbarg et~al.}{2005}]{zinbarg:pm:05} Zinbarg, R.~E., Revelle, W., Yovel, I., and Li, W. (2005). \newblock Cronbach's {$\alpha$}, {Revelle's} {$\beta$}, and {McDonald's} {$\omega_H$}: Their relations with each other and two alternative conceptualizations of reliability. \newblock {\em Psychometrika}, 70(1):123--133. \bibitem[\protect\astroncite{Zinbarg et~al.}{2006}]{zinbarg:apm:06} Zinbarg, R.~E., Yovel, I., Revelle, W., and McDonald, R.~P. (2006). \newblock Estimating generalizability to a latent variable common to all of a scale's indicators: A comparison of estimators for {$\omega_h$}. \newblock {\em Applied Psychological Measurement}, 30(2):121--144. \end{thebibliography} \printindex \end{document} psychTools/inst/doc/factor.pdf0000644000176200001440000365565514153443017016147 0ustar liggesusers%PDF-1.5 % 2 0 obj << /Type /ObjStm /N 100 /First 814 /Length 1516 /Filter /FlateDecode >> stream xڥWn8W1H$ ݽBkbHo$5N8Y*6ۂ"JHER& #bHx$cHfx$s R"_HI $dJHI'1\R`) L$I7&"f9!<IoeD)Q _)9y $D,"x"6BLDS"^*#/.HieDBUȁc^ya+9KY4!xi&^& <1X\ DA (۰]( &(xHFsQIA(wÁ Rb!   !% N*źYٳ@S9 L\#Se1`@

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X `5`3l  `7쁽80p 8 4i8 <\p .  ܀p f6p=!<3x/%;Y%L3g2d^ n2?pN]i8Gn OnT~|ϽɯY 5ȡ07&xӖm(܅{p)_7Y4$gԵ\d Dss|^@ 2/d^ȼy!B 2/d^ȼy!B 2/d^ȼy!B 2/d^ȼy!B 2/d^ȼy!B 2/d^ȼr(Mk~_D ((PP@A ((PP@A ((PP@A ((PP@A ((PP@A ((PP@AI)PS@M5j )PS@M5j )PS@M5j )PS)PS@M5j )PS@M5j )PS6k Ԇ}͈Zk2e^˼y-ZHgΧ_<K_ |-ZDH~-Zk]G5j )PS@M5j ׿<͟!^Hoi}'^_Xz㥱&MM_zXz>s3>HSp'|+MLiꇿK#iR;-847ci,-G7}2|kU,u҂f6/ci2-8O槅XKSiY,-H ´K1QZ+gZ/Կm2X+`%հ:i`#lͰ6;`'ݰ>) p18'$pbri\cU907fN\W endstream endobj startxref 1000700 %%EOF psychTools/inst/doc/omega.Rnw0000644000176200001440000015450713726243502015743 0ustar liggesusers% \VignetteIndexEntry{How to find Omega} % \VignettePackage{psych} % \VignetteKeywords{multivariate} % \VignetteKeyword{models} % \VignetteKeyword{Hplot} %\VignetteDepends{psych} %\documentclass[doc]{apa} \documentclass[11pt]{article} %\documentclass[11pt]{amsart} \usepackage{geometry} % See geometry.pdf to learn the layout options. There are lots. \geometry{letterpaper} % ... or a4paper or a5paper or ... %\geometry{landscape} % Activate for for rotated page geometry \usepackage[parfill]{parskip} % Activate to begin paragraphs with an empty line rather than an indent \usepackage{graphicx} \usepackage{amssymb} \usepackage{epstopdf} \usepackage{mathptmx} \usepackage{helvet} \usepackage{courier} \usepackage{epstopdf} \usepackage{makeidx} % allows index generation \usepackage[authoryear,round]{natbib} \usepackage{gensymb} \usepackage{longtable} %\usepackage{geometry} \usepackage{amssymb} \usepackage{amsmath} %\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \usepackage{Sweave} %\usepackage{/Volumes/'Macintosh HD'/Library/Frameworks/R.framework/Versions/2.13/Resources/share/texmf/tex/latex/Sweave} %\usepackage[ae]{Rd} %\usepackage[usenames]{color} %\usepackage{setspace} \bibstyle{apacite} \bibliographystyle{apa} %this one plus author year seems to work? %\usepackage{hyperref} \usepackage[colorlinks=true,citecolor=blue]{hyperref} %this makes reference links hyperlinks in pdf! \DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \usepackage{multicol} % used for the two-column index \usepackage[bottom]{footmisc}% places footnotes at page bottom \let\proglang=\textsf \newcommand{\R}{\proglang{R}} %\newcommand{\pkg}[1]{{\normalfont\fontseries{b}\selectfont #1}} \newcommand{\Rfunction}[1]{{\texttt{#1}}} \newcommand{\fun}[1]{{\texttt{#1}\index{#1}\index{R function!#1}}} \newcommand{\pfun}[1]{{\texttt{#1}\index{#1}\index{R function!#1}\index{R function!psych package!#1}}}\newcommand{\Rc}[1]{{\texttt{#1}}} %R command same as Robject \newcommand{\Robject}[1]{{\texttt{#1}}} \newcommand{\Rpkg}[1]{{\textit{#1}\index{#1}\index{R package!#1}}} %different from pkg - which is better? \newcommand{\iemph}[1]{{\emph{#1}\index{#1}}} \newcommand{\wrc}[1]{\marginpar{\textcolor{blue}{#1}}} %bill's comments \newcommand{\wra}[1]{\textcolor{blue}{#1}} %bill's comments \newcommand{\ve}[1]{{\textbf{#1}}} %trying to get a vector command \usepackage{fancyvrb} %this allows fancy boxes \fvset{fontfamily=courier} \DefineVerbatimEnvironment{Routput}{Verbatim} %{fontsize=\scriptsize, xleftmargin=0.6cm} {fontseries=b,fontsize=\scriptsize, xleftmargin=0.1cm} \DefineVerbatimEnvironment{Binput}{Verbatim} {fontseries=b, fontsize=\scriptsize,frame=single, label=\fbox{lavaan model syntax}, framesep=2mm} %\DefineShortVerb{\!} %%% generates error! %change the definition of Sinput from Sweave \DefineVerbatimEnvironment{Sinput}{Verbatim}{fontseries=b, fontsize=\scriptsize, frame=single, label=\fbox{R code},xleftmargin=0pt, framesep=1mm} \DefineVerbatimEnvironment{Rinput}{Verbatim} %{fontsize=\scriptsize, frame=single, label=\fbox{R code}, framesep=1mm} {fontseries=b, fontsize=\scriptsize, frame=single, label=\fbox{R code},xleftmargin=0pt, framesep=1mm} \DefineVerbatimEnvironment{Link}{Verbatim} {fontseries=b, fontsize=\small, formatcom=\color{darkgreen}, xleftmargin=1.0cm} \DefineVerbatimEnvironment{Toutput}{Verbatim} {fontseries=b,fontsize=\tiny, xleftmargin=0.1cm} \DefineVerbatimEnvironment{rinput}{Verbatim} {fontseries=b, fontsize=\tiny, frame=single, label=\fbox{R code}, framesep=1mm} \newcommand{\citeti}[1]{\begin{tiny}\citep{#1}\end{tiny}} \newcommand{\light}[1]{\textcolor{gray}{#1}} \newcommand{\vect}[1]{\boldsymbol{#1}} \let\vec\vect \makeindex % used for the subject index \title{Using \R{} and the \Rpkg{psych} package to find $\omega$} \author{William Revelle\\Department of Psychology\\Northwestern University} %\affiliation{Northwestern University} %\acknowledgements{Written to accompany the psych package. Comments should be directed to William Revelle \\ \url{revelle@northwestern.edu}} %\date{} % Activate to display a given date or no date \begin{document} \maketitle \tableofcontents \newpage \section{Overview of this and related documents} To do basic and advanced personality and psychological research using \R{} is not as complicated as some think. This is one of a set of ``How To'' to do various things using \R{} \citep{R}, particularly using the \Rpkg{psych} \citep{psych} package. The current list of How To's includes: \begin{enumerate} \item An \href{http://personality-project.org/r/psych/intro.pdf}{introduction} (vignette) of the \Rpkg{psych} package \item An \href{http://personality-project.org/r/psych/overview.pdf}{overview} (vignette) of the \Rpkg{psych} package \item \href{http://personality-project.org/r/psych/HowTo/getting_started.pdf}{Installing} \R{} and some useful packages \item Using \R{} and the \Rpkg{psych} package to find \href{http://personality-project.org/r/psych/HowTo/omega.pdf}{$omega_h$} and $\omega_t$ (this document).. \item Using \R{} and the \Rpkg{psych} for \href{http://personality-project.org/r/psych/HowTo/factor.pdf}{factor analysis} and principal components analysis. \item Using the \pfun{scoreItems} function to find \href{http://personality-project.org/r/psych/HowTo/scoring.pdf}{scale scores and scale statistics}. \item Using \pfun{mediate} and \pfun{setCor} to do \href{http://personality-project.org/r/psych/HowTo/mediation.pdf}{mediation, moderation and regression analysis} \end{enumerate} \subsection{$omega_h$ as an estimate of the general factor saturation of a test} Cronbach's coefficient $alpha$ \citep{cronbach:51} is pehaps the most used (and most misused) estimate of the internal consistency of a test. $\alpha$ may be found in the \Rpkg{psych} package using the \pfun{alpha} function. However, two alternative estimates of reliability that take into account the hierarchical structure of the inventory are McDonald's $\omega_h$ and $\omega_t$ \citep{mcdonald:tt}. These may be found in \R{} in one step using one of two functions in the \Rpkg{psych} package: the \pfun{omega} function for an exploratory analysis (See Figure~\ref{fig:omega.9}) or \pfun{omegaSem} for a confirmatory analysis using the \Rpkg{sem} package solution based upon the exploratory solution from \pfun{omega}. This guide explains how to do it for the non or novice \R{} user. These set of instructions are adapted from three different sets of notes that the interested reader might find helpful: A set of slides developed for a \href{http://personality-project.org/r/aps/aps-short.pdf}{ two hour short course} in \R{} given for several years to the Association of Psychological Science as well as a \href{http://personality-project.org/r/}{short guide }to \R{} for psychologists and the \href{http://cran.r-project.org/web/packages/psych/vignettes/overview.pdf}{vignette} for the \Rpkg{psych} package. McDonald has proposed coefficient omega (hierarchical) ($\omega_h$) as an estimate of the general factor saturation of a test. \cite{zinbarg:pm:05} and \cite{rz:09} compare compare McDonald's $\omega_h$ to Cronbach's $\alpha$ and Revelle's $\beta$. They conclude that $\omega_h$ is the best estimate. (See also \cite{zinbarg:apm:06} and \cite{rz:09} as well as \cite{rc:reliability,rc:pa:19}. By following these simple guides, you soon will be able to do such things as find $\omega_{h}$ by issuing just three lines of code: \begin{Rinput} library(psych) my.data <- read.file() omega(my.data) \end{Rinput} The resulting output will be both graphical and textual. This guide helps the naive \R{} user to issue those three lines. Be careful, for once you start using \R, you will want to do more. One way to find $\omega_h$ is to do a factor analysis of the original data set, rotate the factors obliquely, factor that correlation matrix, do a Schmid-Leiman (\pfun{schmid}) transformation to find general factor loadings, and then find $\omega_h$. This is done using the \pfun{omega} function in the \Rpkg{psych} package in \R{}. This requires installing and using both \R{} as well as the \Rpkg{psych} package \citep{psych}. \subsubsection{But what about $\alpha$?} Several statistics were developed in the 1930s-1950s as short cut estimates of reliability \citep{rc:pa:19}. The approaches that consider just one test are collectively known as internal consistency procedures but also borrow from the concepts of domain sampling. Some of these techniques, e.g., \cite{cronbach:51,guttman:45,kuder:37} were developed before advances in computational speed made it trivial to find the factor structure of tests, and were based upon test and item variances. These procedures ($\alpha$, $\lambda_3$, KR20) were essentially short cuts for estimating reliability. To just find Guttman's $\lambda_3$ \citep{guttman:45} which is also known as \emph{coefficient} $\alpha$ \citep{cronbach:51}, you can use the \pfun{alpha} function or the \pfun{scoreItems} function. See the tutorial on how to use the \pfun{scoreItems} function to find \href{http://personality-project.org/r/psych/HowTo/scoring.pdf}{scale scores and scale statistics}. But, with modern computers, we can find \emph{model based} estimates that consider the factor structure of the items. $\omega_h$ and $\omega_t$ are two such model based estimates and are easy to find in \R{}. ~\ <>= library(psych) #make the psych package active library(psychTools) #make psychTools active om <- omega(Thurstone) #do the analysis om #show it @ <>= png('Thurstone.png') omega.diagram(om) dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{Thurstone.png} \caption{$\omega_h$ is a reliability estimate of the general factor of a set of variables. It is based upon the correlation of lower order factors. It may be found in \R{} by using the \pfun{omega} function which is part of the \Rpkg{psych} package. The figure shows a solution for the \pfun{Thurstone} 9 variable data set. Compare this to the solution using the \pfun{omegaDirect} function from \cite{waller:17} (Figure~\ref{fig:direct})} \label{fig:omega.9} \end{center} \end{figure} \newpage To use \R{} obviously requires installing \R{} on your computer. This is very easy to do (see section~\ref{install}) and needs to be done once. (The following sections are elaborated in the \href{https://personality-project.org/r/psych/HowTo/getting_started.pdf}{``getting startedHow To" } . If you need more help in installing \R{} see the longer version.) The power of \R{} is in the supplemental \emph{packages}. There are at least 16,000 packages that have been contributed to the \R{} project. To do any of the analyses discussed in these ``How To's", you will need to install the package \Rpkg{psych} \citep{psych}. To do factor analyses or principal component analyses you will also need the \Rpkg{GPArotation} \citep{GPA} package. With these two packages, you will be be able to find $\omega_{h}$ using Exploratory Factor Analysis. If you want to find to estimate $\omega_h$ using Confirmatory Factor Analysis, you will also need to add the \Rpkg{lavaan} \citep{lavaan} package. To use \Rpkg{psych} to create simulated data sets, you also need the \Rpkg{mnormt} \citep{mnormt} package. For a more complete installation of a number of psychometric packages, you can install and activate a package (\Rpkg{ctv}) that installs a large set of psychometrically relevant packages. As is true for \R{}, you will need to install packages just once. \subsection{Install R for the first time} \begin{enumerate} \item Download from R Cran (\url{http://cran.r-project.org/}) \item Install R (current version is 4.0.2) \item Start \R{}. Note that the \R{} prompt $>$ starts off every line. This is \R{}'s way of indicating that it wants input. In addition, note that almost all commands start and finish with parentheses. \item Add useful packages (just need to do this once) (see section~\ref{installing}) \begin{enumerate} \begin{Rinput} install.packages("psych",dependencies=TRUE) #the minimum requirement or install.packages(c("psych","GPArotation"),dependencies=TRUE) #required for factor analysis \end{Rinput} \item or if you want to do CFA \begin{Rinput} install.packages(c("psych","lavaan"), dependencies=TRUE) \end{Rinput} \item or if you want to install the psychometric task views \begin{Rinput} install.packages("ctv") #this downloads the task view package library(ctv) #this activates the ctv package install.views("Psychometrics") #among others \end{Rinput} \end{enumerate} \item Take a 5 minute break while the packages are loaded. \item Activate the package(s) you want to use (e.g., \Rpkg{psych}) \begin{Rinput} library(psych) #Only need to make psych active once a session \end{Rinput} \Rpkg{psych} will automatically activate the other packages it needs, as long as they are installed. Note that \Rpkg{psych} is updated roughly quarterly, the current version is 2.0.8 Patches and improvements to \Rpkg{psych} (the bleeding edge version) are available from the repository at the personality-project web server and may be installed from there: ~\ \begin{Rinput} install.packages("psych", repos = "https://personality-project.org/r", type="source") \end{Rinput} %\item library(sem) \#will be used for a few examples \item Use \R{} \end{enumerate} \subsubsection{Install R } \label{install} Go to the \href{http://cran.r-project.org}{Comprehensive R Archive Network (CRAN)} at \url{http://cran.r-project.org}: %(Figure~\ref{fig:cran}) %\begin{figure}[htbp] %\begin{center} %%\includegraphics[width=14cm]{../../../tutorials/R_Short_APS.15/rcran3.png} %\includegraphics[width=20cm]{../../../images/CRAN.png} %\caption{The basic \href{http://cran.r-project.org}{CRAN} window allows you choose your operating system. Comprehensive R Archive Network (CRAN) is found at \href{http://cran.r-project.org}{http://cran.r-project.org}:} %\label{fig:cran} %\end{center} %\end{figure} Choose your operating system and then download and install the appropriate version %For a PC: %(Figure~\ref{fig:pc}) %\begin{figure}[htbp] %\begin{center} %%\includegraphics[width=14cm]{../../../tutorials/R_Short_APS.15/cranpc1.png} %\includegraphics[width=19cm]{../../../images/CRAN_pc.pdf} % %\caption{On a PC you want to choose the base system} %\label{fig:pc} %\end{center} %\end{figure} Download and install the appropriate version -- Mac, PC or Unix/Linux %\begin{figure}[htbp] %\begin{center} %%\includegraphics[width=14cm]{../../../tutorials/R_Short_APS.15/cran-pc15.png} %\includegraphics[width=19cm]{../../../images/CRAN_pc_16.png} %\caption{Download the Windows version} %\label{default} %\end{center} %\end{figure} %Starting R on a PC. Once you have installed \R{} you probably, and particularly if you have a PC, will want to download and install the \href{https://www.rstudio.com} {R Studio} program. It is a very nice interface for PCs and Macs that combines four windows into one screen. %\begin{figure}[htbp] %\begin{center} %\includegraphics[width=14cm]{../../../images/RStudio01.png} %\caption{Using R Studio on a PC. } %\label{fig:pcstartup} %\end{center} %\end{figure} % %When using a PC, RStudio is very helpful. (Many like it for Macs as well). % % %\begin{figure}[htbp] %\begin{center} %\includegraphics[width=14cm]{../../../images/RStudio01.png} %\caption{Using R Studio on a PC. } %\label{fig:pcRstudio} %\end{center} %\end{figure} % %\clearpage % %%For a Mac: download and install the appropriate version -- Mac (Figure~\ref{fig:mac}) %\begin{figure}[htbp] %\begin{center} %%\includegraphics[width=14cm]{../../../tutorials/R_Short_APS/cran-mac.png} %\includegraphics[width=19cm]{../../../images/cran_mac.png} %\caption{For the Mac, you want to choose the latest version which includes the GUI as well as the 32 and 64 bit versions.} %\label{fig:mac} %\end{center} %\end{figure} % %\newpage %Start up R and get ready to play (Mac version). %\begin{scriptsize} %\begin{Schunk} %\begin{Soutput} %R version 3.3.0 (2016-05-03) -- "Supposedly Educational" %Copyright (C) 2016 The R Foundation for Statistical Computing %Platform: x86_64-apple-darwin13.4.0 (64-bit) % %R is free software and comes with ABSOLUTELY NO WARRANTY. %You are welcome to redistribute it under certain conditions. %Type 'license()' or 'licence()' for distribution details. % % Natural language support but running in an English locale % %R is a collaborative project with many contributors. %Type 'contributors()' for more information and %'citation()' on how to cite R or R packages in publications. % %Type 'demo()' for some demos, 'help()' for on-line help, or %'help.start()' for an HTML browser interface to help. %Type 'q()' to quit R. % %[R.app GUI 1.68 (7202) x86_64-apple-darwin13.4.0] % %[Workspace restored from /Users/revelle/.RData] %[History restored from /Users/revelle/.Rapp.history] % %> %\end{Soutput} %\end{Schunk} %\end{scriptsize} \subsubsection{Install relevant packages} \label{installing} Once \R{} is installed on your machine, you still need to install a few relevant ``packages''. Packages are what make \R{} so powerful, for they are special sets of functions that are designed for one particular application. In the case of the \Rpkg{psych} package, this is an application for doing the kind of basic data analysis and psychometric analysis that psychologists and many others find particularly useful. \Rpkg{psych} may be thought of a ``Swiss Army Knife" for psychological statistics. While not the best tool for a particular job, it is a useful tool for many jobs. You may either install the minimum set of packages necessary to do the analysis using an Exploratory Factor Analysis (EFA) approach (recommended) or a few more packages to do both an EFA and a CFA approach. It is also possible to add many psychometrically relevant packages all at once by using the ``task views'' approach. A particularly powerful package is the \Rpkg{lavaan} \citep{lavaan} package for doing structural equation modeling. Another useful one is the \Rpkg{sem} pacakge \citep{sem}. \paragraph{Install the minimum set} This may be done by typing into the console or using menu options (e.g., the Package Installer underneath the Packages and Data menu). \begin{Rinput} install.packages(c("psych", "psychTools"), dependencies = TRUE) \end{Rinput} % %\begin{figure}[htbp] %\begin{center} %\includegraphics[width=14cm]{../../../images/RStudio02.PNG} %\caption{Installing packages using R studio on a PC. Use the install menu option.} %\label{fig:installPC} %\end{center} %\end{figure} \paragraph{Install a few more packages } If you want some more functionality for some of the more advanced statistical procedures (e.g., \pfun{omegaSem}) you will need to install a few more packages (e.g., \Rpkg{lavaan}. \begin{Rinput} install.packages(c("psych","GPArotation","lavaan"),dependencies=TRUE) \end{Rinput} \paragraph{Install a ``task view" to get lots of packages} If you know that there are a number of packages that you want to use, it is possible they are listed as a ``task view". For instance, about 50 packages will be installed at once if you install the ``psychometrics'' task view. You can Install all the psychometric packages from the ``psychometrics'' task view by first installing a package (``ctv") that in turn installs many different task views. To see the list of possible task views, go to \url{https://cran.r-project.org/web/views/}. ~\ \begin{Rinput} install.packages("ctv") } #this downloads the task view package library(ctv) #this activates the ctv package install.views("Psychometrics") #one of the many Taskviews \end{Rinput} Take a 5 minute break because you will be installing about 50 packages. \paragraph{For the more adventurous users} The \Rpkg{psych} pacakge is under (sporadic) development with a new release issued to CRAN roughly every 4-6 months. The experimental, development version (prerelease) is always available at the Personality-Project web site and may be installed for Macs or PCs directly: ~\ \begin{Rinput} install.packages("psych", repos= "https://personality-project.org/r", type ="source") \end{Rinput} This development version will have fixed any bugs reported since the last release and will have various new features that are being tested before release to CRAN. After installation, it is necessary to restart \R{} to make the new version active. \paragraph{Make the \Rpkg{psych} package active.} You are almost ready. But first, to use most of the following examples you need to make the \Rpkg{psych} and \Rpkg{psychTools} packages active. You only need to do this once per session. ~\ \begin{Rinput} library(psych) #to do the analyses described here library(psychTools) #for some useful additions such as read.file \end{Rinput} %(If you want to automate this last step, you can create a special command to be run every time you start \R{}. % %\begin{Rinput} %.First <- function() {library(psych)} %\end{Rinput} %Do this when you first start \R. Then quit with the save option. Then restart \R. You will now automatically have loaded the \Rpkg{psych} package every time you start \R{}.) % % \section{Reading in the data for analysis} \subsection{Find a file and read from it} There are of course many ways to enter data into \R. Reading from a local file using \pfun{read.file} is perhaps the most preferred. This will read in most of the standard file types (.csv, .sav, .txt, etc). \pfun{read.file} combines the \fun{file.choose} and \fun{read.table} functions: ~\ \begin{Rinput} my.data <- read.file() #note the open and closing parentheses \end{Rinput} \pfun{read.file} opens a search window on your system just like any open file command does. \pfun{read.file} assumes that the first row of your table has labels for each column. If this is not true, specify names=FALSE, e.g., ~\ \begin{Rinput} my.data <- read.file(names = FALSE) \end{Rinput} If you want to read a remote file, specify the file name and then \pfun{read.file} ~\ \begin{Rinput} datafilename <- "http://personality-project.org/r/datasets/finkel.sav" new.data <- read.file(datafilename) #the data has labels \end{Rinput} \subsection{Or: copy the data from another program using the copy and paste commands of your operating system} However, many users will enter their data in a text editor or spreadsheet program and then want to copy and paste into \R{}. This may be done by using one of the \pfun{read.clipboard} set of functions . \begin{description} \item [\pfun{read.clipboard}] is the base function for reading data from the clipboard. \item [\pfun{read.clipboard.csv}] for reading text that is comma delimited. \item [\pfun{read.clipboard.tab}] for reading text that is tab delimited (e.g., copied directly from an Excel file). \item [\pfun{read.clipboard.lower}] for reading input of a lower triangular matrix with or without a diagonal. The resulting object is a square matrix. \item [\pfun{read.clipboard.upper}] for reading input of an upper triangular matrix. \item[\pfun{read.clipboard.fwf}] for reading in fixed width fields (some very old data sets) \end{description} For example, given a data set copied to the clipboard from a spreadsheet, just enter the command ~\ \begin{Rinput} my.data <- read.clipboard() \end{Rinput} This will work if every data field has a value and even missing data are given some values (e.g., NA or -999). If the data were entered in a spreadsheet and the missing values were just empty cells, then the data should be read in as a tab delimited or by using the \pfun{read.clipboard.tab} function. ~\ \begin{Rinput} my.data <- read.clipboard(sep="\t") #define the tab option, or my.tab.data <- read.clipboard.tab() #just use the alternative function \end{Rinput} For the case of data in fixed width fields (some old data sets tend to have this format), copy to the clipboard and then specify the width of each field (in the example below, the first variable is 5 columns, the second is 2 columns, the next 5 are 1 column the last 4 are 3 columns). ~\ \begin{Rinput} my.data <- read.clipboard.fwf(widths=c(5,2,rep(1,5),rep(3,4)) \end{Rinput} \subsection{Or: import from an SPSS or SAS file} To read data from an SPSS, SAS, or Systat file, you can probably just use the \pfun{read.file} function. \pfun{read.file} examines the suffix of the data file and if it is .sav (from SPSS) or .xpt (from SAS) will attempt to read given various default options. However, if that does not work, use the \Rpkg{foreign} package. This should come with Base \R{} but still need to be loaded using the \Rfunction{library} command. \fun{read.spss} reads a file stored by the SPSS save or export commands. \begin{verbatim}read.spss(file, use.value.labels = TRUE, to.data.frame = FALSE, max.value.labels = Inf, trim.factor.names = FALSE, trim_values = TRUE, reencode = NA, use.missings = to.data.frame) \end{verbatim} The \Rfunction{read.spss} function has many parameters that need to be set. In the example, I have used the parameters that I think are most useful. \begin{description} \item [file] Character string: the name of the file or URL to read. \item [use.value.labels] Convert variables with value labels into R factors with those levels? \item [to.data.frame] return a data frame? Defaults to FALSE, probably should be TRUE in most cases. \item [max.value.labels] Only variables with value labels and at most this many unique values will be converted to factors if use.value.labels $= TRUE$. \item [trim.factor.names] Logical: trim trailing spaces from factor levels? \item [trim\_values] logical: should values and value labels have trailing spaces ignored when matching for use.value.labels $= TRUE $? \item [use.missings] logical: should information on user-defined missing values be used to set the corresponding values to NA? \end{description} The following is an example of reading from a remote SPSS file and then describing the data set to make sure that it looks ok (with thanks to Eli Finkel). ~\ \begin{Rinput} datafilename <- "http://personality-project.org/r/datasets/finkel.sav" eli <-read.file(datafilename) describe(eli,skew=FALSE) \end{Rinput} \begin{Routput} var n mean sd median trimmed mad min max range se USER* 1 69 35.00 20.06 35 35.00 25.20 1 69 68 2.42 HAPPY 2 69 5.71 1.04 6 5.82 0.00 2 7 5 0.13 SOULMATE 3 69 5.09 1.80 5 5.32 1.48 1 7 6 0.22 ENJOYDEX 4 68 6.47 1.01 7 6.70 0.00 2 7 5 0.12 UPSET 5 69 0.41 0.49 0 0.39 0.00 0 1 1 0.06 \end{Routput} \section{Some simple descriptive statistics before you start} Although you probably want to jump right in and find $\omega$, you should first make sure that your data are reasonable. Use the \pfun{describe} function to get some basic descriptive statistics. This next example takes advantage of a built in data set. ~\ \begin{Sinput} my.data <- sat.act #built in example -- replace with your data describe(my.data) \end{Sinput} \begin{Soutput} var n mean sd median trimmed mad min max range skew kurtosis se gender 1 700 1.65 0.48 2 1.68 0.00 1 2 1 -0.61 -1.62 0.02 education 2 700 3.16 1.43 3 3.31 1.48 0 5 5 -0.68 -0.07 0.05 age 3 700 25.59 9.50 22 23.86 5.93 13 65 52 1.64 2.42 0.36 ACT 4 700 28.55 4.82 29 28.84 4.45 3 36 33 -0.66 0.53 0.18 SATV 5 700 612.23 112.90 620 619.45 118.61 200 800 600 -0.64 0.33 4.27 SATQ 6 687 610.22 115.64 620 617.25 118.61 200 800 600 -0.59 -0.02 4.41 \end{Soutput} There are, of course, all kinds of things you could do with your data at this point, but read about them in the \href{http://cran.r-project.org/web/packages/psych/vignettes/intro.pdf}{introductory vignette} and \href{http://cran.r-project.org/web/packages/psychTools/vignettes/overview.pdf}{more advanced vignette} for the \Rpkg{psych} package, \section{Using the \pfun{omega} function to find $\omega$} Two alternative estimates of reliability that take into account the hierarchical structure of the inventory are McDonald's $\omega_h$ and $\omega_t$ \citep{mcdonald:tt,rz:09}. These may be found using the \pfun{omega} function for an exploratory analysis (See Figure~\ref{fig:omega.9}) or \pfun{omegaSem} for a confirmatory analysis using the \Rpkg{sem} based upon the exploratory solution from \pfun{omega}. \subsection{Background on the $\omega$ statistics} \cite{mcdonald:tt} has proposed coefficient omega (hierarchical) ($\omega_h$) as an estimate of the general factor saturation of a test. \href{http://personality-project.org/revelle/publications/zinbarg.revelle.pmet.05.pdf}{\cite{zinbarg:pm:05}} compare McDonald's $\omega_h$ to Cronbach's $\alpha$ and Revelle's $\beta$. They conclude that $\omega_h$ is the best estimate. (See also \cite{zinbarg:apm:06} and \cite{rz:09} ). One way to find $\omega_h$ is to do a factor analysis of the original data set, rotate the factors obliquely, factor that correlation matrix, do a Schmid-Leiman (\pfun{schmid}) transformation to find general factor loadings, and then find $\omega_h$. $\omega_h$ differs slightly as a function of how the factors are estimated. Three options are available, the default will do a minimum residual factor analysis, fm=``pa" does a principal axes factor analysis (\pfun{factor.pa}), and fm=``mle" provides a maximum likelihood solution. For ability items, it is typically the case that all items will have positive loadings on the general factor. However, for non-cognitive items it is frequently the case that some items are to be scored positively, and some negatively. Although probably better to specify which directions the items are to be scored by specifying a key vector, if flip =TRUE (the default), items will be reversed so that they have positive loadings on the general factor. The keys are reported so that scores can be found using the \pfun{score.items} function. Arbitrarily reversing items this way can overestimate the general factor. (See the example with a simulated circumplex). The \pfun{omega} function uses exploratory factor analysis to estimate the $\omega_h$ coefficient. It is important to remember that ``A recommendation that should be heeded, regardless of the method chosen to estimate $\omega_h$, is to always examine the pattern of the estimated general factor loadings prior to estimating $\omega_h$. Such an examination constitutes an informal test of the assumption that there is a latent variable common to all of the scale's indicators that can be conducted even in the context of EFA. If the loadings were salient for only a relatively small subset of the indicators, this would suggest that there is no true general factor underlying the covariance matrix. Just such an informal assumption test would have afforded a great deal of protection against the possibility of misinterpreting the misleading $\omega_h$ estimates occasionally produced in the simulations reported here." \citep[][p 137]{zinbarg:apm:06}. Although $\omega_h$ is uniquely defined only for cases where 3 or more subfactors are extracted, it is sometimes desired to have a two factor solution. By default this is done by forcing the \pfun{schmid} extraction to treat the two subfactors as having equal loadings. There are three possible options for this condition: setting the general factor loadings between the two lower order factors to be ``equal" which will be the $\sqrt{r_{ab}}$ where $r_{ab}$ is the oblique correlation between the factors) or to ``first" or ``second" in which case the general factor is equated with either the first or second group factor. A message is issued suggesting that the model is not really well defined. This solution discussed in Zinbarg et al., 2007. To do this in omega, add the option=``first" or option=``second" to the call. Although obviously not meaningful for a 1 factor solution, it is of course possible to find the sum of the loadings on the first (and only) factor, square them, and compare them to the overall matrix variance. This is done, with appropriate complaints. In addition to $\omega_h$, another of McDonald's coefficients is $\omega_t$. This is an estimate of the total reliability of a test. McDonald's $\omega_t$, which is similar to Guttman's $\lambda_6$, (see \pfun{guttman}) uses the estimates of uniqueness $u^2$ from factor analysis to find $e_j^2$. This is based on a decomposition of the variance of a test score, $V_x$ into four parts: that due to a general factor, $\vec{g}$, that due to a set of group factors, $\vec{f}$, (factors common to some but not all of the items), specific factors, $\vec{s}$ unique to each item, and $\vec{e}$, random error. (Because specific variance can not be distinguished from random error unless the test is given at least twice, some combine these both into error). Letting $\vec{x} = \vec{cg} + \vec{Af} + \vec {Ds} + \vec{e} $ then the communality of item$_j$, based upon general as well as group factors, $h_j^2 = c_j^2 + \sum{f_{ij}^2}$ and the unique variance for the item $u_j^2 = \sigma_j^2 (1-h_j^2)$ may be used to estimate the test reliability. That is, if $h_j^2$ is the communality of item$_j$, based upon general as well as group factors, then for standardized items, $e_j^2 = 1 - h_j^2$ and $$ \omega_t = \frac{\vec{1}\vec{cc'}\vec{1} + \vec{1}\vec{AA'}\vec{1}'}{V_x} = 1 - \frac{\sum(1-h_j^2)}{V_x} = 1 - \frac{\sum u^2}{V_x} $$ Because $h_j^2 \geq r_{smc}^2$, $\omega_t \geq \lambda_6$. It is important to distinguish here between the two $\omega$ coefficients of McDonald, 1978 and Equation 6.20a of McDonald, 1999, $\omega_t$ and $\omega_h$. While the former is based upon the sum of squared loadings on all the factors, the latter is based upon the sum of the squared loadings on the general factor. $$\omega_h = \frac{ \vec{1}\vec{cc'}\vec{1}}{V_x}$$ Another estimate reported is the omega for an infinite length test with a structure similar to the observed test. This is found by $$\omega_{\inf} = \frac{ \vec{1}\vec{cc'}\vec{1}}{\vec{1}\vec{cc'}\vec{1} + \vec{1}\vec{AA'}\vec{1}'}$$ It can be shown In the case of simulated variables, that the amount of variance attributable to a general factor ($\omega_h$) is quite large, and the reliability of the set of items is somewhat greater than that estimated by $\alpha$ or $\lambda_6$. \subsection{Yet another alternative: Coefficient $\beta$} $\beta$, an alternative to $\omega_h$, is defined as the worst split half reliability \citep{revelle:iclust}. It can be estimated by using \pfun{iclust} (Item Cluster analysis: a hierarchical clustering algorithm). For a very complimentary review of why the iclust algorithm is useful in scale construction, see \cite{cooksey:06}. For a discussion of how use \pfun{iclust} see the \href{http://cran.r-project.org/web/packages/psychTools/vignettes/factor.pdf}{factor analysis vignette}. \subsection{Using the \pfun{omega} function} This is \R{}. Just call it. For the next example, we find $\omega$ for a data set from Thurstone. To find it for your data, replace Thurstone with my.data. ~\ <>== omega(Thurstone) @ %\begin{Routput} % %Omega %Call: omega(m = Thurstone) %Alpha: 0.89 %G.6: 0.91 %Omega Hierarchical: 0.74 %Omega H asymptotic: 0.79 %Omega Total 0.93 % %Schmid Leiman Factor loadings greater than 0.2 % g F1* F2* F3* h2 u2 p2 %Sentences 0.71 0.57 0.82 0.18 0.61 %Vocabulary 0.73 0.55 0.84 0.16 0.63 %Sent.Completion 0.68 0.52 0.73 0.27 0.63 %First.Letters 0.65 0.56 0.73 0.27 0.57 %4.Letter.Words 0.62 0.49 0.63 0.37 0.61 %Suffixes 0.56 0.41 0.50 0.50 0.63 %Letter.Series 0.59 0.61 0.72 0.28 0.48 %Pedigrees 0.58 0.23 0.34 0.50 0.50 0.66 %Letter.Group 0.54 0.46 0.53 0.47 0.56 % %With eigenvalues of: % g F1* F2* F3* %3.58 0.96 0.74 0.71 % %general/max 3.71 max/min = 1.35 %mean percent general = 0.6 with sd = 0.05 and cv of 0.09 % %The degrees of freedom are 12 and the fit is 0.01 % %The root mean square of the residuals is 0 %The df corrected root mean square of the residuals is 0.01 % %Compare this with the adequacy of just a general factor and no group factors %The degrees of freedom for just the general factor are 27 and the fit is 1.48 % %The root mean square of the residuals is 0.1 %The df corrected root mean square of the residuals is 0.16 % %Measures of factor score adequacy % g F1* F2* F3* %Correlation of scores with factors 0.86 0.73 0.72 0.75 %Multiple R square of scores with factors 0.74 0.54 0.52 0.56 %Minimum correlation of factor score estimates 0.49 0.08 0.03 0.11 %> % \end{Routput} \subsection{Find three measures of reliability: $\omega_h$, $\alpha$, and $\omega_t$} In a review of various measures of reliability, \cite{rc:pa:19} suggest that one should routinely report 3 estimates of internal consistency ($\omega_h$, $\alpha$, and $\omega_t$). As an example, they use 10 items to measure anxiety taken from the state anxiety data set (\pfun{sai} in the \Rpkg{psychTools} package. First examine the descriptive statistics and then find and summarize the omega for these data. By inspection of the correlation matrix, it seems as if there are two group factors (tension and calmness) as well as an overall general factor of anxiety. We use a two factor solution to better represent the results (Figure~\ref{fig.anxiety}). ~\ <>= anxiety <- sai[c("anxious", "jittery", "nervous" ,"tense", "upset","at.ease" , "calm" , "confident", "content","relaxed")] describe(anxiety) lowerCor(anxiety) om <- omega(anxiety,2) #specify a two factor solution summary(om) #summarize the output @ <>== png('anxiety.png') omega.diagram(om, main="Omega analysis of two factors of anxiety") dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{anxiety} \caption{An \pfun{omega} solution for 10 anxiety items with two group factors. See \cite{rc:pa:19} for more measures of reliability for this data set.} \label{fig.anxiety} \end{center} \end{figure} \subsection{Estimating $\omega_h$ using a direct Schmid-Leiman transformation} The \pfun{omegaDirect} function uses Niels Waller's algorithm for finding a g factor directly without extracting a higher order model \citep{waller:17}. This has the advantage that it will work cleanly for data with just 2 group factors. Unfortunately, it will produce non-zero estimates for omega even if there is no general factor. ~\ <>= om <- omegaDirect(Thurstone) om @ <>== png('direct.png') omega.diagram(om, main="Direct Schmid Leihman solution") dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{direct} \caption{The Direct Schmid Leiman solution is taken from an algorithm by \cite{waller:17}. Compare this solution to Figure~\ref{fig:omega.9}. } \label{fig:direct} \end{center} \end{figure} \subsection{Estimating $\omega_h$ using Confirmatory Factor Analysis} The \pfun{omegaSem} function will do an exploratory analysis and then take the highest loading items on each factor and do a confirmatory factor analysis using the \Rpkg{lavaan} package. These results can produce slightly different estimates of $\omega_h$, primarily because cross loadings are modeled as part of the general factor. We use a classic data set from Holzinger and Swineford, some of the tests of which are included in the \Rpkg{lavaan} package. This analysis allows us to examine the hierarchical structure of these ability tests. The data are taken from the \pfun{holzinger.swineford} data set in the \Rpkg{psychTools} package. ~\ <>= om <- omega(holzinger.swineford[8:31],4) #the exploratory solution omegaSem(holzinger.swineford[8:31],4) #the confirmatory solution @ %\begin{Routput} %Call: omegaSem(m = r9, n.obs = 500) %Omega %Call: omega(m = m, nfactors = nfactors, fm = fm, key = key, flip = flip, % digits = digits, title = title, sl = sl, labels = labels, % plot = plot, n.obs = n.obs, rotate = rotate, Phi = Phi, option = option) %Alpha: 0.75 %G.6: 0.74 %Omega Hierarchical: 0.66 %Omega H asymptotic: 0.84 %Omega Total 0.78 % %Schmid Leiman Factor loadings greater than 0.2 % g F1* F2* F3* h2 u2 p2 %V1 0.70 0.53 0.47 0.93 %V2 0.70 0.52 0.48 0.94 %V3 0.54 0.32 0.68 0.91 %V4 0.53 0.46 0.50 0.50 0.57 %V5 0.44 0.44 0.39 0.61 0.50 %V6 0.40 0.32 0.26 0.74 0.59 %V7 0.31 0.31 0.21 0.79 0.48 %V8 0.34 0.44 0.30 0.70 0.37 %V9 0.24 0.36 0.19 0.81 0.32 % %With eigenvalues of: % g F1* F2* F3* %2.18 0.52 0.08 0.44 % %general/max 4.21 max/min = 6.17 %mean percent general = 0.62 with sd = 0.24 and cv of 0.39 % %The degrees of freedom are 12 and the fit is 0.03 %The number of observations was 500 with Chi Square = 14.23 with prob < 0.29 %The root mean square of the residuals is 0.01 %The df corrected root mean square of the residuals is 0.03 %RMSEA index = 0.02 and the 90 % confidence intervals are NA 0.052 %BIC = -60.35 % %Compare this with the adequacy of just a general factor and no group factors %The degrees of freedom for just the general factor are 27 and the fit is 0.21 %The number of observations was 500 with Chi Square = 103.64 with prob < 6.4e-11 %The root mean square of the residuals is 0.05 %The df corrected root mean square of the residuals is 0.08 % %RMSEA index = 0.076 and the 90 % confidence intervals are 0.06 0.091 %BIC = -64.15 % %Measures of factor score adequacy % g F1* F2* F3* %Correlation of scores with factors 0.86 0.63 0.25 0.59 %Multiple R square of scores with factors 0.74 0.39 0.06 0.35 %Minimum correlation of factor score estimates 0.48 -0.21 -0.88 -0.30 % % Omega Hierarchical from a confirmatory model using sem = 0.68 % Omega Total from a confirmatory model using sem = 0.78 %With loadings of % g F1* F2* F3* h2 u2 %V1 0.73 0.54 0.46 %V2 0.68 0.29 0.54 0.46 %V3 0.51 0.22 0.31 0.69 %V4 0.54 0.47 0.51 0.49 %V5 0.45 0.42 0.38 0.62 %V6 0.39 0.31 0.25 0.75 %V7 0.34 0.34 0.23 0.77 %V8 0.36 0.39 0.28 0.72 %V9 0.26 0.33 0.18 0.82 % %With eigenvalues of: % g F1* F2* F3* %2.21 0.49 0.14 0.38 %\end{Routput} <>= @ \section{Simulating a hierarchical/higher order structure} There are several simulation functions in the \Rpkg{psych} package for creating structures with a general factor. One, \pfun{sim.hierarchical} creates lower level factors which are all correlated with a general factor. The default simulation has the parameters discussed by \cite{jensen:weng}. Another way to simulate a hierarchical structure is to simulate a bifactor model directly using the \pfun{sim.structure} function. The \cite{jensen:weng} model: <>= jen <- sim.hierarchical() #use the default values om <- omega(jen) om @ \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= png('jensen.png' ) omega.diagram(om) dev.off() @ \end{scriptsize} \includegraphics{jensen} \caption{An example of a hierarchical model from Jensen.} \label{fig:outlier} \end{center} \end{figure} %\begin{Routput} %jen <- sim.hierarchical() #use the default values %> om <- omega(jen) %> om %Omega %Call: omega(m = jen) %Alpha: 0.76 %G.6: 0.76 %Omega Hierarchical: 0.69 %Omega H asymptotic: 0.86 %Omega Total 0.8 % %Schmid Leiman Factor loadings greater than 0.2 % g F1* F2* F3* h2 u2 p2 %V1 0.72 0.35 0.64 0.36 0.81 %V2 0.63 0.31 0.49 0.51 0.81 %V3 0.54 0.26 0.36 0.64 0.81 %V4 0.56 0.42 0.49 0.51 0.64 %V5 0.48 0.36 0.36 0.64 0.64 %V6 0.40 0.30 0.25 0.75 0.64 %V7 0.42 0.43 0.36 0.64 0.49 %V8 0.35 0.36 0.25 0.75 0.49 %V9 0.28 0.29 0.16 0.84 0.49 % %With eigenvalues of: % g F1* F2* F3* %2.29 0.28 0.40 0.39 % %general/max 5.78 max/min = 1.4 %mean percent general = 0.65 with sd = 0.14 and cv of 0.21 %Explained Common Variance of the general factor = 0.68 % %The degrees of freedom are 12 and the fit is 0 % %The root mean square of the residuals is 0 %The df corrected root mean square of the residuals is 0 % %Compare this with the adequacy of just a general factor and no group factors %The degrees of freedom for just the general factor are 27 and the fit is 0.18 % %The root mean square of the residuals is 0.06 %The df corrected root mean square of the residuals is 0.07 % %Measures of factor score adequacy % g F1* F2* F3* %Correlation of scores with factors 0.85 0.46 0.57 0.57 %Multiple R square of scores with factors 0.73 0.21 0.32 0.32 %Minimum correlation of factor score estimates 0.46 -0.57 -0.35 -0.35 % % Total, General and Subset omega for each subset % g F1* F2* F3* %Omega total for total scores and subscales 0.80 0.74 0.63 0.50 %Omega general for total scores and subscales 0.69 0.60 0.40 0.25 %Omega group for total scores and subscales 0.11 0.14 0.23 0.26 %> %\end{Routput} \subsubsection{Simulate a bifactor model} Simulate a bifactor model and then compare two ways of finding the solution (normal omega and directOmega). We compare the solutions using the \pfun{fa.congruence} function. \begin{Rinput} fx <- matrix(c(.7,.6,.5,.7,.6,.5,.8,.7,.6, .6,.6,.6,rep(0,9),c(.6,.5,.6),rep(0,9),.6,.6,.6),ncol=4) simx <-sim.structure(fx) lowerMat(simx$model) om <- omega(simx$model) dsl <- omegaDirect(simx$model) summary(om) summary(dsl) fa.congruence(list(om,dsl,fx)) \end{Rinput} <>== fx <- matrix(c(.7,.6,.5,.7,.6,.5,.8,.7,.6, .6,.6,.6,rep(0,9),c(.6,.5,.6),rep(0,9),.6,.6,.6),ncol=4) simx <-sim.structure(fx) om <- omega(simx$model) dsl <- omegaDirect(simx$model) @ \begin{scriptsize} <>= lowerMat(simx$model) summary(om) summary(dsl) fa.congruence(list(om,dsl,fx)) @ \end{scriptsize} %\begin{Routput} %summary(om) %Omega %Alpha: 0.9 %G.6: 0.93 %Omega Hierarchical: 0.74 %Omega H asymptotic: 0.78 %Omega Total 0.95 % %With eigenvalues of: % g F1* F2* F3* %3.67 1.08 1.08 0.97 %The degrees of freedom for the model is 12 and the fit was 0 % %The root mean square of the residuals is 0 %The df corrected root mean square of the residuals is 0 %Explained Common Variance of the general factor = 0.54 % % Total, General and Subset omega for each subset % g F1* F2* F3* %Omega total for total scores and subscales 0.95 0.95 0.89 0.87 %Omega general for total scores and subscales 0.74 0.55 0.45 0.46 %Omega group for total scores and subscales 0.21 0.40 0.44 0.41 %> summary(dsl) %Call: omegaDirect(m = simx$model) %Omega H direct: 0.71 % %With eigenvalues of: % g F1* F2* F3* %3.53 1.22 1.06 0.99 %The degrees of freedom for the model is 12 and the fit was 0 % %The root mean square of the residuals is 0 %The df corrected root mean square of the residuals is 0 % % Total, General and Subset omega for each subset % g F1* F2* F3* %Omega total for total scores and subscales 0.95 0.95 0.89 0.87 %Omega general for total scores and subscales 0.71 0.50 0.45 0.45 %Omega group for total scores and subscales 0.22 0.45 0.43 0.42 %> fa.congruence(list(om,dsl,fx)) % g F1* F2* F3* h2 g F1* F2* F3* %g 1.00 0.64 0.55 0.54 1.00 1.00 0.67 0.56 0.57 1.00 0.54 0.54 0.63 %F1* 0.64 1.00 0.00 0.00 0.65 0.62 1.00 0.02 0.04 0.64 0.00 0.00 1.00 %F2* 0.55 0.00 1.00 0.00 0.55 0.56 0.02 1.00 0.00 0.55 1.00 0.00 0.00 %F3* 0.54 0.00 0.00 1.00 0.52 0.55 0.03 0.00 1.00 0.54 0.00 1.00 0.00 %h2 1.00 0.65 0.55 0.52 1.00 1.00 0.68 0.57 0.55 1.00 0.55 0.52 0.64 %g 1.00 0.62 0.56 0.55 1.00 1.00 0.65 0.58 0.57 1.00 0.56 0.55 0.62 %F1* 0.67 1.00 0.02 0.03 0.68 0.65 1.00 0.04 0.07 0.67 0.02 0.03 1.00 %F2* 0.56 0.02 1.00 0.00 0.57 0.58 0.04 1.00 0.00 0.56 1.00 0.00 0.02 %F3* 0.57 0.04 0.00 1.00 0.55 0.57 0.07 0.00 1.00 0.57 0.00 1.00 0.03 % 1.00 0.64 0.55 0.54 1.00 1.00 0.67 0.56 0.57 1.00 0.54 0.54 0.63 % 0.54 0.00 1.00 0.00 0.55 0.56 0.02 1.00 0.00 0.54 1.00 0.00 0.00 % 0.54 0.00 0.00 1.00 0.52 0.55 0.03 0.00 1.00 0.54 0.00 1.00 0.00 % 0.63 1.00 0.00 0.00 0.64 0.62 1.00 0.02 0.03 0.63 0.00 0.00 1.00 %> %\end{Routput} \section{Summary} In the modern era of computation, there is little justification for continuing with procedures that were developed as \href{https://personality-project.org/revelle/publications/cup.18.final.pdf}{short-cuts 80 years ago} \citep{reh:20}, To find $\omega_h$, $\alpha$, and $\omega_t$ is very easy using the open source statistical system (\R{}) as well as the \pfun{omega} functions in the \Rpkg{psych} package. \section{System Info} When running any \R{} package, it is useful to find out the session information to see if you have the most recent releases. \begin{scriptsize} <>= sessionInfo() @ \end{scriptsize} \newpage \begin{thebibliography}{} \bibitem[\protect\astroncite{Azzalini and Genz}{2016}]{mnormt} Azzalini, A. and Genz, A. (2016). \newblock {\em The {R} package \texttt{mnormt}: The multivariate normal and $t$ distributions (version 1.5-5)}. \bibitem[\protect\astroncite{Bernaards and Jennrich}{2005}]{GPA} Bernaards, C. and Jennrich, R. (2005). \newblock {Gradient projection algorithms and software for arbitrary rotation criteria in factor analysis}. \newblock {\em Educational and Psychological Measurement}, 65(5):676--696. \bibitem[\protect\astroncite{Cooksey and Soutar}{2006}]{cooksey:06} Cooksey, R. and Soutar, G. (2006). \newblock Coefficient beta and hierarchical item clustering - an analytical procedure for establishing and displaying the dimensionality and homogeneity of summated scales. \newblock {\em Organizational Research Methods}, 9:78--98. \bibitem[\protect\astroncite{Cronbach}{1951}]{cronbach:51} Cronbach, L.~J. (1951). \newblock Coefficient alpha and the internal structure of tests. \newblock {\em Psychometrika}, 16:297--334. \bibitem[\protect\astroncite{Fox et~al.}{2013}]{sem} Fox, J., Nie, Z., and Byrnes, J. (2013). \newblock {\em sem: Structural Equation Models}. \newblock R package version 3.1-3. \bibitem[\protect\astroncite{Guttman}{1945}]{guttman:45} Guttman, L. (1945). \newblock A basis for analyzing test-retest reliability. \newblock {\em Psychometrika}, 10(4):255--282. \bibitem[\protect\astroncite{Jensen and Weng}{1994}]{jensen:weng} Jensen, A.~R. and Weng, L.-J. (1994). \newblock What is a good g? \newblock {\em Intelligence}, 18(3):231--258. \bibitem[\protect\astroncite{Kuder and Richardson}{1937}]{kuder:37} Kuder, G. and Richardson, M. (1937). \newblock The theory of the estimation of test reliability. \newblock {\em Psychometrika}, 2(3):151--160. \bibitem[\protect\astroncite{McDonald}{1999}]{mcdonald:tt} McDonald, R.~P. (1999). \newblock {\em Test theory: {A} unified treatment}. \newblock L. Erlbaum Associates, Mahwah, N.J. \bibitem[\protect\astroncite{{R Core Team}}{2020}]{R} {R Core Team} (2020). \newblock {\em R: A Language and Environment for Statistical Computing}. \newblock R Foundation for Statistical Computing, Vienna, Austria. \bibitem[\protect\astroncite{Revelle}{1979}]{revelle:iclust} Revelle, W. (1979). \newblock Hierarchical cluster-analysis and the internal structure of tests. \newblock {\em Multivariate Behavioral Research}, 14(1):57--74. \bibitem[\protect\astroncite{Revelle}{2020}]{psych} Revelle, W. (2020). \newblock {\em psych: Procedures for Personality and Psychological Research}. \newblock Northwestern University, Evanston, https://CRAN.r-project.org/package=psych. \newblock R package version 2.0.8. \bibitem[\protect\astroncite{Revelle and Condon}{2018}]{rc:reliability} Revelle, W. and Condon, D.~M. (2018). \newblock Reliability. \newblock In Irwing, P., Booth, T., and Hughes, D.~J., editors, {\em The {Wiley Handbook of Psychometric Testing:} A Multidisciplinary Reference on Survey, Scale and Test Development}. John Wily \& Sons, London. \bibitem[\protect\astroncite{Revelle and Condon}{2019}]{rc:pa:19} Revelle, W. and Condon, D.~M. (2019). \newblock Reliability from $\alpha$ to $\omega$: A tutorial. \newblock {\em Psychological Assessment} 31 (12) p 1395-1411. \bibitem[\protect\astroncite{Revelle et al.}{2020}]{reh:20} Revelle, W. and Elleman, L.G. and Hall, A. (2020). \newblock Statistical analyses and computer programming in personality. \newblock In Corr, P.J. editor, {\em The {Cambridge University Press Handbook of Personality}}. {Cambridge University Press}. \bibitem[\protect\astroncite{Revelle and Zinbarg}{2009}]{rz:09} Revelle, W. and Zinbarg, R.~E. (2009). \newblock Coefficients alpha, beta, omega and the glb: comments on {Sijtsma}. \newblock {\em Psychometrika}, 74(1):145--154. \bibitem[\protect\astroncite{Rosseel}{2012}]{lavaan} Rosseel, Y. (2012). \newblock {lavaan}: An {R} package for structural equation modeling. \newblock {\em Journal of Statistical Software}, 48(2):1--36. \bibitem[\protect\astroncite{Waller}{2017}]{waller:17} Waller, N.~G. (2017). \newblock Direct {Schmid-Leiman} transformations and rank-deficient loadings matrices. \newblock {\em Psychometrika.} \bibitem[\protect\astroncite{Zinbarg et~al.}{2005}]{zinbarg:pm:05} Zinbarg, R.~E., Revelle, W., Yovel, I., and Li, W. (2005). \newblock Cronbach's {$\alpha$}, {Revelle's} {$\beta$}, and {McDonald's} {$\omega_H$}: Their relations with each other and two alternative conceptualizations of reliability. \newblock {\em Psychometrika}, 70(1):123--133. \bibitem[\protect\astroncite{Zinbarg et~al.}{2006}]{zinbarg:apm:06} Zinbarg, R.~E., Yovel, I., Revelle, W., and McDonald, R.~P. (2006). \newblock Estimating generalizability to a latent variable common to all of a scale's indicators: A comparison of estimators for {$\omega_h$}. \newblock {\em Applied Psychological Measurement}, 30(2):121--144. \end{thebibliography} \end{document} psychTools/inst/doc/mediation.rnw0000644000176200001440000023014714003577154016661 0ustar liggesusers% \VignetteIndexEntry{Overview of the psych package} % \VignettePackage{psych} % \VignetteKeywords{multivariate} % \VignetteKeyword{models} % \VignetteKeyword{Hplot} %\VignetteDepends{psych} %\documentclass[doc]{apa} \documentclass[11pt]{article} %\documentclass[11pt]{amsart} \usepackage{geometry} % See geometry.pdf to learn the layout options. There are lots. \geometry{letterpaper} % ... or a4paper or a5paper or ... %\geometry{landscape} % Activate for for rotated page geometry \usepackage[parfill]{parskip} % Activate to begin paragraphs with an empty line rather than an indent \usepackage{graphicx} \usepackage{amssymb} \usepackage{epstopdf} \usepackage{mathptmx} \usepackage{helvet} \usepackage{courier} \usepackage{epstopdf} \usepackage{makeidx} % allows index generation \usepackage[authoryear,round]{natbib} \usepackage{gensymb} \usepackage{longtable} %\usepackage{geometry} \usepackage{amssymb} \usepackage{amsmath} %\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \usepackage{Sweave} %\usepackage{/Volumes/'Macintosh HD'/Library/Frameworks/R.framework/Versions/2.13/Resources/share/texmf/tex/latex/Sweave} %\usepackage[ae]{Rd} %\usepackage[usenames]{color} %\usepackage{setspace} \bibstyle{apacite} \bibliographystyle{apa} %this one plus author year seems to work? %\usepackage{hyperref} \usepackage[colorlinks=true,citecolor=blue]{hyperref} %this makes reference links hyperlinks in pdf! \DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \usepackage{multicol} % used for the two-column index \usepackage[bottom]{footmisc}% places footnotes at page bottom \let\proglang=\textsf \newcommand{\R}{\proglang{R}} %\newcommand{\pkg}[1]{{\normalfont\fontseries{b}\selectfont #1}} \newcommand{\Rfunction}[1]{{\texttt{#1}}} \newcommand{\fun}[1]{{\texttt{#1}\index{#1}\index{R function!#1}}} \newcommand{\pfun}[1]{{\texttt{#1}\index{#1}\index{R function!#1}\index{R function!psych package!#1}}}\newcommand{\Rc}[1]{{\texttt{#1}}} %R command same as Robject \newcommand{\Robject}[1]{{\texttt{#1}}} \newcommand{\Rpkg}[1]{{\textit{#1}\index{#1}\index{R package!#1}}} %different from pkg - which is better? \newcommand{\iemph}[1]{{\emph{#1}\index{#1}}} \newcommand{\wrc}[1]{\marginpar{\textcolor{blue}{#1}}} %bill's comments \newcommand{\wra}[1]{\textcolor{blue}{#1}} %bill's comments \newcommand{\ve}[1]{{\textbf{#1}}} %trying to get a vector command \usepackage{fancyvrb} %this allows fancy boxes \newcommand{\vect}[1]{\boldsymbol{#1}} \let\vec\vect \fvset{fontfamily=courier} \DefineVerbatimEnvironment{Routput}{Verbatim} %{fontsize=\scriptsize, xleftmargin=0.6cm} {fontseries=b,fontsize=\scriptsize, xleftmargin=0.1cm} \DefineVerbatimEnvironment{Soutput}{Verbatim} %{fontsize=\scriptsize, xleftmargin=0.6cm} {fontseries=b,fontsize=\scriptsize, xleftmargin=0.1cm} \DefineVerbatimEnvironment{Binput}{Verbatim} {fontseries=b, fontsize=\scriptsize,frame=single, label=\fbox{lavaan model syntax}, framesep=2mm} %\DefineShortVerb{\!} %%% generates error! \DefineVerbatimEnvironment{Rinput}{Verbatim} %{fontsize=\scriptsize, frame=single, label=\fbox{R code}, framesep=1mm} {fontseries=b, fontsize=\scriptsize, frame=single, label=\fbox{R code},xleftmargin=0pt, framesep=1mm} \DefineVerbatimEnvironment{Sinput}{Verbatim} %{fontsize=\scriptsize, frame=single, label=\fbox{R code}, framesep=1mm} {fontseries=b, fontsize=\scriptsize, frame=single, label=\fbox{R code},xleftmargin=0pt, framesep=1mm} \DefineVerbatimEnvironment{Link}{Verbatim} {fontseries=b, fontsize=\small, formatcom=\color{darkgreen}, xleftmargin=1.0cm} \DefineVerbatimEnvironment{Toutput}{Verbatim} {fontseries=b,fontsize=\tiny, xleftmargin=0.1cm} \DefineVerbatimEnvironment{rinput}{Verbatim} {fontseries=b, fontsize=\tiny, frame=single, label=\fbox{R code}, framesep=1mm} \newcommand{\citeti}[1]{\begin{tiny}\citep{#1}\end{tiny}} \newcommand{\light}[1]{\textcolor{gray}{#1}} %\newcommand{\vect}[1]{\boldsymbol{#1}} %\let\vec\vect \makeindex % used for the subject index \title{How to use the psych package for mediation/moderation/regression analysis} \author{William Revelle} %the following works only with apaclass \begin{document} \maketitle %\bibliography{all} \tableofcontents \newpage \section{Overview of this and related documents} To do basic and advanced personality and psychological research using \R{} is not as complicated as some think. This is one of a set of ``How To'' to do various things using \R{} \citep{R}, particularly using the \Rpkg{psych} \citep{psych} package. The current list of How To's includes: \begin{enumerate} \item An \href{http://personality-project.org/r/psych/intro.pdf}{introduction} (vignette) of the \Rpkg{psych} package \item An \href{http://personality-project.org/r/psych/overview.pdf}{overview} (vignette) of the \Rpkg{psych} package \item \href{http://personality-project.org/r/psych/HowTo/getting_started.pdf}{Installing} \R{} and some useful packages \item Using \R{} and the \Rpkg{psych} package to find \href{http://personality-project.org/r/psych/HowTo/omega.pdf}{$omega_h$} and $\omega_t$. \item Using \R{} and the \Rpkg{psych} for \href{http://personality-project.org/r/psych/HowTo/factor.pdf}{factor analysis} and principal components analysis. \item Using the \pfun{scoreItems} function to find \href{http://personality-project.org/r/psych/HowTo/scoring.pdf}{scale scores and scale statistics}. \item Using \pfun{mediate} and \pfun{setCor} to do \href{http://personality-project.org/r/psych/HowTo/mediation.pdf}{mediation, moderation and regression analysis} (this document) \end{enumerate} \subsection{Jump starting the \Rpkg{psych} package--a guide for the impatient} You have installed \Rpkg{psych} and you want to use it without reading much more. What should you do? \begin{enumerate} \item Activate the \Rpkg{psych} and \Rpkg{psychTools} packages. <>== library(psych) library(psychTools) @ \item Input your data. If your file name ends in .sav, .text, .txt, .csv, .xpt, .rds, .Rds, .rda, or .RDATA, then just read it in directly using \pfun{read.file}. Or you can go to your friendly text editor or data manipulation program (e.g., Excel) and copy the data to the clipboard. Include a first line that has the variable labels. Paste it into \Rpkg{psych} using the \pfun{read.clipboard.tab} command: \begin{Rinput} myData <- read.file() #this will open a search window on your machine # and read or load the file. #or #first copy your file to your clipboard and then myData <- read.clipboard.tab() #if you have an excel file \end{Rinput} \item Make sure that what you just read is right. Describe it and perhaps look at the first and last few lines. If you want to ``view" the first and last few lines using a spreadsheet like viewer, use \pfun{quickView}. \begin{Rinput} describe(myData) headTail(myData) #or quickView(myData) \end{Rinput} \item Look at the patterns in the data. If you have fewer than about 10 variables, look at the SPLOM (Scatter Plot Matrix) of the data using \pfun{pairs.panels}. \begin{Rinput} pairs.panels(myData) \end{Rinput} \item Find the correlations of all of your data. \begin{itemize} \item Descriptively (just the values) \begin{Rinput} lowerCor(myData) \end{Rinput} \item Graphically \begin{Rinput} corPlot(myData) #does not show the values of the correlations corPlot(myData, numbers=TRUE) #show the numbers,\ #scales the character size by "significance" corPlot(myData, numbers=TRUE,scale=FALSE) #show the numbers, # all characters the same size corPlot(lowerCor(myData), numbers =TRUE) #print the correlations # and show them graphically \end{Rinput} \end{itemize} \end{enumerate} \subsection{For the not impatient} The following pages are meant to lead you through the use of the \pfun{setCor} and \pfun{mediate} functions. The assumption is that you have already made \Rpkg{psych} active and want some example code. \section{Multiple regression and mediation} Mediation and moderation are merely different uses of the linear model $\hat{\vec{Y}}= \mu + \beta_{y.x} \vec{X } $ and are implemented in \Rpkg{psych} with two functions: \pfun{setCor} and \pfun{mediate}. Given a set of predictor variables, $\vec{X}$ and a set of criteria variables, $\vec{Y}$, multiple regression solves the equation $\hat{\vec{Y}} = \mu + \beta_{y.x} \vec{X } $ by finding $\beta_{y.x} = \vec{C_{xx}}^{-1} C_{yx} $ where $\vec{C_{xx}}$ is the covariances of the $\vec{X}$ variables and $\vec{C_{yx}}$ is the covariances of predictors and the criteria. Although typically done using the raw data, clearly this can also be done by using the covariance or correlation matrices. \pfun{setCor} was developed to handle the correlation matrix solution but has been generalized to the case of raw data. In the later case, it assumes a Missing Completely at Random (MCAR) structure, and thus uses all the data and finds pair.wise complete correlations. For complete data sets, the results are identical to using \pfun{lm}. By default, \pfun{setCor} uses standardized variables, but to compare with \pfun{lm}, it can use unstandardized variables. \section{Regression using \pfun{setCor}} Although typically done from a raw data matrix (using the \fun{lm} function), it is sometimes useful to do the regression from a correlation or covariance matrix. \pfun{setCor} was developed for this purpose. From a correlation/covariance matrix, it will do normal regression as well as regression on partialled correlation matrices. With the raw data, it will also do moderated regression (centered or non-centered). In particular, for the raw data, it will work with missing data. An interesting option, if using categorical or dichotomous data is first find the appropriate polychoric, tetrachoric, or poly-serial correlations using \pfun{mixedCor} and then use the resulting correlation matrix for analysis. The resulting correlations and multiple correlations will not match those of the \pfun{lm} analysis. \subsection{Comparison with \pfun{lm} on complete data} We use the \pfun{attitude} data set for our first example. \subsubsection{It is important to know your data by describing it first} <>== psych::describe(attitude) @ \subsubsection{Now do the regressions} <>== #do not standardize mod1 <- setCor(rating ~ complaints + privileges, data=attitude,std=FALSE) mod1 @ Compare this solution with the results of the \pfun{lm} function. <>== summary(lm(rating ~ complaints + privileges, data=attitude)) @ The graphic for the standardized regression is shown in (Figure~\ref{fig:attitude}). <>== png('attitude.png') # standardize by default mod2 <- setCor(rating ~ complaints + privileges, data=attitude) mod2 setCor.diagram(mod2, main="A simple regression model") dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{attitude.png} \caption{A simple multiple regression using the attitude data set (standardized solution is shown).} \label{fig:attitude} \end{center} \end{figure} \subsection{From a correlation matrix} Perhaps most usefully, \pfun{setCor} will find the beta weights between a set of X variables, and a set of Y variables. Consider seven variables in the \pfun{atttitude} data set. We first find the correlation matrix (normally, this could just be supplied by the user). Then we find the regressions from the correlation matrix. Compare this regression to the (standardized) solution shown above. By specifying the number of observations (n.obs), we are able to apply various inferential tests. <>== R <- lowerCor(attitude) setCor(rating ~ complaints + privileges, data=R, n.obs =30) @ Compare this solution (from the correlation matrix) with the \emph{standardized} solution for the raw data. \pfun{setCor} does several things: \begin{itemize} \item Finds the regression weights (betas) between the predictor variables and each of the criterion variables. \item If the number of subjects is specified, or if the raw data are used, it also compares each of these betas to its standard error, finds a $t$ statistic, and reports the probability of the $t > 0$. \item It reports the Multiple R and $R^2$ based upon these beta weights. In addition, following the tradition of the robust beauty of the improper linear models \citep{dawes:79} it also reports the unit weighted multiple correlations. \item The canonical correlations between the two sets \citep{hotelling:36} is reported. \item Cohen's set correlation \citep{cohen:82} as well as the unweighted correlation between the two sets of variables are reported. \end{itemize} \subsection{The Hotelling example} <>== #the second Kelley data from Hotelling kelley <- structure(list(speed = c(1, 0.4248, 0.042, 0.0215, 0.0573), power = c(0.4248, 1, 0.1487, 0.2489, 0.2843), words = c(0.042, 0.1487, 1, 0.6693, 0.4662), symbols = c(0.0215, 0.2489, 0.6693, 1, 0.6915), meaningless = c(0.0573, 0.2843, 0.4662, 0.6915, 1)), .Names = c("speed", "power", "words", "symbols", "meaningless"), class = "data.frame", row.names = c("speed", "power", "words", "symbols", "meaningless")) #first show the correlations lowerMat(kelley) #now find and draw the regression sc <- setCor(power + speed ~ words + symbols + meaningless,data=kelley) #formula mode sc #show it @ %First show the correlation matrix. %\begin{Routput} % %lowerMat(kelley) % speed power words symbl mnngl %speed 1.00 %power 0.42 1.00 %words 0.04 0.15 1.00 %symbols 0.02 0.25 0.67 1.00 %meaningless 0.06 0.28 0.47 0.69 1.00 %\end{Routput} % %Now, use the \pfun{setCor} function. % %\begin{Routput} %Call: setCor(y = power + speed ~ words + symbols + meaningless, data = kelley) % %Multiple Regression from matrix input % % DV = power % slope VIF %words -0.03 1.81 %symbols 0.12 2.72 %meaningless 0.22 1.92 % % Multiple Regression % R R2 Ruw R2uw %power 0.29 0.09 0.26 0.07 % % DV = speed % slope VIF %words 0.05 1.81 %symbols -0.07 2.72 %meaningless 0.08 1.92 % % Multiple Regression % R R2 Ruw R2uw %speed 0.07 0.01 0.05 0 % %Various estimates of between set correlations %Squared Canonical Correlations %[1] 0.0946 0.0035 % % Average squared canonical correlation = 0.05 % Cohen's Set Correlation R2 = 0.1 %Unweighted correlation between the two sets = 0.18 % %\end{Routput} A plot of the regression model is shown as well (Figure~\ref{fig:hotelling}). <>== png('hotelling.png') setCor.diagram(sc, main="The Kelley data set") dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{hotelling.png} \caption{The relationship between three predictors and two criteria from \pfun{setCor}. The data are from the Kelley data set reported by \cite{hotelling:36}.} \label{fig:hotelling} \end{center} \end{figure} %\subsection{From the raw data} % % % %If the data are available, \pfun{setCor} will find the regressions between variables in an X set and those in a Y set. The first analysis (Figure~\ref{fig:2pred}) is perhaps the more typical (one criterion, two predictors), while the second example is more complicated, with three predictors of 3 dependent variables (Figure~\ref{fig:3x3}). % % %<>== %mod2 <- setCor(ACT ~ SATV + SATQ, data=sat.act) %mod2 %@ % %<>== %png('mod2.png') %setCor.diagram(mod2, main="Regressions for sat.act data") %dev.off() %@ % %\begin{Rinput} % %# a typical use of setCor %mod2 <- setCor(ACT ~ SATV + SATQ, data=sat.act) %mod2 %\end{Rinput} %\begin{Routput} %Call: setCor(y = ACT ~ SATV + SATQ, data = sat.act) % %Multiple Regression from raw data % % DV = ACT % slope se t p VIF %SATV 0.31 0.04 8.09 2.7e-15 1.72 %SATQ 0.39 0.04 10.08 0.0e+00 1.72 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %ACT 0.63 0.4 0.63 0.4 0.4 0.03 234.26 2 697 0 %\end{Routput} %\begin{figure}[htbp] %\begin{center} %\includegraphics{mod2.png} %\caption{The relationship between two predictors and one criterion from \pfun{setCor}. The data are from the \pfun{sat.act} data set } %\label{fig:2pred} %\end{center} %\end{figure} % But, we can also do multiple predictors \emph{and} multiple criteria in the same call: <>== png('satact.png') mod3 <- setCor(SATV + SATQ + ACT ~ gender + education + age, data = sat.act) dev.off() @ %<>== %png('satact.png') %setCor.diagram(mod3, main="Three predictors, 3 criteria") %dev.off() @ %\begin{Rinput} %mod3 <- setCor(SATV + SATQ + ACT ~ gender + education + age, data = sat.act) % %\end{Rinput} % %\begin{Routput} %Multiple Regression from raw data % % DV = SATV % slope se t p VIF %gender -0.03 0.04 -0.79 0.430 1.01 %education 0.10 0.05 2.29 0.022 1.45 %age -0.10 0.05 -2.21 0.028 1.44 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %SATV 0.1 0.01 0.05 0 0.01 0.01 2.26 3 696 0.0808 % % DV = SATQ % slope se t p VIF %gender -0.18 0.04 -4.71 3.0e-06 1.01 %education 0.10 0.04 2.25 2.5e-02 1.45 %age -0.09 0.04 -2.08 3.8e-02 1.44 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %SATQ 0.19 0.04 0.11 0.01 0.03 0.01 8.63 3 696 1.24e-05 % % DV = ACT % slope se t p VIF %gender -0.05 0.04 -1.28 0.2000 1.01 %education 0.14 0.05 3.14 0.0017 1.45 %age 0.03 0.04 0.71 0.4800 1.44 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %ACT 0.16 0.03 0.15 0.02 0.02 0.01 6.49 3 696 0.000248 % %Various estimates of between set correlations %Squared Canonical Correlations %[1] 0.050 0.033 0.008 %Chisq of canonical correlations %[1] 35.8 23.1 5.6 % % Average squared canonical correlation = 0.03 % Cohen's Set Correlation R2 = 0.09 % Shrunken Set Correlation R2 = 0.08 % F and df of Cohen's Set Correlation 7.26 9 1681.86 %Unweighted correlation between the two sets = 0.01 % %\end{Routput} % \begin{figure}[htbp] \begin{center} \includegraphics{satact.png} \caption{The relationship between three predictors and three criteria from \pfun{setCor}. The data are from the \pfun{sat.act} data set.} \label{fig:3x3} \end{center} \end{figure} \subsection{Moderated multiple regression} If we have the raw data, we can also find interactions (known as moderated multiple regression). This is done by zero centering the data \citep{cohen:03} and then multiplying the two terms of the interaction. As an option, we can not zero center the data \citep{hayes:13} which results in different ``main effects" but the same interaction term. To show the equivalence of the interaction terms, we also must not standardize the results. <>== mod <- setCor(SATQ ~ SATV*gender + ACT, data=sat.act, std =FALSE, plot=FALSE) mod mod0 <- setCor(SATQ ~ SATV*gender + ACT, data=sat.act, zero=FALSE, std=FALSE,plot=FALSE) mod0 @ <>== png('moderation.png') setCor.diagram(mod, main="not zero centered") dev.off() @ <>== png('moderation0.png') setCor.diagram(mod0, main="zero centered") dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{moderation.png} \caption{Showing a moderated multiple regression using \pfun{setCor}. The data are from the \pfun{sat.act} data set.} \label{fig:mod} \end{center} \end{figure} %\begin{Routput} %Call: setCor(y = SATQ ~ SATV * gender + ACT, data = sat.act, std = FALSE) % %Multiple Regression from raw data % % DV = SATQ % slope se t p VIF %SATV 0.47 0.03 14.47 0.0e+00 1.46 %gender -35.08 6.40 -5.48 6.0e-08 1.00 %ACT 7.72 0.77 10.05 0.0e+00 1.47 %SATV*gender -0.03 0.06 -0.47 6.4e-01 1.01 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %SATQ 0.72 0.51 70.33 4946.43 0.51 0.03 183.23 4 695 0 % % %Call: setCor(y = SATQ ~ SATV * gender + ACT, data = sat.act, std = FALSE, % zero = FALSE) % %Multiple Regression from raw data % % DV = SATQ % slope se t p VIF %SATV 0.52 0.10 5.20 2.7e-07 13.52 %gender -18.71 35.31 -0.53 6.0e-01 30.44 %ACT 7.72 0.77 10.05 0.0e+00 1.47 %SATV*gender -0.03 0.06 -0.47 6.4e-01 41.50 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %SATQ 0.72 0.51 40.02 1601.68 0.51 0.03 183.23 4 695 0 % %\end{Routput} \begin{figure}[htbp] \begin{center} \includegraphics[width=7cm]{moderation.png} \includegraphics[width=7cm]{moderation0.png} \caption{The difference between 0 and not 0 centering \pfun{setCor}. The data are from the \pfun{sat.act} data set. In both cases, the data are not standarized.} \label{default} \end{center} \end{figure} \subsection{Plotting the interactions} To visualize the effect of zero (mean) centering, it is useful to plot the various elements that go into the linear model. \pfun{setCor} returns the product terms as well as the original data. I combine the two datasets to make it clearer. Note that the correlations of the centered SATQ, SATV, and gender with the uncentered are 1.0, but that the correlations with the product terms depend upon centering versus not. I drop the ACT variables from the figure for clarity (Figure~\ref{fig:splom}). <>== both <- cbind(mod$data[-1],mod0$data[-1]) png('splom.png') pairs.panels(both[,-c(4,9)],gap=0) #show the mean centered data dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{splom.png} \caption{The effect of not mean centering versus mean centering on the product terms. The first four variables were not zero centered, the second four were. } \label{fig:splom} \end{center} \end{figure} \subsection{Comparisons to \fun{lm}} The \pfun{setCor} function duplicates the functionality of the \fun{lm} function for complete data, although \fun{lm} does not zero center and \pfun{setCor} will (by default). In addition, \pfun{setCor} finds correlations based upon pair.wise deletion of missing data, while \fun{lm} does case.wise deletion. We compare the \fun{lm} and \pfun{setCor} results for complete data by setting the \texttt{ use = "complete"} option. <>== lm(SATQ ~ SATV*gender + ACT, data=sat.act) mod <- setCor(SATQ ~ SATV*gender + ACT, data=(sat.act), zero=FALSE, std=FALSE,use="complete") print(mod,digits=5) @ % lm(SATQ ~ SATV*gender + ACT, data=sat.act) %Call: %lm(formula = SATQ ~ SATV * gender + ACT, data = sat.act) % %Coefficients: %(Intercept) SATV gender ACT SATV:gender % 138.52395 0.50280 -22.24995 7.71702 -0.01984 % %> mod <- setCor(SATQ ~ SATV*gender + ACT, data=(sat.act), zero=FALSE, std=FALSE,use="complete") % %print(mod,digits=5) %Call: setCor(y = SATQ ~ SATV * gender + ACT, data = (sat.act), use = "complete", % std = FALSE, zero = FALSE) % %Multiple Regression from raw data % % DV = SATQ % slope se t p VIF %SATV 0.50280 0.09936 5.06050 5.3589e-07 13.43994 %gender -22.24995 35.25783 -0.63106 5.2821e-01 30.29663 %ACT 7.71702 0.76977 10.02511 0.0000e+00 1.46678 %SATV*gender -0.01984 0.05652 -0.35105 7.2566e-01 41.25607 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %SATQ 0.71414 0.51 39.93879 1595.107 0.50718 0.02621 180.8401 4 695 0 %\end{Routput} \section{Mediation using the \pfun{mediate} function} Mediation analysis is just linear regression reorganized slightly to show the direct effects of an X variable upon Y, partialling out the effect of a ``mediator" (Figure~\ref{fig:mediation}). Although the statistical ``significance" of the (c) path and the (c') path are both available from standard regression, the mediation effect (ab) is best found by boot strapping the regression model and displaying the empirical confidence intervals. \begin{figure}[htbp] \begin{center} \begin{picture}(200,200) \put(10,50){\framebox(20,20){$X_{1}$}} \put(85,123){\framebox(20,20){$M_{1}$}} \put(160,50){\framebox(20,20){$Y_{1}$}} \put(30,70){\vector(1,1){54}} \put(105,123){\vector(1,-1){54}} \put(30,60){\vector(1,0){130}} \put(50,98){a} \put(134,98){b} \put(95,65){c} \put(78,51){c'= c - ab} \end{picture} \caption{The classic mediation model. The Direct Path from X -> Y (c) is said to be mediated by the indirect path (a) to the mediator (X -> M) and (b) from the mediator to Y (M -> Y). The mediation effect is (ab). } \label{fig:mediation} \end{center} \end{figure} A number of papers discuss how to test for the effect of mediation and there are some very popular `macros' for SPSS and SAS to do so \citep{hayes:13,preacher:04,preacher:07,preacher:15}. A useful discussion of mediation and moderation with sample data sets is found in \cite{hayes:13}. More recently, the \Rpkg{processR} package \citep{processR} has been released with these data sets. Although these data used to be be available from \href{"http://www.afhayes.com/public/hayes2018data.zip"}{http://www.afhayes.com/public/hayes2018data.zip} this now longer seems to be case.\footnote{The Hayes data sets (2018) do not correspond exactly with those from the 2013 book. Those data files were at \href{"http://www.afhayes.com/public/hayes2013data.zip"}{http://www.afhayes.com/public/hayes2013data.zip}.}. I use these for comparisons with the results in \cite{hayes:13}. Four of these data sets are now included in the \Rpkg{psych} package with the kind permission of their authors: \pfun{Garcia} is from \cite{garcia:10}, and \pfun{Tal\_Or} is from \cite{talor:10}, The \pfun{Pollack} correlation matrix is taken from an article by \cite{pollack}. The \pfun{globalWarm} data set is the \pfun{glbwarm} data set in the \Rpkg{processR} package and added to \Rpkg{psychTools} with the kind permission of the original author, Erik Nisbet. To find the confidence intervals of the effect of mediation (the reduction between the c and c' paths, where c' = c - ab), I bootstrap the results by randomly sampling from the data with replacement (n.iter = 5000) times. For these examples, the data files \pfun{Garcia} \citep{garcia:10} and \pfun{Tal\_Or} \citep{talor:10} are included in the \pfun{psych} package. The \pfun{estrss} data set and \pfun{glbwarm} were originally downloaded from the \cite{hayes:13} data seta and stored in a local directory on my computer. They are now available from the \Rpkg{processR} package \cite{processR}. The syntax is that $ y \sim x + (m) $ where m is the mediating variable. By default the output is to two decimals, as is the graphic output. This can be increased by returning the output to an object and then printing that object with the desired number of decimals. \subsection{Simple mediation} The first example \citep[mod.4.5]{hayes:13} is taken from \citep{talor:10} and examines the mediating effect of ``Presumed Media Influence'' (pmi) on the intention to act (reaction) based upon the importance of a message (import). The data are in the \pfun{Tal\_Or} data set in \Rpkg{psych} (with the kind permission of Nurit Tal-Or, Jonanathan Cohen, Yariv Tasfati, and Albert Gunther). In the \cite{hayes:13} book, this is the \pfun{pmi} data set. <>== data(Tal.Or) psych::describe(Tal_Or) #descriptive statistics mod4.4 <- mediate(reaction ~ cond + (pmi), data =Tal_Or) mod4.4 #print(mod4.4, digits = 4) # in order to get the precision of the Hayes (2013) p 99 example @ %\begin{Routput} %data(Tal_Or) %describe(Tal_Or) #descriptive statistics % vars n mean sd median trimmed mad min max range skew kurtosis se %cond 1 123 0.47 0.50 0.00 0.46 0.00 0 1 1 0.11 -2.00 0.05 %pmi 2 123 5.60 1.32 6.00 5.78 1.48 1 7 6 -1.17 1.30 0.12 %import 3 123 4.20 1.74 4.00 4.26 1.48 1 7 6 -0.26 -0.89 0.16 %reaction 4 123 3.48 1.55 3.25 3.44 1.85 1 7 6 0.21 -0.90 0.14 %gender 5 123 1.65 0.48 2.00 1.69 0.00 1 2 1 -0.62 -1.62 0.04 %age 6 123 24.63 5.80 24.00 23.76 1.48 18 61 43 4.71 24.76 0.52 % % % mod4.4 <- mediate(reaction ~ cond + (pmi), data =Tal_Or) %> mod4.4 %Mediation/Moderation Analysis %Call: mediate(y = reaction ~ cond + (pmi), data = Tal_Or) % %The DV (Y) was reaction . The IV (X) was cond . The mediating variable(s) = pmi . % %Total effect(c) of cond on reaction = 0.5 S.E. = 0.28 t = 1.79 df= 120 with p = 0.077 %Direct effect (c') of cond on reaction removing pmi = 0.25 S.E. = 0.26 % t = 0.99 df= 120 with p = 0.32 %Indirect effect (ab) of cond on reaction through pmi = 0.24 %Mean bootstrapped indirect effect = 0.24 with standard error = 0.13 Lower CI = 0 Upper CI = 0.52 %R = 0.45 R2 = 0.21 F = 15.56 on 2 and 120 DF p-value: 9.83e-07 % % To see the longer output, specify short = FALSE in the print statement or ask for the summary % % Full output % Total effect estimates (c) % reaction se t df Prob %cond 0.5 0.28 1.79 120 0.0766 % %Direct effect estimates (c') % reaction se t df Prob %cond 0.25 0.26 0.99 120 3.22e-01 %pmi 0.51 0.10 5.22 120 7.66e-07 % %R = 0.45 R2 = 0.21 F = 15.56 on 2 and 120 DF p-value: 9.83e-07 % % 'a' effect estimates % pmi se t df Prob %cond 0.48 0.24 2.02 121 0.0454 % % 'b' effect estimates % reaction se t df Prob %pmi 0.51 0.1 5.22 120 7.66e-07 % % 'ab' effect estimates % reaction boot sd lower upper %cond 0.24 0.24 0.13 0 0.52 % %\end{Routput} <>== png('mediate99.png') mediate.diagram(mod4.4) dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{mediate99.png} \caption{A simple mediation model \citep[p 99] {hayes:13} with data derived from \cite{talor:10}. The effect of a salience manipulation (cond) on the intention to buy a product (reaction) is mediated through the presumed media influence (pmi).} \label{default} \end{center} \end{figure} A second example from \citep{hayes:13} is an example of moderated mediated effect. The data are from \citep{garcia:10} and report on the effect of protest on reactions to a case of sexual discrimination. <>== data(GSBE) #alias to Garcia data set #compare two models (bootstrapping n.iter set to 50 for speed # 1) mean center the variables prior to taking product terms mod1 <- mediate(respappr ~ prot2 * sexism +(sexism),data=Garcia,n.iter=50 ,main="Moderated mediation (mean centered)") # 2) do not mean center mod2 <- mediate(respappr ~ prot2 * sexism +(sexism),data=Garcia,zero=FALSE, n.iter=50, main="Moderated mediation (not centered") summary(mod1) summary(mod2) @ %A second example of simple mediation from \cite[p 118-121]{hayes:13} is the effect of economic stress. The original data are from a study by \cite{pollack:12} and are available from the \cite{hayes:13} website. Is the effect of economic stress (estress) on subsequent disengagement from entreprenuerial activities (withdraw) mediated through depressed affect (affect)? % %\begin{Rinput} %estress <- read.file() #read the external file %describe(estress) %mod4.5 <- mediate(withdraw ~ estress + (affect), data =estress) %mod4.5 #normal printing is to 2 decimals %# and show the graphic to 2 decimals %#print(mod4.5, digits=4) #print to four decimals to confirm output with Hayes %#mediate.diagram(mod4.5,digits=3) to show the graphic to 3 decimals % %\end{Rinput} % %\begin{Routput} %estress <- read.file() %re-encoding from CP1252 %Data from the SPSS sav file" % /Users/WR/Box Sync/pmc_folder/tutorials/HowTo/mediation/hayes2018data/estress/estress.sav has been loaded. %> describe(estress) % vars n mean sd median trimmed mad min max range skew kurtosis se %tenure 1 262 5.93 6.58 4.00 4.73 5.07 0.00 33 33.00 1.64 2.67 0.41 %estress 2 262 4.62 1.42 4.50 4.66 1.48 1.00 7 6.00 -0.27 -0.49 0.09 %affect 3 262 1.60 0.72 1.33 1.46 0.49 1.00 5 4.00 1.97 4.57 0.04 %withdraw 4 262 2.32 1.25 2.00 2.19 1.48 1.00 7 6.00 0.70 -0.17 0.08 %sex 5 262 0.62 0.49 1.00 0.65 0.00 0.00 1 1.00 -0.48 -1.77 0.03 %age 6 262 43.79 10.36 44.00 43.78 11.86 23.00 71 48.00 -0.01 -0.82 0.64 %ese 7 262 5.61 0.94 5.73 5.67 1.08 2.53 7 4.47 -0.55 -0.13 0.06 % %\end{Routput} %\begin{Toutput} %mod4.5 % %Mediation/Moderation Analysis %Call: mediate(y = withdraw ~ estress + (affect), data = estress) % %The DV (Y) was withdraw . The IV (X) was estress . The mediating variable(s) = affect . % %Total effect(c) of estress on withdraw = 0.06 S.E. = 0.05 t = 1.04 df= 259 with p = 0.3 %Direct effect (c') of estress on withdraw removing affect = -0.08 S.E. = 0.05 t = -1.47 df= 259 with p = 0.14 %Indirect effect (ab) of estress on withdraw through affect = 0.13 %Mean bootstrapped indirect effect = 0.13 with standard error = 0.03 Lower CI = 0.07 Upper CI = 0.2 %R = 0.42 R2 = 0.18 F = 28.49 on 2 and 259 DF p-value: 6.53e-12 %\end{Toutput} %\begin{Routput} % Full output % Total effect estimates (c) % withdraw se t df Prob %estress 0.06 0.05 1.04 259 0.302 % %Direct effect estimates (c') % withdraw se t df Prob %estress -0.08 0.05 -1.47 259 1.44e-01 %affect 0.77 0.10 7.46 259 1.29e-12 % %R = 0.42 R2 = 0.18 F = 28.49 on 2 and 259 DF p-value: 6.53e-12 % % 'a' effect estimates % affect se t df Prob %estress 0.17 0.03 5.83 260 1.63e-08 % % 'b' effect estimates % withdraw se t df Prob %affect 0.77 0.1 7.46 259 1.29e-12 % % 'ab' effect estimates % withdraw boot sd lower upper %estress 0.13 0.13 0.03 0.07 0.2 %\end{Routput} % %\begin{figure}[htbp] %\begin{center} % %\includegraphics{mediate118.pdf} %\caption{A simple mediation model \citep[p 118] {hayes:13}. The data are from \cite{pollack:12} taken from the \cite{hayes:13} website. Is the effect of economic stress (estress) on subsequent disengagement from entreprenuerial activities (withdraw) mediated through depressed affect (affect)?} %\label{default} %\end{center} %\end{figure} \subsection{Multiple mediators} It is trivial to show the effect of multiple mediators. I do this by adding the second (or third) mediator into our equation. I use the \fun{Tal\_Or} data set \citep{talor:10} again. I show the graphical representation in Figure~\ref{fig:2m}. <>== mod5.4 <- mediate(reaction ~ cond + (import) + (pmi), data = Tal_Or) print(mod5.4, digits=4) #to compare with Hayes @ <>== png('mediate131.png') mediate.diagram(mod5.4, digits=3, main="Hayes example 5.3") dev.off() @ %\begin{Toutput} %Call: mediate(y = reaction ~ cond + (import) + (pmi), data = Tal_Or) % %The DV (Y) was reaction . The IV (X) was cond . The mediating variable(s) = import pmi . Variable(s) partialled out were % %Total Direct effect(c) of cond on reaction = 0.4957 S.E. = 0.2775 t direct = 1.786 with probability = 0.07661 %Direct effect (c') of cond on reaction removing import pmi = 0.1034 S.E. = 0.2391 t direct = 0.4324 with probability = 0.6662 %Indirect effect (ab) of cond on reaction through import pmi = 0.3923 %Mean bootstrapped indirect effect = 0.3964 with standard error = 0.1658 Lower CI = 0.0895 Upper CI = 0.7317 %R2 of model = 0.3251 % To see the longer output, specify short = FALSE in the print statement % % %\end{Toutput} %\begin{Routput} % Full output % % Total effect estimates (c) % reaction se t Prob %cond 0.4957 0.2775 1.786 0.076608 % %Direct effect estimates (c') % reaction se t Prob %cond 0.1034 0.2391 0.4324 6.6622e-01 %import 0.3244 0.0707 4.5857 1.1267e-05 %pmi 0.3965 0.0930 4.2645 4.0383e-05 % % 'a' effect estimates % cond se t Prob %import 0.6268 0.3098 2.0234 0.045235 %pmi 0.4765 0.2357 2.0218 0.045401 % % 'b' effect estimates % reaction se t Prob %import 0.3244 0.0707 4.5857 1.1267e-05 %pmi 0.3965 0.0930 4.2645 4.0383e-05 % % 'ab' effect estimates % reaction boot sd lower upper %cond 0.3923 0.3965 0.1645 0.0896 0.7392 %> \end{Routput} % \begin{figure}[htbp] \begin{center} \includegraphics{mediate131.png} \caption{A mediation model with two mediators \citep[p 131] {hayes:13}. The data are data derived from \cite{talor:10}. The effect of a salience manipulation (cond) on the intention to buy a product (reaction) is mediated through the presumed media influence (pmi) and importance of the message (import).} \label{fig:2m} \end{center} \end{figure} \subsection{Serial mediators} The example from \cite{hayes:13} for two mediators, where one effects the second, is a bit more complicated and currently can be done by combining two separate analyses. The first is just model 5.4, the second is the effect of cond on pmi mediated by import. Combining the two results leads to the output found on \cite[page 153]{hayes:13}. <>== png('mediate131.png') mediate.diagram(mod5.4, digits=3, main="Hayes example 5.3") dev.off() @ <>== #model 5.4 + mod5.7 is the two chained mediator model mod5.7 <- mediate(pmi ~ cond + (import) , data = Tal_Or) summary(mod5.7, digits=4) @ % %\begin{Routput} %Call: mediate(y = pmi ~ cond + (import), data = Tal_Or) % % Total effect estimates (c) % pmi se t df Prob %cond 0.4765 0.2357 2.0218 120 0.045419 % %Direct effect estimates (c') % pmi se t df Prob %cond 0.3536 0.2325 1.5207 120 0.1309600 %import 0.1961 0.0671 2.9228 120 0.0041467 % %R = 0.3114 R2 = 0.097 F = 6.4428 on 2 and 120 DF p-value: 0.0021989 % % 'a' effect estimates % import se t df Prob %cond 0.6268 0.3098 2.0234 121 0.045235 % % 'b' effect estimates % pmi se t df Prob %import 0.1961 0.0671 2.9228 120 0.0041467 % % 'ab' effect estimates % pmi boot sd lower upper %cond 0.1229 0.1226 0.0825 -0.0017 0.3152 %> % %\end{Routput} % \subsection{Single mediators, multiple covariates} The \fun{Pollack} data set \citep{pollack:12} is used as an example of multiple covariates (included in \Rpkg{psychTools} as a correlation matrix). The raw data are available from the \Rpkg{processR} package as \pfun{estress}. Confidence in executive decision making (``Entrepeneurial self-effiicacy), gender (sex), and length of time in business (tenure) are used as covariates. There are two ways of doing this: enter them as predictors of the criterion or to partial them out. The first approach estimates their effects, the second just removes them. <>== lowerMat(Pollack) mod6.2 <- mediate(withdrawal ~ economic.stress + self.efficacy + sex + tenure + (depression), data=Pollack, n.obs=262) summary(mod6.2) @ <>== png('mediate177.png') mediate.diagram(mod6.2, digits=3, main = "Simple mediation, 3 covariates") dev.off() @ The graphical output (Figure~\ref{fig:3cov}) looks a bit more complicated than the figure in \cite[p 177]{hayes:13} because I are showing the covariates as causal paths. %\begin{Toutput} %Call: mediate(y = withdraw ~ estress + ese + sex + tenure + (affect), % data = estress) % %The DV (Y) was withdraw . The IV (X) was estress ese sex tenure . The mediating variable(s) = affect . % %Total effect(c) of estress on withdraw = 0.02 S.E. = 0.05 t = 0.35 df= 256 with p = 0.72 %Direct effect (c') of estress on withdraw removing affect = -0.09 S.E. = 0.05 t = -1.78 df= 256 with p = 0.077 %Indirect effect (ab) of estress on withdraw through affect = 0.11 %Mean bootstrapped indirect effect = 0.11 with standard error = 0.03 Lower CI = 0.06 Upper CI = 0.17 % %Total effect(c) of ese on withdraw = -0.32 S.E. = 0.08 t = -3.98 df= 256 with p = 9e-05 %Direct effect (c') of ese on NA removing affect = -0.21 S.E. = 0.08 t = -2.78 df= 256 with p = 0.0059 %Indirect effect (ab) of ese on withdraw through affect = -0.11 %Mean bootstrapped indirect effect = 0.11 with standard error = 0.03 Lower CI = -0.19 Upper CI = -0.03 % %Total effect(c) of sex on withdraw = 0.14 S.E. = 0.16 t = 0.88 df= 256 with p = 0.38 %Direct effect (c') of sex on NA removing affect = 0.13 S.E. = 0.14 t = 0.88 df= 256 with p = 0.38 %Indirect effect (ab) of sex on withdraw through affect = 0.01 %Mean bootstrapped indirect effect = 0.11 with standard error = 0.03 Lower CI = -0.09 Upper CI = 0.15 % %Total effect(c) of tenure on withdraw = -0.01 S.E. = 0.01 t = -0.85 df= 256 with p = 0.4 %Direct effect (c') of tenure on NA removing affect = 0 S.E. = 0.01 t = -0.19 df= 256 with p = 0.85 %Indirect effect (ab) of tenure on withdraw through affect = -0.01 %Mean bootstrapped indirect effect = 0.11 with standard error = 0.03 Lower CI = -0.02 Upper CI = 0 %R = 0.45 R2 = 0.21 F = 13.28 on 5 and 256 DF p-value: 1.63e-11 %\end{Toutput} % %\begin{Routput} % Full output % % % Total effect estimates (c) % withdraw se t df Prob %estress 0.02 0.05 0.35 256 7.24e-01 %ese -0.32 0.08 -3.98 256 9.02e-05 %sex 0.14 0.16 0.88 256 3.78e-01 %tenure -0.01 0.01 -0.85 256 3.97e-01 % %Direct effect estimates (c') % withdraw se t df Prob %estress -0.09 0.05 -1.78 256 0.07710 %ese -0.21 0.08 -2.78 256 0.00589 %sex 0.13 0.14 0.88 256 0.37800 %tenure 0.00 0.01 -0.19 256 0.84600 % %R = 0.45 R2 = 0.21 F = 13.28 on 5 and 256 DF p-value: 1.63e-11 % % 'a' effect estimates % affect se t df Prob %estress 0.16 0.03 5.36 257 1.84e-07 %ese -0.15 0.04 -3.49 257 5.70e-04 %sex 0.01 0.09 0.17 257 8.63e-01 %tenure -0.01 0.01 -1.72 257 8.61e-02 % % 'b' effect estimates % withdraw se t df Prob %affect 0.71 0.1 6.74 256 1.03e-10 % % 'ab' effect estimates % withdraw boot sd lower upper %estress 0.11 0.11 0.03 0.06 0.17 %ese -0.11 -0.11 0.04 -0.19 -0.03 %sex 0.01 0.02 0.06 -0.09 0.15 %tenure -0.01 -0.01 0.00 -0.02 0.00 %> %\end{Routput} \begin{figure}[htbp] \begin{center} \includegraphics{mediate177.png} \caption{A mediation model with three covariates \citep[p 177] {hayes:13}. Compare this to the solution in which they are partialled out. (Figure~\ref{fig:mod6.2a}).} \label{fig:3cov} \end{center} \end{figure} \subsection{Single predictor, single criterion, multiple covariates} An alternative way to display the previous results is to remove the three covariates from the mediation model. I do this by partialling out the covariates. This is represented in the \pfun{mediate} code by a negative sign. (Figure~\ref{fig:mod6.2a}) <>== mod6.2a <- mediate(withdrawal ~ economic.stress -self.efficacy - sex - tenure + (depression), data=Pollack, n.obs=262) summary(mod6.2a) @ %\begin{Rinput} %mod6.2a <- mediate(withdraw ~ estress - ese - sex - tenure + (affect), data=estress) %mod6.2a #give the output %\end{Rinput} %\begin{Toutput} % %The DV (Y) was withdraw . The IV (X) was estress . The mediating variable(s) = affect . Variable(s) partialled out were ese sex tenure % %Total effect(c) of estress on withdraw = 0.02 S.E. = 0.05 t = 0.36 df= 256 with p = 0.72 %Direct effect (c') of estress on withdraw removing affect = -0.09 S.E. = 0.05 t = -1.77 df= 256 with p = 0.078 %Indirect effect (ab) of estress on withdraw through affect = 0.11 %Mean bootstrapped indirect effect = 0.11 with standard error = 0.03 Lower CI = 0.06 Upper CI = 0.17 %R = 0.39 R2 = 0.15 F = 22.8 on 2 and 256 DF p-value: 7.71e-10 % %\end{Toutput} %\begin{Routput} % Full output %Total effect estimates (c) % withdraw se t df Prob %estress 0.02 0.05 0.36 256 0.722 % %Direct effect estimates (c') % withdraw se t df Prob %estress -0.09 0.05 -1.77 256 7.78e-02 %affect 0.71 0.11 6.72 256 1.14e-10 % %R = 0.39 R2 = 0.15 F = 22.8 on 2 and 256 DF p-value: 7.71e-10 % % 'a' effect estimates % affect se t df Prob %estress 0.16 0.03 5.39 257 1.58e-07 % % 'b' effect estimates % withdraw se t df Prob %affect 0.71 0.11 6.72 256 1.14e-10 % % 'ab' effect estimates % withdraw boot sd lower upper %estress 0.11 0.11 0.03 0.06 0.17 % % %\end{Routput} % <>== png('mod62partial.png') mediate.diagram(mod6.2a, digits=3, main = "Simple mediation, 3 covariates (partialled out)") dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{mod62partial.png} \caption{Show the mediation model from Figure~\ref{fig:3cov} with the covariates (ese, sex, tenure) removed.} \label{fig:mod6.2a} \end{center} \end{figure} \subsection{Multiple predictors, single criterion} It is straightforward to use multiple predictors see \cite[p196]{hayes:13} and in fact I did so in the previous example where the predictors were treated as \emph{covariates}. \pfun{mediate} also allows for multiple criteria. \section{Mediation and moderation} We already saw how to do moderation in the discussion of \pfun{setCor}. Combining the concepts of mediation with moderation is done in \pfun{mediate}. That is, I can find the linear model of product terms as they are associated with dependent variables and regressed on the mediating variables. The \fun{Garcia} data set \citep{garcia:10} can be used for an example of moderation. (This was taken from \citep{hayes:13} but is used with kind permission of Donna M. Garcia, Michael T. Schmitt, Nyla R. Branscombe, and Naomi Ellemers.) Just as \pfun{setCor} and \fun{lm} will find the interaction term by forming a product, so will \pfun{mediate}. Notice that by default, \pfun{setCor} reports zero centered and standardized regressions, \pfun{mediate} reports zero centered but not standardized regressions, and some of the examples from \cite{hayes:13} do not zero center the data. Thus, I specify zero=FALSE to get the \cite{hayes:13} results. It is important to note that the \fun{protest} data set discussed here is from the 2013 examples and not the more recent 2018 examples available from \href{http://afhayes.com}{afhayes.com}. The 2013 data have a dichotomous protest variable, while the 2018 data set has three levels for the protest variable. The \pfun{Garcia} data set is composed of the 2018 data set with the addition of a dichotomous variable (prot2) to match the 2013 exampes. We consider how the interaction of sexism with protest affects the mediation effect of sexism \citep[p 362]{hayes:13}, I contrast the \fun{lm}, \pfun{setCor} and \pfun{mediate} approaches. For reasons to be discussed in the next section, I do not zero center the variables. The graphic output is in Figure~\ref{fig:modmed} and the output is below. For comparison purposes, I show the results from the \fun{lm} as well as \pfun{setCor} and \pfun{mediate}. <>== summary(lm(respappr ~ prot2 * sexism,data = Garcia)) #show the lm results for comparison #show the setCor analysis setCor(respappr ~ prot2* sexism ,data=Garcia,zero=FALSE,main="Moderation",std=FALSE) #then show the mediate results modgarcia <-mediate(respappr ~ prot2 * sexism +(sexism),data=Garcia,zero=FALSE,main="Moderated mediation") summary(modgarcia) @ <>== png('moderatedmediation.png') mediate.diagram(modgarcia, main= "An example of moderated mediation") dev.off() @ %lm(formula = respappr ~ prot2 * sexism, data = Garcia) % %Residuals: % Min 1Q Median 3Q Max %-3.4984 -0.7540 0.0801 0.8301 3.1853 % %Coefficients: % Estimate Std. Error t value Pr(>|t|) %(Intercept) 6.5667 1.2095 5.429 2.83e-07 *** %prot2 -2.6866 1.4515 -1.851 0.06654 . %sexism -0.5290 0.2359 -2.243 0.02668 * %prot2:sexism 0.8100 0.2819 2.873 0.00478 ** %--- %Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1 % %Residual standard error: 1.144 on 125 degrees of freedom %Multiple R-squared: 0.2962, Adjusted R-squared: 0.2793 %F-statistic: 17.53 on 3 and 125 DF, p-value: 1.456e-09 % %setCor(respappr ~ prot2* sexism ,data=Garcia,zero=FALSE,main="Moderation",std=FALSE) %Call: setCor(y = respappr ~ prot2 * sexism, data = Garcia, std = FALSE, % main = "Moderation", zero = FALSE) % %Multiple Regression from raw data % % DV = respappr % slope se t p VIF %prot2 -2.69 1.45 -1.85 0.0670 44.99 %sexism -0.53 0.24 -2.24 0.0270 3.34 %prot2*sexism 0.81 0.28 2.87 0.0048 48.14 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %respappr 0.54 0.3 0.65 0.43 0.28 0.06 17.53 3 125 1.46e-09 % %> summary( mediate(respappr ~ prot2 * sexism +(sexism),data=Garcia,zero=FALSE,main="Moderated mediation")) %Call: mediate(y = respappr ~ prot2 * sexism + (sexism), data = Garcia, % zero = FALSE, main = "Moderated mediation") % % Total effect estimates (c) % respappr se t df Prob %prot2 0.00 0.84 0.00 125 0.9960 %prot2*sexism 0.28 0.16 1.79 125 0.0756 % %Direct effect estimates (c') % respappr se t df Prob %prot2 -2.69 1.45 -1.85 125 0.06650 %prot2*sexism 0.81 0.28 2.87 125 0.00478 % %R = 0.54 R2 = 0.3 F = 17.53 on 3 and 125 DF p-value: 1.46e-09 % % 'a' effect estimates % sexism se t df Prob %prot2 -5.07 0.31 -16.33 126 6.81e-33 %prot2*sexism 1.00 0.06 17.15 126 9.41e-35 % % 'b' effect estimates % respappr se t df Prob %sexism -0.53 0.24 -2.24 125 0.0267 % % 'ab' effect estimates % respappr boot sd lower upper %prot2 2.68 2.65 1.60 -0.69 5.60 %prot2*sexism -0.53 -0.52 0.32 -1.11 0.14 %\end{Routput} \begin{figure}[htbp] \begin{center} \includegraphics{moderatedmediation.png} \caption{Moderated mediation from \citep[p 362]{hayes:13}. The data are from \cite{garcia:10}.} \label{fig:modmed} \end{center} \end{figure} \subsection{To center or not to center, that is the question} We have discussed the difference between zero centering and not zero centering. Although \cite{hayes:13} seems to prefer not centering, some of his examples are in fact centered. So, when we examine Table 8.2 and try to replicate the regression, we need to zero center the data. With the global warming data from \cite{hayes:13}, the default (uncentered) regression does not reproduce his Table, but zero centering does. To this in \fun{lm} requires two steps, but we can do this in \pfun{setCor} with the zero=TRUE or zero=FALSE option. <>== lm(govact ~ age * negemot + posemot + ideology + sex, data=globalWarm) # but zero center and try again glbwarmc <-data.frame(scale(globalWarm,scale=FALSE)) lm(govact ~ age * negemot + posemot + ideology + sex, data=glbwarmc) mod.glb <- setCor(govact ~ age * negemot + posemot + ideology + sex, data=globalWarm,zero=FALSE,std=FALSE) print(mod.glb,digits=6) mod.glb0 <- setCor(govact ~ age * negemot + posemot + ideology + sex, data=globalWarm,std=FALSE) print(mod.glb0,digits=6) @ %\begin{Routput} %> lm(govact ~ age * negemot + posemot + ideology + sex, data=glbwarm) %Call: %lm(formula = govact ~ age * negemot + posemot + ideology + sex, % data = glbwarm) %Coefficients: %(Intercept) age negemot posemot ideology sex age:negemot % 5.173849 -0.023879 0.119583 -0.021419 -0.211515 -0.011191 0.006331 %> # but zero center and try again %> glbwarmc <-data.frame(scale(glbwarm,scale=FALSE)) %> lm(govact ~ age * negemot + posemot + ideology + sex, data=glbwarmc) % %Call: %lm(formula = govact ~ age * negemot + posemot + ideology + sex, % data = glbwarmc) % %Coefficients: %(Intercept) age negemot posemot ideology sex age:negemot % 0.008979 -0.001354 0.433184 -0.021419 -0.211515 -0.011191 0.006331 % %> mod.glb <- setCor(govact ~ age * negemot + posemot + ideology + sex, data=glbwarm,zero=FALSE,std=FALSE) %> print(mod.glb,digits=6) %Call: setCor(y = govact ~ age * negemot + posemot + ideology + sex, % data = glbwarm, std = FALSE, zero = FALSE) % %Multiple Regression from raw data % % DV = govact % slope se t p VIF %age -0.023879 0.005980 -3.992944 7.12038e-05 6.949401 %negemot 0.119583 0.082535 1.448881 1.47759e-01 11.594520 %posemot -0.021419 0.027904 -0.767597 4.42951e-01 1.028663 %ideology -0.211515 0.026833 -7.882678 1.02141e-14 1.198910 %sex -0.011191 0.076003 -0.147240 8.82979e-01 1.052907 %age*negemot 0.006331 0.001543 4.103542 4.48155e-05 16.455422 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %govact 0.633093 0.400806 0.571703 0.326844 0.396357 0.026299 90.07983 6 808 0 % %> mod.glb0 <- setCor(govact ~ age * negemot + posemot + ideology + sex, data=glbwarm,std=FALSE) %> print(mod.glb0,digits=6) %Call: setCor(y = govact ~ age * negemot + posemot + ideology + sex, % data = glbwarm, std = FALSE) % %Multiple Regression from raw data % % DV = govact % slope se t p VIF %age -0.001354 0.002348 -0.576864 5.64192e-01 1.071058 %negemot 0.433184 0.026243 16.506679 0.00000e+00 1.172207 %posemot -0.021419 0.027904 -0.767597 4.42951e-01 1.028663 %ideology -0.211515 0.026833 -7.882678 1.02141e-14 1.198910 %sex -0.011191 0.076003 -0.147240 8.82979e-01 1.052907 %age*negemot 0.006331 0.001543 4.103542 4.48155e-05 1.014744 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %govact 0.633093 0.400806 0.34298 0.117635 0.396357 0.026299 90.07983 6 808 0 % %\end{Routput} % So, when we do the mediated moderation model, we need to use the zero centered option to match the \cite{hayes:13} results from Figure 8.5. <>== #by default, mediate zero centers before finding the products mod.glb <- mediate(govact ~ age * negemot + posemot + ideology + sex + (age), data=globalWarm,zero=TRUE) summary(mod.glb,digits=4) @ Compare this output to that of Table 8.2 and Figure 8.5 (p 258 - 259). %\begin{Routput} %Call: mediate(y = govact ~ age * negemot + posemot + ideology + sex + % (age), data = glbwarm, zero = TRUE) % % Total effect estimates (c) % govact se t df Prob %negemot 0.4328 0.0262 16.5043 808 5.9317e-53 %posemot -0.0220 0.0279 -0.7890 808 4.3036e-01 %ideology -0.2145 0.0263 -8.1510 808 1.3712e-15 %sex -0.0173 0.0752 -0.2304 808 8.1783e-01 %age*negemot 0.0063 0.0015 4.1025 808 4.5004e-05 % %Direct effect estimates (c') % govact se t df Prob %negemot 0.4332 0.0262 16.5067 808 5.7578e-53 %posemot -0.0214 0.0279 -0.7676 808 4.4295e-01 %ideology -0.2115 0.0268 -7.8827 808 1.0360e-14 %sex -0.0112 0.0760 -0.1472 808 8.8298e-01 %age*negemot 0.0063 0.0015 4.1035 808 4.4816e-05 % %R = 0.6331 R2 = 0.4008 F = 90.0798 on 6 and 808 DF p-value: 1.8246e-86 % % 'a' effect estimates % age se t df Prob %negemot 0.2757 0.3929 0.7017 809 4.8305e-01 %posemot 0.4232 0.4176 1.0135 809 3.1112e-01 %ideology 2.2079 0.3943 5.6002 809 2.9334e-08 %sex 4.5345 1.1269 4.0238 809 6.2643e-05 %age*negemot 0.0031 0.0231 0.1346 809 8.9294e-01 % % 'b' effect estimates % govact se t df Prob %age -0.0014 0.0023 -0.5769 808 0.56419 % % 'ab' effect estimates % govact boot sd lower upper %negemot -0.0004 -0.0004 0.0012 -0.0033 0.0016 %posemot -0.0006 -0.0005 0.0014 -0.0038 0.0021 %ideology -0.0030 -0.0029 0.0051 -0.0136 0.0070 %sex -0.0061 -0.0057 0.0106 -0.0273 0.0150 %age*negemot 0.0000 0.0000 0.0001 -0.0002 0.0002 % \end{Routput} % \subsection{Another example of moderated medation} The \pfun{Garcia} data set (\pfun{protest} in \cite{hayes:13}) is another example of a moderated analysis. We can use either \pfun{setCor} or \pfun{mediate} to examine this data set. The defaults for these two differ, in that \pfun{setCor} assumes we want to zero center \emph{and} standardize, while \pfun{mediate} defaults to not standardizing but also defaults to zero (mean) centering. Note that in the next examples we specify we do not want to standardize nor to mean center. <>== psych::describe(Garcia) lm(liking ~ prot2* sexism + respappr, data=Garcia) setCor(liking ~ prot2* sexism + respappr, data = Garcia, zero=FALSE,std=FALSE) mod7.4 <- mediate(liking ~ prot2 * sexism +respappr, data = Garcia, zero=FALSE) summary(mod7.4) @ %\begin{Routput} % describe(Garcia) % vars n mean sd median trimmed mad min max range skew kurtosis se %protest 1 129 1.03 0.82 1.00 1.04 1.48 0.00 2 2.00 -0.06 -1.52 0.07 %sexism 2 129 5.12 0.78 5.12 5.10 0.74 2.87 7 4.13 0.12 -0.32 0.07 %anger 3 129 2.12 1.66 1.00 1.84 0.00 1.00 7 6.00 1.29 0.26 0.15 %liking 4 129 5.64 1.05 5.83 5.73 0.99 1.00 7 6.00 -1.15 2.48 0.09 %respappr 5 129 4.87 1.35 5.25 4.98 1.11 1.50 7 5.50 -0.75 -0.18 0.12 %prot2 6 129 0.68 0.47 1.00 0.72 0.00 0.00 1 1.00 -0.77 -1.41 0.04 % % % %Call: %lm(formula = liking ~ prot2 * sexism + respappr, data = Garcia) % %Coefficients: % (Intercept) prot2 sexism respappr prot2:sexism % 5.3471 -2.8075 -0.2824 0.3593 0.5426 % %> setCor(liking ~ prot2* sexism + respappr, data = Garcia, zero=FALSE,std=FALSE) %Call: setCor(y = liking ~ prot2 * sexism + respappr, data = Garcia, % std = FALSE, zero = FALSE) % %Multiple Regression from raw data % % DV = liking % slope se t p VIF %prot2 -2.81 1.16 -2.42 1.7e-02 46.22 %sexism -0.28 0.19 -1.49 1.4e-01 3.47 %respappr 0.36 0.07 5.09 1.3e-06 1.42 %prot2*sexism 0.54 0.23 2.36 2.0e-02 51.32 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %liking 0.53 0.28 0.39 0.15 0.26 0.06 12.26 4 124 1.99e-08 %> mod7.4m <- mediate(liking ~ protest * sexism, data = protest, zero=FALSE) %> mod7.4m %Call: mediate(y = liking ~ prot2 * sexism + respappr, data = Garcia, % zero = FALSE) % %The DV (Y) was liking . The IV (X) was prot2 sexism respappr prot2*sexism . The mediating variable(s) = . % DV = liking % slope se t p %prot2 -2.81 1.16 -2.42 1.7e-02 %sexism -0.28 0.19 -1.49 1.4e-01 %respappr 0.36 0.07 5.09 1.3e-06 %prot2*sexism 0.54 0.23 2.36 2.0e-02 % %With R2 = 0.28 %R = 0.53 R2 = 0.28 F = 12.26 on 4 and 124 DF p-value: 1.99e-08 %\end{Routput} <>== png('moderation.png') mediate.diagram(mod7.4, main= "Another example of moderated mediation") dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{moderation.png} \caption{A simple moderated regression analysis of the \fun{protest} data set. The data were not zero centered. This shows the strength of the three regressions. Figure~\ref{fig:garcia} shows the actual data and the three regression lines. } \label{fig:moderation} \end{center} \end{figure} \subsection{Graphic Displays of Interactions} In order to graphically display interactions, particularly if one of the variable is categorical, we can plot separate regression lines for each value of the categorical variable. We do this for the \pfun{Garcia} data set to show the interaction of protest with sexism. (see Figure~\ref{fig:garcia}). This is just an example of how to use Core-R to do graphics and is not a feature of \Rpkg{psych}. <>== png('garciainteraction.png') plot(respappr ~ sexism, pch = 23- protest, bg = c("black","red", "blue")[protest], data=Garcia, main = "Response to sexism varies as type of protest") by(Garcia,Garcia$protest, function(x) abline(lm(respappr ~ sexism, data =x),lty=c("solid","dashed","dotted")[x$protest+1])) text(6.5,3.5,"No protest") text(3,3.9,"Individual") text(3,5.2,"Collective") dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics{garciainteraction.png} \caption{Showing the interaction between type of protest and sexism from the \fun{Garcia} data set. The strength of the regression effects is shown in Fig~\ref{fig:moderation}.} \label{fig:garcia} \end{center} \end{figure} \section{Partial Correlations} Although not strickly speaking part of mediation or moderation, the use of \emph{partial correlations} can be addressed here. \subsection{Partial some variables from the rest of the variables} Given a set of X variables and a set of Y variables, we can control for an additional set of Z variables when we find the correlations between X and Y. This is effectively what happens when we want to add covariates into a model. We see this when we compare the regression model for government action as a function of the iteraction of ideology and age with some covariates, or when we partial them out first. <>== #first, the more complicated model mod.glb <- setCor(govact ~ age * negemot + posemot + ideology + sex, data=globalWarm,std=FALSE) print(mod.glb,digits=3) # compare this to the partialled model mod.glb.partialled <- setCor(govact ~ age * negemot - posemot - ideology - sex,data = globalWarm) @ % %\begin{Routput} % print(mod.glb,digits=3) %Call: setCor(y = govact ~ age * negemot + posemot + ideology + sex, % data = glbwarm, std = FALSE) % %Multiple Regression from raw data % % DV = govact % slope se t p VIF %age -0.001 0.002 -0.577 5.64e-01 1.071 %negemot 0.433 0.026 16.507 0.00e+00 1.172 %posemot -0.021 0.028 -0.768 4.43e-01 1.029 %ideology -0.212 0.027 -7.883 1.02e-14 1.199 %sex -0.011 0.076 -0.147 8.83e-01 1.053 %age*negemot 0.006 0.002 4.104 4.48e-05 1.015 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %govact 0.633 0.401 0.343 0.118 0.396 0.026 90.08 6 808 0 %> %mod.glb.partialled <- setCor(govact ~ age * negemot - posemot - ideology - sex, %+ data=glbwarm,std=FALSE) % %mod.glb.partialled %Call: setCor(y = govact ~ age * negemot - posemot - ideology - sex, % data = glbwarm) % %Multiple Regression from raw data % % DV = govact % slope se t p VIF %age -0.02 0.03 -0.54 0.59000 1.00 %negemot 0.49 0.03 16.19 0.00000 1.01 %age*negemot 0.11 0.03 3.75 0.00019 1.01 % % Multiple Regression % R R2 Ruw R2uw Shrunken R2 SE of R2 overall F df1 df2 p %govact 0.52 0.27 0.33 0.11 0.27 0.03 100.5 3 811 0 %> %\end{Routput} Note how the beta weights for the age, negemot and interaction terms are identical. \subsection{Partial everything from everything} Sometimes we want to examine just the independent effects of all our variables. That is to say, we want to partial all the variables from all the other variables. I do this with the \pfun{partial.r} function. To show the results, I compare the partialed rs to the original rs. I show the lower off diagonal matrix using \pfun{lowerMat}. Then to compare the partial matrix to the original matrix, I form the square matrix where the lower off diagonal is the original matrix and the upper off diagonal is the partial matrix. <>== upper <-partial.r(globalWarm) lowerMat(upper) #show it lower <- lowerCor(globalWarm) lowup <- lowerUpper(lower,upper) @ %\begin{Routput} %upper <-partial.r(glbwarm) %> lowerMat(upper) #show it % govct posmt negmt idlgy age sex prtyd %govact 1.00 %posemot -0.03 1.00 %negemot 0.50 0.13 1.00 %ideology -0.19 0.00 -0.07 1.00 %age -0.02 0.04 0.03 0.14 1.00 %sex 0.00 0.08 -0.07 0.04 0.14 1.00 %partyid -0.08 -0.01 -0.09 0.53 0.03 0.02 1.00 %> lower <- lowerCor(glbwarm) % govct posmt negmt idlgy age sex prtyd %govact 1.00 %posemot 0.04 1.00 %negemot 0.58 0.13 1.00 %ideology -0.42 -0.03 -0.35 1.00 %age -0.10 0.04 -0.06 0.21 1.00 %sex -0.10 0.07 -0.12 0.13 0.17 1.00 %partyid -0.36 -0.04 -0.32 0.62 0.15 0.11 1.00 % %\end{Routput} <>== png('partials.png') psych::corPlot(lowup,numbers = TRUE) dev.off() @ \begin{figure}[htbp] \begin{center} \includegraphics[width=9cm]{partials.png} \caption{Correlations (below diagonal) and partial correlations (above the diagonal) } \label{default} \end{center} \end{figure} \section{Related packages} \pfun{mediate} and \pfun{setCor} are just two functions in the \Rpkg{psych} package. There are several additional packages available in \R{} to do mediation. The \Rpkg{mediation} package \citep{mediation} seems the most powerful, in that is tailor made for mediation. \Rpkg{MBESS} \citep{MBESS} has a mediation function. Steven Short has a nice tutorial on mediation analysis available for download \href{http://docs.wixstatic.com/ugd/bb3887\_73181065d7c744c4a0925844302cf813.pdf}{that discusses how to use R for mediation.} And, of course, the \Rpkg{lavaan} package \citep{lavaan} is the recommended package to do SEM and path models. \newpage \section{Development version and a users guide} The \Rpkg{psych} package is available from the CRAN repository. However, the most recent development version of the \Rpkg{psych} package is available as a source file at the repository maintained at \href{ href="http://personality-project.org/r"}{\url{http://personality-project.org/r}}. That version will have removed the most recently discovered bugs (but perhaps introduced other, yet to be discovered ones). To install this development version, either for PCs or Macs, \begin{Rinput} install.packages("psych", repos = "http://personality-project.org/r", type = "source") \end{Rinput} After doing this, it is important to restart \R{} to get the new package. Although the individual help pages for the \Rpkg{psych} package are available as part of \R{} and may be accessed directly (e.g. ?psych) , the full manual for the \pfun{psych} package is also available as a pdf at \url{http://personality-project.org/r/psych_manual.pdf} %psych\_manual.pdf. News and a history of changes are available in the NEWS and CHANGES files in the source files. To view the most recent news, \begin{Schunk} \begin{Sinput} > news(Version >-= "2.0.12",package="psych") \end{Sinput} \end{Schunk} \section{Psychometric Theory} The \Rpkg{psych} package has been developed to help psychologists do basic research. Many of the functions were developed to supplement a book (\url{http://personality-project.org/r/book} An introduction to Psychometric Theory with Applications in \R{} \citep{revelle:intro} More information about the use of some of the functions may be found in the book . For more extensive discussion of the use of \Rpkg{psych} in particular and \R{} in general, consult \url{http://personality-project.org/r/r.guide.html} A short guide to R. \section{SessionInfo} This document was prepared using the following settings. \begin{tiny} <>== sessionInfo() @ \end{tiny} \ \newpage %\bibliography{../../../all} \begin{thebibliography}{} \bibitem[\protect\astroncite{Cohen}{1982}]{cohen:82} Cohen, J. (1982). \newblock Set correlation as a general mulitivariate data-analytic method. \newblock {\em Multivariate Behavioral Research}, 17(3):301--341. \bibitem[\protect\astroncite{Cohen et~al.}{2003}]{cohen:03} Cohen, J., Cohen, P., West, S.~G., and Aiken, L.~S. (2003). \newblock {\em Applied multiple regression/correlation analysis for the behavioral sciences}. \newblock L. Erlbaum Associates, Mahwah, N.J., 3rd ed edition. \bibitem[\protect\astroncite{Dawes}{1979}]{dawes:79} Dawes, R.~M. (1979). \newblock The robust beauty of improper linear models in decision making. \newblock {\em American Psychologist}, 34(7):571--582. \bibitem[\protect\astroncite{Garcia et~al.}{2010}]{garcia:10} Garcia, D.~M., Schmitt, M.~T., Branscombe, N.~R., and Ellemers, N. (2010). \newblock Women's reactions to ingroup members who protest discriminatory treatment: The importance of beliefs about inequality and response appropriateness. \newblock {\em European Journal of Social Psychology}, 40(5):733--745. \bibitem[\protect\astroncite{Hayes}{2013}]{hayes:13} Hayes, A.~F. (2013). \newblock {\em Introduction to mediation, moderation, and conditional process analysis: A regression-based approach}. \newblock Guilford Press, New York. \bibitem[\protect\astroncite{Hotelling}{1936}]{hotelling:36} Hotelling, H. (1936). \newblock Relations between two sets of variates. \newblock {\em Biometrika}, 28(3/4):321--377. \bibitem[\protect\astroncite{Kelley}{2017}]{MBESS} Kelley, K. (2017). \newblock {\em {MBESS: The MBESS R} Package}. \newblock R package version 4.4.1. \bibitem[\protect\astroncite{Pollack et~al.}{2012}]{pollack:12} Pollack, J.~M., Vanepps, E.~M., and Hayes, A.~F. (2012). \newblock The moderating role of social ties on entrepreneurs' depressed affect and withdrawal intentions in response to economic stress. \newblock {\em Journal of Organizational Behavior}, 33(6):789--810. \bibitem[\protect\astroncite{Preacher}{2015}]{preacher:15} Preacher, K.~J. (2015). \newblock Advances in mediation analysis: A survey and synthesis of new developments. \newblock {\em Annual Review of Psychology}, 66:825--852. \bibitem[\protect\astroncite{Preacher and Hayes}{2004}]{preacher:04} Preacher, K.~J. and Hayes, A.~F. (2004). \newblock {SPSS and SAS} procedures for estimating indirect effects in simple mediation models. \newblock {\em Behavior Research Methods, Instruments, \& Computers}, 36(4):717--731. \bibitem[\protect\astroncite{Preacher et~al.}{2007}]{preacher:07} Preacher, K.~J., Rucker, D.~D., and Hayes, A.~F. (2007). \newblock Addressing moderated mediation hypotheses: Theory, methods, and prescriptions. \newblock {\em Multivariate behavioral research}, 42(1):185--227. \bibitem[\protect\astroncite{{R Core Team}}{2020}]{R} {R Core Team} (2020). \newblock {\em R: A Language and Environment for Statistical Computing}. \newblock R Foundation for Statistical Computing, Vienna, Austria. \bibitem[\protect\astroncite{Revelle}{2020}]{psych} Revelle, W. (2020). \newblock {\em \href{https://cran.r-project.org/web/packages/psych/index.html}{psych}: Procedures for Personality and Psychological Research}. \newblock Northwestern University, Evanston, https://CRAN.r-project.org/package=psych. \newblock R package version 2.0.9. \bibitem[\protect\astroncite{Revelle}{prep}]{revelle:intro} Revelle, W. ({in prep}). \newblock {\em An introduction to psychometric theory with applications in {R}}. \newblock Springer. \bibitem[\protect\astroncite{Rosseel}{2012}]{lavaan} Rosseel, Y. (2012). \newblock {lavaan}: An {R} package for structural equation modeling. \newblock {\em Journal of Statistical Software}, 48(2):1--36. \bibitem[\protect\astroncite{Tal-Or et~al.}{2010}]{talor:10} Tal-Or, N., Cohen, J., Tsfati, Y., and Gunther, A.~C. (2010). \newblock Testing causal direction in the influence of presumed media influence. \newblock {\em Communication Research}, 37(6):801--824. \bibitem[\protect\astroncite{Tingley et~al.}{2014}]{mediation} Tingley, D., Yamamoto, T., Hirose, K., Keele, L., and Imai, K. (2014). \newblock {mediation}: {R} package for causal mediation analysis. \newblock {\em Journal of Statistical Software}, 59(5):1--38. \end{thebibliography} \begin{thebibliography}{} \bibitem[\protect\astroncite{Cohen}{1982}]{cohen:82} Cohen, J. (1982). \newblock Set correlation as a general mulitivariate data-analytic method. \newblock {\em Multivariate Behavioral Research}, 17(3):301--341. \bibitem[\protect\astroncite{Cohen et~al.}{2003}]{cohen:03} Cohen, J., Cohen, P., West, S.~G., and Aiken, L.~S. (2003). \newblock {\em Applied multiple regression/correlation analysis for the behavioral sciences}. \newblock L. Erlbaum Associates, Mahwah, N.J., 3rd ed edition. \bibitem[\protect\astroncite{Dawes}{1979}]{dawes:79} Dawes, R.~M. (1979). \newblock The robust beauty of improper linear models in decision making. \newblock {\em American Psychologist}, 34(7):571--582. \bibitem[\protect\astroncite{Garcia et~al.}{2010}]{garcia:10} Garcia, D.~M., Schmitt, M.~T., Branscombe, N.~R., and Ellemers, N. (2010). \newblock Women's reactions to ingroup members who protest discriminatory treatment: The importance of beliefs about inequality and response appropriateness. \newblock {\em European Journal of Social Psychology}, 40(5):733--745. \bibitem[\protect\astroncite{Hayes}{2013}]{hayes:13} Hayes, A.~F. (2013). \newblock {\em Introduction to mediation, moderation, and conditional process analysis: A regression-based approach}. \newblock Guilford Press, New York. \bibitem[\protect\astroncite{Hotelling}{1936}]{hotelling:36} Hotelling, H. (1936). \newblock Relations between two sets of variates. \newblock {\em Biometrika}, 28(3/4):321--377. \bibitem[\protect\astroncite{Kelley}{2017}]{MBESS} Kelley, K. (2017). \newblock {\em {MBESS: The MBESS R} Package}. \newblock R package version 4.4.1. \bibitem[\protect\astroncite{Pollack et~al.}{2012}]{pollack:12} Pollack, J.~M., Vanepps, E.~M., and Hayes, A.~F. (2012). \newblock The moderating role of social ties on entrepreneurs' depressed affect and withdrawal intentions in response to economic stress. \newblock {\em Journal of Organizational Behavior}, 33(6):789--810. \bibitem[\protect\astroncite{Preacher}{2015}]{preacher:15} Preacher, K.~J. (2015). \newblock Advances in mediation analysis: A survey and synthesis of new developments. \newblock {\em Annual Review of Psychology}, 66:825--852. \bibitem[\protect\astroncite{Preacher and Hayes}{2004}]{preacher:04} Preacher, K.~J. and Hayes, A.~F. (2004). \newblock {SPSS and SAS} procedures for estimating indirect effects in simple mediation models. \newblock {\em Behavior Research Methods, Instruments, \& Computers}, 36(4):717--731. \bibitem[\protect\astroncite{Preacher et~al.}{2007}]{preacher:07} Preacher, K.~J., Rucker, D.~D., and Hayes, A.~F. (2007). \newblock Addressing moderated mediation hypotheses: Theory, methods, and prescriptions. \newblock {\em Multivariate behavioral research}, 42(1):185--227. \bibitem[\protect\astroncite{{R Core Team}}{2020}]{R} {R Core Team} (2020). \newblock {\em R: A Language and Environment for Statistical Computing}. \newblock R Foundation for Statistical Computing, Vienna, Austria. \bibitem[\protect\astroncite{{Moon}}{2020}]{processR} {Keon-Woong Moon} (2020). \newblock {\em processR: Implementation of the 'PROCESS' Macro}. \newblock https://CRAN.R-project.org/package=processR \bibitem[\protect\astroncite{Revelle}{2020}]{psych} Revelle, W. (2020). \newblock {\em \href{https://cran.r-project.org/web/packages/psych/index.html}{psych}: Procedures for Personality and Psychological Research}. \newblock Northwestern University, Evanston, https://CRAN.r-project.org/package=psych. \newblock R package version 2.0.12. \bibitem[\protect\astroncite{Revelle}{prep}]{revelle:intro} Revelle, W. ({in prep}). \newblock {\em An introduction to psychometric theory with applications in {R}}. \newblock Springer. \bibitem[\protect\astroncite{Rosseel}{2012}]{lavaan} Rosseel, Y. (2012). \newblock {lavaan}: An {R} package for structural equation modeling. \newblock {\em Journal of Statistical Software}, 48(2):1--36. \bibitem[\protect\astroncite{Tal-Or et~al.}{2010}]{talor:10} Tal-Or, N., Cohen, J., Tsfati, Y., and Gunther, A.~C. (2010). \newblock Testing causal direction in the influence of presumed media influence. \newblock {\em Communication Research}, 37(6):801--824. \bibitem[\protect\astroncite{Tingley et~al.}{2014}]{mediation} Tingley, D., Yamamoto, T., Hirose, K., Keele, L., and Imai, K. (2014). \newblock {mediation}: {R} package for causal mediation analysis. \newblock {\em Journal of Statistical Software}, 59(5):1--38. \end{thebibliography} %\printindex \end{document} psychTools/inst/doc/factor.Rnw0000644000176200001440000045246513720521015016125 0ustar liggesusers% \VignetteIndexEntry{Using the psych package for factor analysis} % \VignettePackage{psych} % \VignetteKeywords{multivariate} % \VignetteKeyword{models} % \VignetteKeyword{Hplot} %\VignetteDepends{psych} %\documentclass[doc]{apa} \documentclass[11pt]{article} %\documentclass[11pt]{amsart} \usepackage{geometry} % See geometry.pdf to learn the layout options. There are lots. \geometry{letterpaper} % ... or a4paper or a5paper or ... %\geometry{landscape} % Activate for for rotated page geometry \usepackage[parfill]{parskip} % Activate to begin paragraphs with an empty line rather than an indent \usepackage{graphicx} \usepackage{amssymb} \usepackage{epstopdf} \usepackage{mathptmx} \usepackage{helvet} \usepackage{courier} \usepackage{epstopdf} \usepackage{makeidx} % allows index generation \usepackage[authoryear,round]{natbib} \usepackage{gensymb} \usepackage{longtable} %\usepackage{geometry} \usepackage{amssymb} \usepackage{amsmath} %\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \usepackage{Sweave} %\usepackage{/Volumes/'Macintosh HD'/Library/Frameworks/R.framework/Versions/2.13/Resources/share/texmf/tex/latex/Sweave} %\usepackage[ae]{Rd} %\usepackage[usenames]{color} %\usepackage{setspace} \bibstyle{apacite} \bibliographystyle{apa} %this one plus author year seems to work? %\usepackage{hyperref} \usepackage[colorlinks=true,citecolor=blue]{hyperref} %this makes reference links hyperlinks in pdf! \DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \usepackage{multicol} % used for the two-column index \usepackage[bottom]{footmisc}% places footnotes at page bottom \let\proglang=\textsf \newcommand{\R}{\proglang{R}} %\newcommand{\pkg}[1]{{\normalfont\fontseries{b}\selectfont #1}} \newcommand{\Rfunction}[1]{{\texttt{#1}}} \newcommand{\fun}[1]{{\texttt{#1}\index{#1}\index{R function!#1}}} \newcommand{\pfun}[1]{{\texttt{#1}\index{#1}\index{R function!#1}\index{R function!psych package!#1}}}\newcommand{\Rc}[1]{{\texttt{#1}}} %R command same as Robject \newcommand{\Robject}[1]{{\texttt{#1}}} \newcommand{\Rpkg}[1]{{\textit{#1}\index{#1}\index{R package!#1}}} %different from pkg - which is better? \newcommand{\iemph}[1]{{\emph{#1}\index{#1}}} \newcommand{\wrc}[1]{\marginpar{\textcolor{blue}{#1}}} %bill's comments \newcommand{\wra}[1]{\textcolor{blue}{#1}} %bill's comments \newcommand{\ve}[1]{{\textbf{#1}}} %trying to get a vector command \usepackage{fancyvrb} %this allows fancy boxes \fvset{fontfamily=courier} \DefineVerbatimEnvironment{Routput}{Verbatim} %{fontsize=\scriptsize, xleftmargin=0.6cm} {fontseries=b,fontsize=\scriptsize, xleftmargin=0.1cm} \DefineVerbatimEnvironment{Binput}{Verbatim} {fontseries=b, fontsize=\scriptsize,frame=single, label=\fbox{lavaan model syntax}, framesep=2mm} %\DefineShortVerb{\!} %%% generates error! \DefineVerbatimEnvironment{Rinput}{Verbatim} %{fontsize=\scriptsize, frame=single, label=\fbox{R code}, framesep=1mm} {fontseries=b, fontsize=\scriptsize, frame=single, label=\fbox{R code},xleftmargin=0pt, framesep=1mm} \DefineVerbatimEnvironment{Link}{Verbatim} {fontseries=b, fontsize=\small, formatcom=\color{darkgreen}, xleftmargin=1.0cm} \DefineVerbatimEnvironment{Toutput}{Verbatim} {fontseries=b,fontsize=\tiny, xleftmargin=0.1cm} \DefineVerbatimEnvironment{rinput}{Verbatim} {fontseries=b, fontsize=\tiny, frame=single, label=\fbox{R code}, framesep=1mm} \newcommand{\citeti}[1]{\begin{tiny}\citep{#1}\end{tiny}} \newcommand{\light}[1]{\textcolor{gray}{#1}} \newcommand{\vect}[1]{\boldsymbol{#1}} \let\vec\vect \makeindex % used for the subject index \title{How To: Use the psych package for Factor Analysis and data reduction} \author{William Revelle\\Department of Psychology\\Northwestern University} %\affiliation{Northwestern University} %\acknowledgements{Written to accompany the psych package. Comments should be directed to William Revelle \\ \url{revelle@northwestern.edu}} %\date{} % Activate to display a given date or no date \begin{document} \SweaveOpts{concordance=TRUE} \maketitle \tableofcontents \newpage \section{Overview of this and related documents} To do basic and advanced personality and psychological research using \R{} is not as complicated as some think. This is one of a set of ``How To'' to do various things using \R{} \citep{R}, particularly using the \Rpkg{psych} \citep{psych} package. The current list of How To's includes: \begin{enumerate} \item \href{http://personality-project.org/r/psych/HowTo/getting_started.pdf}{Installing} \R{} and some useful packages \item Using \R{} and the \Rpkg{psych} package to find \href{http://personality-project.org/r/psych/HowTo/omega.pdf}{$omega_h$} and $\omega_t$. \item Using \R{} and the \Rpkg{psych} for \href{http://personality-project.org/r/psych/HowTo/factor.pdf}{factor analysis} and principal components analysis. (This document). \item Using the \pfun{score.items} function to find \href{http://personality-project.org/r/psych/HowTo/scoring.pdf}{scale scores and scale statistics}. \item An \href{http://personality-project.org/r/psych/overview.pdf}{overview} (vignette) of the \Rpkg{psych} package Several functions are meant to do multiple regressions, either from the raw data or from a variance/covariance matrix, or a correlation matrix. This is discussed in more detail in \item How to do mediation and moderation analysis using \pfun{mediate} and \pfun{setCor} is discuseded in the \href{https://personality-project.org/r/psych/HowTo/mediation.pdf}{mediation, moderation and regression analysis} tutorial. \end{enumerate} \subsection{Jump starting the \Rpkg{psych} package--a guide for the impatient} You have installed \Rpkg{psych} (section \ref{sect:starting}) and you want to use it without reading much more. What should you do? \begin{enumerate} \item Activate the \Rpkg{psych} package: \begin{Rinput} library(psych) library(psychTools) \end{Rinput} \item Input your data (section \ref{sect:read}). Go to your friendly text editor or data manipulation program (e.g., Excel) and copy the data to the clipboard. Include a first line that has the variable labels. Paste it into \Rpkg{psych} using the \pfun{read.clipboard.tab} command: \begin{Rinput} myData <- read.clipboard.tab() \end{Rnput} \item Make sure that what you just read is right. Describe it (section~\ref{sect:describe}) and perhaps look at the first and last few lines: \begin{Rinput} describe(myData) headTail(myData) \end{Rinput} \item Look at the patterns in the data. If you have fewer than about 10 variables, look at the SPLOM (Scatter Plot Matrix) of the data using \pfun{pairs.panels} (section~\ref{sect:pairs}). \begin{Rinput} pairs.panels(myData) \end{Rinput} %\item Note that you have some weird subjects, probably due to data entry errors. Either edit the data by hand (use the \fun{edit} command) or just \pfun{scrub} the data (section \ref{sect:scrub}). %\begin{scriptsize} %\begin{Schunk} %\begin{Sinput} %cleaned <- scrub(myData, max=9) #e.g., change anything great than 9 to NA %\end{Sinput} %\end{Schunk} %\end{scriptsize} %\item Graph the data with error bars for each variable (section \ref{sect:errorbars}). %\begin{scriptsize} %\begin{Schunk} %\begin{Sinput} %error.bars(myData) %\end{Sinput} %\end{Schunk} %\end{scriptsize} \item Find the correlations of all of your data. \begin{itemize} \item Descriptively (just the values) (section \ref{sect:lowerCor}) \begin{Rinput} lowerCor(myData) \end{Rinput} \item Graphically (section \ref{sect:corplot}) \begin{Rinput} corPlot(r) \end{Rinput} \end{itemize} % %\item Inferentially (the values, the ns, and the p values) (section \ref{sect:corr.test}) %\begin{scriptsize} %\begin{Schunk} %\begin{Sinput} %corr.test(myData) % %\end{Sinput} %\end{Schunk} %\end{scriptsize} %\end{itemize} \item Test for the number of factors in your data using parallel analysis (\pfun{fa.parallel}, section \ref{sect:fa.parallel}) or Very Simple Structure (\pfun{vss}, \ref{sect:vss}) . \begin{Rinput} fa.parallel(myData) vss(myData) \end{Rinput} \item Factor analyze (see section \ref{sect:fa}) the data with a specified number of factors (the default is 1), the default method is minimum residual, the default rotation for more than one factor is oblimin. There are many more possibilities (see sections \ref{sect:minres}-\ref{sect:wls}). Compare the solution to a hierarchical cluster analysis using the ICLUST algorithm \citep{revelle:iclust} (see section \ref{sect:iclust}). Also consider a hierarchical factor solution to find coefficient $\omega$ (see \ref{sect:omega}). Yet another option is to do a series of factor analyses in what is known as the ``bass akward" procedure \citep{goldberg:06} which considers the correlation between factors at multiple levels of analysis (see \ref{sect:bassAckward}). \begin{Rinput} fa(myData) iclust(myData) omega(myData) bassAckward(myData) \end{Rinput} \item Some people like to find coefficient $\alpha$ as an estimate of reliability. This may be done for a single scale using the \pfun{alpha} function (see \ref{sect:alpha}). Perhaps more useful is the ability to create several scales as unweighted averages of specified items using the \pfun{scoreIems} function (see \ref{sect:score}) and to find various estimates of internal consistency for these scales, find their intercorrelations, and find scores for all the subjects. \begin{Rinput} alpha(myData) #score all of the items as part of one scale. myKeys <- make.keys(nvar=20,list(first = c(1,-3,5,-7,8:10),second=c(2,4,-6,11:15,-16))) my.scores <- scoreItems(myKeys,myData) #form several scales my.scores #show the highlights of the results \end{Rinput} \end{enumerate} At this point you have had a chance to see the highlights of the \Rpkg{psych} package and to do some basic (and advanced) data analysis. You might find reading the entire \href{http://personality-project.org/r/psych/overview.pdf}{overview} vignette helpful to get a broader understanding of what can be done in \R{} using the \Rpkg{psych}. Remember that the help command (?) is available for every function. Try running the examples for each help page. \newpage \section{Overview of this and related documents} The \Rpkg{psych} package \citep{psych} has been developed at Northwestern University since 2005 to include functions most useful for personality, psychometric, and psychological research. The package is also meant to supplement a text on psychometric theory \citep{revelle:intro}, a draft of which is available at \url{http://personality-project.org/r/book/}. Some of the functions (e.g., \pfun{read.clipboard}, \pfun{describe}, \pfun{pairs.panels}, \pfun{scatter.hist}, \pfun{error.bars}, \pfun{multi.hist}, \pfun{bi.bars}) are useful for basic data entry and descriptive analyses. Psychometric applications emphasize techniques for dimension reduction including factor analysis, cluster analysis, and principal components analysis. The \pfun{fa} function includes five methods of \iemph{factor analysis} (\iemph{minimum residual}, \iemph{principal axis}, \iemph{weighted least squares}, \iemph{generalized least squares} and \iemph{maximum likelihood} factor analysis). Determining the number of factors or components to extract may be done by using the Very Simple Structure \citep{revelle:vss} (\pfun{vss}), Minimum Average Partial correlation \citep{velicer:76} (\pfun{MAP}) or parallel analysis (\pfun{fa.parallel}) criteria. Item Response Theory (IRT) models for dichotomous or polytomous items may be found by factoring \pfun{tetrachoric} or \pfun{polychoric} correlation matrices and expressing the resulting parameters in terms of location and discrimination using \pfun{irt.fa}. Bifactor and hierarchical factor structures may be estimated by using Schmid Leiman transformations \citep{schmid:57} (\pfun{schmid}) to transform a hierarchical factor structure into a \iemph{bifactor} solution \citep{holzinger:37}. Scale construction can be done using the Item Cluster Analysis \citep{revelle:iclust} (\pfun{iclust}) function to determine the structure and to calculate reliability coefficients $\alpha$ \citep{cronbach:51}(\pfun{alpha}, \pfun{scoreItems}, \pfun{score.multiple.choice}), $\beta$ \citep{revelle:iclust,rz:09} (\pfun{iclust}) and McDonald's $\omega_h$ and $\omega_t$ \citep{mcdonald:tt} (\pfun{omega}). Guttman's six estimates of internal consistency reliability (\cite{guttman:45}, as well as additional estimates \citep{rz:09} are in the \pfun{guttman} function. The six measures of Intraclass correlation coefficients (\pfun{ICC}) discussed by \cite{shrout:79} are also available. Graphical displays include Scatter Plot Matrix (SPLOM) plots using \pfun{pairs.panels}, correlation ``heat maps'' (\pfun{cor.plot}) factor, cluster, and structural diagrams using \pfun{fa.diagram}, \pfun{iclust.diagram}, \pfun{structure.diagram}, as well as item response characteristics and item and test information characteristic curves \pfun{plot.irt} and \pfun{plot.poly}. %This vignette is meant to give an overview of the \Rpkg{psych} package. That is, it is meant to give a summary of the main functions in the \Rpkg{psych} package with examples of how they are used for data description, dimension reduction, and scale construction. The extended user manual at \url{psych_manual.pdf} includes examples of graphic output and more extensive demonstrations than are found in the help menus. (Also available at \url{http://personality-project.org/r/psych_manual.pdf}). The vignette, psych for sem, at \url{psych_for_sem.pdf}, discusses how to use psych as a front end to the \Rpkg{sem} package of John Fox \citep{sem}. (The vignette is also available at \href{"http://personality-project.org/r/book/psych_for_sem.pdf"}{\url{http://personality-project.org/r/book/psych_for_sem.pdf}}). % %For a step by step tutorial in the use of the psych package and the base functions in R for basic personality research, see the guide for using \R{} for personality research at \url{http://personalitytheory.org/r/r.short.html}. For an \iemph{introduction to psychometric theory with applications in \R{}}, see the draft chapters at \url{http://personality-project.org/r/book}). % % % \section{Getting started} \label{sect:starting} Some of the functions described in this overview require other packages. Particularly useful for rotating the results of factor analyses (from e.g., \pfun{fa} or \pfun {principal}) or hierarchical factor models using \pfun{omega} or \pfun{schmid}, is the \Rpkg{GPArotation} package. These and other useful packages may be installed by first installing and then using the task views (\Rpkg{ctv}) package to install the ``Psychometrics" task view, but doing it this way is not necessary. % %\begin{Schunk} %\begin{Sinput} %install.packages("ctv") %library(ctv) %task.views("Psychometrics") %\end{Sinput} %\end{Schunk} % %The ``Psychometrics'' task view will install a large number of useful packages. To install the bare minimum for the examples in this vignette, it is necessary to install just 3 packages: % %\begin{Schunk} %\begin{Sinput} %install.packages(list(c("GPArotation","mvtnorm","MASS") %\end{Sinput} %\end{Schunk} % % %Because of the difficulty of installing the package \Rpkg{Rgraphviz}, alternative graphics have been developed and are available as \iemph{diagram} functions. If \Rpkg{Rgraphviz} is available, some functions will take advantage of it. An alternative is to use ``dot'' output of commands for any external graphics package that uses the dot language. % \section{Basic data analysis} A number of \Rpkg{psych} functions facilitate the entry of data and finding basic descriptive statistics. Remember, to run any of the \Rpkg{psych} functions, it is necessary to make the package active by using the \fun{library} command: \begin{Rinput} library(psych) library(psychTools) \end{Rinput} The other packages, once installed, will be called automatically by \Rpkg{psych}. It is possible to automatically load \Rpkg{psych} and other functions by creating and then saving a ``.First" function: e.g., \begin{Rinput} .First <- function(x) {library(psych)} \end{Rinput} \subsection{Data input from a local or remote file} \label{sect:read} Find and read standard files using \pfun{read.file}. This will open a search window for your operating system which you can use to find the file. If the file has a suffix of .text, .txt, .TXT, .csv, ,dat, .data, .sav, .xpt, .XPT, .r, .R, .rds, .Rds, .rda, .Rda, .rdata, Rdata, or .RData, then the file will be opened and the data will be read in (or loaded in the case of Rda files) \begin{scriptsize} \begin{Schunk} \begin{Sinput} myData <- read.file() # find the appropriate file using your normal operating system \end{Sinput} \end{Schunk} \end{scriptsize} Alternatively, if you have a file name for a remote file, you can read it using \pfun{read.file} as well. \begin{scriptsize} \begin{Schunk} \begin{Sinput} myData <- read.file(fn) # where file name is the the remote address of the file \end{Sinput} \end{Schunk} \end{scriptsize} \subsection{Data input from the clipboard} There are of course many ways to enter data into \R. Reading from a local file using \fun{read.file} is perhaps the most preferred. However, many users will enter their data in a text editor or spreadsheet program and then want to copy and paste into \R{}. This may be done by using \fun{read.table} and specifying the input file as ``clipboard" (PCs) or ``pipe(pbpaste)" (Macs). Alternatively, the \pfun{read.clipboard} set of functions are perhaps more user friendly: \begin{description} \item [\pfun{read.clipboard}] is the base function for reading data from the clipboard. \item [\pfun{read.clipboard.csv}] for reading text that is comma delimited. \item [\pfun{read.clipboard.tab}] for reading text that is tab delimited (e.g., copied directly from an Excel file). \item [\pfun{read.clipboard.lower}] for reading input of a lower triangular matrix with or without a diagonal. The resulting object is a square matrix. \item [\pfun{read.clipboard.upper}] for reading input of an upper triangular matrix. \item[\pfun{read.clipboard.fwf}] for reading in fixed width fields (some very old data sets) \end{description} For example, given a data set copied to the clipboard from a spreadsheet, just enter the command \begin{Rinput} my.data <- read.clipboard() \end{Rinput} This will work if every data field has a value and even missing data are given some values (e.g., NA or -999). If the data were entered in a spreadsheet and the missing values were just empty cells, then the data should be read in as a tab delimited or by using the \pfun{read.clipboard.tab} function. \begin{Rinput} my.data <- read.clipboard(sep="\t") #define the tab option, or my.tab.data <- read.clipboard.tab() #just use the alternative function \end{Rinput} For the case of data in fixed width fields (some old data sets tend to have this format), copy to the clipboard and then specify the width of each field (in the example below, the first variable is 5 columns, the second is 2 columns, the next 5 are 1 column the last 4 are 3 columns). \begin{Rinput} my.data <- read.clipboard.fwf(widths=c(5,2,rep(1,5),rep(3,4)) \end{Rinput} \subsection{Basic descriptive statistics} \label{sect:describe} Once the data are read in, then \pfun{describe} will provide basic descriptive statistics arranged in a data frame format. Consider the data set \pfun{sat.act} which includes data from 700 web based participants on 3 demographic variables and 3 ability measures. \begin{description} \item[\pfun{describe}] reports means, standard deviations, medians, min, max, range, skew, kurtosis and standard errors for integer or real data. Non-numeric data, although the statistics are meaningless, will be treated as if numeric (based upon the categorical coding of the data), and will be flagged with an *. \end{description} It is very important to describe your data before you continue on doing more complicated multivariate statistics. The problem of outliers and bad data can not be overemphasized. \begin{scriptsize} <>= options(width=160) library(psych) library(psychTools) data(sat.act) describe(sat.act) #basic descriptive statistics @ \end{scriptsize} %These data may then be analyzed by groups defined in a logical statement or by some other variable. E.g., break down the descriptive data for males or females. These descriptive data can also be seen graphically using the \pfun{error.bars.by} function (Figure~\ref{fig:error.bars}). By setting skew=FALSE and ranges=FALSE, the output is limited to the most basic statistics. % %\begin{scriptsize} %<>= % #basic descriptive statistics by a grouping variable. %describeBy(sat.act,sat.act$gender,skew=FALSE,ranges=FALSE) %@ %\end{scriptsize} % % %The output from the \pfun{describeBy} function can be forced into a matrix form for easy analysis by other programs. In addition, describeBy can group by several grouping variables at the same time. % %\begin{scriptsize} %<>= %sa.mat <- describeBy(sat.act,list(sat.act$gender,sat.act$education), % skew=FALSE,ranges=FALSE,mat=TRUE) %headTail(sa.mat) %@ %\end{scriptsize} %\subsubsection{Basic data cleaning using \pfun{scrub}} %\label{sect:scrub} %If, after describing the data it is apparent that there were data entry errors that need to be globally replaced with NA, or only certain ranges of data will be analyzed, the data can be ``cleaned" using the \pfun{scrub} function. % %Consider a data set of 10 rows of 12 columns with values from 1 - 120. All values of columns 3 - 5 that are less than 30, 40, or 50 respectively, or greater than 70 in any of the three columns will be replaced with NA. In addition, any value exactly equal to 45 will be set to NA. (max and isvalue are set to one value here, but they could be a different value for every column). % %\begin{scriptsize} %<>= %x <- matrix(1:120,ncol=10,byrow=TRUE) %colnames(x) <- paste('V',1:10,sep='') %new.x <- scrub(x,3:5,min=c(30,40,50),max=70,isvalue=45,newvalue=NA) %new.x %@ %\end{scriptsize} %Note that the number of subjects for those columns has decreased, and the minimums have gone up but the maximums down. Data cleaning and examination for outliers should be a routine part of any data analysis. % %\subsubsection{Recoding categorical variables into dummy coded variables} %Sometimes categorical variables (e.g., college major, occupation, ethnicity) are to be analyzed using correlation or regression. To do this, one can form ``dummy codes'' which are merely binary variables for each category. This may be done using \pfun{dummy.code}. Subsequent analyses using these dummy coded variables may be using \pfun{biserial} or point biserial (regular Pearson r) to show effect sizes and may be plotted in e.g., \pfun{spider} plots. \subsection{Simple descriptive graphics} Graphic descriptions of data are very helpful both for understanding the data as well as communicating important results. Scatter Plot Matrices (SPLOMS) using the \pfun{pairs.panels} function are useful ways to look for strange effects involving outliers and non-linearities. \pfun{error.bars.by} will show group means with 95\% confidence boundaries. \subsubsection{Scatter Plot Matrices} Scatter Plot Matrices (SPLOMS) are very useful for describing the data. The \pfun{pairs.panels} function, adapted from the help menu for the \fun{pairs} function produces xy scatter plots of each pair of variables below the diagonal, shows the histogram of each variable on the diagonal, and shows the \iemph{lowess} locally fit regression line as well. An ellipse around the mean with the axis length reflecting one standard deviation of the x and y variables is also drawn. The x axis in each scatter plot represents the column variable, the y axis the row variable (Figure~\ref{fig:pairs.panels}). When plotting many subjects, it is both faster and cleaner to set the plot character (pch) to be '.'. (See Figure~\ref{fig:pairs.panels} for an example.) \begin{description} \label{sect:pairs} \item[\pfun{pairs.panels} ] will show the pairwise scatter plots of all the variables as well as histograms, locally smoothed regressions, and the Pearson correlation. When plotting many data points (as in the case of the sat.act data, it is possible to specify that the plot character is a period to get a somewhat cleaner graphic. \end{description} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= png( 'pairspanels.png' ) pairs.panels(sat.act,pch='.') dev.off() @ \end{scriptsize} \includegraphics{pairspanels} \caption{Using the \pfun{pairs.panels} function to graphically show relationships. The x axis in each scatter plot represents the column variable, the y axis the row variable. Note the extreme outlier for the ACT. The plot character was set to a period (pch='.') in order to make a cleaner graph. } \label{fig:pairs.panels} \end{center} \end{figure} %Another example of \pfun{pairs.panels} is to show differences between experimental groups. Consider the data in the \pfun{affect} data set. The scores reflect post test scores on positive and negative affect and energetic and tense arousal. The colors show the results for four movie conditions: depressing, frightening movie, neutral, and a comedy. % %\begin{figure}[htbp] %\begin{center} %\begin{scriptsize} %<>= %png('affect.png') %pairs.panels(affect[14:17],bg=c("red","black","white","blue")[affect$Film],pch=21, % main="Affect varies by movies ") %dev.off() %@ %\end{scriptsize} %\includegraphics{affect} %\caption{Using the \pfun{pairs.panels} function to graphically show relationships. The x axis in each scatter plot represents the column variable, the y axis the row variable. The coloring represent four different movie conditions. } %\label{fig:pairs.panels2} %\end{center} %\end{figure} % %\subsubsection{Means and error bars} %\label{sect:errorbars} %Additional descriptive graphics include the ability to draw \iemph{error bars} on sets of data, as well as to draw error bars in both the x and y directions for paired data. These are the functions % %\begin{description} %\item [\pfun{error.bars}] show the 95 \% confidence intervals for each variable in a data frame or matrix. These errors are based upon normal theory and the standard errors of the mean. Alternative options include +/- one standard deviation or 1 standard error. If the data are repeated measures, the error bars will be reflect the between variable correlations. %\item [\pfun{error.bars.by}] does the same, but grouping the data by some condition. %\item [\pfun{error.crosses}] draw the confidence intervals for an x set and a y set of the same size. %\end{description} % %The use of the \pfun{error.bars.by} function allows for graphic comparisons of different groups (see Figure~\ref{fig:error.bars}). Five personality measures are shown as a function of high versus low scores on a ``lie" scale. People with higher lie scores tend to report being more agreeable, conscientious and less neurotic than people with lower lie scores. The error bars are based upon normal theory and thus are symmetric rather than reflect any skewing in the data. % %\begin{figure}[htbp] %\begin{center} %\begin{scriptsize} %<>= %data(epi.bfi) %error.bars.by(epi.bfi[,6:10],epi.bfi$epilie<4) %@ %\end{scriptsize} %\caption{Using the \pfun{error.bars.by} function shows that self reported personality scales on the Big Five Inventory vary as a function of the Lie scale on the EPI. } %\label{fig:error.bars} %\end{center} %\end{figure} % %Although not recommended, it is possible to use the \pfun{error.bars} function to draw bar graphs with associated error bars. (This kind of`\iemph{dynamite plot} (Figure~\ref{fig:dynamite}) can be very misleading in that the scale is arbitrary. Go to a discussion of the problems in presenting data this way at \url{http://emdbolker.wikidot.com/blog:dynamite}. In the example shown, note that the graph starts at 0, although is out of the range. This is a function of using bars, which always are assumed to start at zero. Consider other ways of showing your data. % %\begin{figure}[htbp] %\begin{center} %\begin{scriptsize} %<>= %error.bars.by(sat.act[5:6],sat.act$gender,bars=TRUE, % labels=c("Male","Female"),ylab="SAT score",xlab="") %@ %\end{scriptsize} %\caption{A ``Dynamite plot" of SAT scores as a function of gender is one way of misleading the reader. By using a bar graph, the range of scores is ignored. Bar graphs start from 0. } %\label{fig:dynamite} %\end{center} %\end{figure} % % %\subsubsection{Two dimensional displays of means and errors} %Yet another way to display data for different conditions is to use the \pfun{errorCrosses} function. For instance, the effect of various movies on both ``Energetic Arousal'' and ``Tense Arousal'' can be seen in one graph and compared to the same movie manipulations on ``Positive Affect'' and ``Negative Affect''. Note how Energetic Arousal is increased by three of the movie manipulations, but that Positive Affect increases following the Happy movie only. % % %\begin{figure}[htbp] %\begin{center} %\begin{scriptsize} %<>= %op <- par(mfrow=c(1,2)) % data(affect) %colors <- c("black","red","white","blue") % films <- c("Sad","Horror","Neutral","Happy") %affect.stats <- errorCircles("EA2","TA2",data=affect,group="Film",labels=films,xlab="Energetic Arousal",ylab="Tense Arousal",ylim=c(10,22),xlim=c(8,20),pch=16,cex=2,col=colors, % main =' Movies effect on arousal') % errorCircles("PA2","NA2",data=affect.stats,labels=films,xlab="Positive Affect",ylab="Negative Affect",pch=16,cex=2,col=colors, % main ="Movies effect on affect") %op <- par(mfrow=c(1,1)) %@ %\end{scriptsize} %\caption{The use of the \pfun{errorCircles} function allows for two dimensional displays of means and error bars. The first call to \pfun{errorCircles} finds descriptive statistics for the \iemph{affect} data.frame based upon the grouping variable of Film. These data are returned and then used by the second call which examines the effect of the same grouping variable upon different measures. The size of the circles represent the relative sample sizes for each group. The data are from the PMC lab and reported in \cite{smillie:jpsp}.} %\label{fig:errorCircles} %\end{center} %\end{figure} % %\clearpage %\subsubsection{Back to back histograms} %The \pfun{bi.bars} function summarize the characteristics of two groups (e.g., males and females) on a second variable (e.g., age) by drawing back to back histograms (see Figure~\ref{fig:bibars}). %\begin{figure}[htbp] %\begin{center} %\begin{scriptsize} %<>= %data(bfi) %with(bfi,{bi.bars(age,gender,ylab="Age",main="Age by males and females")}) %@ %\end{scriptsize} %\caption{A bar plot of the age distribution for males and females shows the use of \pfun{bi.bars}. The data are males and females from 2800 cases collected using the \iemph{SAPA} procedure and are available as part of the \pfun{bfi} data set. } %\label{fig:bibars} %\end{center} %\end{figure} % %\clearpage \subsubsection{Correlational structure} \label{sect:lowerCor} There are many ways to display correlations. Tabular displays are probably the most common. The output from the \fun{cor} function in core R is a rectangular matrix. \pfun{lowerMat} will round this to (2) digits and then display as a lower off diagonal matrix. \pfun{lowerCor} calls \fun{cor} with \emph{use=`pairwise', method=`pearson'} as default values and returns (invisibly) the full correlation matrix and displays the lower off diagonal matrix. \begin{scriptsize} <>= lowerCor(sat.act) @ \end{scriptsize} When comparing results from two different groups, it is convenient to display them as one matrix, with the results from one group below the diagonal, and the other group above the diagonal. Use \pfun{lowerUpper} to do this: \begin{scriptsize} <>= female <- subset(sat.act,sat.act$gender==2) male <- subset(sat.act,sat.act$gender==1) lower <- lowerCor(male[-1]) upper <- lowerCor(female[-1]) both <- lowerUpper(lower,upper) round(both,2) @ \end{scriptsize} It is also possible to compare two matrices by taking their differences and displaying one (below the diagonal) and the difference of the second from the first above the diagonal: \begin{scriptsize} <>= diffs <- lowerUpper(lower,upper,diff=TRUE) round(diffs,2) @ \end{scriptsize} \subsubsection{Heatmap displays of correlational structure} \label{sect:corplot} Perhaps a better way to see the structure in a correlation matrix is to display a \emph{heat map} of the correlations. This is just a matrix color coded to represent the magnitude of the correlation. This is useful when considering the number of factors in a data set. Consider the \pfun{Thurstone} data set which has a clear 3 factor solution (Figure~\ref{fig:cor.plot}) or a simulated data set of 24 variables with a circumplex structure (Figure~\ref{fig:cor.plot.circ}). The color coding represents a ``heat map'' of the correlations, with darker shades of red representing stronger negative and darker shades of blue stronger positive correlations. As an option, the value of the correlation can be shown. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= png('corplot.png') cor.plot(Thurstone,numbers=TRUE,main="9 cognitive variables from Thurstone") dev.off() @ \end{scriptsize} \includegraphics{corplot.png} \caption{The structure of correlation matrix can be seen more clearly if the variables are grouped by factor and then the correlations are shown by color. By using the 'numbers' option, the values are displayed as well. } \label{fig:cor.plot} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= png('circplot.png') circ <- sim.circ(24) r.circ <- cor(circ) cor.plot(r.circ,main='24 variables in a circumplex') dev.off() @ \end{scriptsize} \includegraphics{circplot.png} \caption{Using the cor.plot function to show the correlations in a circumplex. Correlations are highest near the diagonal, diminish to zero further from the diagonal, and the increase again towards the corners of the matrix. Circumplex structures are common in the study of affect.} \label{fig:cor.plot.circ} \end{center} \end{figure} %Yet another way to show structure is to use ``spider'' plots. Particularly if variables are ordered in some meaningful way (e.g., in a circumplex), a spider plot will show this structure easily. This is just a plot of the magnitude of the correlation as a radial line, with length ranging from 0 (for a correlation of -1) to 1 (for a correlation of 1). (See Figure~\ref{fig:cor.plot.spider}). % %\begin{figure}[htbp] %\begin{center} %\begin{scriptsize} %<>= %png('spider.png') %op<- par(mfrow=c(2,2)) %spider(y=c(1,6,12,18),x=1:24,data=r.circ,fill=TRUE,main="Spider plot of 24 circumplex variables") %op <- par(mfrow=c(1,1)) %dev.off() %@ %\end{scriptsize} %\includegraphics{spider.png} %\caption{A spider plot can show circumplex structure very clearly. Circumplex structures are common in the study of affect.} %\label{fig:cor.plot.spider} %\end{center} %\end{figure} % %\subsection{Testing correlations} %\label{sect:corr.test} %Correlations are wonderful descriptive statistics of the data but some people like to test whether these correlations differ from zero, or differ from each other. The \fun{cor.test} function (in the \Rpkg{stats} package) will test the significance of a single correlation, and the \fun{rcorr} function in the \Rpkg{Hmisc} package will do this for many correlations. In the \Rpkg{psych} package, the \pfun{corr.test} function reports the correlation (Pearson, Spearman, or Kendall) between all variables in either one or two data frames or matrices, as well as the number of observations for each case, and the (two-tailed) probability for each correlation. Unfortunately, these probability values have not been corrected for multiple comparisons and so should be taken with a great deal of salt. Thus, in \pfun{corr.test} and \pfun{corr.p} the raw probabilities are reported below the diagonal and the probabilities adjusted for multiple comparisons using (by default) the Holm correction are reported above the diagonal (Table~\ref{tab:corr.test}). (See the \fun{p.adjust} function for a discussion of \cite{holm:79} and other corrections.) % %\begin{table}[htpb] %\caption{The \pfun{corr.test} function reports correlations, cell sizes, and raw and adjusted probability values. \pfun{corr.p} reports the probability values for a correlation matrix. By default, the adjustment used is that of \cite{holm:79}.} %\begin{scriptsize} %<>= %corr.test(sat.act) %@ %\end{scriptsize} %\label{tab:corr.test} %\end{table}% % % %Testing the difference between any two correlations can be done using the \pfun{r.test} function. The function actually does four different tests (based upon an article by \cite{steiger:80b}, depending upon the input: % %1) For a sample size n, find the t and p value for a single correlation as well as the confidence interval. %\begin{scriptsize} %<>= %r.test(50,.3) %@ %\end{scriptsize} % %2) For sample sizes of n and n2 (n2 = n if not specified) find the z of the difference between the z transformed correlations divided by the standard error of the difference of two z scores. %\begin{scriptsize} %<>= %r.test(30,.4,.6) %@ %\end{scriptsize} % % %3) For sample size n, and correlations ra= r12, rb= r23 and r13 specified, test for the difference of two dependent correlations (Steiger case A). %\begin{scriptsize} %<>= %r.test(103,.4,.5,.1) %@ %\end{scriptsize} % %4) For sample size n, test for the difference between two dependent correlations involving different variables. (Steiger case B). %\begin{scriptsize} %<>= %r.test(103,.5,.6,.7,.5,.5,.8) #steiger Case B %@ %\end{scriptsize} % % %To test whether a matrix of correlations differs from what would be expected if the population correlations were all zero, the function \pfun{cortest} follows \cite{steiger:80b} who pointed out that the sum of the squared elements of a correlation matrix, or the Fisher z score equivalents, is distributed as chi square under the null hypothesis that the values are zero (i.e., elements of the identity matrix). This is particularly useful for examining whether correlations in a single matrix differ from zero or for comparing two matrices. Although obvious, \pfun{cortest} can be used to test whether the \pfun{sat.act} data matrix produces non-zero correlations (it does). This is a much more appropriate test when testing whether a residual matrix differs from zero. % %\begin{scriptsize} %<>= %cortest(sat.act) %@ %\end{scriptsize} % \subsection{Polychoric, tetrachoric, polyserial, and biserial correlations} The Pearson correlation of dichotomous data is also known as the $\phi$ coefficient. If the data, e.g., ability items, are thought to represent an underlying continuous although latent variable, the $\phi$ will underestimate the value of the Pearson applied to these latent variables. One solution to this problem is to use the \pfun{tetrachoric} correlation which is based upon the assumption of a bivariate normal distribution that has been cut at certain points. The \pfun{draw.tetra} function demonstrates the process (Figure~\ref{fig:tetra}). A simple generalization of this to the case of the multiple cuts is the \pfun{polychoric} correlation. % %\begin{figure}[htbp] %\begin{center} %\begin{scriptsize} %<>= %draw.tetra() %@ %\end{scriptsize} %\caption{The tetrachoric correlation estimates what a Pearson correlation would be given a two by two table of observed values assumed to be sampled from a bivariate normal distribution. The $\phi$ correlation is just a Pearson r performed on the observed values.} %\label{fig:tetra} %\end{center} %\end{figure} Other estimated correlations based upon the assumption of bivariate normality with cut points include the \pfun{biserial} and \pfun{polyserial} correlation. If the data are a mix of continuous, polytomous and dichotomous variables, the \pfun{mixed.cor} function will calculate the appropriate mixture of Pearson, polychoric, tetrachoric, biserial, and polyserial correlations. The correlation matrix resulting from a number of tetrachoric or polychoric correlation matrix sometimes will not be positive semi-definite. This will also happen if the correlation matrix is formed by using pair-wise deletion of cases. The \pfun{cor.smooth} function will adjust the smallest eigen values of the correlation matrix to make them positive, rescale all of them to sum to the number of variables, and produce a ``smoothed'' correlation matrix. An example of this problem is a data set of \pfun{burt} which probably had a typo in the original correlation matrix. Smoothing the matrix corrects this problem. %\subsection{Multiple regression from data or correlation matrices} % %The typical application of the \fun{lm} function is to do a linear model of one Y variable as a function of multiple X variables. Because \fun{lm} is designed to analyze complex interactions, it requires raw data as input. It is, however, sometimes convenient to do \iemph{multiple regression} from a correlation or covariance matrix. The \pfun{setCor} function will do this, taking a set of y variables predicted from a set of x variables, perhaps with a set of z covariates removed from both x and y. Consider the \iemph{Thurstone} correlation matrix and find the multiple correlation of the last five variables as a function of the first 4. % %\begin{scriptsize} %<>= %setCor(y = 5:9,x=1:4,data=Thurstone) %@ %\end{scriptsize} % %By specifying the number of subjects in correlation matrix, appropriate estimates of standard errors, t-values, and probabilities are also found. The next example finds the regressions with variables 1 and 2 used as covariates. The $\hat{\beta}$ weights for variables 3 and 4 do not change, but the multiple correlation is much less. It also shows how to find the residual correlations between variables 5-9 with variables 1-4 removed. % %\begin{scriptsize} %<>= %sc <- setCor(y = 5:9,x=3:4,data=Thurstone,z=1:2) %round(sc$residual,2) %@ %\end{scriptsize} \section{Item and scale analysis} The main functions in the \Rpkg{psych} package are for analyzing the structure of items and of scales and for finding various estimates of scale reliability. These may be considered as problems of dimension reduction (e.g., factor analysis, cluster analysis, principal components analysis) and of forming and estimating the reliability of the resulting composite scales. \subsection{Dimension reduction through factor analysis and cluster analysis} \label{sect:fa} Parsimony of description has been a goal of science since at least the famous dictum commonly attributed to William of Ockham to not multiply entities beyond necessity\footnote{Although probably neither original with Ockham nor directly stated by him \citep{thornburn:1918}, Ockham's razor remains a fundamental principal of science.}. The goal for parsimony is seen in psychometrics as an attempt either to describe (components) or to explain (factors) the relationships between many observed variables in terms of a more limited set of components or latent factors. The typical data matrix represents multiple items or scales usually thought to reflect fewer underlying constructs\footnote{\cite{cattell:fa78} as well as \cite{maccallum:07} argue that the data are the result of many more factors than observed variables, but are willing to estimate the major underlying factors.}. At the most simple, a set of items can be be thought to represent a random sample from one underlying domain or perhaps a small set of domains. The question for the psychometrician is how many domains are represented and how well does each item represent the domains. Solutions to this problem are examples of \iemph{factor analysis} (\iemph{FA}), \iemph{principal components analysis} (\iemph{PCA}), and \iemph{cluster analysis} (\emph{CA}). All of these procedures aim to reduce the complexity of the observed data. In the case of FA, the goal is to identify fewer underlying constructs to explain the observed data. In the case of PCA, the goal can be mere data reduction, but the interpretation of components is frequently done in terms similar to those used when describing the latent variables estimated by FA. Cluster analytic techniques, although usually used to partition the subject space rather than the variable space, can also be used to group variables to reduce the complexity of the data by forming fewer and more homogeneous sets of tests or items. At the data level the data reduction problem may be solved as a \iemph{Singular Value Decomposition} of the original matrix, although the more typical solution is to find either the \iemph{principal components} or \iemph{factors} of the covariance or correlation matrices. Given the pattern of regression weights from the variables to the components or from the factors to the variables, it is then possible to find (for components) individual \index{component scores} \emph{component} or \iemph{cluster scores} or estimate (for factors) \iemph{factor scores}. Several of the functions in \Rpkg{psych} address the problem of data reduction. \begin{description} \item[\pfun{fa}] incorporates five alternative algorithms: \iemph{minres factor analysis}, \iemph{principal axis factor analysis}, \iemph{weighted least squares factor analysis}, \iemph{generalized least squares factor analysis} and \iemph{maximum likelihood factor analysis}. That is, it includes the functionality of three other functions that will be eventually phased out. \item[\pfun(bassAckward)] will do multiple factor and principal components analyses and then show the relationship between factor levels by finding the interfactor correlations. \item[\pfun{fa.extend}] will extend the factor solution for an X set of variables into a Y set (perhaps of criterion variables). %\item [\pfun{factor.minres}] Minimum residual factor analysis is a least squares, iterative solution to the factor problem. minres attempts to minimize the residual (off-diagonal) correlation matrix. It produces solutions similar to maximum likelihood solutions, but will work even if the matrix is singular. % %\item [\pfun{factor.pa}] Principal Axis factor analysis is a least squares, iterative solution to the factor problem. PA will work for cases where maximum likelihood techniques (\fun{factanal}) will not work. The original communality estimates are either the squared multiple correlations (\pfun{smc}) for each item or 1. % %\item [\pfun{factor.wls}] Weighted least squares factor analysis is a least squares, iterative solution to the factor problem. It minimizes the (weighted) squared residual matrix. The weights are based upon the independent contribution of each variable. % \item [\pfun{principal}] Principal Components Analysis reports the largest n eigen vectors rescaled by the square root of their eigen values. \item [\pfun{factor.congruence}] The congruence between two factors is the cosine of the angle between them. This is just the cross products of the loadings divided by the sum of the squared loadings. This differs from the correlation coefficient in that the mean loading is not subtracted before taking the products. \pfun{factor.congruence} will find the cosines between two (or more) sets of factor loadings. \item [\pfun{vss}] Very Simple Structure \cite{revelle:vss} applies a goodness of fit test to determine the optimal number of factors to extract. It can be thought of as a quasi-confirmatory model, in that it fits the very simple structure (all except the biggest c loadings per item are set to zero where c is the level of complexity of the item) of a factor pattern matrix to the original correlation matrix. For items where the model is usually of complexity one, this is equivalent to making all except the largest loading for each item 0. This is typically the solution that the user wants to interpret. The analysis includes the \pfun{MAP} criterion of \cite{velicer:76} and a $\chi^2$ estimate. \item [\pfun{fa.parallel}] The parallel factors technique compares the observed eigen values of a correlation matrix with those from random data. \item [\pfun{fa.plot}] will plot the loadings from a factor, principal components, or cluster analysis (just a call to plot will suffice). If there are more than two factors, then a SPLOM of the loadings is generated. \item[\pfun{nfactors}] A number of different tests for the number of factors problem are run. \item[\pfun{fa.diagram}] replaces \pfun{fa.graph} and will draw a path diagram representing the factor structure. It does not require Rgraphviz and thus is probably preferred. \item[\pfun{fa.graph}] requires \fun{Rgraphviz} and will draw a graphic representation of the factor structure. If factors are correlated, this will be represented as well. \item[\pfun{iclust} ] is meant to do item cluster analysis using a hierarchical clustering algorithm specifically asking questions about the reliability of the clusters \citep{revelle:iclust}. Clusters are formed until either coefficient $\alpha$ \cite{cronbach:51} or $\beta$ \cite{revelle:iclust} fail to increase. \end{description} \subsubsection{Minimum Residual Factor Analysis} \label{sect:minres} The factor model is an approximation of a correlation matrix by a matrix of lower rank. That is, can the correlation matrix, $\vec{_nR_n}$ be approximated by the product of a factor matrix, $\vec{_nF_k}$ and its transpose plus a diagonal matrix of uniqueness. \begin{equation} R = FF' + U^2 \end{equation} The maximum likelihood solution to this equation is found by \fun{factanal} in the \Rpkg{stats} package. Five alternatives are provided in \Rpkg{psych}, all of them are included in the \pfun{fa} function and are called by specifying the factor method (e.g., fm=``minres", fm=``pa", fm=``"wls", fm="gls" and fm="ml"). In the discussion of the other algorithms, the calls shown are to the \pfun{fa} function specifying the appropriate method. \pfun{factor.minres} attempts to minimize the off diagonal residual correlation matrix by adjusting the eigen values of the original correlation matrix. This is similar to what is done in \fun{factanal}, but uses an ordinary least squares instead of a maximum likelihood fit function. The solutions tend to be more similar to the MLE solutions than are the \pfun{factor.pa} solutions. \iemph{min.res} is the default for the \pfun{fa} function. A classic data set, collected by \cite{thurstone:41} and then reanalyzed by \cite{bechtoldt:61} and discussed by \cite{mcdonald:tt}, is a set of 9 cognitive variables with a clear bi-factor structure \cite{holzinger:37}. The minimum residual solution was transformed into an oblique solution using the default option on rotate which uses an oblimin transformation (Table~\ref{tab:factor.minres}). Alternative rotations and transformations include ``none", ``varimax", ``quartimax", ``bentlerT", and ``geominT" (all of which are orthogonal rotations). as well as ``promax", ``oblimin", ``simplimax", ``bentlerQ, and``geominQ" and ``cluster" which are possible oblique transformations of the solution. The default is to do a oblimin transformation, although prior versions defaulted to varimax. The measures of factor adequacy reflect the multiple correlations of the factors with the best fitting linear regression estimates of the factor scores \citep{grice:01}. \begin{table}[htpb] \caption{Three correlated factors from the Thurstone 9 variable problem. By default, the solution is transformed obliquely using oblimin. The extraction method is (by default) minimum residual.} \begin{scriptsize} \begin{center} <>= f3t <- fa(Thurstone,3,n.obs=213) f3t @ \end{center} \end{scriptsize} \label{tab:factor.minres} \end{table}% \subsubsection{Principal Axis Factor Analysis} An alternative, least squares algorithm, \pfun{factor.pa}, does a Principal Axis factor analysis by iteratively doing an eigen value decomposition of the correlation matrix with the diagonal replaced by the values estimated by the factors of the previous iteration. This OLS solution is not as sensitive to improper matrices as is the maximum likelihood method, and will sometimes produce more interpretable results. It seems as if the SAS example for PA uses only one iteration. Setting the max.iter parameter to 1 produces the SAS solution. The solutions from the \pfun{fa}, the \pfun{factor.minres} and \pfun{factor.pa} as well as the \pfun{principal} functions can be rotated or transformed with a number of options. Some of these call the \Rpkg{GPArotation} package. Orthogonal rotations are \fun{varimax} and \fun{quartimax}. Oblique transformations include \fun{oblimin}, \fun{quartimin} and then two targeted rotation functions \pfun{Promax} and \pfun{target.rot}. The latter of these will transform a loadings matrix towards an arbitrary target matrix. The default is to transform towards an independent cluster solution. Using the Thurstone data set, three factors were requested and then transformed into an independent clusters solution using \pfun{target.rot} (Table~\ref{tab:Thurstone}). \begin{table}[htpb] \caption{The 9 variable problem from Thurstone is a classic example of factoring where there is a higher order factor, g, that accounts for the correlation between the factors. The extraction method was principal axis. The transformation was a targeted transformation to a simple cluster solution.} \begin{center} \begin{scriptsize} <>= f3 <- fa(Thurstone,3,n.obs = 213,fm="pa") f3o <- target.rot(f3) f3o @ \end{scriptsize} \end{center} \label{tab:Thurstone} \end{table} \subsubsection{Weighted Least Squares Factor Analysis} \label{sect:wls} Similar to the minres approach of minimizing the squared residuals, factor method ``wls" weights the squared residuals by their uniquenesses. This tends to produce slightly smaller overall residuals. In the example of weighted least squares, the output is shown by using the \pfun{print} function with the cut option set to 0. That is, all loadings are shown (Table~\ref{tab:Thurstone.wls}). \begin{table}[htpb] \caption{The 9 variable problem from Thurstone is a classic example of factoring where there is a higher order factor, g, that accounts for the correlation between the factors. The factors were extracted using a weighted least squares algorithm. All loadings are shown by using the cut=0 option in the \pfun{print.psych} function.} \begin{scriptsize} <>= f3w <- fa(Thurstone,3,n.obs = 213,fm="wls") print(f3w,cut=0,digits=3) @ \end{scriptsize} \label{tab:Thurstone.wls} \end{table} The unweighted least squares solution may be shown graphically using the \pfun{fa.plot} function which is called by the generic \fun{plot} function (Figure~\ref{fig:thurstone}. Factors were transformed obliquely using a oblimin. These solutions may be shown as item by factor plots (Figure~\ref{fig:thurstone} or by a structure diagram (Figure~\ref{fig:thurstone.diagram}. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= plot(f3t) @ \end{scriptsize} \caption{A graphic representation of the 3 oblique factors from the Thurstone data using \pfun{plot}. Factors were transformed to an oblique solution using the oblimin function from the GPArotation package.} \label{fig:thurstone} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= fa.diagram(f3t) @ \end{scriptsize} \caption{A graphic representation of the 3 oblique factors from the Thurstone data using \pfun{fa.diagram}. Factors were transformed to an oblique solution using oblimin.} \label{fig:thurstone.diagram} \end{center} \end{figure} A comparison of these three approaches suggests that the minres solution is more similar to a maximum likelihood solution and fits slightly better than the pa or wls solutions. Comparisons with SPSS suggest that the pa solution matches the SPSS OLS solution, but that the minres solution is slightly better. At least in one test data set, the weighted least squares solutions, although fitting equally well, had slightly different structure loadings. Note that the rotations used by SPSS will sometimes use the ``Kaiser Normalization''. By default, the rotations used in psych do not normalize, but this can be specified as an option in \pfun{fa}. \subsubsection{Principal Components analysis (PCA)} An alternative to factor analysis, which is unfortunately frequently confused with \iemph{factor analysis}, is \iemph{principal components analysis}. Although the goals of \iemph{PCA} and \iemph{FA} are similar, PCA is a descriptive model of the data, while FA is a structural model. Psychologists typically use PCA in a manner similar to factor analysis and thus the \pfun{principal} function produces output that is perhaps more understandable than that produced by \fun{princomp} in the \Rpkg{stats} package. Table~\ref{tab:pca} shows a PCA of the Thurstone 9 variable problem rotated using the \pfun{Promax} function. Note how the loadings from the factor model are similar but smaller than the principal component loadings. This is because the PCA model attempts to account for the entire variance of the correlation matrix, while FA accounts for just the \iemph{common variance}. This distinction becomes most important for small correlation matrices. Also note how the goodness of fit statistics, based upon the residual off diagonal elements, is much worse than the \pfun{fa} solution. \begin{table}[htpb] \caption{The Thurstone problem can also be analyzed using Principal Components Analysis. Compare this to Table~\ref{tab:Thurstone}. The loadings are higher for the PCA because the model accounts for the unique as well as the common variance.The fit of the off diagonal elements, however, is much worse than the \pfun{fa} results.} \begin{center} \begin{scriptsize} <>= p3p <-principal(Thurstone,3,n.obs = 213,rotate="Promax") p3p @ \end{scriptsize} \end{center} \label{tab:pca} \end{table} \subsubsection{Hierarchical and bi-factor solutions} \label{sect:omega} For a long time structural analysis of the ability domain have considered the problem of factors that are themselves correlated. These correlations may themselves be factored to produce a higher order, general factor. An alternative \citep{holzinger:37,jensen:weng} is to consider the general factor affecting each item, and then to have group factors account for the residual variance. Exploratory factor solutions to produce a hierarchical or a bifactor solution are found using the \pfun{omega} function. This technique has more recently been applied to the personality domain to consider such things as the structure of neuroticism (treated as a general factor, with lower order factors of anxiety, depression, and aggression). Consider the 9 Thurstone variables analyzed in the prior factor analyses. The correlations between the factors (as shown in Figure~\ref{fig:thurstone.diagram} can themselves be factored. This results in a higher order factor model (Figure~\ref{fig:omega}). An an alternative solution is to take this higher order model and then solve for the general factor loadings as well as the loadings on the residualized lower order factors using the \iemph{Schmid-Leiman} procedure. (Figure ~\ref{fig:omega.2}). Yet another solution is to use structural equation modeling to directly solve for the general and group factors. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= om.h <- omega(Thurstone,n.obs=213,sl=FALSE) op <- par(mfrow=c(1,1)) @ \end{scriptsize} \caption{A higher order factor solution to the Thurstone 9 variable problem} \label{fig:omega} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= om <- omega(Thurstone,n.obs=213) @ \end{scriptsize} \caption{A bifactor factor solution to the Thurstone 9 variable problem} \label{fig:omega.2} \end{center} \end{figure} Yet another approach to the bifactor structure is do use the \pfun{bifactor} rotation function in either \Rpkg{psych} or in \Rpkg{GPArotation}. This does the rotation discussed in \cite{jennrich:11}. \subsubsection{Item Cluster Analysis: iclust} \label{sect:iclust} An alternative to factor or components analysis is \iemph{cluster analysis}. The goal of cluster analysis is the same as factor or components analysis (reduce the complexity of the data and attempt to identify homogeneous subgroupings). Mainly used for clustering people or objects (e.g., projectile points if an anthropologist, DNA if a biologist, galaxies if an astronomer), clustering may be used for clustering items or tests as well. Introduced to psychologists by \cite{tryon:39} in the 1930's, the cluster analytic literature exploded in the 1970s and 1980s \citep{blashfield:80,blashfield:88,everitt:74,hartigan:75}. Much of the research is in taxonmetric applications in biology \citep{sneath:73,sokal:63} and marketing \citep{cooksey:06} where clustering remains very popular. It is also used for taxonomic work in forming clusters of people in family \citep{henry:05} and clinical psychology \citep{martinent:07,mun:08}. Interestingly enough it has has had limited applications to psychometrics. This is unfortunate, for as has been pointed out by e.g. \citep{tryon:35,loevinger:53}, the theory of factors, while mathematically compelling, offers little that the geneticist or behaviorist or perhaps even non-specialist finds compelling. \cite{cooksey:06} reviews why the \pfun{iclust} algorithm is particularly appropriate for scale construction in marketing. \emph{Hierarchical cluster analysis} \index{hierarchical cluster analysis} forms clusters that are nested within clusters. The resulting \iemph{tree diagram} (also known somewhat pretentiously as a \iemph{rooted dendritic structure}) shows the nesting structure. Although there are many hierarchical clustering algorithms in \R{} (e.g., \fun{agnes}, \fun{hclust}, and \pfun{iclust}), the one most applicable to the problems of scale construction is \pfun{iclust} \citep{revelle:iclust}. \begin{enumerate} \item Find the proximity (e.g. correlation) matrix, \item Identify the most similar pair of items \item Combine this most similar pair of items to form a new variable (cluster), \item Find the similarity of this cluster to all other items and clusters, \item Repeat steps 2 and 3 until some criterion is reached (e.g., typicallly, if only one cluster remains or in \pfun{iclust} if there is a failure to increase reliability coefficients $\alpha$ or $\beta$). \item Purify the solution by reassigning items to the most similar cluster center. \end{enumerate} \pfun{iclust} forms clusters of items using a hierarchical clustering algorithm until one of two measures of internal consistency fails to increase \citep{revelle:iclust}. The number of clusters may be specified a priori, or found empirically. The resulting statistics include the average split half reliability, $\alpha$ \citep{cronbach:51}, as well as the worst split half reliability, $\beta$ \citep{revelle:iclust}, which is an estimate of the general factor saturation of the resulting scale (Figure~\ref{fig:iclust}). Cluster loadings (corresponding to the structure matrix of factor analysis) are reported when printing (Table~\ref{tab:iclust}). The pattern matrix is available as an object in the results. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= data(bfi) ic <- iclust(bfi[1:25]) @ \end{scriptsize} \caption{Using the \pfun{iclust} function to find the cluster structure of 25 personality items (the three demographic variables were excluded from this analysis). When analyzing many variables, the tree structure may be seen more clearly if the graphic output is saved as a pdf and then enlarged using a pdf viewer.} \label{fig:iclust} \end{center} \end{figure} \begin{table}[htpb] \caption{The summary statistics from an iclust analysis shows three large clusters and smaller cluster.} \begin{center} \begin{scriptsize} <>= summary(ic) #show the results @ \end{scriptsize} \end{center} \label{tab:iclust} \end{table}% The previous analysis (Figure~\ref{fig:iclust}) was done using the Pearson correlation. A somewhat cleaner structure is obtained when using the \pfun{polychoric} function to find polychoric correlations (Figure~\ref{fig:iclust.poly}). Note that the first time finding the polychoric correlations some time, but the next three analyses were done using that correlation matrix (r.poly\$rho). When using the console for input, \pfun{polychoric} will report on its progress while working using \pfun{progressBar}. \begin{table}[htpb] \caption{The \pfun{polychoric} and the \pfun{tetrachoric} functions can take a long time to finish and report their progress by a series of dots as they work. The dots are suppressed when creating a Sweave document.} \begin{center} \begin{tiny} <>= data(bfi) r.poly <- polychoric(bfi[1:25]) #the ... indicate the progress of the function @ \end{tiny} \end{center} \label{tab:bad} \end{table}% \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= ic.poly <- iclust(r.poly$rho,title="ICLUST using polychoric correlations") iclust.diagram(ic.poly) @ \end{scriptsize} \caption{ICLUST of the BFI data set using polychoric correlations. Compare this solution to the previous one (Figure~\ref{fig:iclust}) which was done using Pearson correlations. } \label{fig:iclust.poly} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= ic.poly <- iclust(r.poly$rho,5,title="ICLUST using polychoric correlations for nclusters=5") iclust.diagram(ic.poly) @ \end{scriptsize} \caption{ICLUST of the BFI data set using polychoric correlations with the solution set to 5 clusters. Compare this solution to the previous one (Figure~\ref{fig:iclust.poly}) which was done without specifying the number of clusters and to the next one (Figure~\ref{fig:iclust.3}) which was done by changing the beta criterion. } \label{fig:iclust.5} \end{center} \end{figure} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= ic.poly <- iclust(r.poly$rho,beta.size=3,title="ICLUST beta.size=3") @ \end{scriptsize} \caption{ICLUST of the BFI data set using polychoric correlations with the beta criterion set to 3. Compare this solution to the previous three (Figure~\ref{fig:iclust},~\ref{fig:iclust.poly}, \ref{fig:iclust.5}).} \label{fig:iclust.3} \end{center} \end{figure} \begin{table}[htpb] \caption{The output from \pfun{iclust}includes the loadings of each item on each cluster. These are equivalent to factor structure loadings. By specifying the value of cut, small loadings are suppressed. The default is for cut=0.su } \begin{center} \begin{scriptsize} <>= print(ic,cut=.3) @ \end{scriptsize} \end{center} \label{tab:iclust} \end{table}% A comparison of these four cluster solutions suggests both a problem and an advantage of clustering techniques. The problem is that the solutions differ. The advantage is that the structure of the items may be seen more clearly when examining the clusters rather than a simple factor solution. \subsection{Confidence intervals using bootstrapping techniques} Exploratory factoring techniques are sometimes criticized because of the lack of statistical information on the solutions. Overall estimates of goodness of fit including $\chi^{2}$ and RMSEA are found in the \pfun{fa} and \pfun{omega} functions. Confidence intervals for the factor loadings may be found by doing multiple bootstrapped iterations of the original analysis. This is done by setting the n.iter parameter to the desired number of iterations. This can be done for factoring of Pearson correlation matrices as well as polychoric/tetrachoric matrices (See Table~\ref{tab:bootstrap}). Although the example value for the number of iterations is set to 20, more conventional analyses might use 1000 bootstraps. This will take much longer. \begin{table}[htpb] \caption{An example of bootstrapped confidence intervals on 10 items from the Big 5 inventory. The number of bootstrapped samples was set to 20. More conventional bootstrapping would use 100 or 1000 replications. } \begin{tiny} \begin{center} <>= fa(bfi[1:10],2,n.iter=20) @ \end{center} \end{tiny} \label{tab:bootstrap} \end{table}% \subsection{Comparing factor/component/cluster solutions} Cluster analysis, factor analysis, and principal components analysis all produce structure matrices (matrices of correlations between the dimensions and the variables) that can in turn be compared in terms of the \iemph{congruence coefficient} which is just the cosine of the angle between the dimensions $$c_{f_{i}f_{j}} = \frac{\sum_{k=1}^{n}{f_{ik}f_{jk}}} {\sum{f_{ik}^{2}}\sum{f_{jk}^{2}}}.$$ Consider the case of a four factor solution and four cluster solution to the Big Five problem. \begin{scriptsize} <>= f4 <- fa(bfi[1:25],4,fm="pa") factor.congruence(f4,ic) @ \end{scriptsize} A more complete comparison of oblique factor solutions (both minres and principal axis), bifactor and component solutions to the Thurstone data set is done using the \pfun{factor.congruence} function. (See table~\ref{tab:congruence}). \begin{table}[htpb] \caption{Congruence coefficients for oblique factor, bifactor and component solutions for the Thurstone problem.} \begin{scriptsize} <>= factor.congruence(list(f3t,f3o,om,p3p)) @ \end{scriptsize} \label{tab:congruence} \end{table}% \subsubsection{Factor correlations} Factor congruences may be found between any two sets of factor loadings. If given the same data set/correlation matrix, factor correlations may be found using \pfun{faCor} which finds the correlations between the factors. This procedure is also used in the \pfun{bassAckward} function which compares multiple solutions with a different number of factors. Consider the correlation of three versus five factors of the \pfun{bfi} data set. \begin{table}[htpb] \caption{Factor correlations and factor congruences between ``minres" factor analysis and ``pca" principal components using ``oblimin" rotation for both.} \begin{center} \begin{scriptsize} <>= faCor(Thurstone,c(3,3),fm=c("minres","pca"), rotate=c("oblimin","oblimin")) @ \end{scriptsize} \end{center} \label{tab:faCor} \end{table} \subsection{Determining the number of dimensions to extract.} How many dimensions to use to represent a correlation matrix is an unsolved problem in psychometrics. There are many solutions to this problem, none of which is uniformly the best. Henry Kaiser once said that ``a solution to the number-of factors problem in factor analysis is easy, that he used to make up one every morning before breakfast. But the problem, of course is to find \emph{the} solution, or at least a solution that others will regard quite highly not as the best" \cite{horn:79}. Techniques most commonly used include 1) Extracting factors until the chi square of the residual matrix is not significant. 2) Extracting factors until the change in chi square from factor n to factor n+1 is not significant. 3) Extracting factors until the eigen values of the real data are less than the corresponding eigen values of a random data set of the same size (parallel analysis) \pfun{fa.parallel} \citep{horn:65}. 4) Plotting the magnitude of the successive eigen values and applying the scree test (a sudden drop in eigen values analogous to the change in slope seen when scrambling up the talus slope of a mountain and approaching the rock face \citep{cattell:scree}. 5) Extracting factors as long as they are interpretable. 6) Using the Very Structure Criterion (\pfun{vss}) \citep{revelle:vss}. 7) Using Wayne Velicer's Minimum Average Partial (\pfun{MAP}) criterion \citep{velicer:76}. 8) Extracting principal components until the eigen value < 1. Each of the procedures has its advantages and disadvantages. Using either the chi square test or the change in square test is, of course, sensitive to the number of subjects and leads to the nonsensical condition that if one wants to find many factors, one simply runs more subjects. Parallel analysis is partially sensitive to sample size in that for large samples the eigen values of random factors will be very small. The scree test is quite appealing but can lead to differences of interpretation as to when the scree``breaks". Extracting interpretable factors means that the number of factors reflects the investigators creativity more than the data. vss, while very simple to understand, will not work very well if the data are very factorially complex. (Simulations suggests it will work fine if the complexities of some of the items are no more than 2). The eigen value of 1 rule, although the default for many programs, seems to be a rough way of dividing the number of variables by 3 and is probably the worst of all criteria. An additional problem in determining the number of factors is what is considered a factor. Many treatments of factor analysis assume that the residual correlation matrix after the factors of interest are extracted is composed of just random error. An alternative concept is that the matrix is formed from major factors of interest but that there are also numerous minor factors of no substantive interest but that account for some of the shared covariance between variables. The presence of such minor factors can lead one to extract too many factors and to reject solutions on statistical grounds of misfit that are actually very good fits to the data. This problem is partially addressed later in the discussion of simulating complex structures using \pfun{sim.structure} and of small extraneous factors using the \pfun{sim.minor} function. \subsubsection{Very Simple Structure} \label{sect:vss} The \pfun{vss} function compares the fit of a number of factor analyses with the loading matrix ``simplified" by deleting all except the c greatest loadings per item, where c is a measure of factor complexity \cite{revelle:vss}. Included in \pfun{vss} is the MAP criterion (Minimum Absolute Partial correlation) of \cite{velicer:76}. Using the Very Simple Structure criterion for the bfi data suggests that 4 factors are optimal (Figure~\ref{fig:vss}). However, the MAP criterion suggests that 5 is optimal. \begin{figure}[htbp] \begin{center} <>= vss <- vss(bfi[1:25],title="Very Simple Structure of a Big 5 inventory") @ \caption{The Very Simple Structure criterion for the number of factors compares solutions for various levels of item complexity and various numbers of factors. For the Big 5 Inventory, the complexity 1 and 2 solutions both achieve their maxima at four factors. This is in contrast to parallel analysis which suggests 6 and the MAP criterion which suggests 5. } \label{fig:vss} \end{center} \end{figure} \begin{scriptsize} <>= vss @ \end{scriptsize} \subsubsection{Parallel Analysis} \label{sect:fa.parallel} An alternative way to determine the number of factors is to compare the solution to random data with the same properties as the real data set. If the input is a data matrix, the comparison includes random samples from the real data, as well as normally distributed random data with the same number of subjects and variables. For the BFI data, parallel analysis suggests that 6 factors might be most appropriate (Figure~\ref{fig:parallel}). It is interesting to compare \pfun{fa.parallel} with the \fun{paran} from the \Rpkg{paran} package. This latter uses smcs to estimate communalities. Simulations of known structures with a particular number of major factors but with the presence of trivial, minor (but not zero) factors, show that using smcs will tend to lead to too many factors. \begin{figure}[htbp] \begin{scriptsize} \begin{center} <>= fa.parallel(bfi[1:25],main="Parallel Analysis of a Big 5 inventory") @ \caption{Parallel analysis compares factor and principal components solutions to the real data as well as resampled data. Although vss suggests 4 factors, MAP 5, parallel analysis suggests 6. One more demonstration of Kaiser's dictum.} \label{fig:parallel} \end{center} \end{scriptsize} \end{figure} A more tedious problem in terms of computation is to do parallel analysis of \iemph{polychoric} correlation matrices. This is done by \pfun{fa.parallel.poly} or \pfun{fa.parallel} with the cor option="poly". By default the number of replications is 20. This is appropriate when choosing the number of factors from dicthotomous or polytomous data matrices. \subsection{Factor extension} Sometimes we are interested in the relationship of the factors in one space with the variables in a different space. One solution is to find factors in both spaces separately and then find the structural relationships between them. This is the technique of structural equation modeling in packages such as \Rpkg{sem} or \Rpkg{lavaan}. An alternative is to use the concept of \iemph{factor extension} developed by \citep{dwyer:37}. Consider the case of 16 variables created to represent one two dimensional space. If factors are found from eight of these variables, they may then be extended to the additional eight variables (See Figure~\ref{fig:fa.extension}). \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= v16 <- sim.item(16) s <- c(1,3,5,7,9,11,13,15) f2 <- fa(v16[,s],2) fe <- fa.extension(cor(v16)[s,-s],f2) fa.diagram(f2,fe=fe) @ \end{scriptsize} \caption{Factor extension applies factors from one set (those on the left) to another set of variables (those on the right). \pfun{fa.extension} is particularly useful when one wants to define the factors with one set of variables and then apply those factors to another set. \pfun{fa.diagram} is used to show the structure. } \label{fig:fa.extension} \end{center} \end{figure} Factor extension may also be used to see the validity of a certain factor solution compared to a set of criterion variables. Consider the case of 5 factors from the 25 items of the \pfun{bfi} data set and how they predict gender, age, and education (See Figure~\ref{fig:fa:extend}). \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= fe <- fa.extend(bfi,5,ov=1:25,ev=26:28) extension.diagram(fe) @ \end{scriptsize} \caption{Factor extension applies factors from one set (those on the left) to another set of variables (those on the right). \pfun{fa.extend} is particularly useful when one wants to define the factors with one set of variables and then apply those factors to another set. \pfun{diagram} is used to show the structure. } \label{fig:fa.extend} \end{center} \end{figure} Another way to examine the overlap between two sets is the use of \iemph{set correlation} found by \pfun{setCor} (discussed later). \subsection{Comparing multiple solutions} A procedure dubbed ``bass Ackward" by Lew Goldberg \citep{goldberg:06} compares solutions at multiple levels of complexity. Here we show a 2, 3, 4 and 5 dimensional solution to the \pfun{bfi} data set. (Figure~\ref{fig:bass.ack}). This is done by finding the factor correlations between solutions (see \pfun{faCor}) and then organizing them sequentially. The factor correlations for two solutions from the same correlation matrix, $\vec{R}$ , $\vec{F_1} $ and $\vec{F_2}$ are found by using the two weights matrices, $\vec{W_1}$ and $\vec{W_2}$ (for finding factor scores for the first and second model) and then finding the factor covariances, $C = \vec{W_1' R W_2} $ which may then be converted to factor correlations by dividing by the square root of the diagonal of $\vec{C}$. By default \pfun{bassAckward} uses the correlation preserving weights discussed by \cite{tenBerge.99}, although other options (e.g. regression weights) may also be used. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= ba5 <- bassAckward(bfi[1:25], nfactors =c(2,3,4,5),plot=FALSE) baf <- bassAckward.diagram(ba5) @ \end{scriptsize} \caption{\pfun{bassAckward} compares solutions at multiple levels by successive factoring and the finding the factor correlations across levels. Compare the three factor solution to the five factor solution. The dimensions of social approach, withdrawal, and constraint seen at the three factor level become the more traditional CANOE factors at the five factor level. } \label{fig:bass.ack} \end{center} \end{figure} And we show the items associated with this solution by using \pfun{fa.lookup} (Table~\ref{tab:bfi}) \begin{table}[htpb] \caption{bfi items sorted in the order of the five factors from \pfun{bassAckward}} \begin{center} \begin{scriptsize} <>= # fa.lookup(baf$bass.ack[[5]],dictionary=bfi.dictionary[2]) @ \end{scriptsize} \end{center} \label{tab:bfi} \end{table} \section{Classical Test Theory and Reliability} Surprisingly, 107 years after \cite{spearman:rho} introduced the concept of reliability to psychologists, there are still multiple approaches for measuring it. Although very popular, Cronbach's $\alpha$ \citep{cronbach:51} underestimates the reliability of a test and over estimates the first factor saturation \citep{rz:09}. $\alpha$ \citep{cronbach:51} is the same as Guttman's $\lambda3$ \citep{guttman:45} and may be found by $$ \lambda_3 = \frac{n}{n-1}\Bigl(1 - \frac{tr(\vec{V})_x}{V_x}\Bigr) = \frac{n}{n-1} \frac{V_x - tr(\vec{V}_x)}{V_x} = \alpha $$ Perhaps because it is so easy to calculate and is available in most commercial programs, alpha is without doubt the most frequently reported measure of internal consistency reliability. Alpha is the mean of all possible spit half reliabilities (corrected for test length). For a unifactorial test, it is a reasonable estimate of the first factor saturation, although if the test has any microstructure (i.e., if it is ``lumpy") coefficients $\beta$ \citep{revelle:iclust} (see \pfun{iclust}) and $\omega_h$ (see \pfun{omega}) are more appropriate estimates of the general factor saturation. $\omega_t$is a better estimate of the reliability of the total test. Guttman's $\lambda _6$ (G6) considers the amount of variance in each item that can be accounted for the linear regression of all of the other items (the squared multiple correlation or smc), or more precisely, the variance of the errors, $e_j^2$, and is $$ \lambda_6 = 1 - \frac{\sum e_j^2}{V_x} = 1 - \frac{\sum(1-r_{smc}^2)}{V_x}. $$ The squared multiple correlation is a lower bound for the item communality and as the number of items increases, becomes a better estimate. G6 is also sensitive to lumpiness in the test and should not be taken as a measure of unifactorial structure. For lumpy tests, it will be greater than alpha. For tests with equal item loadings, alpha > G6, but if the loadings are unequal or if there is a general factor, G6 > alpha. G6 estimates item reliability by the squared multiple correlation of the other items in a scale. A modification of G6, G6*, takes as an estimate of an item reliability the smc with all the items in an inventory, including those not keyed for a particular scale. This will lead to a better estimate of the reliable variance of a particular item. Alpha, G6 and G6* are positive functions of the number of items in a test as well as the average intercorrelation of the items in the test. When calculated from the item variances and total test variance, as is done here, raw alpha is sensitive to differences in the item variances. Standardized alpha is based upon the correlations rather than the covariances. More complete reliability analyses of a single scale can be done using the \pfun{omega} function which finds $\omega_h$ and $\omega_t$ based upon a hierarchical factor analysis. Alternative functions \pfun{scoreItems} and \pfun{cluster.cor} will also score multiple scales and report more useful statistics. ``Standardized" alpha is calculated from the inter-item correlations and will differ from raw alpha. Functions for examining the reliability of a single scale or a set of scales include: \begin{description} \item [alpha] Internal consistency measures of reliability range from $\omega_h$ to $\alpha$ to $\omega_t$. The \pfun{alpha} function reports two estimates: Cronbach's coefficient $\alpha$ and Guttman's $\lambda_6$. Also reported are item - whole correlations, $\alpha$ if an item is omitted, and item means and standard deviations. \item [guttman] Eight alternative estimates of test reliability include the six discussed by \cite{guttman:45}, four discussed by ten Berge and Zergers (1978) ($\mu_0 \dots \mu_3$) as well as $\beta$ \citep[the worst split half,][]{revelle:iclust}, the glb (greatest lowest bound) discussed by Bentler and Woodward (1980), and $\omega_h$ and$\omega_t$ (\citep{mcdonald:tt,zinbarg:pm:05}. \item [omega] Calculate McDonald's omega estimates of general and total factor saturation. (\cite{rz:09} compare these coefficients with real and artificial data sets.) \item [cluster.cor] Given a n x c cluster definition matrix of -1s, 0s, and 1s (the keys) , and a n x n correlation matrix, find the correlations of the composite clusters. \item [scoreItems] Given a matrix or data.frame of k keys for m items (-1, 0, 1), and a matrix or data.frame of items scores for m items and n people, find the sum scores or average scores for each person and each scale. If the input is a square matrix, then it is assumed that correlations or covariances were used, and the raw scores are not available. In addition, report Cronbach's alpha, coefficient G6*, the average r, the scale intercorrelations, and the item by scale correlations (both raw and corrected for item overlap and scale reliability). Replace missing values with the item median or mean if desired. Will adjust scores for reverse scored items. \item [score.multiple.choice] Ability tests are typically multiple choice with one right answer. score.multiple.choice takes a scoring key and a data matrix (or data.frame) and finds total or average number right for each participant. Basic test statistics (alpha, average r, item means, item-whole correlations) are also reported. \end{description} \subsection{Reliability of a single scale} \label{sect:alpha} A conventional (but non-optimal) estimate of the internal consistency reliability of a test is coefficient $\alpha$ \citep{cronbach:51}. Alternative estimates are Guttman's $\lambda_6$, Revelle's $\beta$, McDonald's $\omega_h$ and $\omega_t$. Consider a simulated data set, representing 9 items with a hierarchical structure and the following correlation matrix. Then using the \pfun{alpha} function, the $\alpha$ and $\lambda_6$ estimates of reliability may be found for all 9 items, as well as the if one item is dropped at a time. \begin{scriptsize} <>= set.seed(17) r9 <- sim.hierarchical(n=500,raw=TRUE)$observed round(cor(r9),2) alpha(r9) @ \end{scriptsize} Some scales have items that need to be reversed before being scored. Rather than reversing the items in the raw data, it is more convenient to just specify which items need to be reversed scored. This may be done in \pfun{alpha} by specifying a \iemph{keys} vector of 1s and -1s. (This concept of keys vector is more useful when scoring multiple scale inventories, see below.) As an example, consider scoring the 7 attitude items in the attitude data set. Assume a conceptual mistake in that item 2 is to be scored (incorrectly) negatively. \begin{scriptsize} <>= keys <- c(1,-1,1,1,1,1,1) alpha(attitude,keys) @ \end{scriptsize} Note how the reliability of the 7 item scales with an incorrectly reversed item is very poor, but if the item 2 is dropped then the reliability is improved substantially. This suggests that item 2 was incorrectly scored. Doing the analysis again with item 2 positively scored produces much more favorable results. \begin{scriptsize} <>= keys <- c(1,1,1,1,1,1,1) alpha(attitude,keys) @ \end{scriptsize} It is useful when considering items for a potential scale to examine the item distribution. This is done in \pfun{scoreItems} as well as in \pfun{alpha}. \begin{scriptsize} <>= items <- sim.congeneric(N=500,short=FALSE,low=-2,high=2,categorical=TRUE) #500 responses to 4 discrete items alpha(items$observed) #item response analysis of congeneric measures @ \end{scriptsize} \subsection{Using \pfun{omega} to find the reliability of a single scale} Two alternative estimates of reliability that take into account the hierarchical structure of the inventory are McDonald's $\omega_h$ and $\omega_t$. These may be found using the \pfun{omega} function for an exploratory analysis (See Figure~\ref{fig:omega.9}) or \pfun{omegaSem} for a confirmatory analysis using the \Rpkg{sem} based upon the exploratory solution from \pfun{omega}. McDonald has proposed coefficient omega (hierarchical) ($\omega_h$) as an estimate of the general factor saturation of a test. \cite{zinbarg:pm:05} \url{http://personality-project.org/revelle/publications/zinbarg.revelle.pmet.05.pdf} compare McDonald's $\omega_h$ to Cronbach's $\alpha$ and Revelle's $\beta$. They conclude that $\omega_h$ is the best estimate. (See also \cite{zinbarg:apm:06} and \cite{rz:09} \url{http://personality-project.org/revelle/publications/revelle.zinbarg.08.pdf} ). One way to find $\omega_h$ is to do a factor analysis of the original data set, rotate the factors obliquely, factor that correlation matrix, do a Schmid-Leiman (\pfun{schmid}) transformation to find general factor loadings, and then find $\omega_h$. $\omega_h$ differs slightly as a function of how the factors are estimated. Four options are available, the default will do a minimum residual factor analysis, fm=``pa" does a principal axes factor analysis (\pfun{factor.pa}), fm=``mle" uses the factanal function, and fm=``pc" does a principal components analysis (\pfun{principal}). For ability items, it is typically the case that all items will have positive loadings on the general factor. However, for non-cognitive items it is frequently the case that some items are to be scored positively, and some negatively. Although probably better to specify which directions the items are to be scored by specifying a key vector, if flip =TRUE (the default), items will be reversed so that they have positive loadings on the general factor. The keys are reported so that scores can be found using the \pfun{scoreItems} function. Arbitrarily reversing items this way can overestimate the general factor. (See the example with a simulated circumplex). $\beta$, an alternative to $\omega$, is defined as the worst split half reliability. It can be estimated by using \pfun{iclust} (Item Cluster analysis: a hierarchical clustering algorithm). For a very complimentary review of why the iclust algorithm is useful in scale construction, see \cite{cooksey:06}. The \pfun{omega} function uses exploratory factor analysis to estimate the $\omega_h$ coefficient. It is important to remember that ``A recommendation that should be heeded, regardless of the method chosen to estimate $\omega_h$, is to always examine the pattern of the estimated general factor loadings prior to estimating $\omega_h$. Such an examination constitutes an informal test of the assumption that there is a latent variable common to all of the scale's indicators that can be conducted even in the context of EFA. If the loadings were salient for only a relatively small subset of the indicators, this would suggest that there is no true general factor underlying the covariance matrix. Just such an informal assumption test would have afforded a great deal of protection against the possibility of misinterpreting the misleading $\omega_h$ estimates occasionally produced in the simulations reported here." \citep[][p 137]{zinbarg:apm:06}. Although $\omega_h$ is uniquely defined only for cases where 3 or more subfactors are extracted, it is sometimes desired to have a two factor solution. By default this is done by forcing the \pfun{schmid} extraction to treat the two subfactors as having equal loadings. There are three possible options for this condition: setting the general factor loadings between the two lower order factors to be ``equal" which will be the $\sqrt{r_{ab}}$ where $r_{ab}$ is the oblique correlation between the factors) or to ``first" or ``second" in which case the general factor is equated with either the first or second group factor. A message is issued suggesting that the model is not really well defined. This solution discussed in Zinbarg et al., 2007. To do this in omega, add the option=``first" or option=``second" to the call. Although obviously not meaningful for a 1 factor solution, it is of course possible to find the sum of the loadings on the first (and only) factor, square them, and compare them to the overall matrix variance. This is done, with appropriate complaints. In addition to $\omega_h$, another of McDonald's coefficients is $\omega_t$. This is an estimate of the total reliability of a test. McDonald's $\omega_t$, which is similar to Guttman's $\lambda_6$, (see \pfun{guttman}) uses the estimates of uniqueness $u^2$ from factor analysis to find $e_j^2$. This is based on a decomposition of the variance of a test score, $V_x$ into four parts: that due to a general factor, $\vec{g}$, that due to a set of group factors, $\vec{f}$, (factors common to some but not all of the items), specific factors, $\vec{s}$ unique to each item, and $\vec{e}$, random error. (Because specific variance can not be distinguished from random error unless the test is given at least twice, some combine these both into error). Letting $\vec{x} = \vec{cg} + \vec{Af} + \vec {Ds} + \vec{e} $ then the communality of item$_j$, based upon general as well as group factors, $h_j^2 = c_j^2 + \sum{f_{ij}^2}$ and the unique variance for the item $u_j^2 = \sigma_j^2 (1-h_j^2)$ may be used to estimate the test reliability. That is, if $h_j^2$ is the communality of item$_j$, based upon general as well as group factors, then for standardized items, $e_j^2 = 1 - h_j^2$ and $$ \omega_t = \frac{\vec{1}\vec{cc'}\vec{1} + \vec{1}\vec{AA'}\vec{1}'}{V_x} = 1 - \frac{\sum(1-h_j^2)}{V_x} = 1 - \frac{\sum u^2}{V_x} $$ Because $h_j^2 \geq r_{smc}^2$, $\omega_t \geq \lambda_6$. It is important to distinguish here between the two $\omega$ coefficients of McDonald, 1978 and Equation 6.20a of McDonald, 1999, $\omega_t$ and $\omega_h$. While the former is based upon the sum of squared loadings on all the factors, the latter is based upon the sum of the squared loadings on the general factor. $$\omega_h = \frac{ \vec{1}\vec{cc'}\vec{1}}{V_x}$$ Another estimate reported is the omega for an infinite length test with a structure similar to the observed test. This is found by $$\omega_{\inf} = \frac{ \vec{1}\vec{cc'}\vec{1}}{\vec{1}\vec{cc'}\vec{1} + \vec{1}\vec{AA'}\vec{1}'}$$ \begin{figure}[htbp] \begin{center} <>= om.9 <- omega(r9,title="9 simulated variables") @ \caption{A bifactor solution for 9 simulated variables with a hierarchical structure. } \label{fig:omega.9} \end{center} \end{figure} In the case of these simulated 9 variables, the amount of variance attributable to a general factor ($\omega_h$) is quite large, and the reliability of the set of 9 items is somewhat greater than that estimated by $\alpha$ or $\lambda_6$. \begin{scriptsize} <>= om.9 @ \end{scriptsize} \subsection{Estimating $\omega_h$ using Confirmatory Factor Analysis} The \pfun{omegaSem} function will do an exploratory analysis and then take the highest loading items on each factor and do a confirmatory factor analysis using the \Rpkg{sem} package. These results can produce slightly different estimates of $\omega_h$, primarily because cross loadings are modeled as part of the general factor. \begin{scriptsize} <>= omegaSem(r9,n.obs=500) @ \end{scriptsize} \subsubsection{Other estimates of reliability} Other estimates of reliability are found by the \pfun{splitHalf} function. These are described in more detail in \cite{rz:09}. They include the 6 estimates from Guttman, four from TenBerge, and an estimate of the greatest lower bound. \begin{scriptsize} <>= splitHalf(r9) @ \end{scriptsize} \subsection{Reliability and correlations of multiple scales within an inventory} \label{sect:score} A typical research question in personality involves an inventory of multiple items purporting to measure multiple constructs. For example, the data set \pfun{bfi} includes 25 items thought to measure five dimensions of personality (Extraversion, Emotional Stability, Conscientiousness, Agreeableness, and Openness). The data may either be the raw data or a correlation matrix (\pfun{scoreItems}) or just a correlation matrix of the items ( \pfun{cluster.cor} and \pfun{cluster.loadings}). When finding reliabilities for multiple scales, item reliabilities can be estimated using the squared multiple correlation of an item with all other items, not just those that are keyed for a particular scale. This leads to an estimate of G6*. \subsubsection{Scoring from raw data} To score these five scales from the 25 items, use the \pfun{scoreItems} function with the helper function \pfun{make.keys}. Logically, scales are merely the weighted composites of a set of items. The weights used are -1, 0, and 1. 0 implies do not use that item in the scale, 1 implies a positive weight (add the item to the total score), -1 a negative weight (subtract the item from the total score, i.e., reverse score the item). Reverse scoring an item is equivalent to subtracting the item from the maximum + minimum possible value for that item. The minima and maxima can be estimated from all the items, or can be specified by the user. There are two different ways that scale scores tend to be reported. Social psychologists and educational psychologists tend to report the scale score as the \emph{average item score} while many personality psychologists tend to report the \emph{total item score}. The default option for \pfun{scoreItems} is to report item averages (which thus allows interpretation in the same metric as the items) but totals can be found as well. Personality researchers should be encouraged to report scores based upon item means and avoid using the total score although some reviewers are adamant about the following the tradition of total scores. The printed output includes coefficients $\alpha$ and G6*, the average correlation of the items within the scale (corrected for item overlap and scale relliability), as well as the correlations between the scales (below the diagonal, the correlations above the diagonal are corrected for attenuation. As is the case for most of the \Rpkg{psych} functions, additional information is returned as part of the object. First, create keys matrix using the \pfun{make.keys} function. (The keys matrix could also be prepared externally using a spreadsheet and then copying it into \R{}). Although not normally necessary, show the keys to understand what is happening. Note that the number of items to specify in the \pfun{make.keys} function is the total number of items in the inventory. That is, if scoring just 5 items from a 25 item inventory, \pfun{make.keys} should be told that there are 25 items. \pfun{make.keys} just changes a list of items on each scale to make up a scoring matrix. Because the \pfun{bfi} data set has 25 items as well as 3 demographic items, the number of variables is specified as 28. \begin{scriptsize} <>= keys <- make.keys(nvars=28,list(Agree=c(-1,2:5),Conscientious=c(6:8,-9,-10), Extraversion=c(-11,-12,13:15),Neuroticism=c(16:20), Openness = c(21,-22,23,24,-25)), item.labels=colnames(bfi)) keys @ \end{scriptsize} The use of multiple key matrices for different inventories is facilitated by using the \pfun{superMatrix} function to combine two or more matrices. This allows convenient scoring of large data sets combining multiple inventories with keys based upon each individual inventory. Pretend for the moment that the big 5 items were made up of two inventories, one consisting of the first 10 items, the second the last 18 items. (15 personality items + 3 demographic items.) Then the following code would work: \begin{scriptsize} <>= keys.1<- make.keys(10,list(Agree=c(-1,2:5),Conscientious=c(6:8,-9,-10))) keys.2 <- make.keys(15,list(Extraversion=c(-1,-2,3:5),Neuroticism=c(6:10), Openness = c(11,-12,13,14,-15))) keys.25 <- superMatrix(list(keys.1,keys.2)) @ \end{scriptsize} The resulting keys matrix is identical to that found above except that it does not include the extra 3 demographic items. This is useful when scoring the raw items because the response frequencies for each category are reported, and for the demographic data, This use of making multiple key matrices and then combining them into one super matrix of keys is particularly useful when combining demographic information with items to be scores. A set of demographic keys can be made and then these can be combined with the keys for the particular scales. Now use these keys in combination with the raw data to score the items, calculate basic reliability and intercorrelations, and find the item-by scale correlations for each item and each scale. By default, missing data are replaced by the median for that variable. \begin{scriptsize} <>= scores <- scoreItems(keys,bfi) scores @ \end{scriptsize} To see the additional information (the raw correlations, the individual scores, etc.), they may be specified by name. Then, to visualize the correlations between the raw scores, use the \pfun{pairs.panels} function on the scores values of scores. (See figure~\ref{fig:scores} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= png('scores.png') pairs.panels(scores$scores,pch='.',jiggle=TRUE) dev.off() @ \end{scriptsize} \includegraphics{scores} \caption{A graphic analysis of the Big Five scales found by using the scoreItems function. The pair.wise plot allows us to see that some participants have reached the ceiling of the scale for these 5 items scales. Using the pch='.' option in pairs.panels is recommended when plotting many cases. The data points were ``jittered'' by setting jiggle=TRUE. Jiggling this way shows the density more clearly. To save space, the figure was done as a png. For a clearer figure, save as a pdf.} \label{fig:scores} \end{center} \end{figure} \subsubsection{Forming scales from a correlation matrix} There are some situations when the raw data are not available, but the correlation matrix between the items is available. In this case, it is not possible to find individual scores, but it is possible to find the reliability and intercorrelations of the scales. This may be done using the \pfun{cluster.cor} function or the \pfun{scoreItems} function. The use of a keys matrix is the same as in the raw data case. Consider the same \pfun{bfi} data set, but first find the correlations, and then use \pfun{cluster.cor}. \begin{scriptsize} <>= r.bfi <- cor(bfi,use="pairwise") scales <- cluster.cor(keys,r.bfi) summary(scales) @ \end{scriptsize} To find the correlations of the items with each of the scales (the ``structure" matrix) or the correlations of the items controlling for the other scales (the ``pattern" matrix), use the \pfun{cluster.loadings} function. To do both at once (e.g., the correlations of the scales as well as the item by scale correlations), it is also possible to just use \pfun{scoreItems}. \subsection{Scoring Multiple Choice Items} Some items (typically associated with ability tests) are not themselves mini-scales ranging from low to high levels of expression of the item of interest, but are rather multiple choice where one response is the correct response. Two analyses are useful for this kind of item: examining the response patterns to all the alternatives (looking for good or bad distractors) and scoring the items as correct or incorrect. Both of these operations may be done using the \pfun{score.multiple.choice} function. Consider the 16 example items taken from an online ability test at the Personality Project: \url{http://test.personality-project.org}. This is part of the Synthetic Aperture Personality Assessment (\iemph{SAPA}) study discussed in \cite{rcw:methods,rwr:sapa}. \begin{scriptsize} <>= data(iqitems) iq.keys <- c(4,4,4, 6,6,3,4,4, 5,2,2,4, 3,2,6,7) score.multiple.choice(iq.keys,iqitems) #just convert the items to true or false iq.tf <- score.multiple.choice(iq.keys,iqitems,score=FALSE) describe(iq.tf) #compare to previous results @ \end{scriptsize} Once the items have been scored as true or false (assigned scores of 1 or 0), they made then be scored into multiple scales using the normal \pfun{scoreItems} function. \subsection{Item analysis} Basic item analysis starts with describing the data (\pfun{describe}, finding the number of dimensions using factor analysis (\pfun{fa}) and cluster analysis \pfun{iclust} perhaps using the Very Simple Structure criterion (\pfun{vss}), or perhaps parallel analysis \pfun{fa.parallel}. Item whole correlations may then be found for scales scored on one dimension (\pfun{alpha} or many scales simultaneously (\pfun{scoreItems}). Scales can be modified by changing the keys matrix (i.e., dropping particular items, changing the scale on which an item is to be scored). This analysis can be done on the normal Pearson correlation matrix or by using polychoric correlations. Validities of the scales can be found using multiple correlation of the raw data or based upon correlation matrices using the \pfun{setCor} function. However, more powerful item analysis tools are now available by using Item Response Theory approaches. Although the \pfun{response.frequencies} output from \pfun{score.multiple.choice} is useful to examine in terms of the probability of various alternatives being endorsed, it is even better to examine the pattern of these responses as a function of the underlying latent trait or just the total score. This may be done by using \pfun{irt.responses} (Figure~\ref{fig:irt.response}). \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= data(iqitems) iq.keys <- c(4,4,4, 6,6,3,4,4, 5,2,2,4, 3,2,6,7) scores <- score.multiple.choice(iq.keys,iqitems,score=TRUE,short=FALSE) #note that for speed we can just do this on simple item counts rather than IRT based scores. op <- par(mfrow=c(2,2)) #set this to see the output for multiple items irt.responses(scores$scores,iqitems[1:4],breaks=11) @ \end{scriptsize} \caption{ The pattern of responses to multiple choice ability items can show that some items have poor distractors. This may be done by using the the \pfun{irt.responses} function. A good distractor is one that is negatively related to ability.} \label{fig:irt.response} \end{center} \end{figure} \section{Item Response Theory analysis} The use of Item Response Theory has become is said to be the ``new psychometrics". The emphasis is upon item properties, particularly those of item difficulty or location and item discrimination. These two parameters are easily found from classic techniques when using factor analyses of correlation matrices formed by \pfun{polychoric} or \pfun{tetrachoric} correlations. The \pfun{irt.fa} function does this and then graphically displays item discrimination and item location as well as item and test information (see Figure~\ref{fig:irt}). \subsection{Factor analysis and Item Response Theory} If the correlations of all of the items reflect one underlying latent variable, then factor analysis of the matrix of tetrachoric correlations should allow for the identification of the regression slopes ($\alpha$) of the items on the latent variable. These regressions are, of course just the factor loadings. Item difficulty, $\delta_j$ and item discrimination, $\alpha_j$ may be found from factor analysis of the tetrachoric correlations where $\lambda_j$ is just the factor loading on the first factor and $\tau_j$ is the normal threshold reported by the \pfun{tetrachoric} function. \begin{equation} \delta_j = \frac{D\tau}{\sqrt{1-\lambda_j^2}}, \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\;\; \alpha_j = \frac{\lambda_j}{\sqrt{1-\lambda_j^2}} \label{eq:irt:diff} \end{equation} where D is a scaling factor used when converting to the parameterization of \iemph{logistic} model and is 1.702 in that case and 1 in the case of the normal ogive model. Thus, in the case of the normal model, factor loadings ($\lambda_j$) and item thresholds ($\tau$) are just \begin{equation*} \lambda_j = \frac{\alpha_j}{\sqrt{1+\alpha_j^2}}, \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\;\;\tau_j = \frac{\delta_j}{\sqrt{1+\alpha_j^2}}. \end{equation*} Consider 9 dichotomous items representing one factor but differing in their levels of difficulty \begin{scriptsize} <>= set.seed(17) d9 <- sim.irt(9,1000,-2.,2.,mod="normal") #dichotomous items test <- irt.fa(d9$items) test @ \end{scriptsize} \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= op <- par(mfrow=c(3,1)) plot(test,type="ICC") plot(test,type="IIC") plot(test,type="test") op <- par(mfrow=c(1,1)) @ \end{scriptsize} \caption{A graphic analysis of 9 dichotomous (simulated) items. The top panel shows the probability of item endorsement as the value of the latent trait increases. Items differ in their location (difficulty) and discrimination (slope). The middle panel shows the information in each item as a function of latent trait level. An item is most informative when the probability of endorsement is 50\%. The lower panel shows the total test information. These items form a test that is most informative (most accurate) at the middle range of the latent trait.} \label{fig:irt} \end{center} \end{figure} Similar analyses can be done for polytomous items such as those of the bfi extraversion scale: \begin{scriptsize} <>= data(bfi) e.irt <- irt.fa(bfi[11:15]) e.irt @ \end{scriptsize} The item information functions show that not all of items are equally good (Figure~\ref{fig:e.irt}): \begin{figure}[htbp] \begin{center} <>= e.info <- plot(e.irt,type="IIC") @ \caption{A graphic analysis of 5 extraversion items from the bfi. The curves represent the amount of information in the item as a function of the latent score for an individual. That is, each item is maximally discriminating at a different part of the latent continuum. Print e.info to see the average information for each item.} \label{fig:e.irt} \end{center} \end{figure} These procedures can be generalized to more than one factor by specifying the number of factors in \pfun{irt.fa}. The plots can be limited to those items with discriminations greater than some value of cut. An invisible object is returned when plotting the output from \pfun{irt.fa} that includes the average information for each item that has loadings greater than cut. \begin{scriptsize} <>= print(e.info,sort=TRUE) @ \end{scriptsize} More extensive IRT packages include the \Rpkg{ltm} and \Rpkg{eRm} and should be used for serious Item Response Theory analysis. \subsection{Speeding up analyses} Finding tetrachoric or polychoric correlations is very time consuming. Thus, to speed up the process of analysis, the original correlation matrix is saved as part of the output of both \pfun{irt.fa} and \pfun{omega}. Subsequent analyses may be done by using this correlation matrix. This is done by doing the analysis not on the original data, but rather on the output of the previous analysis. For example, taking the output from the 16 ability items from the \iemph{SAPA} project when scored for True/False using \pfun{score.multiple.choice} we can first do a simple IRT analysis of one factor (Figure~\ref{fig:iq.irt}) and then use that correlation matrix to do an \pfun{omega} analysis to show the sub-structure of the ability items . \begin{figure}[htbp] \begin{tiny} \begin{center} <>= iq.irt <- irt.fa(iq.tf) @ \end{center} \end{tiny} \caption{A graphic analysis of 16 ability items sampled from the \iemph{SAPA} project. The curves represent the amount of information in the item as a function of the latent score for an individual. That is, each item is maximally discriminating at a different part of the latent continuum. Print iq.irt to see the average information for each item. Partly because this is a power test (it is given on the web) and partly because the items have not been carefully chosen, the items are not very discriminating at the high end of the ability dimension.} \label{fig:iq.irt} \end{figure} \begin{scriptsize} <>= iq.irt @ \end{scriptsize} \begin{figure}[htbp] \begin{center} <>= om <- omega(iq.irt$rho,4) @ \caption{An Omega analysis of 16 ability items sampled from the SAPA project. The items represent a general factor as well as four lower level factors. The analysis is done using the tetrachoric correlations found in the previous \pfun{irt.fa} analysis. The four matrix items have some serious problems, which may be seen later when examine the item response functions.} \label{fig:iq.irt} \end{center} \end{figure} \subsection{IRT based scoring} The primary advantage of IRT analyses is examining the item properties (both difficulty and discrimination). With complete data, the scores based upon simple total scores and based upon IRT are practically identical (this may be seen in the examples for \pfun{scoreIrt}). However, when working with data such as those found in the Synthetic Aperture Personality Assessment (\iemph{SAPA}) project, it is advantageous to use IRT based scoring. \iemph{SAPA} data might have 2-3 items/person sampled from scales with 10-20 items. Simply finding the average of the three (classical test theory) fails to consider that the items might differ in either discrimination or in difficulty. The \pfun{scoreIrt} function applies basic IRT to this problem. Consider 1000 randomly generated subjects with scores on 9 true/false items differing in difficulty. Selectively drop the hardest items for the 1/3 lowest subjects, and the 4 easiest items for the 1/3 top subjects (this is a crude example of what tailored testing would do). Then score these subjects: \begin{scriptsize} <>= v9 <- sim.irt(9,1000,-2.,2.,mod="normal") #dichotomous items items <- v9$items test <- irt.fa(items) total <- rowSums(items) ord <- order(total) items <- items[ord,] #now delete some of the data - note that they are ordered by score items[1:333,5:9] <- NA items[334:666,3:7] <- NA items[667:1000,1:4] <- NA scores <- scoreIrt(test,items) unitweighted <- scoreIrt(items=items,keys=rep(1,9)) scores.df <- data.frame(true=v9$theta[ord],scores,unitweighted) colnames(scores.df) <- c("True theta","irt theta","total","fit","rasch","total","fit") @ \end{scriptsize} These results are seen in Figure~\ref{fig:scoreIrt.pdf}. \begin{figure}[htbp] \begin{center} \caption{IRT based scoring and total test scores for 1000 simulated subjects. True theta values are reported and then the IRT and total scoring systems. } <>= pairs.panels(scores.df,pch='.',gap=0) title('Comparing true theta for IRT, Rasch and classically based scoring',line=3) @ \label{fig:scoreIrt.pdf} \end{center} \end{figure} \section{Multilevel modeling} Correlations between individuals who belong to different natural groups (based upon e.g., ethnicity, age, gender, college major, or country) reflect an unknown mixture of the pooled correlation within each group as well as the correlation of the means of these groups. These two correlations are independent and do not allow inferences from one level (the group) to the other level (the individual). When examining data at two levels (e.g., the individual and by some grouping variable), it is useful to find basic descriptive statistics (means, sds, ns per group, within group correlations) as well as between group statistics (over all descriptive statistics, and overall between group correlations). Of particular use is the ability to decompose a matrix of correlations at the individual level into correlations within group and correlations between groups. \subsection{Decomposing data into within and between level correlations using \pfun{statsBy}} There are at least two very powerful packages (\Rpkg{nlme} and \Rpkg{multilevel}) which allow for complex analysis of hierarchical (multilevel) data structures. \pfun{statsBy} is a much simpler function to give some of the basic descriptive statistics for two level models. This follows the decomposition of an observed correlation into the pooled correlation within groups (rwg) and the weighted correlation of the means between groups which is discussed by \cite{pedhazur:97} and by \cite{bliese:09} in the multilevel package. \begin{equation} r_{xy} = \eta_{x_{wg}} * \eta_{y_{wg}} * r_{xy_{wg}} + \eta_{x_{bg}} * \eta_{y_{bg}} * r_{xy_{bg} } \end{equation} where $r_{xy} $ is the normal correlation which may be decomposed into a within group and between group correlations $r_{xy_{wg}}$ and $r_{xy_{bg}} $ and $\eta$ (eta) is the correlation of the data with the within group values, or the group means. \subsection{Generating and displaying multilevel data} \pfun{withinBetween} is an example data set of the mixture of within and between group correlations. The within group correlations between 9 variables are set to be 1, 0, and -1 while those between groups are also set to be 1, 0, -1. These two sets of correlations are crossed such that V1, V4, and V7 have within group correlations of 1, as do V2, V5 and V8, and V3, V6 and V9. V1 has a within group correlation of 0 with V2, V5, and V8, and a -1 within group correlation with V3, V6 and V9. V1, V2, and V3 share a between group correlation of 1, as do V4, V5 and V6, and V7, V8 and V9. The first group has a 0 between group correlation with the second and a -1 with the third group. See the help file for \pfun{withinBetween} to display these data. \pfun{sim.multilevel} will generate simulated data with a multilevel structure. The \pfun{statsBy.boot} function will randomize the grouping variable ntrials times and find the statsBy output. This can take a long time and will produce a great deal of output. This output can then be summarized for relevant variables using the \pfun{statsBy.boot.summary} function specifying the variable of interest. Consider the case of the relationship between various tests of ability when the data are grouped by level of education (statsBy(sat.act)) or when affect data are analyzed within and between an affect manipulation (statsBy(affect) ). \section{Set Correlation and Multiple Regression from the correlation matrix} An important generalization of multiple regression and multiple correlation is \iemph{set correlation} developed by \cite{cohen:set} and discussed by \cite{cohen:03}. Set correlation is a multivariate generalization of multiple regression and estimates the amount of variance shared between two sets of variables. Set correlation also allows for examining the relationship between two sets when controlling for a third set. This is implemented in the \pfun{setCor} function. Set correlation is $$R^{2} = 1 - \prod_{i=1}^n(1-\lambda_{i})$$ where $\lambda_{i}$ is the ith eigen value of the eigen value decomposition of the matrix $$R = R_{xx}^{-1}R_{xy}R_{xx}^{-1}R_{xy}^{-1}.$$ Unfortunately, there are several cases where set correlation will give results that are much too high. This will happen if some variables from the first set are highly related to those in the second set, even though most are not. In this case, although the set correlation can be very high, the degree of relationship between the sets is not as high. In this case, an alternative statistic, based upon the average canonical correlation might be more appropriate. \pfun{setCor} has the additional feature that it will calculate multiple and partial correlations from the correlation or covariance matrix rather than the original data. Consider the correlations of the 6 variables in the \pfun{sat.act} data set. First do the normal multiple regression, and then compare it with the results using \pfun{setCor}. Two things to notice. \pfun{setCor} works on the \emph{correlation} or \emph{covariance} or \emph{raw data} matrix, and thus if using the correlation matrix, will report standardized $\hat{\beta}$ weights. Secondly, it is possible to do several multiple regressions simultaneously. If the number of observations is specified, or if the analysis is done on raw data, statistical tests of significance are applied. For this example, the analysis is done on the correlation matrix rather than the raw data. \begin{scriptsize} <>= C <- cov(sat.act,use="pairwise") model1 <- lm(ACT~ gender + education + age, data=sat.act) summary(model1) @ Compare this with the output from \pfun{setCor}. <>= #compare with mat.regress setCor(c(4:6),c(1:3),C, n.obs=700) @ \end{scriptsize} Note that the \pfun{setCor} analysis also reports the amount of shared variance between the predictor set and the criterion (dependent) set. This set correlation is symmetric. That is, the $R^{2}$ is the same independent of the direction of the relationship. For a much more detailed discussion of \pfun{setCor} see the \href{https://personality-project.org/r/psych/HowTo/mediation.pdf}{mediation, moderation and regression analysis} tutorial. \section{Simulation functions} It is particularly helpful, when trying to understand psychometric concepts, to be able to generate sample data sets that meet certain specifications. By knowing ``truth" it is possible to see how well various algorithms can capture it. Several of the \pfun{sim} functions create artificial data sets with known structures. A number of functions in the psych package will generate simulated data. These functions include \pfun{sim} for a factor simplex, and \pfun{sim.simplex} for a data simplex, \pfun{sim.circ} for a circumplex structure, \pfun{sim.congeneric} for a one factor factor congeneric model, \pfun{sim.dichot} to simulate dichotomous items, \pfun{sim.hierarchical} to create a hierarchical factor model, \pfun{sim.item} is a more general item simulation, \pfun{sim.minor} to simulate major and minor factors, \pfun{sim.omega} to test various examples of omega, \pfun{sim.parallel} to compare the efficiency of various ways of determining the number of factors, \pfun{sim.rasch} to create simulated rasch data, \pfun{sim.irt} to create general 1 to 4 parameter IRT data by calling \pfun{sim.npl} 1 to 4 parameter logistic IRT or \pfun{sim.npn} 1 to 4 paramater normal IRT, \pfun{sim.structural} a general simulation of structural models, and \pfun{sim.anova} for ANOVA and lm simulations, and \pfun{sim.vss}. Some of these functions are separately documented and are listed here for ease of the help function. See each function for more detailed help. \begin{description} \item [\pfun{sim}] The default version is to generate a four factor simplex structure over three occasions, although more general models are possible. \item [\pfun{sim.simple}] Create major and minor factors. The default is for 12 variables with 3 major factors and 6 minor factors. \item [\pfun{sim.structure}] To combine a measurement and structural model into one data matrix. Useful for understanding structural equation models. \item [\pfun{sim.hierarchical}] To create data with a hierarchical (bifactor) structure. \item [\pfun{sim.congeneric}] To create congeneric items/tests for demonstrating classical test theory. This is just a special case of sim.structure. \item [\pfun{sim.circ}] To create data with a circumplex structure. \item [\pfun{sim.item}]To create items that either have a simple structure or a circumplex structure. \item [\pfun{sim.dichot}] Create dichotomous item data with a simple or circumplex structure. \item[\pfun{sim.rasch}] Simulate a 1 parameter logistic (Rasch) model. \item[\pfun{sim.irt}] Simulate a 2 parameter logistic (2PL) or 2 parameter Normal model. Will also do 3 and 4 PL and PN models. \item[\pfun{sim.multilevel}] Simulate data with different within group and between group correlational structures. \end{description} Some of these functions are described in more detail in the companion vignette: \href{"psych_for_sem.pdf"}{psych for sem}. The default values for \pfun{sim.structure} is to generate a 4 factor, 12 variable data set with a simplex structure between the factors. Two data structures that are particular challenges to exploratory factor analysis are the simplex structure and the presence of minor factors. Simplex structures \pfun{sim.simplex} will typically occur in developmental or learning contexts and have a correlation structure of r between adjacent variables and $r^n$ for variables n apart. Although just one latent variable (r) needs to be estimated, the structure will have nvar-1 factors. Many simulations of factor structures assume that except for the major factors, all residuals are normally distributed around 0. An alternative, and perhaps more realistic situation, is that the there are a few major (big) factors and many minor (small) factors. The challenge is thus to identify the major factors. \pfun{sim.minor} generates such structures. The structures generated can be thought of as having a a major factor structure with some small correlated residuals. Although coefficient $\omega_h$ is a very useful indicator of the general factor saturation of a unifactorial test (one with perhaps several sub factors), it has problems with the case of multiple, independent factors. In this situation, one of the factors is labelled as ``general'' and the omega estimate is too large. This situation may be explored using the \pfun{sim.omega} function. The four irt simulations, \pfun{sim.rasch}, \pfun{sim.irt}, \pfun{sim.npl} and \pfun{sim.npn}, simulate dichotomous items following the Item Response model. \pfun{sim.irt} just calls either \pfun{sim.npl} (for logistic models) or \pfun{sim.npn} (for normal models) depending upon the specification of the model. The logistic model is \begin{equation} P(x | \theta_i, \delta_j, \gamma_j, \zeta_j )= \gamma_j + \frac{\zeta_j - \gamma_j}{1+e^{\alpha_j(\delta_j - \theta_i}}. \end{equation} where $\gamma$ is the lower asymptote or guessing parameter, $\zeta$ is the upper asymptote (normally 1), $\alpha_j$ is item discrimination and $\delta_j$ is item difficulty. For the 1 Paramater Logistic (Rasch) model, gamma=0, zeta=1, alpha=1 and item difficulty is the only free parameter to specify. (Graphics of these may be seen in the demonstrations for the logistic function.) The normal model (\pfun{irt.npn} calculates the probability using \fun{pnorm} instead of the logistic function used in \pfun{irt.npl}, but the meaning of the parameters are otherwise the same. With the a = $\alpha$ parameter = 1.702 in the logiistic model the two models are practically identical. \section{Graphical Displays} Many of the functions in the \Rpkg{psych} package include graphic output and examples have been shown in the previous figures. After running \pfun{fa}, \pfun{iclust}, \pfun{omega}, \pfun{irt.fa}, plotting the resulting object is done by the \pfun{plot.psych} function as well as specific diagram functions. e.g., (but not shown) \begin{scriptsize} \begin{Schunk} \begin{Sinput} f3 <- fa(Thurstone,3) plot(f3) fa.diagram(f3) c <- iclust(Thurstone) plot(c) #a pretty boring plot iclust.diagram(c) #a better diagram c3 <- iclust(Thurstone,3) plot(c3) #a more interesting plot data(bfi) e.irt <- irt.fa(bfi[11:15]) plot(e.irt) ot <- omega(Thurstone) plot(ot) omega.diagram(ot) \end{Sinput} \end{Schunk} \end{scriptsize} The ability to show path diagrams to represent factor analytic and structural models is discussed in somewhat more detail in the accompanying vignette, \href{"psych_for_sem.pdf"}{psych for sem}. Basic routines to draw path diagrams are included in the \pfun{dia.rect} and accompanying functions. These are used by the \pfun{fa.diagram}, \pfun{structure.diagram} and \pfun{iclust.diagram} functions. \begin{figure}[htbp] \begin{center} \begin{scriptsize} <>= xlim=c(0,10) ylim=c(0,10) plot(NA,xlim=xlim,ylim=ylim,main="Demontration of dia functions",axes=FALSE,xlab="",ylab="") ul <- dia.rect(1,9,labels="upper left",xlim=xlim,ylim=ylim) ll <- dia.rect(1,3,labels="lower left",xlim=xlim,ylim=ylim) lr <- dia.ellipse(9,3,"lower right",xlim=xlim,ylim=ylim) ur <- dia.ellipse(9,9,"upper right",xlim=xlim,ylim=ylim) ml <- dia.ellipse(3,6,"middle left",xlim=xlim,ylim=ylim) mr <- dia.ellipse(7,6,"middle right",xlim=xlim,ylim=ylim) bl <- dia.ellipse(1,1,"bottom left",xlim=xlim,ylim=ylim) br <- dia.rect(9,1,"bottom right",xlim=xlim,ylim=ylim) dia.arrow(from=lr,to=ul,labels="right to left") dia.arrow(from=ul,to=ur,labels="left to right") dia.curved.arrow(from=lr,to=ll$right,labels ="right to left") dia.curved.arrow(to=ur,from=ul$right,labels ="left to right") dia.curve(ll$top,ul$bottom,"double") #for rectangles, specify where to point dia.curved.arrow(mr,ur,"up") #but for ellipses, just point to it. dia.curve(ml,mr,"across") dia.arrow(ur,lr,"top down") dia.curved.arrow(br$top,lr$bottom,"up") dia.curved.arrow(bl,br,"left to right") dia.arrow(bl,ll$bottom) dia.curved.arrow(ml,ll$right) dia.curved.arrow(mr,lr$top) @ \end{scriptsize} \caption{The basic graphic capabilities of the dia functions are shown in this figure.} \label{fig:dia} \end{center} \end{figure} \section{Miscellaneous functions} A number of functions have been developed for some very specific problems that don't fit into any other category. The following is an incomplete list. Look at the \iemph{Index} for \Rpkg{psych} for a list of all of the functions. \begin{description} \item [\pfun{block.random}] Creates a block randomized structure for n independent variables. Useful for teaching block randomization for experimental design. \item [\pfun{df2latex}] is useful for taking tabular output (such as a correlation matrix or that of \pfun{describe} and converting it to a \LaTeX{} table. May be used when Sweave is not convenient. \item [\pfun{cor2latex}] Will format a correlation matrix in APA style in a \LaTeX{} table. See also \pfun{fa2latex} and \pfun{irt2latex}. \item [\pfun{cosinor}] One of several functions for doing \iemph{circular statistics}. This is important when studying mood effects over the day which show a diurnal pattern. See also \pfun{circadian.mean}, \pfun{circadian.cor} and \pfun{circadian.linear.cor} for finding circular means, circular correlations, and correlations of circular with linear data. \item[\pfun{fisherz}] Convert a correlation to the corresponding Fisher z score. \item [\pfun{geometric.mean}] also \pfun{harmonic.mean} find the appropriate mean for working with different kinds of data. \item [\pfun{ICC}] and \pfun{cohen.kappa} are typically used to find the reliability for raters. \item [\pfun{headtail}] combines the \fun{head} and \fun{tail} functions to show the first and last lines of a data set or output. \item [\pfun{topBottom}] Same as headtail. Combines the \fun{head} and \fun{tail} functions to show the first and last lines of a data set or output, but does not add ellipsis between. \item [\pfun{mardia}] calculates univariate or multivariate (Mardia's test) skew and kurtosis for a vector, matrix, or data.frame \item [\pfun{p.rep}] finds the probability of replication for an F, t, or r and estimate effect size. \item [\pfun{partial.r}] partials a y set of variables out of an x set and finds the resulting partial correlations. (See also \pfun{setCor}.) \item [\pfun{rangeCorrection}] will correct correlations for restriction of range. \item [\pfun{reverse.code}] will reverse code specified items. Done more conveniently in most \Rpkg{psych} functions, but supplied here as a helper function when using other packages. \item [\pfun{superMatrix}] Takes two or more matrices, e.g., A and B, and combines them into a ``Super matrix'' with A on the top left, B on the lower right, and 0s for the other two quadrants. A useful trick when forming complex keys, or when forming example problems. \end{description} \section{Data sets} A number of data sets for demonstrating psychometric techniques are included in the \Rpkg{psych} package. These include six data sets showing a hierarchical factor structure (five cognitive examples, \pfun{Thurstone}, \pfun{Thurstone.33}, \pfun{Holzinger}, \pfun{Bechtoldt.1}, \pfun{Bechtoldt.2}, and one from health psychology \pfun{Reise}). One of these (\pfun{Thurstone}) is used as an example in the \Rpkg{sem} package as well as \cite{mcdonald:tt}. The original data are from \cite{thurstone:41} and reanalyzed by \cite{bechtoldt:61}. Personality item data representing five personality factors on 25 items (\pfun{bfi}) or 13 personality inventory scores (\pfun{epi.bfi}), and 14 multiple choice iq items (\pfun{iqitems}). The \pfun{vegetables} example has paired comparison preferences for 9 vegetables. This is an example of Thurstonian scaling used by \cite{guilford:54} and \cite{nunnally:67}. Other data sets include \pfun{cubits}, \pfun{peas}, and \pfun{heights} from Galton. \begin{description} \item[Thurstone] Holzinger-Swineford (1937) introduced the bifactor model of a general factor and uncorrelated group factors. The Holzinger correlation matrix is a 14 * 14 matrix from their paper. The Thurstone correlation matrix is a 9 * 9 matrix of correlations of ability items. The Reise data set is 16 * 16 correlation matrix of mental health items. The Bechtholdt data sets are both 17 x 17 correlation matrices of ability tests. \item [bfi] 25 personality self report items taken from the International Personality Item Pool (ipip.ori.org) were included as part of the Synthetic Aperture Personality Assessment (\iemph{SAPA}) web based personality assessment project. The data from 2800 subjects are included here as a demonstration set for scale construction, factor analysis and Item Response Theory analyses. \item [sat.act] Self reported scores on the SAT Verbal, SAT Quantitative and ACT were collected as part of the Synthetic Aperture Personality Assessment (\iemph{SAPA}) web based personality assessment project. Age, gender, and education are also reported. The data from 700 subjects are included here as a demonstration set for correlation and analysis. \item [epi.bfi] A small data set of 5 scales from the Eysenck Personality Inventory, 5 from a Big 5 inventory, a Beck Depression Inventory, and State and Trait Anxiety measures. Used for demonstrations of correlations, regressions, graphic displays. \item [iq] 14 multiple choice ability items were included as part of the Synthetic Aperture Personality Assessment (\iemph{SAPA}) web based personality assessment project. The data from 1000 subjects are included here as a demonstration set for scoring multiple choice inventories and doing basic item statistics. \item [galton] Two of the earliest examples of the correlation coefficient were Francis Galton's data sets on the relationship between mid parent and child height and the similarity of parent generation peas with child peas. \pfun{galton} is the data set for the Galton height. \pfun{peas} is the data set Francis Galton used to ntroduce the correlation coefficient with an analysis of the similarities of the parent and child generation of 700 sweet peas. \item[Dwyer] \cite{dwyer:37} introduced a method for \emph{factor extension} (see \pfun{fa.extension} that finds loadings on factors from an original data set for additional (extended) variables. This data set includes his example. \item [miscellaneous] \pfun{cities} is a matrix of airline distances between 11 US cities and may be used for demonstrating multiple dimensional scaling. \pfun{vegetables} is a classic data set for demonstrating Thurstonian scaling and is the preference matrix of 9 vegetables from \cite{guilford:54}. Used by \cite{guilford:54,nunnally:67,nunnally:bernstein:84}, this data set allows for examples of basic scaling techniques. \end{description} \section{Development version and a users guide} The most recent development version is available as a source file at the repository maintained at \href{ href="http://personality-project.org/r"}{\url{http://personality-project.org/r}}. That version will have removed the most recently discovered bugs (but perhaps introduced other, yet to be discovered ones). To download and install that version for either Macs or PCs: \begin{Rinput} install.packages("psych",repos="http://personality-project.org/r", type="source") \end{Rinput} Although the individual help pages for the \Rpkg{psych} package are available as part of \R{} and may be accessed directly (e.g. ?psych) , the full manual for the \pfun{psych} package is also available as a pdf at \url{http://personality-project.org/r/psych_manual.pdf} %psych\_manual.pdf. News and a history of changes are available in the NEWS and CHANGES files in the source files. To view the most recent news, \begin{Rinput} news(Version > "1.2.8",package="psych") \end{Rinput} \section{Psychometric Theory} The \Rpkg{psych} package has been developed to help psychologists do basic research. Many of the functions were developed to supplement a book (\url{http://personality-project.org/r/book} An introduction to Psychometric Theory with Applications in \R{} \citep{revelle:intro} More information about the use of some of the functions may be found in the book . For more extensive discussion of the use of \Rpkg{psych} in particular and \R{} in general, consult \url{http://personality-project.org/r/r.guide.html} A short guide to R. \section{SessionInfo} This document was prepared using the following settings. \begin{tiny} <>= sessionInfo() @ \end{tiny} \newpage %\bibliography{/Volumes/WR/Documents/Active/book/all} %\bibliography{../../../../all} \begin{thebibliography}{} \bibitem[\protect\astroncite{Bechtoldt}{1961}]{bechtoldt:61} Bechtoldt, H. (1961). \newblock An empirical study of the factor analysis stability hypothesis. \newblock {\em Psychometrika}, 26(4):405--432. \bibitem[\protect\astroncite{Blashfield}{1980}]{blashfield:80} Blashfield, R.~K. 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(1997). \newblock {\em Multiple regression in behavioral research: explanation and prediction}. \newblock Harcourt Brace College Publishers. \bibitem[\protect\astroncite{{R Core Team}}{2019}]{R} {R Core Team} (2019). \newblock {\em R: A Language and Environment for Statistical Computing}. \newblock R Foundation for Statistical Computing, Vienna, Austria. \bibitem[\protect\astroncite{Revelle}{1979}]{revelle:iclust} Revelle, W. (1979). \newblock Hierarchical cluster-analysis and the internal structure of tests. \newblock {\em Multivariate Behavioral Research}, 14(1):57--74. \bibitem[\protect\astroncite{Revelle}{2020}]{psych} Revelle, W. (2020). \newblock {\em psych: Procedures for Personality and Psychological Research}. \newblock Northwestern University, Evanston, https://CRAN.r-project.org/package=psych. \newblock R package version 2.0.8 \bibitem[\protect\astroncite{Revelle}{prep}]{revelle:intro} Revelle, W. ({in prep}). \newblock {\em An introduction to psychometric theory with applications in {R}}. \newblock Springer. \bibitem[\protect\astroncite{Revelle et~al.}{2011}]{rcw:methods} Revelle, W., Condon, D., and Wilt, J. (2011). \newblock Methodological advances in differential psychology. \newblock In Chamorro-Premuzic, T., Furnham, A., and von Stumm, S., editors, {\em Handbook of Individual Differences}, chapter~2, pages 39--73. Wiley-Blackwell. \bibitem[\protect\astroncite{Revelle and Rocklin}{1979}]{revelle:vss} Revelle, W. and Rocklin, T. (1979). \newblock {Very Simple Structure} - alternative procedure for estimating the optimal number of interpretable factors. \newblock {\em Multivariate Behavioral Research}, 14(4):403--414. \bibitem[\protect\astroncite{Revelle et~al.}{2010}]{rwr:sapa} Revelle, W., Wilt, J., and Rosenthal, A. (2010). \newblock Individual differences in cognition: New methods for examining the personality-cognition link. \newblock In Gruszka, A., Matthews, G., and Szymura, B., editors, {\em Handbook of Individual Differences in Cognition: Attention, Memory and Executive Control}, chapter~2, pages 27--49. Springer, New York, N.Y. \bibitem[\protect\astroncite{Revelle and Zinbarg}{2009}]{rz:09} Revelle, W. and Zinbarg, R.~E. (2009). \newblock Coefficients alpha, beta, omega and the glb: comments on {Sijtsma}. \newblock {\em Psychometrika}, 74(1):145--154. \bibitem[\protect\astroncite{Schmid and Leiman}{1957}]{schmid:57} Schmid, J.~J. and Leiman, J.~M. (1957). \newblock The development of hierarchical factor solutions. \newblock {\em Psychometrika}, 22(1):83--90. \bibitem[\protect\astroncite{Shrout and Fleiss}{1979}]{shrout:79} Shrout, P.~E. and Fleiss, J.~L. (1979). \newblock Intraclass correlations: Uses in assessing rater reliability. \newblock {\em Psychological Bulletin}, 86(2):420--428. \bibitem[\protect\astroncite{Sneath and Sokal}{1973}]{sneath:73} Sneath, P. H.~A. and Sokal, R.~R. (1973). \newblock {\em Numerical taxonomy: the principles and practice of numerical classification}. \newblock A Series of books in biology. W. H. Freeman, San Francisco. \bibitem[\protect\astroncite{Sokal and Sneath}{1963}]{sokal:63} Sokal, R.~R. and Sneath, P. H.~A. (1963). \newblock {\em Principles of numerical taxonomy}. \newblock A Series of books in biology. W. H. Freeman, San Francisco. \bibitem[\protect\astroncite{Spearman}{1904}]{spearman:rho} Spearman, C. (1904). \newblock The proof and measurement of association between two things. \newblock {\em The American Journal of Psychology}, 15(1):72--101. \bibitem[\protect\astroncite{ten Berge et~al.}{1999}]{tenBerge.99} ten Berge, J.~M., Krijnen, W.~P., Wansbeek, T., and Shapiro, A. (1999). \newblock Some new results on correlation-preserving factor scores prediction methods. \newblock {\em Linear Algebra and its Applications}, 289(1-3):311 -- 318. \bibitem[\protect\astroncite{Thorburn}{1918}]{thornburn:1918} Thorburn, W.~M. (1918). \newblock The myth of {Occam's} razor. \newblock {\em Mind}, 27:345--353. \bibitem[\protect\astroncite{Thurstone and Thurstone}{1941}]{thurstone:41} Thurstone, L.~L. and Thurstone, T.~G. (1941). \newblock {\em Factorial studies of intelligence}. \newblock The University of Chicago press, Chicago, Ill. \bibitem[\protect\astroncite{Tryon}{1935}]{tryon:35} Tryon, R.~C. (1935). \newblock A theory of psychological components--an alternative to "mathematical factors.". \newblock {\em Psychological Review}, 42(5):425--454. \bibitem[\protect\astroncite{Tryon}{1939}]{tryon:39} Tryon, R.~C. (1939). \newblock {\em Cluster analysis}. \newblock Edwards Brothers, Ann Arbor, Michigan. \bibitem[\protect\astroncite{Velicer}{1976}]{velicer:76} Velicer, W. (1976). \newblock Determining the number of components from the matrix of partial correlations. \newblock {\em Psychometrika}, 41(3):321--327. \bibitem[\protect\astroncite{Zinbarg et~al.}{2005}]{zinbarg:pm:05} Zinbarg, R.~E., Revelle, W., Yovel, I., and Li, W. (2005). \newblock Cronbach's {$\alpha$}, {Revelle's} {$\beta$}, and {McDonald's} {$\omega_H$}: Their relations with each other and two alternative conceptualizations of reliability. \newblock {\em Psychometrika}, 70(1):123--133. \bibitem[\protect\astroncite{Zinbarg et~al.}{2006}]{zinbarg:apm:06} Zinbarg, R.~E., Yovel, I., Revelle, W., and McDonald, R.~P. (2006). \newblock Estimating generalizability to a latent variable common to all of a scale's indicators: A comparison of estimators for {$\omega_h$}. \newblock {\em Applied Psychological Measurement}, 30(2):121--144. \end{thebibliography} \printindex \end{document} psychTools/inst/doc/mediation.R0000644000176200001440000002572314153443050016247 0ustar liggesusers### R code from vignette source 'mediation.rnw' ################################################### ### code chunk number 1: library ################################################### library(psych) library(psychTools) ################################################### ### code chunk number 2: attitude ################################################### psych::describe(attitude) ################################################### ### code chunk number 3: attitude ################################################### #do not standardize mod1 <- setCor(rating ~ complaints + privileges, data=attitude,std=FALSE) mod1 ################################################### ### code chunk number 4: attitudelm ################################################### summary(lm(rating ~ complaints + privileges, data=attitude)) ################################################### ### code chunk number 5: attitude ################################################### png('attitude.png') # standardize by default mod2 <- setCor(rating ~ complaints + privileges, data=attitude) mod2 setCor.diagram(mod2, main="A simple regression model") dev.off() ################################################### ### code chunk number 6: attitudeR ################################################### R <- lowerCor(attitude) setCor(rating ~ complaints + privileges, data=R, n.obs =30) ################################################### ### code chunk number 7: kelley ################################################### #the second Kelley data from Hotelling kelley <- structure(list(speed = c(1, 0.4248, 0.042, 0.0215, 0.0573), power = c(0.4248, 1, 0.1487, 0.2489, 0.2843), words = c(0.042, 0.1487, 1, 0.6693, 0.4662), symbols = c(0.0215, 0.2489, 0.6693, 1, 0.6915), meaningless = c(0.0573, 0.2843, 0.4662, 0.6915, 1)), .Names = c("speed", "power", "words", "symbols", "meaningless"), class = "data.frame", row.names = c("speed", "power", "words", "symbols", "meaningless")) #first show the correlations lowerMat(kelley) #now find and draw the regression sc <- setCor(power + speed ~ words + symbols + meaningless,data=kelley) #formula mode sc #show it ################################################### ### code chunk number 8: kelly ################################################### png('hotelling.png') setCor.diagram(sc, main="The Kelley data set") dev.off() ################################################### ### code chunk number 9: satact ################################################### png('satact.png') mod3 <- setCor(SATV + SATQ + ACT ~ gender + education + age, data = sat.act) dev.off() ################################################### ### code chunk number 10: moderation ################################################### mod <- setCor(SATQ ~ SATV*gender + ACT, data=sat.act, std =FALSE, plot=FALSE) mod mod0 <- setCor(SATQ ~ SATV*gender + ACT, data=sat.act, zero=FALSE, std=FALSE,plot=FALSE) mod0 ################################################### ### code chunk number 11: modplot ################################################### png('moderation.png') setCor.diagram(mod, main="not zero centered") dev.off() ################################################### ### code chunk number 12: modplot ################################################### png('moderation0.png') setCor.diagram(mod0, main="zero centered") dev.off() ################################################### ### code chunk number 13: plotting ################################################### both <- cbind(mod$data[-1],mod0$data[-1]) png('splom.png') pairs.panels(both[,-c(4,9)],gap=0) #show the mean centered data dev.off() ################################################### ### code chunk number 14: setcorvslm ################################################### lm(SATQ ~ SATV*gender + ACT, data=sat.act) mod <- setCor(SATQ ~ SATV*gender + ACT, data=(sat.act), zero=FALSE, std=FALSE,use="complete") print(mod,digits=5) ################################################### ### code chunk number 15: Tak_Or ################################################### data(Tal.Or) psych::describe(Tal_Or) #descriptive statistics mod4.4 <- mediate(reaction ~ cond + (pmi), data =Tal_Or) mod4.4 #print(mod4.4, digits = 4) # in order to get the precision of the Hayes (2013) p 99 example ################################################### ### code chunk number 16: ned99 ################################################### png('mediate99.png') mediate.diagram(mod4.4) dev.off() ################################################### ### code chunk number 17: garcia ################################################### data(GSBE) #alias to Garcia data set #compare two models (bootstrapping n.iter set to 50 for speed # 1) mean center the variables prior to taking product terms mod1 <- mediate(respappr ~ prot2 * sexism +(sexism),data=Garcia,n.iter=50 ,main="Moderated mediation (mean centered)") # 2) do not mean center mod2 <- mediate(respappr ~ prot2 * sexism +(sexism),data=Garcia,zero=FALSE, n.iter=50, main="Moderated mediation (not centered") summary(mod1) summary(mod2) ################################################### ### code chunk number 18: Tal.or ################################################### mod5.4 <- mediate(reaction ~ cond + (import) + (pmi), data = Tal_Or) print(mod5.4, digits=4) #to compare with Hayes ################################################### ### code chunk number 19: ned131 ################################################### png('mediate131.png') mediate.diagram(mod5.4, digits=3, main="Hayes example 5.3") dev.off() ################################################### ### code chunk number 20: ned131 ################################################### png('mediate131.png') mediate.diagram(mod5.4, digits=3, main="Hayes example 5.3") dev.off() ################################################### ### code chunk number 21: Tal.or54 ################################################### #model 5.4 + mod5.7 is the two chained mediator model mod5.7 <- mediate(pmi ~ cond + (import) , data = Tal_Or) summary(mod5.7, digits=4) ################################################### ### code chunk number 22: Pollack ################################################### lowerMat(Pollack) mod6.2 <- mediate(withdrawal ~ economic.stress + self.efficacy + sex + tenure + (depression), data=Pollack, n.obs=262) summary(mod6.2) ################################################### ### code chunk number 23: Pollackgraph ################################################### png('mediate177.png') mediate.diagram(mod6.2, digits=3, main = "Simple mediation, 3 covariates") dev.off() ################################################### ### code chunk number 24: Pollack ################################################### mod6.2a <- mediate(withdrawal ~ economic.stress -self.efficacy - sex - tenure + (depression), data=Pollack, n.obs=262) summary(mod6.2a) ################################################### ### code chunk number 25: Pollackgraph ################################################### png('mod62partial.png') mediate.diagram(mod6.2a, digits=3, main = "Simple mediation, 3 covariates (partialled out)") dev.off() ################################################### ### code chunk number 26: interacions ################################################### summary(lm(respappr ~ prot2 * sexism,data = Garcia)) #show the lm results for comparison #show the setCor analysis setCor(respappr ~ prot2* sexism ,data=Garcia,zero=FALSE,main="Moderation",std=FALSE) #then show the mediate results modgarcia <-mediate(respappr ~ prot2 * sexism +(sexism),data=Garcia,zero=FALSE,main="Moderated mediation") summary(modgarcia) ################################################### ### code chunk number 27: interacionsplot ################################################### png('moderatedmediation.png') mediate.diagram(modgarcia, main= "An example of moderated mediation") dev.off() ################################################### ### code chunk number 28: zeri ################################################### lm(govact ~ age * negemot + posemot + ideology + sex, data=globalWarm) # but zero center and try again glbwarmc <-data.frame(scale(globalWarm,scale=FALSE)) lm(govact ~ age * negemot + posemot + ideology + sex, data=glbwarmc) mod.glb <- setCor(govact ~ age * negemot + posemot + ideology + sex, data=globalWarm,zero=FALSE,std=FALSE) print(mod.glb,digits=6) mod.glb0 <- setCor(govact ~ age * negemot + posemot + ideology + sex, data=globalWarm,std=FALSE) print(mod.glb0,digits=6) ################################################### ### code chunk number 29: izero2 ################################################### #by default, mediate zero centers before finding the products mod.glb <- mediate(govact ~ age * negemot + posemot + ideology + sex + (age), data=globalWarm,zero=TRUE) summary(mod.glb,digits=4) ################################################### ### code chunk number 30: garcia2t ################################################### psych::describe(Garcia) lm(liking ~ prot2* sexism + respappr, data=Garcia) setCor(liking ~ prot2* sexism + respappr, data = Garcia, zero=FALSE,std=FALSE) mod7.4 <- mediate(liking ~ prot2 * sexism +respappr, data = Garcia, zero=FALSE) summary(mod7.4) ################################################### ### code chunk number 31: modertionplot ################################################### png('moderation.png') mediate.diagram(mod7.4, main= "Another example of moderated mediation") dev.off() ################################################### ### code chunk number 32: modertionplot ################################################### png('garciainteraction.png') plot(respappr ~ sexism, pch = 23- protest, bg = c("black","red", "blue")[protest], data=Garcia, main = "Response to sexism varies as type of protest") by(Garcia,Garcia$protest, function(x) abline(lm(respappr ~ sexism, data =x),lty=c("solid","dashed","dotted")[x$protest+1])) text(6.5,3.5,"No protest") text(3,3.9,"Individual") text(3,5.2,"Collective") dev.off() ################################################### ### code chunk number 33: oartial ################################################### #first, the more complicated model mod.glb <- setCor(govact ~ age * negemot + posemot + ideology + sex, data=globalWarm,std=FALSE) print(mod.glb,digits=3) # compare this to the partialled model mod.glb.partialled <- setCor(govact ~ age * negemot - posemot - ideology - sex,data = globalWarm) ################################################### ### code chunk number 34: partial.all ################################################### upper <-partial.r(globalWarm) lowerMat(upper) #show it lower <- lowerCor(globalWarm) lowup <- lowerUpper(lower,upper) ################################################### ### code chunk number 35: partial.plot ################################################### png('partials.png') psych::corPlot(lowup,numbers = TRUE) dev.off() ################################################### ### code chunk number 36: sessionInfo ################################################### sessionInfo() psychTools/inst/doc/overview.R0000644000176200001440000004040114153443073016137 0ustar liggesusers### R code from vignette source 'overview.Rnw' ################################################### ### code chunk number 1: overview.Rnw:448-449 ################################################### if(!require('GPArotation')) {stop('GPArotation must be installed to do rotations')} ################################################### ### code chunk number 2: overview.Rnw:457-462 ################################################### if(!require('GPArotation')) {stop('GPArotation must be installed to do rotations')} else { library(psych) library(psychTools) f3t <- fa(Thurstone,3,n.obs=213) f3t } ################################################### ### code chunk number 3: overview.Rnw:483-487 ################################################### if(!require('GPArotation')) {stop('GPArotation must be installed to do rotations')} else { f3 <- fa(Thurstone,3,n.obs = 213,fm="pa") f3o <- target.rot(f3) f3o} ################################################### ### code chunk number 4: overview.Rnw:510-512 ################################################### f3w <- fa(Thurstone,3,n.obs = 213,fm="wls") print(f3w,cut=0,digits=3) ################################################### ### code chunk number 5: overview.Rnw:525-526 ################################################### plot(f3t) ################################################### ### code chunk number 6: overview.Rnw:538-539 ################################################### fa.diagram(f3t) ################################################### ### code chunk number 7: overview.Rnw:558-560 ################################################### p3p <-principal(Thurstone,3,n.obs = 213,rotate="Promax") p3p ################################################### ### code chunk number 8: overview.Rnw:579-581 ################################################### om.h <- omega(Thurstone,n.obs=213,sl=FALSE) op <- par(mfrow=c(1,1)) ################################################### ### code chunk number 9: overview.Rnw:592-593 ################################################### om <- omega(Thurstone,n.obs=213) ################################################### ### code chunk number 10: overview.Rnw:626-628 ################################################### data(bfi) ic <- iclust(bfi[1:25]) ################################################### ### code chunk number 11: overview.Rnw:640-641 ################################################### summary(ic) #show the results ################################################### ### code chunk number 12: overview.Rnw:654-656 ################################################### data(bfi) r.poly <- polychoric(bfi[1:25],correct=0) #the ... indicate the progress of the function ################################################### ### code chunk number 13: overview.Rnw:668-670 ################################################### ic.poly <- iclust(r.poly$rho,title="ICLUST using polychoric correlations") iclust.diagram(ic.poly) ################################################### ### code chunk number 14: overview.Rnw:681-683 ################################################### ic.poly <- iclust(r.poly$rho,5,title="ICLUST using polychoric correlations for nclusters=5") iclust.diagram(ic.poly) ################################################### ### code chunk number 15: overview.Rnw:694-695 ################################################### ic.poly <- iclust(r.poly$rho,beta.size=3,title="ICLUST beta.size=3") ################################################### ### code chunk number 16: overview.Rnw:707-708 ################################################### print(ic,cut=.3) ################################################### ### code chunk number 17: overview.Rnw:731-733 ################################################### fa(bfi[1:10],2,n.iter=20) ################################################### ### code chunk number 18: overview.Rnw:746-748 ################################################### f4 <- fa(bfi[1:25],4,fm="pa") factor.congruence(f4,ic) ################################################### ### code chunk number 19: overview.Rnw:757-758 ################################################### factor.congruence(list(f3t,f3o,om,p3p)) ################################################### ### code chunk number 20: overview.Rnw:802-803 ################################################### vss <- vss(bfi[1:25],title="Very Simple Structure of a Big 5 inventory") ################################################### ### code chunk number 21: overview.Rnw:811-812 ################################################### vss ################################################### ### code chunk number 22: overview.Rnw:822-823 ################################################### fa.parallel(bfi[1:25],main="Parallel Analysis of a Big 5 inventory") ################################################### ### code chunk number 23: overview.Rnw:843-848 ################################################### v16 <- sim.item(16) s <- c(1,3,5,7,9,11,13,15) f2 <- fa(v16[,s],2) fe <- fa.extension(cor(v16)[s,-s],f2) fa.diagram(f2,fe=fe) ################################################### ### code chunk number 24: overview.Rnw:864-871 ################################################### fx <-matrix(c( .9,.8,.6,rep(0,4),.6,.8,-.7),ncol=2) fy <- matrix(c(.6,.5,.4),ncol=1) rownames(fx) <- c("V","Q","A","nach","Anx") rownames(fy)<- c("gpa","Pre","MA") Phi <-matrix( c(1,0,.7,.0,1,.7,.7,.7,1),ncol=3) gre.gpa <- sim.structural(fx,Phi,fy) print(gre.gpa) ################################################### ### code chunk number 25: overview.Rnw:877-879 ################################################### esem.example <- esem(gre.gpa$model,varsX=1:5,varsY=6:8,nfX=2,nfY=1,n.obs=1000,plot=FALSE) esem.example ################################################### ### code chunk number 26: overview.Rnw:884-885 ################################################### esem.diagram(esem.example) ################################################### ### code chunk number 27: overview.Rnw:938-942 ################################################### set.seed(17) r9 <- sim.hierarchical(n=500,raw=TRUE)$observed round(cor(r9),2) alpha(r9) ################################################### ### code chunk number 28: overview.Rnw:949-952 ################################################### alpha(attitude,keys=c("complaints","critical")) ################################################### ### code chunk number 29: overview.Rnw:959-961 ################################################### alpha(attitude) ################################################### ### code chunk number 30: overview.Rnw:968-970 ################################################### items <- sim.congeneric(N=500,short=FALSE,low=-2,high=2,categorical=TRUE) #500 responses to 4 discrete items alpha(items$observed) #item response analysis of congeneric measures ################################################### ### code chunk number 31: overview.Rnw:1023-1024 ################################################### om.9 <- omega(r9,title="9 simulated variables") ################################################### ### code chunk number 32: overview.Rnw:1035-1036 ################################################### om.9 ################################################### ### code chunk number 33: overview.Rnw:1044-1045 ################################################### omegaSem(r9,n.obs=500,lavaan=TRUE) ################################################### ### code chunk number 34: overview.Rnw:1054-1055 ################################################### splitHalf(r9) ################################################### ### code chunk number 35: overview.Rnw:1069-1089 ################################################### #the newer way is probably preferred keys.list <- list(agree=c("-A1","A2","A3","A4","A5"), conscientious=c("C1","C2","C2","-C4","-C5"), extraversion=c("-E1","-E2","E3","E4","E5"), neuroticism=c("N1","N2","N3","N4","N5"), openness = c("O1","-O2","O3","O4","-O5")) #this can also be done by location-- keys.list <- list(Agree=c(-1,2:5),Conscientious=c(6:8,-9,-10), Extraversion=c(-11,-12,13:15),Neuroticism=c(16:20), Openness = c(21,-22,23,24,-25)) #These two approaches can be mixed if desired keys.list <- list(agree=c("-A1","A2","A3","A4","A5"),conscientious=c("C1","C2","C3","-C4","-C5"), extraversion=c("-E1","-E2","E3","E4","E5"), neuroticism=c(16:20),openness = c(21,-22,23,24,-25)) keys.list ################################################### ### code chunk number 36: overview.Rnw:1111-1113 ################################################### scores <- scoreItems(keys.list,bfi) scores ################################################### ### code chunk number 37: scores ################################################### png('scores.png') pairs.panels(scores$scores,pch='.',jiggle=TRUE) dev.off() ################################################### ### code chunk number 38: overview.Rnw:1139-1142 ################################################### r.bfi <- cor(bfi,use="pairwise") scales <- scoreItems(keys.list,r.bfi) summary(scales) ################################################### ### code chunk number 39: overview.Rnw:1152-1158 ################################################### data(iqitems) iq.keys <- c(4,4,4, 6,6,3,4,4, 5,2,2,4, 3,2,6,7) score.multiple.choice(iq.keys,iqitems) #just convert the items to true or false iq.tf <- score.multiple.choice(iq.keys,iqitems,score=FALSE) describe(iq.tf) #compare to previous results ################################################### ### code chunk number 40: overview.Rnw:1176-1182 ################################################### data(iqitems) iq.keys <- c(4,4,4, 6,6,3,4,4, 5,2,2,4, 3,2,6,7) scores <- score.multiple.choice(iq.keys,iqitems,score=TRUE,short=FALSE) #note that for speed we can just do this on simple item counts rather than IRT based scores. op <- par(mfrow=c(2,2)) #set this to see the output for multiple items irt.responses(scores$scores,iqitems[1:4],breaks=11) ################################################### ### code chunk number 41: overview.Rnw:1194-1196 ################################################### m <- colMeans(bfi,na.rm=TRUE) item.lookup(scales$item.corrected[,1:3],m,dictionary=bfi.dictionary[1:2]) ################################################### ### code chunk number 42: overview.Rnw:1204-1206 ################################################### data(bfi) bestScales(bfi,criteria=c("gender","education","age"),cut=.1,dictionary=bfi.dictionary[,1:3]) ################################################### ### code chunk number 43: overview.Rnw:1230-1234 ################################################### set.seed(17) d9 <- sim.irt(9,1000,-2.0,2.0,mod="normal") #dichotomous items test <- irt.fa(d9$items,correct=0) test ################################################### ### code chunk number 44: overview.Rnw:1241-1246 ################################################### op <- par(mfrow=c(3,1)) plot(test,type="ICC") plot(test,type="IIC") plot(test,type="test") op <- par(mfrow=c(1,1)) ################################################### ### code chunk number 45: overview.Rnw:1257-1260 ################################################### data(bfi) e.irt <- irt.fa(bfi[11:15]) e.irt ################################################### ### code chunk number 46: overview.Rnw:1267-1268 ################################################### e.info <- plot(e.irt,type="IIC") ################################################### ### code chunk number 47: overview.Rnw:1279-1280 ################################################### print(e.info,sort=TRUE) ################################################### ### code chunk number 48: overview.Rnw:1309-1310 ################################################### iq.irt <- irt.fa(ability) ################################################### ### code chunk number 49: overview.Rnw:1322-1323 ################################################### plot(iq.irt,type='test') ################################################### ### code chunk number 50: overview.Rnw:1334-1335 ################################################### iq.irt ################################################### ### code chunk number 51: overview.Rnw:1341-1342 ################################################### om <- omega(iq.irt$rho,4) ################################################### ### code chunk number 52: overview.Rnw:1356-1370 ################################################### v9 <- sim.irt(9,1000,-2.,2.,mod="normal") #dichotomous items items <- v9$items test <- irt.fa(items) total <- rowSums(items) ord <- order(total) items <- items[ord,] #now delete some of the data - note that they are ordered by score items[1:333,5:9] <- NA items[334:666,3:7] <- NA items[667:1000,1:4] <- NA scores <- scoreIrt(test,items) unitweighted <- scoreIrt(items=items,keys=rep(1,9)) scores.df <- data.frame(true=v9$theta[ord],scores,unitweighted) colnames(scores.df) <- c("True theta","irt theta","total","fit","rasch","total","fit") ################################################### ### code chunk number 53: overview.Rnw:1379-1381 ################################################### pairs.panels(scores.df,pch='.',gap=0) title('Comparing true theta for IRT, Rasch and classically based scoring',line=3) ################################################### ### code chunk number 54: overview.Rnw:1393-1409 ################################################### keys.list <- list(agree=c("-A1","A2","A3","A4","A5"), conscientious=c("C1","C2","C3","-C4","-C5"), extraversion=c("-E1","-E2","E3","E4","E5"), neuroticism=c("N1","N2","N3","N4","N5"), openness = c("O1","-O2","O3","O4","-O5")) item.list <- list(agree=c("A1","A2","A3","A4","A5"), conscientious=c("C1","C2","C3","C4","C5"), extraversion=c("E1","E2","E3","E4","E5"), neuroticism=c("N1","N2","N3","N4","N5"), openness = c("O1","O2","O3","O4","O5")) bfi.1pl <- scoreIrt.1pl(keys.list,bfi) #the one parameter solution bfi.2pl <- scoreIrt.2pl(item.list,bfi) #the two parameter solution bfi.ctt <- scoreFast(keys.list,bfi) # fast scoring function ################################################### ### code chunk number 55: overview.Rnw:1414-1418 ################################################### #compare the solutions using the cor2 function cor2(bfi.1pl,bfi.ctt) cor2(bfi.2pl,bfi.ctt) cor2(bfi.2pl,bfi.1pl) ################################################### ### code chunk number 56: overview.Rnw:1482-1486 ################################################### C <- cov(sat.act,use="pairwise") model1 <- lm(ACT~ gender + education + age, data=sat.act) summary(model1) ################################################### ### code chunk number 57: overview.Rnw:1489-1491 ################################################### #compare with setCor setCor(gender + education + age ~ ACT + SATV + SATQ, data = C, n.obs=700) ################################################### ### code chunk number 58: overview.Rnw:1574-1598 ################################################### xlim=c(0,10) ylim=c(0,10) plot(NA,xlim=xlim,ylim=ylim,main="Demonstration of dia functions",axes=FALSE,xlab="",ylab="") ul <- dia.rect(1,9,labels="upper left",xlim=xlim,ylim=ylim) ll <- dia.rect(1,3,labels="lower left",xlim=xlim,ylim=ylim) lr <- dia.ellipse(9,3,"lower right",xlim=xlim,ylim=ylim,e.size=.09) ur <- dia.ellipse(7,9,"upper right",xlim=xlim,ylim=ylim,e.size=.1) ml <- dia.ellipse(3,6,"middle left",xlim=xlim,ylim=ylim,e.size=.1) mr <- dia.ellipse(7,6,"middle right",xlim=xlim,ylim=ylim,e.size=.08) bl <- dia.ellipse(1,1,"bottom left",xlim=xlim,ylim=ylim,e.size=.08) br <- dia.rect(9,1,"bottom right",xlim=xlim,ylim=ylim) dia.arrow(from=lr,to=ul,labels="right to left") dia.arrow(from=ul,to=ur,labels="left to right") dia.curved.arrow(from=lr,to=ll$right,labels ="right to left") dia.curved.arrow(to=ur,from=ul$right,labels ="left to right") dia.curve(ll$top,ul$bottom,"double",-1) #for rectangles, specify where to point dia.curved.arrow(mr,ur,"up") #but for ellipses, just point to it. dia.curve(ml,mr,"across") dia.curved.arrow(ur,lr,"top down") dia.curved.arrow(br$top,lr$bottom,"up") dia.curved.arrow(bl,br,"left to right") dia.arrow(bl$top,ll$bottom) dia.curved.arrow(ml,ll$top,scale=-1) dia.curved.arrow(mr,lr$top) ################################################### ### code chunk number 59: overview.Rnw:1719-1720 ################################################### sessionInfo() psychTools/inst/doc/omega.R0000644000176200001440000000630614153443055015367 0ustar liggesusers### R code from vignette source 'omega.Rnw' ################################################### ### code chunk number 1: omega.Rnw:162-166 ################################################### library(psych) #make the psych package active library(psychTools) #make psychTools active om <- omega(Thurstone) #do the analysis om #show it ################################################### ### code chunk number 2: Thurstone ################################################### png('Thurstone.png') omega.diagram(om) dev.off() ################################################### ### code chunk number 3: omega ################################################### omega(Thurstone) ################################################### ### code chunk number 4: anxiety ################################################### anxiety <- sai[c("anxious", "jittery", "nervous" ,"tense", "upset","at.ease" , "calm" , "confident", "content","relaxed")] describe(anxiety) lowerCor(anxiety) om <- omega(anxiety,2) #specify a two factor solution summary(om) #summarize the output ################################################### ### code chunk number 5: anxietyplot ################################################### png('anxiety.png') omega.diagram(om, main="Omega analysis of two factors of anxiety") dev.off() ################################################### ### code chunk number 6: direct ################################################### om <- omegaDirect(Thurstone) om ################################################### ### code chunk number 7: drawdirect ################################################### png('direct.png') omega.diagram(om, main="Direct Schmid Leihman solution") dev.off() ################################################### ### code chunk number 8: omega.Rnw:734-737 ################################################### om <- omega(holzinger.swineford[8:31],4) #the exploratory solution omegaSem(holzinger.swineford[8:31],4) #the confirmatory solution ################################################### ### code chunk number 9: holzinger ################################################### ################################################### ### code chunk number 10: omega.Rnw:831-834 ################################################### jen <- sim.hierarchical() #use the default values om <- omega(jen) om ################################################### ### code chunk number 11: jensen ################################################### png('jensen.png' ) omega.diagram(om) dev.off() ################################################### ### code chunk number 12: Simulate1 ################################################### fx <- matrix(c(.7,.6,.5,.7,.6,.5,.8,.7,.6, .6,.6,.6,rep(0,9),c(.6,.5,.6),rep(0,9),.6,.6,.6),ncol=4) simx <-sim.structure(fx) om <- omega(simx$model) dsl <- omegaDirect(simx$model) ################################################### ### code chunk number 13: Simulate.2 ################################################### lowerMat(simx$model) summary(om) summary(dsl) fa.congruence(list(om,dsl,fx)) ################################################### ### code chunk number 14: omega.Rnw:1014-1015 ################################################### sessionInfo() psychTools/inst/doc/mediation.pdf0000644000176200001440000213612114153443050016614 0ustar liggesusers%PDF-1.5 % 152 0 obj << /Length 1524 /Filter /FlateDecode >> stream xYKs6WH΄|\6I3xeO$ӥ(, (?6cJ Z|a"#ӣo 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TmzH}(|a\,z2鱗yї\٦뭗3g(+ hx$YxGKTl0ǽصBc*⮜>uF~?MT99duO '1yMYɿ+StkY G(ƽM悌.Ή@!pKn0.P6uXLV9c9LȘs`.VL5nQSqq,Hjao J<- ɔ)қx^sGI+f'4$~wVw|`#'>0w_$>:W&>&ρS#2G|H:#fGu|(oRZ>ztg[/Z-sՐܽ[Ww endstream endobj 886 0 obj << /Length 1848 /Filter /FlateDecode >> stream xڽY[o6~0 YR؆0 b;,'ix.HIY l^CF̮gb?<{%BVWBVa!bi׻cP;O9kFqlvR.:mm~-M-4T;"B( EqOeQf_6&9"Pb'AFrKxG!5'P~wGB3L)̏vJ9ͧľc&t0-sw_@Zoa8:1`6Q05jPhY:5ԔT:c4(gGXR],pDDx=;u?U@ JG@iXEPP|]QptZ{z;¥3\*3 } 5).'k~FjŐ.IzFL $ u<yBojZ&(p|ܛ4Sror72eG)oBjRv\NRq7攝D8cQ} ւ`gTڞˌ΢ڑ!B_X<|l!pP#g-1Gw"ti\?P!N}w:6P6A VVG1pع"=m 6'+s eHXSsg R M%^7ͤ b !.HubjOn9,;Nt-sܺhFeKP(8^vG.XT ]fՄaGLz/&l ts'hۈuT:+FS\AsYW\~eh]"WPQ ڨbLť=;U[h:Wj١vf5m:0`t-'ni3T.x`+դ.й3 gN/|ʾ7 QN<\p7^ –\S3*-x xd'B& )ݪ STR-a CM%cAoPDFV ߋt$lyܫ_>|/y>%:/:5_$}kd_uIw-ۯN4Lg?*\2/dzan)ɗ=MB:b'b"^N.@;|C>{n"\ DƄ˘pi ҐD*uM!LL}&A3F8JIVoxHx;Ɗ<\raHJVbH"'SNI7MHOՌH^zQ$; X8I7wS^ < endstream endobj 903 0 obj << /Length 3519 /Filter /FlateDecode >> stream xZIsϯ`bj7,qAjeJ)WxXmǹ;kw#'+ ᫅TCm\RnkJ6ˍ!䙫wpKP88ys2tETOdNX{@OM?qp gzn$C"thyKPe^cML$rvGj7'5 8kYa >e2g븑He@UpZLdǥUzHz$Av%"eV΃tR:93Pe-vEHz5G=GVC7[@U_@0 UiTC gm[R)#o^\w,UsYI9`gEG'UaCDxYYߡ2 WH8ZbC}z:$A7,2VGNEtp`!-W6+~_ZHv̨$p1'-uii ;T?#9[gRYcݡ$p-ދL-gqeFd,:]=>L!=5Y-p5kTwWX6 C+ =oj5zС@=6E݇C;? fufEJ+p.2hmUd۾EǵZ---oQ]^={> ֓fGVfn1j+QfMga/eϠ%MzE,C'`X*2GXܐ8Vչ5Y_U*҄f-jC5s)j&l>͂_Cʢ^V4I'=Pߐyus7 M\8nR2lޭ,$' >߽7։=iF~c B?2z 6JxmT'6B幗_pG&d-mn_od\^ M'[r0=<h#|]~Btr^:y:eVꑧ#E#x׾KqiF]zd hb[a[&U@f~D_B-R]{rrw7@^ e+=|R*q|F#amN>90B3y`W<ᇕ&A} j?a(Mֿ(_&;,<>/K\*l2U8I#;r'G#WK }ȅT+% +lqG 6);l[N6\OXMnl+&G{;A.UnN ț?F߬bu;yJf9m{¯nϢ]-\GR"P^ׯ& <6=$#XB0AciI7[bֆz:]8Nil-m1x.gacX;#{KZ-ilR/yfs `7kB>kaʰB8vlɩ SσLi?FfCvKFq+OOWX)Q< ?%(~sE>A#^fҦ٣]C2Fٸ&1{9;m9xm5Ɓv$uΉV z5z9Ta8It(ٹ1]2$wttq2d>e͘C.ӺJb}۬>8rt?sQ_"lĝGpPxG"IuSCr,d̽w}-na  lA(<<-u ȭIY-y׻a8m_'&kBszϹ)pb ^jEۘ^`CLÏ+U9]P2@h]NR-kIrC?ҟC4(\6`@I$t&Iiw\^i2'\ܑ mKF pߛ`c)^ҍ|t.ތd!!"l{$1 #>~#[J!x҆65KHqxC㝗#yw|#|@TTp~}e*^R!\wX[pEa4AMH&78># 5N,N728<dQK`CVy$;7|'E{Nr2t<8$bوɌk_s%"T%Bp8%FӀG];]ٕh3{EU<{̠07% a>4U"(5sDUɾy~#LS9U1ŕTt{Cze*IkX7<=_9`_Wy2Mο=} Tn>Sge$~Єr|#p8v#5CN*;9/~S@sw>~4Z=IN./Uf>-gsɌ2+ԇEZj"mdg0S$_!՜x27]x,C@H|j3|dˋ"D ?DG 7jGQ<uU]Ud}=a,MTu^$gq#w/r]ܜQmL&7ޏ3nDL+L1d ބ|ML+>1YgMEܗ>؆\VpetM T4a:͔Mv0t p]V@`ڢ) JB^ -3DNQ@FO<&t4g7|.ȭrtA/')blAE +qtfm:!TOn#(H͚C"-vUnYWouhvRK'C .jnc/+:@bγZ/]6+}çDlX{^˹5h1> Ljwb/|K#/Dv";hL%>qXdkXmbf|PpfOpx$P`QJgd3}q v)Nf 3I6\$?s::̬,R;6_^;S endstream endobj 914 0 obj << /Length 4099 /Filter /FlateDecode >> stream xڽ[KPb$@W.?ʩd*ęQJEiwC~{؞+lݍF L9oߔ͛?}*ln֪•jָfv0?.unoKm罴q ;k?,L/GC~}ǟoJ|wgHn@n#m%9@[xgPRZV e xhaGm t;`#KL"xYV4g.Ut"$s9#G1_{=4YޞG dhU٭m0zDԋ#Ku^~|&B>,ziWBB0& P Np@M~ 7rC,M45*04eKt %>$iM?-$C +Fh3+_I0Ι#5 xق6:@񎁢1 ̡WVrʀol{Xdd@Q#ͻY-|[weKRy 6w;v?%0pˆ$qtt| |O1И*qPSjr\p%{WCdd8C7AT8p  2U6[D+vGAl,5l+y%Үp*igݛ~,gk,*]>ntazo*i 4Af`:`Ll ]Illv*LjEtb~otqguQҨ1hlCGQ&39R6[UӁשQl/:!Cxy`,( 1y#FxN Ce ·#'x "7{ebcYeϔJ-'~heb)p.B\ _pJLr:F| eۡ&o^:rfl{bUEa8 PT?Q|~7 Gy/*B>["µEnn("t0jn$ Fwo[NiXB)o|7d.ԫL/䠬Kld=HywP ͉/ DY_r4VB'Ş[$"7)`[sS[H|xڡb[۴ j`G! 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(2021) ", "psychTools:Tools to Accompany the 'psych' Package for Psychological Research ", "Northwestern University, Evanston, Illinois, USA, ", "https://CRAN.R-project.org/package=psychTools", "Version = 2.1.6,", ".",sep="") ) psychTools/inst/NEWS.Rd0000644000176200001440000001445114153442765014461 0ustar liggesusers\name{NEWS} \title{News for Package 'psychTools'} \section{Changes in psychTools version 2.1.12 (2021-12-06)}{ \subsection{Introduction}{ \itemize{ \item Version 2.2.1 is the development release of the psychTools package. It is available as a source file for Macs or PCs in the repository at \url{https://personality-project.org/r/}. The released version on CRAN is 2.1.12 The second digit reflects the year (i.e., 2021), the third set the month (i.e., 1.8.3 was released in March of 2018, the last two digits of development versions reflect either an minor change or the day of any modifications, e.g. 1.8.3.3 was the third attempt to get 1.8.3 released. 1.7.8 was released in August, 2017. \item To install this development version, use the command: install.packages("psychTools", repos="https://personality-project.org/r/", type="source"). Remember to restart R and library(psych) to make the new version active. \item The psychTools package includes functions and data sets to accompany the psych package which does classic and modern psychometrics and to analyze personality and experimental psychological data sets. The psych package has been developed as a supplement to courses in research methods in psychology, personality research, and graduate level psychometric theory. The functions are a supplement to the text (in progress): An introduction to psychometric theory with applications in R. \item These data sets are meant to be useful adjuncts to teaching and research. \item Additional functions are added sporadically. \item This NEWS file reports changes that have been made as the package has been developed. \item To report bugs, send email to \url{mailto:revelle@northwestern.edu} using bug.report. Remember to include the systemInfo() information. } } } \section{Changes in psychTools version 2.1.12 (2021-12-06 )}{ \subsection{Additions}{ \itemize{ \item Added the eminence data set from Simonton and del Giudice as a nice example of misinterpretation of beta weights in the presence of highly colinear predictors. \item Updated this news file for version 2.1.6 to reflect datasets added. } } } \section{Changes in psychTools version 2.1.6 (2021-06-20 )}{ \subsection{Additions}{ \itemize{ \item Added the GERAS dataset from Gruber et al. (2020) to show Mahalobnis distances in scatterHist. \item Added the Spengeler and Damian data sets from Project Talent (nice example of mediation) \item Added the USAF data set of anthropometric measurements (to help understand what a g factor does or does not mean) \item Added the globalWarm data set from Erik Nisbett (to help in mediation) \item Added the ability to do long tables in df2latex (requested by Lizz Dworak) } } } \section{Changes in psychTools version 2.1.3 (2021-03-14)}{ \subsection{Additions}{ \itemize{ \item Moved four vignettes over from psych to make psych smaller. \item Current vignettes here include overview, omega, factor, and mediation \item Checked against psych_2.0.12 and psych_2.1.3 \item Added B5 and L27 columns to the spi.dictionary. } } \subsection{Bugs Fixed}{ \itemize{ \item Nothing yet } } } \section{Changes in psychTools version 2.0.9 (2020-09-14)}{ \subsection{Additions}{ \itemize{ \item Added the BFI 100 items as a data set } } \subsection{Bugs Fixed}{ \itemize{ \item .dat and RData files were not being read in read.file following a clean up in code. } } } \section{Changes in psychTools version 2.0.8 (2020-08-04)}{ \subsection{Additions}{ \itemize{\item ability.keys to the ability.rda file \item Modified df2latex to allow mixed numeric and character with rounding of numeric } } \subsection{Bugs Fixed}{ \itemize{ \item None yet } } } \section{Changes in psychTools version 2.0.6 (2020-1-12)}{ \subsection{Additions}{ \itemize{ \item Modified df2latex to allow mixed numeric and character with rounding of numeric } } \subsection{Bugs Fixed}{ \itemize{ \item Fixed dfOrder to handle Null data \item Fixed holzinger.swineford and holzinger.raw for two cases (180 and 231) where K. Widaman had provided incorrect values (thanks to Keith for finding this). } } } \section{Changes in psychTools version 1.9.10 (2018-06-24)}{ \subsection{Additions}{ \itemize{ \item Added the holzinger.raw, holzinger.swineford and holzinger.dictionary data sets. The data come from Keith Widaman. } } \subsection{Bugs Fixed}{ \itemize{ \item None yet. } } } \section{Changes in psychTools version 1.9.6 (2018-06-24)}{ \subsection{Additions}{ \itemize{ \item Added bfi.keys to the bfi data set \item Added examples to the sai data set to match Revelle and Condon 2019 \item Added spengler data set } } \subsection{Bugs Fixed}{ \itemize{ \item Minor correction to the cities help file } } } \section{Changes in psychTools version 1.9.5 (2018-05-25)}{ \subsection{Additions}{ \itemize{ \item Data sets and a few helper functions switched over from psych to psychTools to make psych a smaller package. \item Data sets included are: ability, bfi, epi.bfi,income, iqitems, msq, msqR, neo, sai, spi, and tai. \item Helper functions include the df2latex set, dfOrder, and the various file utilities such as read.clipboard. \item Version number increased to 1.9.5.18 as we work through minor fixes to the submission to meet the newly enforced more stringent requirements of CRAN \item Changed cat and print in interactive functions (fileCreate) to message() following request from CRAN \item Following yet another request from CRAN, changed the read.file function to not automatically load an .rda file, but rather suggest how to load it. \item Changed the use of \%in\% to is.element to get around some problems in the msqR help file \item Changed the examples in read.clipboard to donttest instead of dontrun because they are interactive \item Changed all dontrun to donttest following request from CRAN. } } \subsection{Bugs Fixed}{ \itemize{ \item None yet } } }