metap/0000755000176200001440000000000013612367615011372 5ustar liggesusersmetap/NAMESPACE0000644000176200001440000000212613442163526012606 0ustar liggesusersexport(allmetap) export(invchisq) export(invt) export(logitp) export(meanp) export(meanz) export(maximump) export(minimump) export(plotp) export(schweder) export(sumlog) export(sump) export(sumz) export(truncated) export(two2one) export(votep) export(wilkinsonp) S3method(print,allmetap) S3method(print,invchisq) S3method(print,invt) S3method(print,logitp) S3method(print,meanp) S3method(print,meanz) S3method(print,sumlog) S3method(print,sump) S3method(print,sumz) S3method(print,votep) S3method(print,maximump) S3method(print,minimump) S3method(print,truncated) S3method(summary,truncated) S3method(plot,truncated) S3method(print,wilkinsonp) S3method(plot,metap) importFrom("graphics", "abline", "plot") importFrom("stats", "binom.test", "coef", "lm", "na.fail", "pbeta", "pchisq", "pnorm", "ppoints", "pt", "qbeta", "qchisq", "qnorm", "qqline", "qqplot", "qunif", "qt", "sd") importFrom("utils", "read.csv") importFrom("lattice", "xyplot") importFrom("Rdpack", "reprompt") importFrom("TFisher", "p.tpm", "stat.tpm") importFrom("mutoss", "ranktruncated") metap/data/0000755000176200001440000000000013611774102012273 5ustar liggesusersmetap/data/rosenthal.rda0000644000176200001440000000035012037557252014766 0ustar liggesusers r0b```b`@& `bN./N+Ha`` T1g.ۂ׹vn yr;h|=S~PZJAitE-5y?δD~)ɽ']w44%PǂK 4PO`yAjېX &ȕXVa(]metap/data/cholest.rda0000644000176200001440000000033012162554176014427 0ustar liggesusers r0b```b`@& `bN fOI-.2X?e[[v7g@cߥ3Dtq?+,a |cjkc0l3yO?[ w j }o@/n{q+XT:([nz?͝KFmetap/data/naep.rda0000644000176200001440000000114012174004104013671 0ustar liggesusers r0b```b`@& `bN'/1YH+A`āXEAjXXdK$1< fbi bCxXY81 ^! Es+[NjYjN1&+} 8+ rvr cy<],?; ggyYp6:o+ !0V\6.gE8krNb1,ѐ\_dF봥팂|nUOۅu\f9|/qq*7įhjUO\\3w?& py`d $B9hZ8`@HD!XJI,IK+j?x`A:metap/data/validity.rda0000644000176200001440000000034712162554176014623 0ustar liggesusers r0b```b`@& `bN (KL,Xľ_צRWþR+g|WzdWKb?䶟H~ğ3]Td~/z~gKٱn:&N kcf<sN۝*.xb%\oq~u'ؗ2g~W'y/Xf\\pD?Ѝ|?metap/data/teachexpect.rda0000644000176200001440000000031612037557252015266 0ustar liggesusers r0b```b`@& `bN .IMLH(HM.rXVoOWvbˉE'$^y)s.ۂ׹TEؿ]x@P*5j/T=t_'RmsW >>~)Q2k@M;"1\Sa˫metap/data/mourning.rda0000644000176200001440000000050712174004104014612 0ustar liggesusers r0b```b`@& `bN /-Kg`` yrL>X-#,ԲԜb Kl_BqIb^ _PWsJRЌcMI,#-1$K\ńBmlFPrwK]C6#uf]ѱo\sW[,#/fmϾݶ'cطa\g__c0KMEQ6Pp&”%8hp Sw~JI,IK+jnFmetap/data/edgington.rda0000644000176200001440000000012112301311464014724 0ustar liggesusers r0b```b`@& `bNNMIK/rLe- M88?=pmetap/data/beckerp.rda0000644000176200001440000000014612037557252014405 0ustar liggesusers r0b```b`@& `bN fOJMN-*2~B累+oT{Yq ``u܎7[Dɬ Jj^metap/man/0000755000176200001440000000000013611774102012135 5ustar liggesusersmetap/man/sumlog.Rd0000644000176200001440000000337713363362512013746 0ustar liggesusers\name{sumlog} \alias{sumlog} \alias{Fisher's method} \alias{chi-square (2) method} \alias{print.sumlog} \title{Combine p-values by the sum of logs (Fisher's) method } \description{Combine \eqn{p}-values by the sum of logs method, also known as Fisher's method, and sometimes as the chi-square (2) method.} \usage{ sumlog(p) \method{print}{sumlog}(x, ...) } \arguments{ \item{p}{\sigvec} \item{x}{An object of class \sQuote{\code{sumlog}}} \item{...}{Other arguments to be passed through} } \details{ The method relies on the fact that \deqn{\sum_{i=1}^k - 2 \log p_i}{sum (-2 log p)} is a chi-squared with \eqn{2 k}{2 * k} df where \eqn{k} is the number of studies \insertCite{fisher25}{metap}. \insertNoCite{becker94}{metap} \insertNoCite{rosenthal78}{metap} \insertNoCite{sutton00}{metap} \ltle{two} \plotmethod \nocancel } \value{ An object of class \sQuote{\code{sumlog}} and \sQuote{\code{metap}}, a list with entries \item{chisq}{Value of chi-squared statistic} \item{df}{Associated degrees of freedom} \item{p}{Associated \eqn{p}-value} \item{validp}{The input vector with the illegal values removed} %% ... } \references{ \insertAllCited{} } \author{Michael Dewey} \seealso{ See also \code{\link{plotp}} } \examples{ data(teachexpect) sumlog(teachexpect) # chisq = 69.473, df = 38, p = 0.0014, from Becker data(beckerp) sumlog(beckerp) # chisq = 18.533, df = 10, sig data(rosenthal) sumlog(rosenthal$p) # chisq = 22.97, df = 10, p = 0.006 one sided data(cholest) sumlog(cholest) # chisq = 58.62, df = 68, p = 0.78 data(validity) sumlog(validity) # chisq = 159.82, df = 40, p = 2.91 * 10^{-16} sumlog(c(0.0001, 0.0001, 0.9999, 0.9999)) # is significant } % Add one or more standard keywords, see file 'KEYWORDS' in the % R documentation directory. \keyword{ htest } metap/man/invt.Rd0000644000176200001440000000273413556605134013420 0ustar liggesusers\name{invt} \alias{invt} \alias{print.invt} \title{Combine p values using inverse t method} \description{Combine p values using the inverse t method} \usage{ invt(p, k, data = NULL, subset = NULL, na.action = na.fail) \method{print}{invt}(x, ...) } \arguments{ \item{p}{\sigvec} \item{k}{A vector of degrees of freedom to use, see details} \item{data}{Optional data frame containing variables} \item{subset}{Optional vector of logicals to specify a subset of the \eqn{p}-values} \item{na.action}{A function indicating what should happen when data contains NAs} \item{x}{An object of class \sQuote{\code{invt}}} \item{...}{Other arguments to be passed through} } \details{ Defined as \deqn{\frac{\sum_{i=1}^n t_{k_i}(p_i)}% {\sqrt{\sum_{i=1}^n \frac{k_i}{k_i - 2}}}% > z(\alpha)% }{\sum t (p) /% \sqrt{\sum k / (k - 2)}% > z(\alpha)} If \code{k} is a single value it is used for all the \code{p} of which there are \code{n}. If any value of \eqn{k_i\le2} then the corresponding \eqn{p_i} is not included. \ltle{two} \plotmethod } \value{ An object of class \sQuote{\code{invt}} and \sQuote{\code{metap}}, a list with entries \item{z}{Value of z} \item{p}{Associated p-value} \item{validp}{The input vector with the illegal values removed} } \references{ \insertRef{becker94}{metap} } \author{Michael Dewey} \seealso{ See also \code{\link{plotp}} } \examples{ data(beckerp) invt(beckerp, 50) } \keyword{ htest } metap/man/truncated.Rd0000644000176200001440000000540713561032366014426 0ustar liggesusers\name{truncated} \alias{truncated} \alias{print.truncated} \alias{summary.truncated} \alias{plot.truncated} \title{Truncated product method routine} \description{Implements two procedures for truncated versions of Fisher's method} \usage{ truncated(p, ptrunc = NULL, rtrunc = NULL, ...) \method{print}{truncated}(x, ...) \method{summary}{truncated}(object, ...) \method{plot}{truncated}(x, pparams = list(pchs = c(16, 1), pcols = c("black", "black")), ...) } \arguments{ \item{p}{\sigvec} \item{ptrunc}{Value of \eqn{p} to truncate at. See Details} \item{rtrunc}{Which rank \eqn{p} to truncate at. See Details} \item{x}{An object of class \sQuote{\code{truncated}}} \item{object}{An object of class \sQuote{\code{truncated}}} \item{\dots}{Other arguments to be passed through} \item{pparams}{A list containing a vector \code{pchs} of plotting symbols and a vector \code{pcols} of colours} } \details{Uses routines from TFisher to implement truncated Fisher \insertCite{zaykin07,zhang18}{metap} and from mutoss to implement rank-truncated Fisher \insertCite{dudbridge03}{metap} The print method prints the statistic and its associated \eqn{p}-value. The summary method also prints how many of the input values were actually used. The plot method provides an index plot of the \eqn{p}-values. By default used \eqn{p}-values are plotted with a black filled circle and those not used with a black unfilled circle. The colours and plotting symbols can be changed in \code{pparams}. The first element of each vector is for the used \eqn{p}-values and the second for the unused ones. } \value{An object of class \sQuote{\code{truncated}} a list with entries \item{p}{The resultant overall \eqn{p}-value} \item{stat}{The test statistic} \item{validp}{The input vector with illegal values removed} \item{ptrunc}{The \eqn{p}-value used for truncation} \item{trunc}{The rank value used for truncation} \item{n}{A vector containing \code{incl}, the number of valid values included, \code{excl}, the number of valid values not used} \item{method}{The method used} } \note{ Only one of \code{ptrunc} or \code{rtrunc} is returned as appropriate. If the function encounters an error it returns NA for the value of \code{p} and gives a warning message. The effect of specifying both \code{ptrunc} and \code{rtrunc} is undefined. } \references{ \insertAllCited{} } \author{ Michael Dewey } %% ~Make other sections like Warning with \section{Warning }{....} ~ \seealso{ \code{\link{ranktruncated}}, \code{\link{stat.tpm}}, and \code{\link{p.tpm}} } \examples{ data(validity) truncated(validity, ptrunc = 0.5) truncated(validity, rtrunc = 5) } \keyword{htest1} metap/man/schweder.Rd0000644000176200001440000000675413325603672014251 0ustar liggesusers\name{schweder} \alias{schweder} \title{Schweder and Spjotvoll plot} \description{Produces the plot suggested by Schweder and Spjotvoll to display a collection of \eqn{p}-values and also optionally draws the lowest slope line suggested by Benjamini and Hochberg } \usage{ schweder(p, xlab = "Rank of p", ylab = "p", drawline = NULL, bh.lwd = 1, bh.lty = "solid", bh.col = "black", ls.control = list(frac = NULL), ls.lwd = 1, ls.lty = "dotted", ls.col = "black", ab.control = list(a = NULL, b = NULL), ab.lwd = 1, ab.lty = "dashed", ab.col = "black", ...) } %- maybe also 'usage' for other objects documented here. \arguments{ \item{p}{A vector of \eqn{p}-values} \item{xlab}{Label for plot \eqn{x}-axis} \item{ylab}{Label for plot \eqn{y}-axis} \item{drawline}{Which line, if any, to draw. See Details} \item{bh.lwd}{Width of the lowest slope line} \item{bh.lty}{Line type of the lowest slope line} \item{bh.col}{Colour of the lowest slope line} \item{ls.control}{A list containing a named element \code{frac}. See Details} \item{ls.lwd}{Width of the least squares line} \item{ls.lty}{Line type of the least squares line} \item{ls.col}{Colour of the least squares line} \item{ab.control}{A list containing two elements \code{a} and \code{b}. See Details} \item{ab.lwd}{Width of the specified line} \item{ab.lty}{Line type of the specified line} \item{ab.col}{Colour of the specified line} \item{\dots}{Further parameters to be passed through to plot} } \details{After removing invalid \eqn{p}-values plots them. Any graphics parameters passed in \code{...} affect this part of the plot only. Note that the axes are chosen according to the scheme of Banjamini and Hochberg, Schweder and Spjotvoll used a different system. By setting \code{drawline} appropriately up to three lines may be drawn. If \code{drawline} includes the string \code{"bh"} the lowest slope line of Benjaimin and Hochberg is drawn. No further parameters are needed here but the characteristics of the line may be set: width, line type, and colour. If \code{drawline} includes the string \code{"ls"} a least squares line is drawn passing through the point \eqn{k+1, 1}. The parameter \code{frac} specified what fraction of the values be used for this and may need experimentation to obtain a suitable line. The characteristics of the line may be set: width, line type, and colour. If \code{drawline} includes the string \code{"ab"} a user specified line is drawn. The parameters \code{a} and \code{b} specify the intercept and slope. The characteristics of the line may be set: width, line type, and colour. } \value{ A list containing \item{p}{The values of \eqn{p} plotted} \item{bh.params}{A list containing at least items \code{a} and \code{b} the intercept and slope of the line drawn. This is only included if the lowest slope line was drawn} \item{ls.params}{A list containing at least items \code{a} and \code{b} the intercept and slope of the line drawn. This is only included if the least squares line was drawn} \item{ab.params}{A list containing at least items \code{a} and \code{b} the intercept and slope of the line drawn. This is only included if the user specified line was drawn} } \references{ \insertRef{becker94}{metap} \insertRef{benjamini00}{metap} \insertRef{schweder82}{metap} } \author{Michael Dewey} \examples{ data(teachexpect) schweder(teachexpect) } \keyword{ hplot } metap/man/allmetap.Rd0000644000176200001440000000257113352100116014216 0ustar liggesusers\name{allmetap} \alias{allmetap} \alias{print.allmetap} \title{Carry out all or some of the methods in the package} \description{Enables all or a selected number of the package methods to be executed on the same dataset} \usage{ allmetap(p, method = NULL) \method{print}{allmetap}(x, digits = 5, ...) } \arguments{ \item{p}{A vector of \eqn{p}-values} \item{method}{A vector of character containing names of methods or \sQuote{\code{all}}} \item{x}{An object of class \sQuote{\code{allmetap}}} \item{digits}{How many digits to print?} \item{...}{Other arguments to be passed through} } \details{The function uses each of the methods specified on the data-set in \code{p}. If method contains \sQuote{\code{all}} then this over--rides any specific method(s) named. } \value{Returns an object of class \sQuote{\code{allmetap}} and \sQuote{\code{data.frame}} containing columns \item{p}{The value of p returned} \item{valid}{The number of valid p values used for that method} \item{eponym}{The eponym, if any, for the method} The row names are the names of the methods used. If any of the methods cannot be applied to that data--set NA are returned for \code{p} and \code{valid}. } \author{Michael Dewey} \examples{ data(beckerp) allmetap(beckerp, method = "all") allmetap(beckerp, method = c("sumz", "sumlog")) } \keyword{htest} metap/man/meanz.Rd0000644000176200001440000000313313577121100013531 0ustar liggesusers\name{meanz} \alias{meanz} \alias{print.meanz} \title{Combine p values using mean z method} \description{Combines p values using the mean of z method} \usage{ meanz(p) \method{print}{meanz}(x, ...) } %- maybe also 'usage' for other objects documented here. \arguments{ \item{p}{\sigvec} \item{x}{An object of class \sQuote{\code{meanz}}} \item{...}{Other arguments to be passed through} } \details{ Let \deqn{\bar{z} = \sum_{i=1}^k \frac{z(p_i)}{k}}{\bar{z} = \sum(z(p) / k)} and \deqn{s_{\bar{z}} = \frac{s_z}{\sqrt{k}}}{s_{\bar{z}} = s_z / \sqrt k } Defined as \deqn{% \frac{\bar{z}}{s_{\bar{z}}} > t_{k-1}(\alpha) }{((\bar{z} / s_\bar{z}) > t_{k-1}(\alpha)} \lele{two} As can be seen if all the \eqn{p_i} are equal or close to equal this gives a \eqn{t=\pm\infty} leading to a returned value of 0 or 1. A set of \eqn{p} values with small variance will necessarily give a small \eqn{p} value which may be smaller than that for another set all of whose primary values are less than any in the first set. See examples for a demonstration. \plotmethod } \value{An object of class \sQuote{\code{meanz}} and \sQuote{\code{metap}}, a list with entries \item{z}{The value of the mean \eqn{z} statistic} \item{p}{The associated \eqn{p} value} \item{validp}{The input vector with illegal values removed} } \references{ \insertRef{becker94}{metap} } \author{Michael Dewey} \seealso{ See also \code{\link{plotp}} } \examples{ data(beckerp) meanz(beckerp) meanz(c(0.1, 0.2)) # greater than next example meanz(c(0.3, 0.31)) # less than above } \keyword{ htest } metap/man/sumz.Rd0000644000176200001440000000425513363364240013432 0ustar liggesusers\name{sumz} \alias{sumz} \alias{Stouffer's method} \alias{Sum of normals method} \alias{print.sumz} \title{Combine p-values using the sum of z (Stouffer's) method} \description{Combine \eqn{p}-values using the sum z method} \usage{ sumz(p, weights = NULL, data = NULL, subset = NULL, na.action = na.fail) \method{print}{sumz}(x, ...) } %- maybe also 'usage' for other objects documented here. \arguments{ \item{p}{\sigvec} \item{weights}{A vector of weights} \item{data}{Optional data frame containing variables} \item{subset}{Optional vector of logicals to specify a subset of the \eqn{p}-values} \item{na.action}{A function indicating what should happen when data contains NAs} \item{x}{An object of class \sQuote{\code{sumz}}} \item{...}{Other arguments to be passed through} } \details{ Defined as \deqn{\frac{\sum_{i=1}^k w_i z(p_i)}{\sqrt {\sum_{i=1}^k w_i ^ 2}}}{sum (w * z(p)) / sqrt(sum (w * w))} is a \eqn{z} where \eqn{k} is the number of studies and \eqn{w} are the weights \insertCite{stouffer49}{metap}. By default the weights are equal. In the absence of effect sizes (in which case a method for combining effect sizes would be more appropriate anyway) best results are believed to be obtained with weights proportional to the square root of the sample sizes \insertCite{zaykin11}{metap} \insertNoCite{becker94}{metap} \insertNoCite{rosenthal78}{metap} \ltlt{two} If the omitted \eqn{p} values had supplied weights a further warning is issued. \plotmethod } \value{An object of class \sQuote{\code{sumz}} and \sQuote{\code{metap}}, a list with entries \item{z}{Transformed sum of \eqn{z} values} \item{p}{Associated \eqn{p}-value} \item{validp}{The input vector with illegal values removed} \item{weights}{The weight vector corresponding to \code{validp}} } \references{ \insertAllCited{} } \author{Michael Dewey} \seealso{ See also \code{\link{plotp}} } \examples{ data(teachexpect) sumz(teachexpect) # z = 2.435, p = 0.0074, from Becker data(beckerp) sumz(beckerp) # z = 1.53, NS, from Beckerp data(rosenthal) sumz(rosenthal$p) # 2.39, p = 0.009 sumz(p, df, rosenthal) # 3.01, p = 0.0013 data(validity) sumz(validity) # z = 8.191, p = 1.25 * 10^{-16} } \keyword{ htest } metap/man/meanp.Rd0000644000176200001440000000232113352115234013517 0ustar liggesusers\name{meanp} \alias{meanp} \alias{print.meanp} %- Also NEED an '\alias' for EACH other topic documented here. \title{Combine p values by the mean p method} \description{Combine \eqn{p} values by the mean p method} \usage{ meanp(p) \method{print}{meanp}(x, ...) } %- maybe also 'usage' for other objects documented here. \arguments{ \item{p}{\sigvec} \item{x}{An object of class \sQuote{\code{meanp}}} \item{...}{Other arguments to be passed through} } \details{ Defined as \deqn{z = (0.5 - \bar{p}) \sqrt{12k}}{z = (0.5 - mean(p)) * sqrt(12k)} which is a standard normal and due to \insertCite{edgington72b;textual}{metap} \insertNoCite{becker94}{metap}\insertNoCite{rosenthal78}{metap} \lele{four} \plotmethod } \value{ An object of class \sQuote{\code{meanp}} and \sQuote{\code{metap}}, a list with entries \item{z}{Value of \eqn{z}} \item{p}{Associated \eqn{p}-value} \item{validp}{The input vector with illegal values removed} } \references{ \insertAllCited{} } \author{Michael Dewey} \seealso{ See also \code{\link{plotp}} } \examples{ data(rosenthal) meanp(rosenthal$p) # 2.17, p = 0.015 one tailed } % Add one or more standard keywords, see file 'KEYWORDS' in the % R documentation directory. \keyword{ htest } metap/man/votep.Rd0000644000176200001440000000272413352152022013557 0ustar liggesusers\name{votep} \alias{votep} \alias{print.votep} \title{Combine p-values by the vote counting method} \description{ Combine \eqn{p}-values by the vote counting method } \usage{ votep(p, alpha = c(0.5, 0.5)) \method{print}{votep}(x, ...) } %- maybe also 'usage' for other objects documented here. \arguments{ \item{p}{\sigvec} \item{alpha}{A vector of length 2 defining the significance value limits} \item{x}{An object of class \sQuote{\code{votep}}} \item{...}{Other arguments to be passed through} } \details{By default splits the \eqn{p}-values at 0.5 with those below counting as positive and those above counting as negative. However setting alpha allows for a different choice of cut-off and also by specifying two different values for a neutral zone. If either value of alpha is greater than unity it is assumed to be a percentage. \ltlt{two} \plotmethod } \value{An object of class \sQuote{\code{votep}} and \sQuote{\code{votep}}, a list with entries \item{p}{\eqn{p} value} \item{pos}{Number of positives} \item{neg}{Number of negatives} \item{alpha}{A vector of length 2 defining the significance values and lying in the range 0 to 1} \item{validp}{The input vector with illegal values removed} } \references{ \insertRef{becker94}{metap} } \author{Michael Dewey} \seealso{ See also \code{\link{plotp}} } \examples{ data(beckerp) votep(beckerp) } % Add one or more standard keywords, see file 'KEYWORDS' in the % R documentation directory. \keyword{ htest } metap/man/two2one.Rd0000644000176200001440000000233713352151664014032 0ustar liggesusers\name{two2one} \alias{two2one} %- Also NEED an '\alias' for EACH other topic documented here. \title{Convert two-sided p-values to one-sided} \description{Convert two-sided \eqn{p}-values to one-sided} \usage{ two2one(p, two = NULL, invert = NULL) } %- maybe also 'usage' for other objects documented here. \arguments{ \item{p}{A vector of p-values} \item{two}{A logical vector defining which \eqn{p}-values are two-sided and to be converted} \item{invert}{A logical vector defining which \eqn{p}-values are to be inverted} } \details{The \eqn{p}-values supplied to the other functions should be one-sided and all in the same direction. This convenience function will convert two-sided to one-sided and invert as necessary. By default it is assumed that all \eqn{p}-values are two-sided and the function converts them all to one-sided. Use \code{invert} to specify if some of the tests resulted in outcomes in the other direction. So for example a value of 0.05 will be converted to 0.025 unless \code{invert} is TRUE in which case it is converted to 0.975.} \value{A vector of one-sided p-values all in the correct direction } \author{Michael Dewey} \examples{ data(rosenthal) twop <- with(rosenthal, (pt(t, df))) } \keyword{ manip } metap/man/package-data.Rd0000644000176200001440000000600113352116154014722 0ustar liggesusers\name{package-data} \alias{beckerp} \alias{cholest} \alias{edgington} \alias{naep} \alias{mourning} \alias{rosenthal} \alias{teachexpect} \alias{validity} \docType{data} \title{Example data} \description{ The package contains the following datasets: \code{beckerp}, \code{cholest}, \code{edgington}, \code{mourning}, \code{naep}, \code{rosenthal}, \code{teachexpect}, and \code{validity}. } \usage{data(beckerp) data(cholest) data(edgington) data(mourning) data(naep) data(rosenthal) data(teachexpect) data(validity) } \format{ \describe{ \item{\code{beckerp}}{ A vector of length 5 } \item{\code{cholest}}{ A vector of length 34 } \item{\code{edgington}}{ A vector of length 7 } \item{\code{naep}}{A data frame with 34 observations on the following 2 variables. \describe{ \item{\code{state}}{a factor with levels \code{AL}, \code{AR}, \code{AZ}, \code{CA}, \code{CO}, \code{CT}, \code{DE}, \code{FL}, \code{GA}, \code{HI}, \code{IA}, \code{ID}, \code{IN}, \code{KY}, \code{LA}, \code{MD}, \code{MI}, \code{MN}, \code{NC}, \code{ND}, \code{NE}, \code{NH}, \code{NJ}, \code{NM}, \code{NY}, \code{OH}, \code{OK}, \code{PA}, \code{RI}, \code{TX}, \code{VA}, \code{WI}, \code{WV}, \code{WY},} \item{\code{p}}{a numeric vector} }} \item{\code{mourning}}{ A data frame with 9 observations on the following 3 variables. \describe{ \item{\code{stance}}{a factor with levels \code{No stand}, \code{Opponent}, \code{Supporter}} \item{\code{grade}}{a factor with levels \code{G11-12}, \code{G7-8}, \code{G9-10}} \item{\code{p}}{a numeric vector} }} \item{\code{rosenthal}}{ A data frame with 5 observations on the following 3 variables. \describe{ \item{\code{t}}{A numeric vector of values of \eqn{t}} \item{\code{df}}{a numeric vector of degrees of freedom} \item{\code{p}}{a numeric vector of one sided \eqn{p} values} } } \item{\code{teachexpect}}{A vector of length 19 } \item{\code{validity}}{A vector of length 20 } } } \details{ \describe{ \item{\code{beckerp}}{Hypothetical \eqn{p} values from \insertCite{becker94;textual}{metap}} \item{\code{cholest}}{\eqn{p}-values from studies of cholesterol lowering from \insertCite{sutton00;textual}{metap}} \item{\code{edgington}}{Hypothetical \eqn{p} values from \insertCite{edgington72a;textual}{metap}} \item{\code{mourning}}{Results from a study of mourning practices of Israeli youth from \insertCite{benjamini00;textual}{metap}} \item{\code{naep}}{Results on the National Assessment of Educational Progress from \insertCite{benjamini00;textual}{metap}} \item{\code{rosenthal}}{Hypothetical example from \insertCite{rosenthal78;textual}{metap}} \item{\code{teachexpect}}{\eqn{p}-values from studies of the effect of manipulating teacher expectancy from \insertCite{becker94;textual}{metap}} \item{\code{validity}}{\eqn{p}-values from studies of validity of student ratings from \insertCite{becker94;textual}{metap}} } } \references{ \insertAllCited{} } \keyword{datasets} metap/man/macros/0000755000176200001440000000000013472726173013433 5ustar liggesusersmetap/man/macros/macros.Rd0000644000176200001440000000223513556527035015207 0ustar liggesusers\newcommand{\nocancel}{Inspection of the distribution of \eqn{p}-values is highly recommended as extreme values in opposite directions do not cancel out. See last example. This may not be what you want.} \newcommand{\plotmethod}{The plot method for class \sQuote{\code{metap}} calls \code{\link{plotp}} on the valid \eqn{p}-values.} \newcommand{\sigvec}{A vector of significance values} \newcommand{\ltle}{The values of \eqn{p_i} should be such that \eqn{0 1}{sum p > 1}. There seems no particular need to use this method but it is returned as the value of \code{conservativep} for use in checking published values. \lele{two} A warning is given when the internal calculations are likely to have been subject to numerical error and an alternative method should be used to check the result. \plotmethod } \value{An object of class \sQuote{\code{sump}} and \sQuote{\code{metap}}, a list with entries \item{p}{The transformed sum of the \eqn{p}-values} \item{conservativep}{See details} \item{validp}{The input vector with illegal values removed} } \references{ \insertAllCited{} } \author{Michael Dewey} \seealso{ See also \code{\link{plotp}} } \examples{ data(edgington) sump(edgington) # p = 0.097 } \keyword{ htest } metap/man/metap-package.Rd0000644000176200001440000000211513325603720015121 0ustar liggesusers\name{metap-package} \alias{metap-package} \alias{metap} \docType{package} \title{\packageTitle{metap}} \description{\packageDescription{metap}} \details{ %\packageDESCRIPTION{metap} \packageIndices{metap} Provides a number of ways in which significance levels may be combined in a meta-analysis and includes most ot the methods in \insertCite{becker94;textual}{metap}. It includes a number of datasets taken from the literature. It also provides a display and an informal graphical test due to Schweder and Spjotvoll \insertCite{schweder82}{metap} and the lowest slope line of Benjamini and Hochberg \insertCite{benjamini00}{metap}. There is a vignette. Some further documentation and, possibly, a development version may be seen at \url{http://www.dewey.myzen.co.uk/meta/meta.html}. } \author{\packageAuthor{metap} Maintainer: \packageMaintainer{metap} } \references{ \insertAllCited{} } \keyword{ package } \seealso{ The issue of meta-analysis of signficance levels is not completely unconnected with the topic of adjustment for multiple comparisons as in for example \code{\link{p.adjust}} } metap/man/logitp.Rd0000644000176200001440000000272113363360722013727 0ustar liggesusers\name{logitp} \alias{logitp} \alias{print.logitp} %- Also NEED an '\alias' for EACH other topic documented here. \title{Combine p values using logit method} \description{ Combine \eqn{p} values using logit method } \usage{ logitp(p) \method{print}{logitp}(x, ...) } %- maybe also 'usage' for other objects documented here. \arguments{ \item{p}{\sigvec} \item{x}{An object of class \sQuote{\code{logitp}}} \item{...}{Other arguments to be passed through} } \details{ Defined as \deqn{t = - \frac{\sum_{i=1}^k \log\frac{p_i}{1 - p_i}}{C}}{% t = - sum log(p / (1 - p)) / C} where \deqn{C = \sqrt\frac{k \pi^2 (5 k + 2)}{3(5 k + 4)}}{C = sqrt((k pi^2 (5 k + 2)) / (3(5 k + 4)))} and \eqn{k} is the number of studies. \ltlt{two} \plotmethod } \value{ An object of class \sQuote{\code{logitp}} and \sQuote{\code{metap}}, a list with entries \item{t}{Value of Student's \eqn{t}} \item{df}{Associated degrees of freedom} \item{p}{Associated \eqn{p}-value} \item{validp}{The input vector with illegal values removed} } \references{ \insertRef{becker94}{metap} } \author{Michael Dewey} \seealso{ See also \code{\link{plotp}} } \examples{ data(teachexpect) logitp(teachexpect) # t = 2.763, df = 99, p = 0.0034, from Becker data(beckerp) logitp(beckerp) # t = 1.62, df = 29, NS, from Becker data(validity) logitp(validity) # t = 9.521, df = 104, p = 3.89 * 10^{-16} } % Add one or more standard keywords, see file 'KEYWORDS' in the % R documentation directory. \keyword{ htest } metap/man/invchisq.Rd0000644000176200001440000000360713556605776014300 0ustar liggesusers\name{invchisq} \alias{invchisq} \alias{Lancaster's method} \alias{print.invchisq} \title{Combine p values using inverse chi squared method} \description{Combine \eqn{p}-values by the inverse chi-squared method, also known as Lancaster's method, } \usage{ invchisq(p, k, data = NULL, subset = NULL, na.action = na.fail) \method{print}{invchisq}(x, ...) } %- maybe also 'usage' for other objects documented here. \arguments{ \item{p}{\sigvec} \item{k}{A vector of degrees of freedom to use, see details} \item{data}{Optional data frame containing variables} \item{subset}{Optional vector of logicals to specify a subset of the \eqn{p}-values} \item{na.action}{A function indicating what should happen when data contains NAs} \item{x}{An object of class \sQuote{\code{invchisq}}} \item{...}{Other arguments to be passed through} } \details{ Defined as \deqn{\sum_{i=1}^n \chi^2_{k_i}(p_i) > \chi^2_{\sum{k_i}}(\alpha)}{\sum \chi^2_k(p) > \chi^2_\sum{k}(\alpha)} If \code{k} is a single value it is used for all the \code{p} of which there are \code{n}. If any value of \eqn{k_i\le0} then the corresponding \eqn{p_i} is not included. \ltle{two} \plotmethod \nocancel } \value{ An object of class \sQuote{\code{invchisq}} and \sQuote{\code{metap}}, a list with entries \item{chisq}{Value of chi-squared statistic} \item{df}{Associated degrees of freedom} \item{p}{Associated p-value} \item{validp}{The input vector with the illegal values removed} } \references{ \insertRef{becker94}{metap} \insertRef{lancaster49}{metap} } \author{ Michael Dewey } \seealso{ See also \code{\link{sumlog}} and \code{\link{plotp}} } \examples{ data(beckerp) invchisq(beckerp, 2) # same as sumlog invchisq(c(0.999, 0.999, 0.001, 0.001), 4) } % Add one or more standard keywords, see file 'KEYWORDS' in the % R documentation directory. \keyword{ htest } metap/man/plotp.Rd0000644000176200001440000000132013343760302013555 0ustar liggesusers\name{plotp} \alias{plotp} \title{Q-Q plot of p-values} \description{ Produces a Q--Q plot of the p-values supplied } \usage{ plotp(pvals, ...) } \arguments{ \item{pvals}{A vector of \eqn{p}--values } \item{\dots}{Other parameters to pass through to \code{qqplot} and \code{qqline} } } \details{After eliminating out of range \eqn{p}--values and missing values produces a Q--Q plot and the line through the quantiles (by default the 0.25 and 0.75). } \value{ Invisibly returns a list conisting of \item{validp}{The valid \eqn{p}--values} } \author{Michael Dewey} \seealso{ \code{\link{qqplot}} and \code{\link{qqline}} } \examples{ data(cholest) plotp(cholest) } \keyword{hplot} metap/DESCRIPTION0000644000176200001440000000201113612367615013072 0ustar liggesusersPackage: metap Type: Package Title: Meta-Analysis of Significance Values Version: 1.3 Date: 2020-01-22 Authors@R: person("Michael", "Dewey", role = c("aut", "cre"), email = "lists@dewey.myzen.co.uk", comment = c(ORCID = "0000-0002-7522-3677") ) Description: The canonical way to perform meta-analysis involves using effect sizes. When they are not available this package provides a number of methods for meta-analysis of significance values including the methods of Edgington, Fisher, Lancaster, Stouffer, Tippett, and Wilkinson; a number of data-sets to replicate published results; and a routine for graphical display. URL: http://www.dewey.myzen.co.uk/meta/meta.html Imports: lattice,Rdpack,TFisher,mutoss RdMacros: Rdpack License: GPL-2 LazyLoad: yes NeedsCompilation: no Packaged: 2020-01-22 07:23:23 UTC; MichaelUser Author: Michael Dewey [aut, cre] () Maintainer: Michael Dewey Repository: CRAN Date/Publication: 2020-01-23 19:10:05 UTC metap/build/0000755000176200001440000000000013611774153012467 5ustar liggesusersmetap/build/vignette.rds0000644000176200001440000000037113611774153015027 0ustar liggesusersuMn0 4U 8,DRMTuѭMblvCw@jo<9!$"q(2AH< ֝aVGu3R{8tZQP]4wT*/)jX'[ͯ0?;v ovFfuh]ɯ|'#[3k:2ET~~~nSYak rQfu_LGg_auTM&sYX s!Umetap/build/partial.rdb0000644000176200001440000015275013611774130014621 0ustar liggesusers}pٖ'%` 6?q ~d ~t7$~ qU zY μ絼^j^GX]>ᘕ,%Y: ڒ,Y?˒w'[UE͛Yۿêd{_GQHI1Ix ?]JzmaK~(]GVmuӡyIRh~ChWο ~TI#K~͆jYK* el}[lu6uh?;5>^ՍB^+Z~qz[Z[-6HбSl<o}8qz {f0j|Weǟ5X_tkzBzBE@}4fbTdG*'5U+! 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These are assumed to be independent. We shall also need the ordered $p$--values: $p_{[1]} \le p_{[2]}, \dots, \le p_{[k]}$ and weights $w_i,\;i = 1, \dots, k$. Logarithms are natural. A function for combining $p$--values is denoted $g$. The size of the test is $\alpha$. We may also need $k$ degrees of freedom, $\nu_i$. The methods are referred to by the name of the function in \func{metap}. Table \ref{funcs} shows other descriptions of each method. \begin{table}[htbp] \begin{tabular}{lll} Function name & \multicolumn{2}{c}{Description(s)} \\[1ex] & \multicolumn{1}{c}{Eponym} \\ \func{invchisq} & Lancaster's method & Inverse chi square \\ \func{invt} & & Inverse t \\ \func{logitp} & & Logistic\\ \func{meanp} \\ \func{meanz} \\ \func{maximump} \\ \func{minimump} & Tippett's method \\ \func{sumlog} & Fisher's method & Chi square (2 df)\\ \func{sump} & Edgington's method & Uniform\\ \func{sumz} & Stouffer's method & Normal\\ \func{truncated} & Truncated Fisher\\ \func{truncated} & & rank--truncated\\ \func{votep} \\ \func{wilkinsonp} & Wilkinson's method \\ \end{tabular} \caption{Methods considered in this document} \label{funcs} \end{table} \section{Theoretical results} There have been various attempts to clarify the problem and to discuss optimality of the methods. A detailed account was provided by \citet{liptak58}. \citet{birnbaum54} considered the property of admissibility. A method is admissible if when it rejects $H_0$ for a set of $p_i$ it will also reject $H_0$ for $P^*_i$ where $p^*_i \le p_i$ for all $i$. He considered that Fisher's and Tippett's method were admissible. See also \citet{owen09}. He also points out the problem is poorly specified. This may account for the number of methods available and their differing behaviour. The null hypothesis $H_0$ is well defined, that all $p_i$ have a uniform distribution on the unit interval. There are two classes of alternative hypothesis \begin{itemize} \item $H_A$: all $p_i$ have the same (unknown) non--uniform, non--increasing density, \item $H_B$: at least one $p_i$ has an (unknown) non--uniform, non--increasing density. \end{itemize} If all the tests being combined come from what are basically replicates then $H_A$ is appropriate whereas if they are of different kinds of test or different conditions then $H_B$ is appropriate. Note that Birnbaum specifically considers the possibility that the tests being combined may be very different for instance some tests of means, some of variances, and so on. \section{The methods} \subsection{Comparison scenarios} To provide a standard of comparison we shall use the following two situations. Some authors have also used the case of exactly two $p_i$. \begin{description} %\subsubsection{What if all $p_i = p$?\label{twopisection}} \item[What if all $p_i = p$?]\label{twopisection} Perhaps surprisingly there are substantial differences here as we shall see when we look at each method. We shall describe how the returned value varies with $p$ and $k$. %\subsubsection{Cancellation} \item[Cancellation] When the collection of primary studies contains a number of values significant in both directions the methods can give very different results. If the intention of the synthesis is to examine a directional hypothesis one would want a method where these cancelled out. The decision between methods should be made on theoretical grounds of course. We shall use the following four values as our example. \end{description} {\codefont <<>>= cancel <- c(0.001, 0.001, 0.999, 0.999) @ } <>= library(metap) data(validity) genvec <- function(pvals, kvals, fun, name) { ps <- length(pvals) ks <- length(kvals) temp <- matrix(-1, nrow = ps, ncol = ks) for(i in 1:ps) for(j in 1:ks) { temp[i, j] <- fun(rep(pvals[i], kvals[j]))$p } temp2 <- as.vector(temp) res <- data.frame(method = rep(name, length(temp2)), p = rep(pvals, ks), k = rep(kvals, each = ps), g = temp2 ) res } @ \subsection{Methods using transformation of the $p$--values} One class of methods relies on transforming the $p$--values first. \begin{table}[htbp] \begin{tabular}{lll} Function name & Definition & Critical value \\[1ex] \func{invchisq} & $\sum_{i=1}^k \chi^2_{\nu_i}(p_i)$ & $\chi^2_{\sum{\nu_i}}(\alpha)$ \\[1ex] \func{invt} & $\frac{\sum_{i=1}^k t_{\nu_i}(p_i)}% {\sqrt{\sum_{i=1}^k \frac{\nu_i}{\nu_i - 2}}}$ & $z(\alpha)$ \\[1ex] \func{logitp} & $\frac{\sum_{i=1}^k \log\frac{p}{1 - p}}{C}$ & $t_{5k+4}$ \\ & where $C = \sqrt\frac{k \pi^2 (5 k + 2)}{3(5 k + 4)}$ & \\[1ex] \func{meanz} & $\frac{\bar{z}}{s_{\bar{z}}}$ & $t_{k-1}(\alpha)$ \\ & where $\bar{z} = \sum_{i=1}^k \frac{z(p_i)}{k}$ \\ & and $s_{\bar{z}} = \frac{s_z}{\sqrt{k}}$ & \\[1ex] \func{sumlog} & $\sum_{i=1}^{k} - 2 \log p_i$ & $\chi_{2k}(\alpha)$ \\[1ex] \func{sumz} & $\frac{\sum_{i=1}^k z(p_i)}{\sqrt{k}}$ & $z(\alpha)$\\ \end{tabular} \caption{Definitions of methods using transformation of the $p$ values} \label{transdefs} \end{table} <>= kvals <- c(4, 5, 6, 8, 10, 15, 20) pvals <- c(0.2, 0.3, 0.3679, 0.4, 0.5, 0.6) dat <- rbind( genvec(pvals, kvals, logitp, "logitp"), genvec(pvals, kvals, meanz, "meanz"), genvec(pvals, kvals, sumlog, "sumlog"), genvec(pvals, kvals, sumz, "sumz") ) @ <>= lattice::xyplot(g ~ k | method, groups = p, type = "l", data = dat, auto.key = list(space = "left", lines = TRUE, title = "p"), ylab = "g(p)" ) @ \subsubsection{The method of summation of logs, Fisher's method} See Table \ref{transdefs} for the definition. This works because $- 2 \log p_i$ is a $\chi^2_2$ and the sum of $\chi^2$ is itself a $\chi^2$ with degrees of freedom equal to the sum of the degrees of freedom of the individual $\chi^2$. Of course the sum of the log of the $p_i$ is also the log of the product of the $p_i$. Fisher's method \citep{fisher25} is provided in \func{sumlog}. <>= set.seed(18122019) temp <- matrix(runif(10000), nrow = 100) fisher <- apply(temp, 1, function(x) sumlog(x)$p) lanc4 <- apply(temp, 1, function(x) invchisq(x, 4)$p) lanc16 <- apply(temp, 1, function(x) invchisq(x, 16)$p) lanc256 <- apply(temp, 1, function(x) invchisq(x, 256)$p) banda <- function(x, y) { res <- data.frame(sum = x + y, diff = (x - y)) res } dat <- data.frame(rbind(banda(fisher, lanc4), banda(fisher, lanc16), banda(fisher, lanc256), banda(lanc4, lanc16), banda(lanc4, lanc256), banda(lanc16, lanc256) ), name = factor(c(rep("FL4", 100), rep("FL16", 100), rep("FL256", 100), rep("L4L16", 100), rep("L4L256", 100), rep("L16L256", 100)), levels = c("FL4", "FL16", "FL256", "L4L16", "L4L256", "L16L256") ) ) @ <>= lattice::xyplot(diff ~ sum | name, data = dat, panel = function(x, y, ...) { lattice::panel.xyplot(x, y, ...) lattice::panel.abline(h = mean(y), lty = 2) lattice::panel.abline(h = mean(y) + 1.96 * sd(y), lty = 3) lattice::panel.abline(h = mean(y) - 1.96 * sd(y), lty = 3) } ) @ <>= stouff <- apply(temp, 1, function(x) sumz(x)$p) invt4 <- apply(temp, 1, function(x) invt(x, 4)$p) invt16 <- apply(temp, 1, function(x) invt(x, 16)$p) invt256 <- apply(temp, 1, function(x) invt(x, 256)$p) banda <- function(x, y) { res <- data.frame(sum = x + y, diff = (x - y)) res } dat <- data.frame(rbind(banda(stouff, invt4), banda(stouff, invt16), banda(stouff, invt256), banda(invt4, invt16), banda(invt4, invt256), banda(invt16, invt256) ), name = factor(c(rep("St4", 100), rep("St16", 100), rep("St256", 100), rep("t4t16", 100), rep("t4t256", 100), rep("t16t256", 100)), levels = c("St4", "St16", "St256", "t4t16", "t4t256", "t16t256") ) ) @ <>= lattice::xyplot(diff ~ sum | name, data = dat, panel = function(x, y, ...) { lattice::panel.xyplot(x, y, ...) lattice::panel.abline(h = mean(y), lty = 2) lattice::panel.abline(h = mean(y) + 1.96 * sd(y), lty = 3) lattice::panel.abline(h = mean(y) - 1.96 * sd(y), lty = 3) } ) @ As can be seen in Figure \ref{equalp} when all the $p_i=p$ \func{sumlog} returns a value which decreases with $k$ when $p<0.32$, increases with $k$ when $p>0.37$, and in between increases with $k$ and then decreases. Some detailed algebra provided in a post to https://stats.stackexchange.com/questions/243003 by Christoph Hanck suggests that the breakpoint is $e^{-1} = 0.3679$. Where the $p_i$ are less than this then for a sufficiently large $k$ (several hundred) the result will be significant and not if above that. Over the range of $k$ we are plotting this bound is not yet closely approached. \mygraph{transeqp}{Behaviour of the methods using transformed $p$ values for $k$ values of $p=p_i$}{equalp} \subsubsection{Inverse $\chi^2$ Lancaster's method} It would of course be possible to generalise Fisher's method to use transformation to $\chi^2$ with any other number of degrees of freedom rather than 2. \citet{lancaster61} suggests that this is highly correlated with \func{sumlog}. Lancaster's method is provided in \func{invchisq}. In fact the resemblance to \func{sumlog} becomes less as the number of degrees of freedom increases. Figure \ref{fishlanc} shows for a small number of selected degrees of freedom how it compares to Fisher's method. \twograph{fishlanc}{Fisher's method and Lancaster's method}{fishlanc}{stouffinvt}{Stouffer's method and inverse $t$}{stouffinvt}{Sum and difference plots of Fisher v Lancaster and Stouffer v inverse $t$}{compfishstouff} \subsubsection{The method of summation of $z$ values, Stouffer's method} The method of summation of $z$ values is provided in \func{sumz} \citep{stouffer49}. See Table \ref{transdefs} for the definition. As can be seen in Figure \ref{equalp} it returns a value for our $p_i=p$ example which decreases with $k$ when $p$ below 0.5 and increases above. A weighted version of Stouffer's method is available %\begin{equation} $\frac{\sum_{i=1}^k w_i z(p_i)}{\sqrt {\sum_{i=1}^k w_i ^ 2}}$ %\end{equation} where $w_i$ are the weights. In the absence of effect sizes (in which case a method using effect sizes would be more appropriate anyway) best results are believed to be obtained with weights proportional to the square root of the sample sizes \citep{zaykin11} following \citet{liptak58}. \subsubsection{Mean of normals method} There is also a method closely related to Stouffer's using the mean of normals provided in \func{meanz} also defined in Table \ref{transdefs} which has very similar properties except that when all the $p_i$ are equal it either gives 0 or 1 as can be seen in Figure \ref{equalp}. <<>>= meanz(c(0.3, 0.31))$p meanz(c(0.1, 0.2))$p @ The method of \func{meanz} also has the unusual property that a set of $p$--values which are all less than those in another set can still give rise to a larger overall $p$. See example above. This is the only method considered here which has this property so if it is a desirable one then that is the only method to consider. \subsubsection{The inverse $t$ method} A closely related method is the inverse $t$ method. See Table \ref{transdefs} for the definition. This method is provided in \func{invt}. As is clear from the definition this method tends to Stouffer's method as $\nu_i \to \infty$. Figure \ref{stouffinvt} shows this for selected degrees of freedom. \subsubsection{The method of summation of logits} See Table \ref{transdefs} for the definition. This method is provided in \func{logitp}. The constant $C$ was arrived at by equating skewness and kurtosis with that of the $t$--distribution \citep{loughin04}. As can be seen in Figure \ref{equalp} this method returns a value for our $p_i=p$ example which decreases with $k$ when $p$ below 0.5 and increases above. \subsubsection{Examples for methods using transformations of the $p$ values} \begin{table}[htbp] \begin{tabular}{lll} Function name & validity & cancel \\[1ex] \func{logitp} & \Sexpr{logitp(validity)$p} & \Sexpr{logitp(cancel)$p} \\ \func{meanz} & \Sexpr{meanz(validity)$p} & \Sexpr{meanz(cancel)$p} \\ \func{sumlog} & \Sexpr{sumlog(validity)$p} & \Sexpr{sumlog(cancel)$p} \\ \func{sumz} & \Sexpr{sumz(validity)$p} & \Sexpr{sumz(cancel)$p}\\ \end{tabular} \caption{Examples of methods using transformation of the $p$ values} \label{transexamples} \end{table} Using the same example dataset which we have already plotted and our cancellation dataset we have the values in Table \ref{transexamples}. As can be seen all the methods cancel except for \func{sumlog}. The agreement for the validity dataset is close. Lancaster's method and inverse $t$ are not shown as they are both infinite families of possible methods and in any event are similar to Fisher's method and Stouffer's method respectively. \subsection{Methods using untransformed $p$--values} \begin{table}[htbp] \begin{tabular}{lll} Function name & Definition & Critical value \\[1ex] \func{meanp} & $\bar p = \frac{\sum_{i=1}^k p_i}{k}$ \\ & $z = (0.5 - \bar{p}) \sqrt{12k}$ & $z(\alpha)$ \\ \func{minimump} & $p_{[1]}$ & $1 - (1 - \alpha)^{\frac{1}{k}}$ \\ \func{maximump} & $p_{[k]}$ & $\alpha^k$ \\ \func{wilkinsonp} & $p_{[r]}$ & $\sum_{s=r}^k {k \choose s}\alpha^s (1 - \alpha)^{k-s}$\\[1ex] \func{sump} & $\frac{(S)^k}{k!}% - {k - 1 \choose 1}\frac{(S - 1)^k}{k!}% + {k - 2 \choose 2}\frac{(S - 2)^k}{k!} - \dots$ & $\alpha$ \\ & where $S = \sum_{i=1}^k p_i$ \\ \end{tabular} \caption{Definitions of methods not using transformation of the $p$ values, % the series for \func{sump} continues until the term in in the numerator $(S-i)$ becomes negative} \label{untransdefs} \end{table} <>= kvals <- c(4, 5, 6, 8, 10, 15, 20) pvals <- c(0.2, 0.3, 0.3679, 0.4, 0.5, 0.6) dat <- rbind( genvec(pvals, kvals, meanp, "meanp"), genvec(pvals, kvals, maximump, "maximump"), genvec(pvals, kvals, minimump, "minimump"), genvec(pvals, kvals, sump, "sump"), genvec(pvals, kvals, votep, "votep") ) @ <>= lattice::xyplot(g ~ k | method, groups = p, type = "l", data = dat, auto.key = list(space = "left", lines = TRUE, title = "p"), ylab = "g(p)" ) @ \mygraph{untranseqp}{Behaviour of the methods using untransformed $p$ values for $k$ values of $p=p_i$}{unequalp} \subsubsection{The method of minimum $p$, maximum $p$, and Wilkinson's method} The methods of minimum $p$ \citep{tippett31}, maximum $p$ and Wilkinson \citep{wilkinson51} are defined in Table \ref{untransdefs}. Wilkinson's method depends on which value (the $r$th) of $p_{[i]}$ is selected. % p is pbeta(p[r], r, k+1-r) % critical p is qbeta(alpha, r, k+1-r) Wilkinson's method is provided in \func{wilkinsonp} and a convenience function \func{minimump} with its own \code{print} method is provided for the minimum $p$ method ($r=1$). It is also possible to use the method for the maximum $p$ (that is $r=k$) and a convenience function \func{maximump} is provided for that purpose. As can be seen in Figure \ref{unequalp} these methods return a value for our $p_i=p$ example which always increases with $k$ which is true for \func{minimump} and which always decreases with $k$ which is true for \func{maximump} \subsubsection{The method of summation of $p$--values, Edgington's method\label{sump}} Defined in Table \ref{untransdefs} \citep{edgington72a}. This method is provided in \func{sump}. As can be seen in Figure \ref{unequalp} this method returns a value for our $p_i=p$ example which decreases with $k$ when $p$ below 0.5 and increases above. Some authors use a simpler version, $\frac{(\sum p)^k}{k!}$, for instance \citet{rosenthal78} in the text although compare his Table 4. This can be very conservative when $\sum p > 1$ There seems no particular need to use this method but it is returned by \func{sump} as the value of \code{conservativep} for use in checking published values. Note also that there can be numerical problems for extreme values of $S$ and in that case recourse might be made to \func{meanp} which has similar properties. \subsubsection{The mean $p$ method} Defined in Table \ref{untransdefs}. Although this method is attributed to Edgington \citep{edgington72b} when the phrase Edgington's method is used it refers to the method of summation of $p$--values described above in Section \ref{sump}. As can be seen in Figure \ref{unequalp} this method returns a value for our $p_i=p$ example which decreases with $k$ when $p$ below 0.5 and increases above. Not surprisingly this method gives very similar results to Edington's other method implemented in \func{sump} and since it does not have the numerical problems of that method it might perhaps be preferred. \subsubsection{Examples for methods using untransformed $p$--values} Using the same example dataset which we have already plotted and our cancellation dataset we have the values in Table \ref{untransexamples}. As can be seen \func{meanp} and \func{sump} cancel but the other two do not. Agreement here is not so good especially for the maximum p method. Wilkinson's method not shown as it depends on the value of $r$. \begin{table}[htbp] \begin{tabular}{lll} Function name & validity & cancel \\[1ex] \func{minimump} & \Sexpr{minimump(validity)$p} & \Sexpr{minimump(cancel)$p} \\ \func{maximump} & \Sexpr{maximump(validity)$p} & \Sexpr{maximump(cancel)$p} \\ \func{meanp} & \Sexpr{meanp(validity)$p} & \Sexpr{meanp(cancel)$p}\\ \func{sump} & \Sexpr{sump(validity)$p} & \Sexpr{sump(cancel)$p} \\ \end{tabular} \caption{Examples for methods using the untransformed $p$ values} \label{untransexamples} \end{table} \subsection{Other methods} \subsubsection{The method of vote--counting} A simple way of looking at the problem is vote counting. Strictly speaking this is not a method which combines $p$--values in the same sense as the other methods. If most of the studies have produced results in favour of the alternative hypothesis irrespective of whether any of them is individually significant then that might be regarded as evidence for that alternative. The numbers for and against may be compared with what would be expected under the null using the binomial distribution. A variation on this would allow for a neutral zone of studies which are considered neither for nor against. For instance one might only count studies which have reached some conventional level of statistical significance in the two different directions. This method returns a value for our $p_i=p$ example which is 1 for $p$ values above 0.5 and otherwise invariant with $p$ but decreases with $k$. This method does cancel significant values in both directions. \begin{table}[htbp] \begin{tabular}{lll} Function name & validity & cancel \\[1ex] \func{votep} & \Sexpr{votep(validity)$p} & \Sexpr{votep(cancel)$p} \\ \end{tabular} \caption{Examples for vote counting} \label{votepexamples} \end{table} \subsubsection{Methods not using all $p$--values} If there is a hypothesis that the signal will be concentrated in only a few $p$--values then alternative methods are available in \func{truncated}. This is a wrapper to two packages available on CRAN: \pkg{TFisher} which provides the truncated Fisher method \citep{zaykin07,zhang18} and \pkg{mutoss} which provides the rank--truncated Fisher method \citep{dudbridge03}. Note that Table \ref{truncatedexamples} only shows results for the validity data--set as, since the methods explicitly only consider results in one direction the cancellation issue does not arise. \begin{table}[htbp] \begin{tabular}{lll} Function name & truncated at $p$ = 0.5 & truncated at rank = 5 \\[1ex] \func{truncated} & \Sexpr{truncated(validity, ptrunc = 0.5)$p} & \Sexpr{truncated(validity, rtrunc = 5)$p} \\ \end{tabular} \caption{Examples for truncated using the validity data--set} \label{truncatedexamples} \end{table} These methods are appropriate for the situation where it is known that many of the $p$--values are noise and there will only be a few signals. \section{Loughin's recommendations} In his simulation study \citet{loughin04} carried out extensive comparisons. Note that he did not consider all the methods implemented here. These omissions are not too important for our purposes. The methods implemented here as \func{invchisq}, \func{invt}, \func{meanp} and \func{meanz} are all very similar to ones which he did study. The truncation methods appeared about the same time as his work but in any case are fundamentally different. Vote counting is arguably not a method of the same sort. As Loghin points out the first thing to consider is whether large $p$--values should cancel small ones. If this is not desired then the only methods to consider are those in \func{sumlog} (Fisher), \func{minimump} (Tippett) and \func{maximump}. \mypdf{struct}{Loughin's recommendations based on structure}{struct}{[height=6cm,width=8cm]} He bases his recommendations on criteria of structure and the arrangement of evidence against $H_0$. Figure \ref{struct} shows a summary of his recommendations about the structure of the evidence. \mypdf{strength}{Loughin's recommendations based on where the strength of the evidence is located}{strength}{[height=8cm,width=12cm]} Figure \ref{strength} summarise his recommendations about the arrangement of evidence. Overall he considered the choice to lie between Stouffer's method, Fisher's method and the logistic method implemented in \func{logitp}. As has already been mentioned Fisher's method cancels whereas the other two do not so if the weak evidence in a small number of $p$--values is not to be over--whelmed by the others then Fisher is the best choice. However where the evidence is more evenly spread Stouffer's method may be preferred. The logistic method represents a compromise between them and is perhaps best suited where the pattern of evidence is not clear in advance. The other methods are not universally ruled out and may be helpful in the specific circumstance outlined in his summaries. \bibliography{metap} \bibliographystyle{plainnat} \end{document} metap/vignettes/metap.Rnw0000644000176200001440000002017213546615554015206 0ustar liggesusers\documentclass[12pt]{article} \usepackage{amsmath,amssymb} %\usepackage{mydef2} \usepackage[round]{natbib} \usepackage{parskip,url} \usepackage{graphicx,subfig} \setlength{\topmargin}{0cm} \addtolength{\textheight}{2cm} %\lhead{} %\input{title} %\VignetteIndexEntry{Introduction to the metap package} \title{Introduction to the \pkg{metap} package} \author{Michael Dewey} \newcommand{\pkg}[1]{\texttt{#1}} \newcommand{\func}[1]{\texttt{#1}} \newcommand{\code}[1]{\texttt{#1}} \newcommand{\codefont}{\footnotesize} \newcommand{\mygraph}[3]{% \begin{figure}[htbp] \includegraphics[height=6cm,width=10cm]{#1} \caption{#2} \label{#3} \end{figure} } \newcommand{\twograph}[8]{% \begin{figure}[htbp] \subfloat[#2\label{#3}]{\includegraphics[height=6cm,width=7cm]{#1}}% \subfloat[#5\label{#6}]{\includegraphics[height=6cm,width=7cm]{#4}} \caption{#7} \label{#8} \end{figure} } \begin{document} \maketitle \section{Introduction} \subsection{What is this document for?} This document describes some methods for the meta--analysis of $p$--values (significance values) and their implementation in the package \pkg{metap}. I welcome feedback about sources of published examples against which I can test the code and any other comments about either the documentation or the code. The problem of meta--analysis of $p$--values is of course not completely unconnected with the more general issue of simultaneous statistical inference. \subsection{Why and when to meta--analyse significance values} The canonical way to meta--analyse a number of primary studies uses estimates of effect sizes from each of them. There are a large number of packages for this purpose available from CRAN and described in the task view \url{http://CRAN.R-project.org/view=MetaAnalysis}. However sometimes the only available information may be $p$--values especially when some of the primary studies were published a long time ago or were published in sources which were less rigorous about insisting on effect sizes. The methods outlined here are designed for this eventuality. The situation may also arise that some of the studies can be combined in a conventional meta--analysis using effect sizes but there are many others which cannot and in that case the conventional meta--analysis of the subset of studies which do have effect sizes may usefully be supplemented by an overall analysis of the $p$--values. Just for the avoidance of doubt, if each study has produced a proportion and the goal is to synthesise them to a common estimate or analyse the differences between them then the standard methods are appropriate not the ones outlined here. The $p$--values in this document are significance levels. The methods are referred to by the name of the function in \func{metap}. Table \ref{funcs} shows other descriptions of each method. \begin{table}[htbp] \begin{tabular}{lll} Function name & \multicolumn{2}{c}{Description(s)} \\[1ex] & \multicolumn{1}{c}{Eponym} \\ \func{invchisq} & Lancaster's method & Inverse chi square \\ \func{invt} & & Inverse t \\ \func{logitp} & & Logistic\\ \func{meanp} \\ \func{meanz} \\ \func{maximump} \\ \func{minimump} & Tippett's method \\ \func{sumlog} & Fisher's method & Chi square (2 df)\\ \func{sump} & Edgington's method & Uniform\\ \func{sumz} & Stouffer's method & Normal\\ \func{truncated} & Truncated Fisher & rank--truncated\\ \func{votep} \\ \func{wilkinsonp} & Wilkinson's method \\ \end{tabular} \caption{Methods considered in this document} \label{funcs} \end{table} \section{Preparation for meta--analysis of $p$--values} \subsection{Preliminaries} I assume you have installed \textsf{R} and \pkg{metap}. You then need to load the package. <<>>= library(metap) @ \subsection{Directionality} It is usual to have a directional hypothesis, for instance that treatment is better than control. For the methods described here a necessary preliminary is to ensure that all the $p$--values refer to the same directional hypothesis. If the value from the primary study is two--sided it needs to be converted. This is not simply a matter of halving the quoted $p$--value as values in the opposite direction need to be reversed. A convenience function \func{two2one} is provided for this. {\codefont <<>>= pvals <- c(0.1, 0.1, 0.9, 0.9, 0.9, 0.9) istwo <- c(TRUE, FALSE, TRUE, FALSE, TRUE, FALSE) toinvert <- c(FALSE, TRUE, FALSE, FALSE, TRUE, TRUE) two2one(pvals, two = istwo, invert = toinvert) @ } Note in particular the way in which $0.9$ is converted under the different scenarios. <>= data(validity) @ \subsection{Plotting} {\codefont <<>>= print(validity) @ } It would be a wise precaution to examine the $p$--values graphically or otherwise before subjecting them to further analysis. Two functions are provided for this purpose: \func{plotp} and \func{schweder}. \subsubsection{Plotting using \func{plotp}} The \func{plotp} provides a Q--Q plot of the $p$--values to detect departure from the uniform distribution. An example is shown in Figure \ref{plotp}. The standard line through through the quartiles is superimposed. This is the function which is called when the \func{plot} method is used on an object returned by any of the meta--analysis functions. <>= plotp(validity) @ \mygraph{metap-plotp}{Q--Q plot from \func{plotp}}{plotp} \subsubsection{Plotting using \func{schweder}} A function \func{schweder} provides plots with a variety of informative lines superimposed. It plots the ordered $p$--values, $p_{[i]}$, against $i$. Although the original motivation for the plot is \citet{schweder82} the function uses a different choice of axes due to \citet{benjamini00}. We will use an example dataset on the validity of student ratings quoted in \citet{becker94}. Figure \ref{simple} shows the plot from \func{schweder} which is the same as from \func{plotp} but without the line. <>= schweder(validity) @ \func{schweder} also offers the possibility of drawing one of a number of straight line summaries. The three possible straight line summaries are shown in Figure \ref{withlines} and are: \begin{itemize} \item the lowest slope line of Benjaimin and Hochberg which is drawn by default as solid, \item a least squares line drawn passing through the point $k+1, 1$ and using a specified fraction of the points which is drawn by default as dotted, \item a line with user specified intercept and slope which is drawn by default as dashed. \end{itemize} <>= schweder(validity, drawline = c("bh", "ls", "ab"), ls.control = list(frac = 0.5), ab.control = list(a = 0, b = 0.01)) @ \twograph{metap-simple}{Simple graph}{simple}{metap-withlines}{With lines}{withlines}{Output from schweder}{schweder} \subsection{Reporting problems in the primary studies} Another issue is what to do with studies which have simply reported on whether a conventional level of significance like 0.05 was achieved or not. If the exact associated $p$ cannot be derived from the statistics quoted in the primary source then the value of the level achieved, in this case 0.05, can be used although this may be conservative. Studies which simply report not significant could be included as having $p=1$ (or $p=0.5$ if it is known that the direction was right) although this is very conservative. The theory of handling $p$--values which have been truncated like this has been developed by \citet{zaykin02} and \func{truncated} provides a convenience wrapper for two methods available in other CRAN packages. \section{Using the methods} All the methods in the package take as their first argument the vector of $p$--values. To use Fisher's method as an example: <<>>= sumlog(validity) @ A few require extra information. Those which rely on inverse transformations often need a vector of degrees of freedom. Currently this applies to \func{invchisq} and \func{invt}. Stouffer's method in \func{sumz} optionally uses weights if a vector of weights is provided. \section{Miscellanea} \begin{description} \item[Extractor functions] The standard \code{print} and \code{plot} methods are provided. \item[Reading] An annotated bibliography is provided by \citet{cousins08} \end{description} \bibliography{metap} \bibliographystyle{plainnat} \end{document} metap/vignettes/struct.pdf0000644000176200001440000007245313576717575015451 0ustar liggesusers%PDF-1.4 % 3 0 obj << /Length 4 0 R /Filter /FlateDecode >> stream xWˊ\7+-J0! 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consideration in psychological research}, journal = {Psychological Bulletin}, year = {1951}, volume = {48}, pages = {156--158}, keywords = {meta-analysis, significance values} } @article{edgington72a, author = {Edgington, E S}, title = {An additive method for combining probability values from independent experiments}, journal = {Journal of Psychology}, year = {1972}, volume = {80}, pages = {351--363}, keywords = {meta-analysis, significance values} } @article{zaykin11, author = {Zaykin, D V}, title = {Optimally weighted $Z$--test is a powerful method for combining probabilities in meta--analysis}, journal = {Journal of Evolutionary Biology}, volume = {24}, year = {2011}, pages = {1836--1841}, keywords = {meta-analysis, significance values} } @incollection{becker94, author = {Becker, B J}, editor = {Cooper, H and Hedges, L V}, title = {Cambining significance levels}, booktitle = {A handbook of research synthesis}, year = {1994}, pages = {215--235}, chapter = {15}, publisher = {Russell Sage}, address = {New York}, keywords = {meta-analysis} } @ARTICLE{schweder82, author = {Schweder, T and Spj{\o}tvoll, E}, year = 1982, title = {Plots of $P$--values to evaluate many tests simultaneously}, journal = {Biometrika}, volume = 69, pages = {493--502}, keywords = {multiple comparison} } @article{benjamini00, author = {Benjamini, Y and Hochberg, Y}, title = {On the adaptive control of the false disovery rate in multiple testing with independent statistics}, journal = {Journal of Educational and Behavioral Statistics}, year = {2000}, volume = {25}, pages = {60--83}, keywords = {multiple comparison} } @article{edgington72b, author = {Edgington, E S}, title = {A normal curve method for combining probability values from independent experiments}, journal = {Journal of Psychology}, year = {1972}, volume = {82}, pages = {85--89}, keywords = {meta-analysis, significance values} } @article{birnbaum54, author = {Birnbaum, A}, title = {Combining independent tests of significance}, journal = {Journal of the American Statistical Association}, year = {1954}, volume = {49}, pages = {559--574}, keywords = {meta-analysis, significance values} } @article{owen09, author = {Owen, A B}, title = {{Karl Pearson's} meta--analysis revisited}, journal = {Annals of Statistics}, year = {2009}, volume = {37}, pages = {3867--3892}, keywords = {meta-analysis, significance values} } @article{liptak58, author = {Lipt\'ak, T}, title = {On the combination of independent tests}, journal = {A Magyar Tudom\'anyos Akad\'emia Matematikai Kutat\'o Int\'ezet\'enek K\"ozlem\'enyi}, year = {1958}, volume = {3}, pages = {171--197}, keywords = {meta-analysis, significance values} } @misc{cousins08, author = {Cousins, R D}, title = {Annotated bibliography of some papers on combining significances or $p$--values}, year = {2008}, note = {arXiv:0705.2209}, keywords = {meta-analysis, significance values} } @article{loughin04, author = {Loughin, T M}, title = {A systematic comparison of methods for combining $p$--values from independent tests}, journal = {Computation Statistics and Data Analysis}, year = {2004}, volume = {47}, pages = {467--485}, keywords = {meta-analysis, significance values} } @article{lancaster61, author = {Lancaster, H}, title = {The combination of probabilities: an application of orthonormal functions}, journal = {Australian Journal of Statistics}, year = {1961}, volume = {3}, pages = {20--33}, keywords = {meta-analysis, significance values} } @article{zaykin02, author = {Zaykin, D V and Zhivotovsky, L A and Westfall, P H and Weir, B S}, title = {Truncated product method for combining $P$--values}, journal = {Genetic Epidemiology}, year = {2002}, volume = {22}, pages = {170--185}, keywords = {meta-analysis, significance values} } @book{fisher25, author = {Fisher, R A}, title = {Statistical methods for research workers}, year = {1925}, publisher = {Oliver and Boyd}, address = {Edinburgh}, } @book{stouffer49, author = {Stouffer, S A and Suchman, E A and DeVinney, L C and Star, S A and Williams, R M Jnr}, title = {The {American} soldier, vol 1: {A}djustment during army life}, year = {1949}, publisher = {Princeton University Press}, address = {Princeton}, } @book{tippett31, author = {Tippett, L H C}, title = {The methods of statistics}, year = {1931}, publisher = {Williams and Norgate}, address = {London}, keywords = {meta-analysis, significance values} } @article{zaykin07, author = {Zaykin, D V and Zhivotovsky, L A and Czika, W and Shao, S and Wolfinger, R D}, title = {Combining p--values in large--scale genomics experiments}, journal = {Pharmaceutical Statistics}, year = {2007}, volume = {6}, pages = {217--236} } @article{dudbridge03, author = {Dudbridge, F and Koeleman, B P C}, title = {Rank truncated product of $P$--values, with application to genomewide association scans}, journal = {Genetic Epidemiology}, year = {2003}, volume = {25}, pages = {360--366} } @article{zhang18, author = {Zhang, H and Tong, T and Landers, J and Wu, Z}, title = {{TF}isher tests: optimal and adaptive thresholding for combining $p$--values}, journal = {arXiv}, year = {2018}, url = {arXiv:1801.04309} } metap/NEWS0000644000176200001440000000300513577121207012062 0ustar liggesusersChanges in version 1.3 - updated documentation - updated vignette - changed to use authors@R Changes in version 1.2 - new function truncated - improved error handling - updated documentation Changes in version 1.1 - new plotting function plotp - split vignette into two Changes in version 1.0 - added new functions: inverse chisq, inverse t, mean z - changed package Rd file to use system macros - changed to using Rdpack for references in Rd files - updated vignette Changes in version 0.9 - added new function allmetap - minor bug in sumz fixed - updated vignette Changes in version 0.8 - updated vignette to include method comparisons Changes in version 0.7-1 - added maximump - updated documentation - this version never uploaded to CRAN Changes in version 0.7 - added vignette - minor bug fixes Changes in version 0.6-3 - test for cancellation error in sump - minor warnings fixes - updated documentation - this version never uploaded to CRAN Changes in version 0.6-2 - importFrom added for CRAN - Description enhanced Changes in version 0.6-1 - new maintainer email address - added CITATION file - this version never uploaded to CRAN Changes in version 0.6 - added weights to sumz - updated documentation Changes in version 0.5-1 - updated DESCRIPTION - First upload to CRAN Changes in version 0.5 - first substantial version metap/R/0000755000176200001440000000000013611774102011563 5ustar liggesusersmetap/R/logitp.R0000644000176200001440000000143013556516551013214 0ustar liggesuserslogitp <- function(p) { keep <- (p > 0) & (p < 1) invalid <- sum(1L * keep) < 2 if(invalid) { warning("Must have at least two valid p values") res <- list(t = NA_real_, df = NA_integer_, p = NA_real_, validp = p[keep]) } else { psum <- sum(log(p[keep] / (1 - p[keep]))) k <- length(p[keep]) mult <- -1 / sqrt(k * pi ^ 2 * (5 * k + 2) / (3 * (5 * k + 4))) if(k != length(p)) { warning("Some studies omitted") } t <- mult * psum df <- (5 * k + 4) res <- list(t = t, df = df, p = pt(t, df, lower.tail = FALSE), validp = p[keep]) } class(res) <- c("logitp", "metap") res } print.logitp <- function(x, ...) { cat("t = ", x$t, " with df = ", x$df, " p = ", x$p, "\n") invisible(x) } metap/R/two2one.R0000644000176200001440000000065212037557252013314 0ustar liggesuserstwo2one <- function(p, two = NULL, invert = NULL) { np <- length(p) if(is.null(two)) { two <- rep(TRUE, np) } if(is.null(invert)) { invert <- rep(FALSE, np) } # check values of p inrange <- sum(1L * ((p >= 0) & (p <= 1))) if(np != inrange) warning("Some p out of range") onep <- ifelse(two, ifelse(invert, (1 - p) + p / 2, p / 2), ifelse(invert, 1 - p, p) ) onep } metap/R/truncated.R0000644000176200001440000001023313561032314013672 0ustar liggesuserstruncated <- function(p, ptrunc = NULL, rtrunc = NULL, ...) { # dop <- function(validp, ptrunc) { if(requireNamespace("TFisher")) { if((ptrunc <= 0) | (ptrunc > 1)) { warning("The value of ptrunc must be between 0 and 1") warning("Setting p to 0.5") ptrunc <- 0.5 } nincl <- sum(1L * (validp < ptrunc)) nexcl <- sum(1L * (validp >= ptrunc)) if(nincl < 1) warning("No p values left after truncation") if(nexcl < 1) message("No studies removed by truncation") stat <- TFisher::stat.tpm(validp, ptrunc) pval <- 1 - TFisher::p.tpm(stat, length(validp), ptrunc) res <- list(p = pval, stat = stat, validp = validp, ptrunc = ptrunc, n = c(incl = nincl, excl = nexcl), method = "truncated") } else { warning("Need to install TFisher for this option") res <- list(p = NA, stat = NA) } res } # dor <- function(validp, rtrunc, ...) { if(requireNamespace("mutoss")) { if((rtrunc < 1) | (rtrunc > length(validp))) { l <- length(validp) warning(paste("The value of rtrunc must be between 1 and "), l) warning(paste("Setting rtrunc to", l %/% 2)) rtrunc <- l %/% 2 } nincl <- rtrunc nexcl <- length(validp) - nincl if(nincl < 1) warning("No p values left after truncation") if(nexcl < 1) message("No studies removed by truncation") temp <- mutoss::ranktruncated(validp, rtrunc, ...) stat <- temp$RTP$Statistic pval <- temp$RTP$p.Value res <- list(p = pval, stat = stat, validp = validp, rtrunc = rtrunc, n = c(incl = nincl, excl = nexcl), method = "rank") } else { warning("Need to install mutoss for this option") res <- list(p = NA, stat = NA) } res } # keep <- (p > 0) & (p <= 1) if(sum(1L * keep) < 2) { warning("Must have at least two valid p values") res <- list(p = NA, stat = NA) } else { validp <- p[keep] if(length(validp) != length(p)) { warning("Some studies omitted") } usetpm <- !is.null(ptrunc) userank <- !is.null(rtrunc) if(usetpm & userank) { warning("Specifying both ptrunc and rtrunc undefined") res <- list(p = NA, stat = NA) } else if(usetpm) { res <- dop(validp, ptrunc) } else if(userank) { res <- dor(validp, rtrunc) } else { warning("Must specify one of ptrunc or rtrunc") res <- list(p = NA, stat = NA) } } class(res) <- c("truncated", "list") res } print.truncated <- function(x, ...) { cat("stat = ", x$stat, " p = ", x$p, "\n") invisible(x) } summary.truncated <- function(object, ...) { with(object, { cat("stat = ", stat, " p = ", p, "\n") if(!is.na(stat)) { cat(paste("Using", n["incl"], "values of", n["incl"] + n["excl"], "valid p values\n")) if(method == "truncated") { cat("Truncated at p= ", ptrunc, "\n") } if(method == "rank") { cat("Truncated at rank= ", rtrunc, "\n") } } # end valid stat }) # end with invisible(object) } plot.truncated <- function(x, pparams = list(pchs = c(16, 1), pcols = c("black", "black")), ...) { if(is.na(x$stat)) { warning("Nothing to plot") } else { with(x, { p <- sort(validp, decreasing = FALSE) nused <- n["incl"] k <- length(p) plot(1:k, c(rep(0, k-1), 1), type = "n", xlab = "Index", ylab = "p", ...) if(nused == k) { points(1:nused, p[1:nused], pch = pparams$pchs[1], col = pparams$pcols[1]) } else if(nused == 0) { points((nused+1):k, p[(nused+1):k], pch = pparams$pchs[2], col = pparams$pcols[2]) } else { points(1:nused, p[1:nused], pch = pparams$pchs[1], col = pparams$pcols[1]) points((nused+1):k, p[(nused+1):k], pch = pparams$pchs[2], col = pparams$pcols[2]) } }) } invisible(x) } metap/R/sumz.R0000644000176200001440000000252513556606413012717 0ustar liggesuserssumz <- function(p, weights = NULL, data = NULL, subset = NULL, na.action = na.fail) { if(is.null(data)) data <- sys.frame(sys.parent()) mf <- match.call() mf$data <- NULL mf$subset <- NULL mf$na.action <- NULL mf[[1]] <- as.name("data.frame") mf <- eval(mf, data) if(!is.null(subset)) mf <- mf[subset,] mf <- na.action(mf) p <- as.numeric(mf$p) weights <- mf$weights noweights <- is.null(weights) if(noweights) weights <- rep(1, length(p)) if(length(p) != length(weights)) warning("Length of p and weights differ") keep <- (p > 0) & (p < 1) invalid <- sum(1L * keep) < 2 if(invalid) { warning("Must have at least two valid p values") res <- list(z = NA_real_, p = NA_real_, validp = p[keep], weights = weights) } else { if(sum(1L * keep) != length(p)) { warning("Some studies omitted") omitw <- weights[!keep] if((sum(1L * omitw) > 0) & !noweights) warning("Weights omitted too") } zp <- (qnorm(p[keep], lower.tail = FALSE) %*% weights[keep]) / sqrt(sum(weights[keep]^2)) res <- list(z = zp, p = pnorm(zp, lower.tail = FALSE), validp = p[keep], weights = weights) } class(res) <- c("sumz", "metap") res } print.sumz <- function(x, ...) { cat("sumz = ", x$z, "p = ", x$p, "\n") invisible(x) } metap/R/sump.R0000644000176200001440000000206513556530135012701 0ustar liggesuserssump <- function(p) { keep <- (p >= 0) & (p <= 1) invalid <- sum(1L * keep) < 2 if(invalid) { warning("Must have at least two valid p values") res <- list(p = NA_real_, conservativep = NA_real_, validp = p[keep]) } else { sigmap <- sum(p[keep]) k <- length(p[keep]) conservativep <- exp( k * log(sigmap) - lgamma(k + 1)) nterm <- floor(sigmap) + 1 # how many values of sump denom <- lfactorial(k) psum <- 0 terms <- vector("numeric", nterm) for (i in 1:nterm) { terms[i] <- lchoose(k, i - 1) + k * log(sigmap - i + 1) - denom pm <- 2 * (i %% 2) - 1 psum <- psum + pm * exp(terms[i]) } if(k != length(p)) { warning("Some studies omitted") } if(sigmap > 20) { warning("Likely to be unreliable, check with another method") } res <- list(p = psum, conservativep = conservativep, validp = p[keep]) } class(res) <- c("sump", "metap") res } print.sump <- function(x, ...) { cat("psum = ", x$p, "\n") invisible(x) } metap/R/allmetap.R0000644000176200001440000000302013326064664013510 0ustar liggesusersallmetap <- function(p, method = NULL) { if(is.null(method)) stop("Must specify a method") mydata <- '"funcs","eponyms" "logitp", "maximump", "meanp", "meanz", "minimump","Tippett" "sumlog","Fisher" "sump","Edgington" "sumz","Stouffer" ' con <- textConnection(mydata) details <- read.csv(con, stringsAsFactors = FALSE) row.names(details) <- c("logitp", "maximump", "meanp", "meanz", "minimump", "sumlog", "sump", "sumz") close(con) # if all specified reset funcnames funcnames <- method if("all" %in% method) funcnames <- row.names(details) # check all funcnames are valid functions keep <- funcnames %in% row.names(details) if(sum(keep * 1L) != length(funcnames)) warning("Some unsupported methods specified") funcnames <- funcnames[keep] if(length(funcnames) < 1) stop("No supported methods specified") helper <- function(x) { thisfunc <- match.fun(row.names(details[x,])) res <- try(thisfunc(p), silent = TRUE) if(inherits(res, "try-error")) { res <- list(p = NA, valid = NA) } else { res <- list(p = res$p, valid = length(res$validp)) } res } res <- as.data.frame(t(sapply(funcnames, helper))) eponym <- vector("character", nrow(res)) for(i in 1:nrow(res)) { eponym[i] <- details[row.names(res[i,]),"eponyms"] } res$eponym <- eponym class(res) <- c("allmetap", "data.frame") res } print.allmetap <- function(x, digits = 5, ...) { print(format(x, digits = digits, ...), ...) invisible(x) } metap/R/meanp.R0000644000176200001440000000120013556527576013022 0ustar liggesusersmeanp <- function(p) { keep <- (p >= 0) & (p <= 1) invalid <- sum(1L * keep) < 4 if(invalid) { warning("Must have at least four valid p values") res <- list(z = NA_real_, p = NA_real_, validp = p[keep]) } else { pi <- mean(p[keep]) k <- length(p[keep]) z <- (0.5 - pi) * sqrt(12 * k) if(k != length(p)) { warning("Some studies omitted") } res <- list(z = z, p = pnorm(z, lower.tail = FALSE), validp = p[keep]) } class(res) <- c("meanp", "metap") res } print.meanp <- function(x, ...) { cat("z = ", x$z, " p = ", x$p, "\n") invisible(x) } metap/R/sumlog.R0000644000176200001440000000132413556515342013223 0ustar liggesuserssumlog <- function(p) { keep <- (p > 0) & (p <= 1) invalid <- sum(1L * keep) < 2 if(invalid) { warning("Must have at least two valid p values") res <- list(chisq = NA_real_, df = NA_integer_, p = NA_real_, validp = p[keep]) } else { lnp <- log(p[keep]) chisq <- (-2) * sum(lnp) df <- 2 * length(lnp) if(length(lnp) != length(p)) { warning("Some studies omitted") } res <- list(chisq = chisq, df = df, p = pchisq(chisq, df, lower.tail = FALSE), validp = p[keep]) } class(res) <- c("sumlog", "metap") res } print.sumlog <- function(x, ...) { cat("chisq = ", x$chisq, " with df = ", x$df, " p = ", x$p, "\n") invisible(x) } metap/R/schweder.R0000644000176200001440000000503313346141432013512 0ustar liggesusersschweder <- function(p, xlab = "Rank of p", ylab = "p", drawline = NULL, # bh.control =list(), bh.lwd = 1, bh.lty = "solid", bh.col = "black", ls.control =list(frac = NULL), ls.lwd = 1, ls.lty = "dotted", ls.col = "black", ab.control =list(a = NULL, b = NULL), ab.lwd = 1, ab.lty = "dashed", ab.col = "black", ...) { bx1y1toa <- function(b, x1, y1) { # find intercept from b, x1, y1 a <- y1 - b * x1 a } bh <- function(p) { # parameters for Benjamini and Hochberg lowet slope line n <- length(p) rankp <- n:1 si <- (1 - p) / (n - rankp + 1) # assumes p in non-ascending order di <- which(diff(si) > 0) ldi <- max(1, di[length(di)]) # in case no slope change b <- si[ldi] res <- list(a = bx1y1toa(b, n + 1, 1), b = b, si = si, di = di, ldi = ldi) res } n <- length(p) stopifnot(n > 1) toplot <- rev(sort(p)) keep <- (toplot >= 0) & (toplot <= 1) toplot <- toplot[keep] if(length(toplot) != n) warning('Some p values omitted') n <- length(toplot) if(n < 1) stop("Not even one to plot") x <- n:1 plot(x, toplot, ylab = ylab, xlab = xlab, ylim = c(0, 1), xlim = c(0, n+1), ...) res <- list(p = toplot) if("bh" %in% drawline) { params <- bh(toplot) abline(a = params$a, b = params$b, lty = bh.lty, lwd = bh.lwd, col = bh.col) res <- c(res, bh.params = list(a = params$a, b = params$b, si = params$si, di = params$di, ldi = params$ldi)) } if("ls" %in% drawline) { if(is.null(ls.control$frac)) { warning("Must specify fraction for ls") } else { frac <- ls.control$frac newy <- toplot - 1 # transform so that line is forced newx <- x - (n + 1) # through (n+1),1 rather than 0,0 plim <- newy[n * frac] fit <- lm(newy ~ newx - 1, subset = newy > plim) b <- coef(fit)[1] # correct slope a <- bx1y1toa(b, (n + 1), 1) # passing through (n+1,1 abline(a = a, b = b, lty = ls.lty, lwd = ls.lwd, col = ls.col) res <- c(res, ls.params = list(a = a, b = b)) } } if ("ab" %in% drawline) { if( is.null(ab.control$a) | is.null(ab.control$b) ) { warning("No parameters for a and b line") } else { abline(a = ab.control$a, b = ab.control$b,lty = ab.lty, lwd = ab.lwd, col = ab.col) res <- c(res, ab.params = list(a = ab.control$a, b = ab.control$b)) } } invisible(res) } metap/R/votep.R0000644000176200001440000000203313556531416013050 0ustar liggesusersvotep <- function(p, alpha = c(0.5, 0.5)) { alpha <- ifelse(alpha > 1, alpha / 100, alpha) # if percent stopifnot(alpha > 0, alpha < 1) stopifnot(alpha[1] <= alpha[2]) keep <- (p >= 0) & (p <= 1) invalid <- sum(1L * keep) < 2 if(invalid) { warning("Must have at least two valid p values") res = list(p = NA_real_, pos = NA_integer_, neg = NA_integer_, alpha = alpha, validp = p[keep]) } else { pi <- p[keep] k <- length(pi) pos <- sum(1L * (pi < alpha[1])) neg <- sum(1L * (pi > alpha[2])) if(k != length(p)) { warning("Some studies omitted") } if((pos + neg) <= 0) { warning("All p values are within specified limits of alpha") p <- 1 } else { p = binom.test(pos, pos + neg, 0.5, alternative = "greater")$p.value } res = list(p = p, pos = pos, neg = neg, alpha = alpha, validp = pi) } class(res) <- c("votep", "metap") res } print.votep <- function(x, ...) { cat("p = ", x$p, "\n") invisible(x) } metap/R/invchisq.R0000644000176200001440000000230713556534067013550 0ustar liggesusersinvchisq <- function(p, k, data = NULL, subset = NULL, na.action = na.fail) { if(is.null(data)) data <- sys.frame(sys.parent()) mf <- match.call() mf$data <- NULL mf$subset <- NULL mf$na.action <- NULL mf[[1]] <- as.name("data.frame") mf <- eval(mf, data) if(!is.null(subset)) mf <- mf[subset,] mf <- na.action(mf) p <- as.numeric(mf$p) k <- as.numeric(mf$k) if(length(k) == 1) { k <- rep(k, length(p)) } else { } keep <- (p > 0) & (p <= 1) & (k > 0) invalid <- sum(1L * keep) < 2 if(invalid) { warning("Must have at least two valid p values") res <- list(chisq = NA_real_, df = NA_integer_, p = NA_real_, validp = p[keep]) } else { pvals <- p[keep] k <- k[keep] if(length(pvals) != length(p)) { warning("Some studies omitted") } chisq <- qchisq(pvals, k, lower.tail = FALSE) chi <- sum(chisq) df <- sum(k) res <- list(chisq = chi, df = df, p = pchisq(chi, df, lower.tail = FALSE), validp = pvals) } class(res) <- c("invchisq", "metap") res } print.invchisq <- function(x, ...) { cat("chisq = ", x$chisq, " with df = ", x$df, " p = ", x$p, "\n") invisible(x) } metap/R/wilkinsonp.R0000644000176200001440000000313413556531545014116 0ustar liggesuserswilkinsonp <- function(p, r = 1, alpha = 0.05) { alpha <- ifelse(alpha > 1, alpha / 100, alpha) # if percent stopifnot(alpha > 0, alpha < 1) alpha <- ifelse(alpha > 0.5, 1 - alpha, alpha) # map to 0 to 0.5 keep <- (p >= 0) & (p <= 1) invalid <- sum(1L * keep) < 2 if (invalid) { warning("Must have at least two valid p values") res <- list(p = NA_real_, pr = NA_real_, r = r, critp = NA_real_, alpha = alpha, validp = p[keep]) } else { pi <- p[keep] k <- length(pi) if(k != length(p)) { warning("Some studies omitted") } if((r < 1) | (r > k)) { r <- 1 warning("Illegal r set to 1") } pi <- sort(pi) pr <- pi[r] res <- list(p = pbeta(pr, r, k + 1 - r), pr = pr, r = r, critp = qbeta(alpha, r, k + 1 - r), alpha = alpha, validp = pi) } class(res) <- c("wilkinsonp", "metap") res } maximump <- function(p, alpha = 0.05) { keep <- (p >= 0) & (p <= 1) validp <- p[keep] k <- length(validp) res <- wilkinsonp(p, r = k, alpha) class(res) <- c("maximump", "metap") res } minimump <- function(p, alpha = 0.05) { res <- wilkinsonp(p, r = 1, alpha) class(res) <- c("minimump", "metap") res } print.wilkinsonp <- function(x, ...) { r <- x$r p <- x$p cat("p = ", p, " using ", paste(r, "th minimum p", sep = ""), "\n") invisible(x) } print.maximump <- function(x, ...) { p <- x$p cat("p = ", p, " using maximum p", "\n") invisible(x) } print.minimump <- function(x, ...) { p <- x$p cat("p = ", p, " using minimum p", "\n") invisible(x) } metap/R/invt.R0000644000176200001440000000213513556604161012674 0ustar liggesusersinvt <- function(p, k, data = NULL, subset = NULL, na.action = na.fail) { if(is.null(data)) data <- sys.frame(sys.parent()) mf <- match.call() mf$data <- NULL mf$subset <- NULL mf$na.action <- NULL mf[[1]] <- as.name("data.frame") mf <- eval(mf, data) if(!is.null(subset)) mf <- mf[subset,] mf <- na.action(mf) p <- as.numeric(mf$p) k <- as.numeric(mf$k) if(length(k) == 1) { k <- rep(k, length(p)) } else { } keep <- (p > 0) & (p <= 1) & (k > 2) invalid <- sum(1L * keep) < 2 if(invalid) { warning("Must have at least two valid p values") res <- list(z = NA_real_, p = NA_real_, validp = p[keep]) } else { pvals <- p[keep] k <- k[keep] if(length(pvals) != length(p)) { warning("Some studies omitted") } z <- sum(qt(pvals, k, lower.tail = FALSE)) / sqrt(sum(k / (k - 2))) res <- list(z = z, p = pnorm(z, lower.tail = FALSE), validp = pvals) } class(res) <- c("invt", "metap") res } print.invt <- function(x, ...) { cat("z = ", x$z, " p = ", x$p, "\n") invisible(x) } metap/R/meanz.R0000644000176200001440000000125013556532516013027 0ustar liggesusersmeanz <- function(p) { keep <- (p > 0) & (p < 1) invalid <- sum(1L * keep) < 2 if(invalid) { warning("Must have at least two valid p values") res <- list(z = NA_real_, p = NA_real_, validp = p[keep]) } else { if(sum(1L * keep) != length(p)) { warning("Some studies omitted") } zvals <- (qnorm(p[keep], lower.tail = FALSE)) zp <- mean(zvals) / (sd(zvals) / sqrt(length(p[keep]))) res <- list(z = zp, p = pnorm(zp, lower.tail = FALSE), validp = p[keep]) } class(res) <- c("meanz", "metap") res } print.meanz <- function(x, ...) { cat("meanz = ", x$z, "p = ", x$p, "\n") invisible(x) } 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{1994}, publisher = {Russell Sage}, address = {New York}, author = {Becker, B J}, title = {Combining significance levels}, chapter = {15}, pages = {215--230}, keywords = {meta--analysis, significance values} } @article{benjamini00, author = {Benjamini, Y and Hochberg, Y}, title = {On the adaptive control of the false discovery rate in multiple testing with independent statistics}, journal = {Journal of Educational and Behavioral Statistics}, year = {2000}, volume = {25}, pages = {60--83}, keywords = {multiple comparison} } @article{rosenthal78, author = {Rosenthal, R}, title = {Combining results of independent studies}, journal = {Psychological Bulletin}, year = {1978}, volume = {85}, pages = {185--193}, keywords = {meta-analysis, significance values} } @ARTICLE{schweder82, author = {Schweder, T and Spjotvoll, E}, year = 1982, title = {Plots of $P$--values to evaluate many tests simultaneously}, journal = {Biometrika}, volume = 69, pages = {493--502}, keywords = {multiple comparison} } @article{lancaster49, author = {Lancaster, H O}, title = {Combination of probabilities arising from data in discrete distributions}, journal = {Biometrika}, year = {1949}, volume = {36}, pages = {370--382}, keywords = {meta-analysis, significance values} } @article{edgington72a, author = {Edgington, E S}, title = {An additive method for combining probability values from independent experiments}, journal = {Journal of Psychology}, year = {1972}, volume = {80}, pages = {351--363}, keywords = {meta-analysis, significance values} } @article{edgington72b, author = {Edgington, E S}, title = {A normal curve method for combining probability values from independent experiments}, journal = {Journal of Psychology}, year = {1972}, volume = {82}, pages = {85--89}, keywords = {meta-analysis, significance values} } @BOOK{sutton00, author = {Sutton, A J and Abrams, K R and Jones, D R and Sheldon, T A and Song, F}, year = 2000, title = {Methods for meta-analysis in medical research}, publisher = {Wiley}, address = {Chichester}, keywords = {meta-analysis, general} } @article{zaykin11, author = {Zaykin, D V}, title = {Optimally weighted $Z$--test is a powerful method for combining probabilities in meta--analysis}, journal = {Journal of Evolutionary Biology}, volume = {24}, year = {2011}, pages = {1836--1841}, keywords = {meta-analysis, significance values} } @book{tippett31, author = {Tippett, L H C}, title = {The methods of statistics}, year = {1931}, publisher = {Williams and Norgate}, address = {London}, keywords = {meta-analysis, significance values} } @article{wilkinson51, author = {Wilkinson, B}, title = {A statistical consideration in psychological research}, journal = {Psychological Bulletin}, year = {1951}, volume = {48}, pages = {156--158}, keywords = {meta-analysis, significance values} } @article{birnbaum54, author = {Birnbaum, A}, title = {Combining independent tests of significance}, journal = {Journal of the American Statistical Association}, year = {1954}, volume = {49}, pages = {559--574}, keywords = {meta-analysis, significance values} } @book{fisher25, author = {Fisher, R A}, title = {Statistical methods for research workers}, year = {1925}, publisher = {Oliver and Boyd}, address = {Edinburgh}, } @book{stouffer49, author = {Stouffer, S A and Suchman, E A and DeVinney, L C and Star, S A and Williams, R M Jnr}, title = {The {American} soldier, vol 1: {A}djustment during army life}, year = {1949}, publisher = {Princeton University Press}, address = {Princeton}, } @article{zaykin07, author = {Zaykin, D V and Zhivotovsky, L A and Czika, W and Shao, S and Wolfinger, R D}, title = {Combining p--values in large--scale genomics experiments}, journal = {Pharmaceutical Statistics}, year = {2007}, volume = {6}, pages = {217--236} } @article{dudbridge03, author = {Dudbridge, F and Koeleman, B P C}, title = {Rank truncated product of $P$--values, with application to genomewide association scans}, journal = {Genetic Epidemiology}, year = {2003}, volume = {25}, pages = {360--366} } @article{zhang18, author = {Zhang, H and Tong, T and Landers, J and Wu, Z}, title = {{TF}isher tests: optimal and adaptive thresholding for combining $p$--values}, journal = {arXiv}, year = {2018}, url = {https://arxiv.org/abs/1801.04309} } metap/inst/doc/0000755000176200001440000000000013611774153013112 5ustar liggesusersmetap/inst/doc/compare.Rnw0000644000176200001440000005637013611774056015245 0ustar liggesusers\documentclass[12pt]{article} \usepackage{amsmath,amssymb} %\usepackage{mydef2} \usepackage[round]{natbib} \usepackage{parskip,url} \usepackage{graphicx,subfig} \setlength{\topmargin}{0cm} \addtolength{\textheight}{2cm} %\lhead{} %\input{title} %\VignetteIndexEntry{Comparison of methods in the metap package} \title{Comparison of methods in the \pkg{metap} package} \author{Michael Dewey} \newcommand{\pkg}[1]{\texttt{#1}} \newcommand{\func}[1]{\texttt{#1}} \newcommand{\code}[1]{\texttt{#1}} \newcommand{\codefont}{\footnotesize} \newcommand{\mygraph}[3]{% \begin{figure}[htbp] \includegraphics[height=6cm,width=10cm]{compare-#1} \caption{#2} \label{#3} \end{figure} } % in mypdf fourth parameter needs [] \newcommand{\mypdf}[4]{% \begin{figure}[htbp] \includegraphics#4{#1.pdf} \caption{#2} \label{#3} \end{figure} } \newcommand{\twograph}[8]{% \begin{figure}[htbp] \subfloat[#2\label{#3}]{\includegraphics[height=6cm,width=7cm]{compare-#1}}% \subfloat[#5\label{#6}]{\includegraphics[height=6cm,width=7cm]{compare-#4}} \caption{#7} \label{#8} \end{figure} } \begin{document} \maketitle \section{Introduction} \subsection{What is this document for?} This document describes some methods for the meta--analysis of $p$--values (significance values) contained in the package \pkg{metap} and contains comments on the performance of the various algorithms under a small number of different scenarios with hints on the choice of method. \subsection{Notation} The $k$ studies give rise to $p$--values, $p_i,\;i = 1, \dots, k$. These are assumed to be independent. We shall also need the ordered $p$--values: $p_{[1]} \le p_{[2]}, \dots, \le p_{[k]}$ and weights $w_i,\;i = 1, \dots, k$. Logarithms are natural. A function for combining $p$--values is denoted $g$. The size of the test is $\alpha$. We may also need $k$ degrees of freedom, $\nu_i$. The methods are referred to by the name of the function in \func{metap}. Table \ref{funcs} shows other descriptions of each method. \begin{table}[htbp] \begin{tabular}{lll} Function name & \multicolumn{2}{c}{Description(s)} \\[1ex] & \multicolumn{1}{c}{Eponym} \\ \func{invchisq} & Lancaster's method & Inverse chi square \\ \func{invt} & & Inverse t \\ \func{logitp} & & Logistic\\ \func{meanp} \\ \func{meanz} \\ \func{maximump} \\ \func{minimump} & Tippett's method \\ \func{sumlog} & Fisher's method & Chi square (2 df)\\ \func{sump} & Edgington's method & Uniform\\ \func{sumz} & Stouffer's method & Normal\\ \func{truncated} & Truncated Fisher\\ \func{truncated} & & rank--truncated\\ \func{votep} \\ \func{wilkinsonp} & Wilkinson's method \\ \end{tabular} \caption{Methods considered in this document} \label{funcs} \end{table} \section{Theoretical results} There have been various attempts to clarify the problem and to discuss optimality of the methods. A detailed account was provided by \citet{liptak58}. \citet{birnbaum54} considered the property of admissibility. A method is admissible if when it rejects $H_0$ for a set of $p_i$ it will also reject $H_0$ for $P^*_i$ where $p^*_i \le p_i$ for all $i$. He considered that Fisher's and Tippett's method were admissible. See also \citet{owen09}. He also points out the problem is poorly specified. This may account for the number of methods available and their differing behaviour. The null hypothesis $H_0$ is well defined, that all $p_i$ have a uniform distribution on the unit interval. There are two classes of alternative hypothesis \begin{itemize} \item $H_A$: all $p_i$ have the same (unknown) non--uniform, non--increasing density, \item $H_B$: at least one $p_i$ has an (unknown) non--uniform, non--increasing density. \end{itemize} If all the tests being combined come from what are basically replicates then $H_A$ is appropriate whereas if they are of different kinds of test or different conditions then $H_B$ is appropriate. Note that Birnbaum specifically considers the possibility that the tests being combined may be very different for instance some tests of means, some of variances, and so on. \section{The methods} \subsection{Comparison scenarios} To provide a standard of comparison we shall use the following two situations. Some authors have also used the case of exactly two $p_i$. \begin{description} %\subsubsection{What if all $p_i = p$?\label{twopisection}} \item[What if all $p_i = p$?]\label{twopisection} Perhaps surprisingly there are substantial differences here as we shall see when we look at each method. We shall describe how the returned value varies with $p$ and $k$. %\subsubsection{Cancellation} \item[Cancellation] When the collection of primary studies contains a number of values significant in both directions the methods can give very different results. If the intention of the synthesis is to examine a directional hypothesis one would want a method where these cancelled out. The decision between methods should be made on theoretical grounds of course. We shall use the following four values as our example. \end{description} {\codefont <<>>= cancel <- c(0.001, 0.001, 0.999, 0.999) @ } <>= library(metap) data(validity) genvec <- function(pvals, kvals, fun, name) { ps <- length(pvals) ks <- length(kvals) temp <- matrix(-1, nrow = ps, ncol = ks) for(i in 1:ps) for(j in 1:ks) { temp[i, j] <- fun(rep(pvals[i], kvals[j]))$p } temp2 <- as.vector(temp) res <- data.frame(method = rep(name, length(temp2)), p = rep(pvals, ks), k = rep(kvals, each = ps), g = temp2 ) res } @ \subsection{Methods using transformation of the $p$--values} One class of methods relies on transforming the $p$--values first. \begin{table}[htbp] \begin{tabular}{lll} Function name & Definition & Critical value \\[1ex] \func{invchisq} & $\sum_{i=1}^k \chi^2_{\nu_i}(p_i)$ & $\chi^2_{\sum{\nu_i}}(\alpha)$ \\[1ex] \func{invt} & $\frac{\sum_{i=1}^k t_{\nu_i}(p_i)}% {\sqrt{\sum_{i=1}^k \frac{\nu_i}{\nu_i - 2}}}$ & $z(\alpha)$ \\[1ex] \func{logitp} & $\frac{\sum_{i=1}^k \log\frac{p}{1 - p}}{C}$ & $t_{5k+4}$ \\ & where $C = \sqrt\frac{k \pi^2 (5 k + 2)}{3(5 k + 4)}$ & \\[1ex] \func{meanz} & $\frac{\bar{z}}{s_{\bar{z}}}$ & $t_{k-1}(\alpha)$ \\ & where $\bar{z} = \sum_{i=1}^k \frac{z(p_i)}{k}$ \\ & and $s_{\bar{z}} = \frac{s_z}{\sqrt{k}}$ & \\[1ex] \func{sumlog} & $\sum_{i=1}^{k} - 2 \log p_i$ & $\chi_{2k}(\alpha)$ \\[1ex] \func{sumz} & $\frac{\sum_{i=1}^k z(p_i)}{\sqrt{k}}$ & $z(\alpha)$\\ \end{tabular} \caption{Definitions of methods using transformation of the $p$ values} \label{transdefs} \end{table} <>= kvals <- c(4, 5, 6, 8, 10, 15, 20) pvals <- c(0.2, 0.3, 0.3679, 0.4, 0.5, 0.6) dat <- rbind( genvec(pvals, kvals, logitp, "logitp"), genvec(pvals, kvals, meanz, "meanz"), genvec(pvals, kvals, sumlog, "sumlog"), genvec(pvals, kvals, sumz, "sumz") ) @ <>= lattice::xyplot(g ~ k | method, groups = p, type = "l", data = dat, auto.key = list(space = "left", lines = TRUE, title = "p"), ylab = "g(p)" ) @ \subsubsection{The method of summation of logs, Fisher's method} See Table \ref{transdefs} for the definition. This works because $- 2 \log p_i$ is a $\chi^2_2$ and the sum of $\chi^2$ is itself a $\chi^2$ with degrees of freedom equal to the sum of the degrees of freedom of the individual $\chi^2$. Of course the sum of the log of the $p_i$ is also the log of the product of the $p_i$. Fisher's method \citep{fisher25} is provided in \func{sumlog}. <>= set.seed(18122019) temp <- matrix(runif(10000), nrow = 100) fisher <- apply(temp, 1, function(x) sumlog(x)$p) lanc4 <- apply(temp, 1, function(x) invchisq(x, 4)$p) lanc16 <- apply(temp, 1, function(x) invchisq(x, 16)$p) lanc256 <- apply(temp, 1, function(x) invchisq(x, 256)$p) banda <- function(x, y) { res <- data.frame(sum = x + y, diff = (x - y)) res } dat <- data.frame(rbind(banda(fisher, lanc4), banda(fisher, lanc16), banda(fisher, lanc256), banda(lanc4, lanc16), banda(lanc4, lanc256), banda(lanc16, lanc256) ), name = factor(c(rep("FL4", 100), rep("FL16", 100), rep("FL256", 100), rep("L4L16", 100), rep("L4L256", 100), rep("L16L256", 100)), levels = c("FL4", "FL16", "FL256", "L4L16", "L4L256", "L16L256") ) ) @ <>= lattice::xyplot(diff ~ sum | name, data = dat, panel = function(x, y, ...) { lattice::panel.xyplot(x, y, ...) lattice::panel.abline(h = mean(y), lty = 2) lattice::panel.abline(h = mean(y) + 1.96 * sd(y), lty = 3) lattice::panel.abline(h = mean(y) - 1.96 * sd(y), lty = 3) } ) @ <>= stouff <- apply(temp, 1, function(x) sumz(x)$p) invt4 <- apply(temp, 1, function(x) invt(x, 4)$p) invt16 <- apply(temp, 1, function(x) invt(x, 16)$p) invt256 <- apply(temp, 1, function(x) invt(x, 256)$p) banda <- function(x, y) { res <- data.frame(sum = x + y, diff = (x - y)) res } dat <- data.frame(rbind(banda(stouff, invt4), banda(stouff, invt16), banda(stouff, invt256), banda(invt4, invt16), banda(invt4, invt256), banda(invt16, invt256) ), name = factor(c(rep("St4", 100), rep("St16", 100), rep("St256", 100), rep("t4t16", 100), rep("t4t256", 100), rep("t16t256", 100)), levels = c("St4", "St16", "St256", "t4t16", "t4t256", "t16t256") ) ) @ <>= lattice::xyplot(diff ~ sum | name, data = dat, panel = function(x, y, ...) { lattice::panel.xyplot(x, y, ...) lattice::panel.abline(h = mean(y), lty = 2) lattice::panel.abline(h = mean(y) + 1.96 * sd(y), lty = 3) lattice::panel.abline(h = mean(y) - 1.96 * sd(y), lty = 3) } ) @ As can be seen in Figure \ref{equalp} when all the $p_i=p$ \func{sumlog} returns a value which decreases with $k$ when $p<0.32$, increases with $k$ when $p>0.37$, and in between increases with $k$ and then decreases. Some detailed algebra provided in a post to https://stats.stackexchange.com/questions/243003 by Christoph Hanck suggests that the breakpoint is $e^{-1} = 0.3679$. Where the $p_i$ are less than this then for a sufficiently large $k$ (several hundred) the result will be significant and not if above that. Over the range of $k$ we are plotting this bound is not yet closely approached. \mygraph{transeqp}{Behaviour of the methods using transformed $p$ values for $k$ values of $p=p_i$}{equalp} \subsubsection{Inverse $\chi^2$ Lancaster's method} It would of course be possible to generalise Fisher's method to use transformation to $\chi^2$ with any other number of degrees of freedom rather than 2. \citet{lancaster61} suggests that this is highly correlated with \func{sumlog}. Lancaster's method is provided in \func{invchisq}. In fact the resemblance to \func{sumlog} becomes less as the number of degrees of freedom increases. Figure \ref{fishlanc} shows for a small number of selected degrees of freedom how it compares to Fisher's method. \twograph{fishlanc}{Fisher's method and Lancaster's method}{fishlanc}{stouffinvt}{Stouffer's method and inverse $t$}{stouffinvt}{Sum and difference plots of Fisher v Lancaster and Stouffer v inverse $t$}{compfishstouff} \subsubsection{The method of summation of $z$ values, Stouffer's method} The method of summation of $z$ values is provided in \func{sumz} \citep{stouffer49}. See Table \ref{transdefs} for the definition. As can be seen in Figure \ref{equalp} it returns a value for our $p_i=p$ example which decreases with $k$ when $p$ below 0.5 and increases above. A weighted version of Stouffer's method is available %\begin{equation} $\frac{\sum_{i=1}^k w_i z(p_i)}{\sqrt {\sum_{i=1}^k w_i ^ 2}}$ %\end{equation} where $w_i$ are the weights. In the absence of effect sizes (in which case a method using effect sizes would be more appropriate anyway) best results are believed to be obtained with weights proportional to the square root of the sample sizes \citep{zaykin11} following \citet{liptak58}. \subsubsection{Mean of normals method} There is also a method closely related to Stouffer's using the mean of normals provided in \func{meanz} also defined in Table \ref{transdefs} which has very similar properties except that when all the $p_i$ are equal it either gives 0 or 1 as can be seen in Figure \ref{equalp}. <<>>= meanz(c(0.3, 0.31))$p meanz(c(0.1, 0.2))$p @ The method of \func{meanz} also has the unusual property that a set of $p$--values which are all less than those in another set can still give rise to a larger overall $p$. See example above. This is the only method considered here which has this property so if it is a desirable one then that is the only method to consider. \subsubsection{The inverse $t$ method} A closely related method is the inverse $t$ method. See Table \ref{transdefs} for the definition. This method is provided in \func{invt}. As is clear from the definition this method tends to Stouffer's method as $\nu_i \to \infty$. Figure \ref{stouffinvt} shows this for selected degrees of freedom. \subsubsection{The method of summation of logits} See Table \ref{transdefs} for the definition. This method is provided in \func{logitp}. The constant $C$ was arrived at by equating skewness and kurtosis with that of the $t$--distribution \citep{loughin04}. As can be seen in Figure \ref{equalp} this method returns a value for our $p_i=p$ example which decreases with $k$ when $p$ below 0.5 and increases above. \subsubsection{Examples for methods using transformations of the $p$ values} \begin{table}[htbp] \begin{tabular}{lll} Function name & validity & cancel \\[1ex] \func{logitp} & \Sexpr{logitp(validity)$p} & \Sexpr{logitp(cancel)$p} \\ \func{meanz} & \Sexpr{meanz(validity)$p} & \Sexpr{meanz(cancel)$p} \\ \func{sumlog} & \Sexpr{sumlog(validity)$p} & \Sexpr{sumlog(cancel)$p} \\ \func{sumz} & \Sexpr{sumz(validity)$p} & \Sexpr{sumz(cancel)$p}\\ \end{tabular} \caption{Examples of methods using transformation of the $p$ values} \label{transexamples} \end{table} Using the same example dataset which we have already plotted and our cancellation dataset we have the values in Table \ref{transexamples}. As can be seen all the methods cancel except for \func{sumlog}. The agreement for the validity dataset is close. Lancaster's method and inverse $t$ are not shown as they are both infinite families of possible methods and in any event are similar to Fisher's method and Stouffer's method respectively. \subsection{Methods using untransformed $p$--values} \begin{table}[htbp] \begin{tabular}{lll} Function name & Definition & Critical value \\[1ex] \func{meanp} & $\bar p = \frac{\sum_{i=1}^k p_i}{k}$ \\ & $z = (0.5 - \bar{p}) \sqrt{12k}$ & $z(\alpha)$ \\ \func{minimump} & $p_{[1]}$ & $1 - (1 - \alpha)^{\frac{1}{k}}$ \\ \func{maximump} & $p_{[k]}$ & $\alpha^k$ \\ \func{wilkinsonp} & $p_{[r]}$ & $\sum_{s=r}^k {k \choose s}\alpha^s (1 - \alpha)^{k-s}$\\[1ex] \func{sump} & $\frac{(S)^k}{k!}% - {k - 1 \choose 1}\frac{(S - 1)^k}{k!}% + {k - 2 \choose 2}\frac{(S - 2)^k}{k!} - \dots$ & $\alpha$ \\ & where $S = \sum_{i=1}^k p_i$ \\ \end{tabular} \caption{Definitions of methods not using transformation of the $p$ values, % the series for \func{sump} continues until the term in in the numerator $(S-i)$ becomes negative} \label{untransdefs} \end{table} <>= kvals <- c(4, 5, 6, 8, 10, 15, 20) pvals <- c(0.2, 0.3, 0.3679, 0.4, 0.5, 0.6) dat <- rbind( genvec(pvals, kvals, meanp, "meanp"), genvec(pvals, kvals, maximump, "maximump"), genvec(pvals, kvals, minimump, "minimump"), genvec(pvals, kvals, sump, "sump"), genvec(pvals, kvals, votep, "votep") ) @ <>= lattice::xyplot(g ~ k | method, groups = p, type = "l", data = dat, auto.key = list(space = "left", lines = TRUE, title = "p"), ylab = "g(p)" ) @ \mygraph{untranseqp}{Behaviour of the methods using untransformed $p$ values for $k$ values of $p=p_i$}{unequalp} \subsubsection{The method of minimum $p$, maximum $p$, and Wilkinson's method} The methods of minimum $p$ \citep{tippett31}, maximum $p$ and Wilkinson \citep{wilkinson51} are defined in Table \ref{untransdefs}. Wilkinson's method depends on which value (the $r$th) of $p_{[i]}$ is selected. % p is pbeta(p[r], r, k+1-r) % critical p is qbeta(alpha, r, k+1-r) Wilkinson's method is provided in \func{wilkinsonp} and a convenience function \func{minimump} with its own \code{print} method is provided for the minimum $p$ method ($r=1$). It is also possible to use the method for the maximum $p$ (that is $r=k$) and a convenience function \func{maximump} is provided for that purpose. As can be seen in Figure \ref{unequalp} these methods return a value for our $p_i=p$ example which always increases with $k$ which is true for \func{minimump} and which always decreases with $k$ which is true for \func{maximump} \subsubsection{The method of summation of $p$--values, Edgington's method\label{sump}} Defined in Table \ref{untransdefs} \citep{edgington72a}. This method is provided in \func{sump}. As can be seen in Figure \ref{unequalp} this method returns a value for our $p_i=p$ example which decreases with $k$ when $p$ below 0.5 and increases above. Some authors use a simpler version, $\frac{(\sum p)^k}{k!}$, for instance \citet{rosenthal78} in the text although compare his Table 4. This can be very conservative when $\sum p > 1$ There seems no particular need to use this method but it is returned by \func{sump} as the value of \code{conservativep} for use in checking published values. Note also that there can be numerical problems for extreme values of $S$ and in that case recourse might be made to \func{meanp} which has similar properties. \subsubsection{The mean $p$ method} Defined in Table \ref{untransdefs}. Although this method is attributed to Edgington \citep{edgington72b} when the phrase Edgington's method is used it refers to the method of summation of $p$--values described above in Section \ref{sump}. As can be seen in Figure \ref{unequalp} this method returns a value for our $p_i=p$ example which decreases with $k$ when $p$ below 0.5 and increases above. Not surprisingly this method gives very similar results to Edington's other method implemented in \func{sump} and since it does not have the numerical problems of that method it might perhaps be preferred. \subsubsection{Examples for methods using untransformed $p$--values} Using the same example dataset which we have already plotted and our cancellation dataset we have the values in Table \ref{untransexamples}. As can be seen \func{meanp} and \func{sump} cancel but the other two do not. Agreement here is not so good especially for the maximum p method. Wilkinson's method not shown as it depends on the value of $r$. \begin{table}[htbp] \begin{tabular}{lll} Function name & validity & cancel \\[1ex] \func{minimump} & \Sexpr{minimump(validity)$p} & \Sexpr{minimump(cancel)$p} \\ \func{maximump} & \Sexpr{maximump(validity)$p} & \Sexpr{maximump(cancel)$p} \\ \func{meanp} & \Sexpr{meanp(validity)$p} & \Sexpr{meanp(cancel)$p}\\ \func{sump} & \Sexpr{sump(validity)$p} & \Sexpr{sump(cancel)$p} \\ \end{tabular} \caption{Examples for methods using the untransformed $p$ values} \label{untransexamples} \end{table} \subsection{Other methods} \subsubsection{The method of vote--counting} A simple way of looking at the problem is vote counting. Strictly speaking this is not a method which combines $p$--values in the same sense as the other methods. If most of the studies have produced results in favour of the alternative hypothesis irrespective of whether any of them is individually significant then that might be regarded as evidence for that alternative. The numbers for and against may be compared with what would be expected under the null using the binomial distribution. A variation on this would allow for a neutral zone of studies which are considered neither for nor against. For instance one might only count studies which have reached some conventional level of statistical significance in the two different directions. This method returns a value for our $p_i=p$ example which is 1 for $p$ values above 0.5 and otherwise invariant with $p$ but decreases with $k$. This method does cancel significant values in both directions. \begin{table}[htbp] \begin{tabular}{lll} Function name & validity & cancel \\[1ex] \func{votep} & \Sexpr{votep(validity)$p} & \Sexpr{votep(cancel)$p} \\ \end{tabular} \caption{Examples for vote counting} \label{votepexamples} \end{table} \subsubsection{Methods not using all $p$--values} If there is a hypothesis that the signal will be concentrated in only a few $p$--values then alternative methods are available in \func{truncated}. This is a wrapper to two packages available on CRAN: \pkg{TFisher} which provides the truncated Fisher method \citep{zaykin07,zhang18} and \pkg{mutoss} which provides the rank--truncated Fisher method \citep{dudbridge03}. Note that Table \ref{truncatedexamples} only shows results for the validity data--set as, since the methods explicitly only consider results in one direction the cancellation issue does not arise. \begin{table}[htbp] \begin{tabular}{lll} Function name & truncated at $p$ = 0.5 & truncated at rank = 5 \\[1ex] \func{truncated} & \Sexpr{truncated(validity, ptrunc = 0.5)$p} & \Sexpr{truncated(validity, rtrunc = 5)$p} \\ \end{tabular} \caption{Examples for truncated using the validity data--set} \label{truncatedexamples} \end{table} These methods are appropriate for the situation where it is known that many of the $p$--values are noise and there will only be a few signals. \section{Loughin's recommendations} In his simulation study \citet{loughin04} carried out extensive comparisons. Note that he did not consider all the methods implemented here. These omissions are not too important for our purposes. The methods implemented here as \func{invchisq}, \func{invt}, \func{meanp} and \func{meanz} are all very similar to ones which he did study. The truncation methods appeared about the same time as his work but in any case are fundamentally different. Vote counting is arguably not a method of the same sort. As Loghin points out the first thing to consider is whether large $p$--values should cancel small ones. If this is not desired then the only methods to consider are those in \func{sumlog} (Fisher), \func{minimump} (Tippett) and \func{maximump}. \mypdf{struct}{Loughin's recommendations based on structure}{struct}{[height=6cm,width=8cm]} He bases his recommendations on criteria of structure and the arrangement of evidence against $H_0$. Figure \ref{struct} shows a summary of his recommendations about the structure of the evidence. \mypdf{strength}{Loughin's recommendations based on where the strength of the evidence is located}{strength}{[height=8cm,width=12cm]} Figure \ref{strength} summarise his recommendations about the arrangement of evidence. Overall he considered the choice to lie between Stouffer's method, Fisher's method and the logistic method implemented in \func{logitp}. As has already been mentioned Fisher's method cancels whereas the other two do not so if the weak evidence in a small number of $p$--values is not to be over--whelmed by the others then Fisher is the best choice. However where the evidence is more evenly spread Stouffer's method may be preferred. The logistic method represents a compromise between them and is perhaps best suited where the pattern of evidence is not clear in advance. The other methods are not universally ruled out and may be helpful in the specific circumstance outlined in his summaries. \bibliography{metap} \bibliographystyle{plainnat} \end{document} metap/inst/doc/struct.gv0000644000176200001440000000175513576717030015005 0ustar liggesusersdigraph test { rankdir = "LR" fontname = "Helvetica" node[fontname = "Helvetica"] edge[fontname = "Helvetica"] structure [label = "What\nis\nstructure?", shape=diamond] small [label = <
sumlog (Fisher)
minimump (Tippett)
>] large [label = <
sump (Edgington)
maximump
>] equal [label = <
sumz (Stouffer)
logitp
>] structure -> small [label = "Emphasise small p-values"] structure -> large [label = "Emphasise large p-values"] structure -> equal [label = "Emphasise p-values equally"] } metap/inst/doc/metap.Rnw0000644000176200001440000002017213546615554014720 0ustar liggesusers\documentclass[12pt]{article} \usepackage{amsmath,amssymb} %\usepackage{mydef2} \usepackage[round]{natbib} \usepackage{parskip,url} \usepackage{graphicx,subfig} \setlength{\topmargin}{0cm} \addtolength{\textheight}{2cm} %\lhead{} %\input{title} %\VignetteIndexEntry{Introduction to the metap package} \title{Introduction to the \pkg{metap} package} \author{Michael Dewey} \newcommand{\pkg}[1]{\texttt{#1}} \newcommand{\func}[1]{\texttt{#1}} \newcommand{\code}[1]{\texttt{#1}} \newcommand{\codefont}{\footnotesize} \newcommand{\mygraph}[3]{% \begin{figure}[htbp] \includegraphics[height=6cm,width=10cm]{#1} \caption{#2} \label{#3} \end{figure} } \newcommand{\twograph}[8]{% \begin{figure}[htbp] \subfloat[#2\label{#3}]{\includegraphics[height=6cm,width=7cm]{#1}}% \subfloat[#5\label{#6}]{\includegraphics[height=6cm,width=7cm]{#4}} \caption{#7} \label{#8} \end{figure} } \begin{document} \maketitle \section{Introduction} \subsection{What is this document for?} This document describes some methods for the meta--analysis of $p$--values (significance values) and their implementation in the package \pkg{metap}. I welcome feedback about sources of published examples against which I can test the code and any other comments about either the documentation or the code. The problem of meta--analysis of $p$--values is of course not completely unconnected with the more general issue of simultaneous statistical inference. \subsection{Why and when to meta--analyse significance values} The canonical way to meta--analyse a number of primary studies uses estimates of effect sizes from each of them. There are a large number of packages for this purpose available from CRAN and described in the task view \url{http://CRAN.R-project.org/view=MetaAnalysis}. However sometimes the only available information may be $p$--values especially when some of the primary studies were published a long time ago or were published in sources which were less rigorous about insisting on effect sizes. The methods outlined here are designed for this eventuality. The situation may also arise that some of the studies can be combined in a conventional meta--analysis using effect sizes but there are many others which cannot and in that case the conventional meta--analysis of the subset of studies which do have effect sizes may usefully be supplemented by an overall analysis of the $p$--values. Just for the avoidance of doubt, if each study has produced a proportion and the goal is to synthesise them to a common estimate or analyse the differences between them then the standard methods are appropriate not the ones outlined here. The $p$--values in this document are significance levels. The methods are referred to by the name of the function in \func{metap}. Table \ref{funcs} shows other descriptions of each method. \begin{table}[htbp] \begin{tabular}{lll} Function name & \multicolumn{2}{c}{Description(s)} \\[1ex] & \multicolumn{1}{c}{Eponym} \\ \func{invchisq} & Lancaster's method & Inverse chi square \\ \func{invt} & & Inverse t \\ \func{logitp} & & Logistic\\ \func{meanp} \\ \func{meanz} \\ \func{maximump} \\ \func{minimump} & Tippett's method \\ \func{sumlog} & Fisher's method & Chi square (2 df)\\ \func{sump} & Edgington's method & Uniform\\ \func{sumz} & Stouffer's method & Normal\\ \func{truncated} & Truncated Fisher & rank--truncated\\ \func{votep} \\ \func{wilkinsonp} & Wilkinson's method \\ \end{tabular} \caption{Methods considered in this document} \label{funcs} \end{table} \section{Preparation for meta--analysis of $p$--values} \subsection{Preliminaries} I assume you have installed \textsf{R} and \pkg{metap}. You then need to load the package. <<>>= library(metap) @ \subsection{Directionality} It is usual to have a directional hypothesis, for instance that treatment is better than control. For the methods described here a necessary preliminary is to ensure that all the $p$--values refer to the same directional hypothesis. If the value from the primary study is two--sided it needs to be converted. This is not simply a matter of halving the quoted $p$--value as values in the opposite direction need to be reversed. A convenience function \func{two2one} is provided for this. {\codefont <<>>= pvals <- c(0.1, 0.1, 0.9, 0.9, 0.9, 0.9) istwo <- c(TRUE, FALSE, TRUE, FALSE, TRUE, FALSE) toinvert <- c(FALSE, TRUE, FALSE, FALSE, TRUE, TRUE) two2one(pvals, two = istwo, invert = toinvert) @ } Note in particular the way in which $0.9$ is converted under the different scenarios. <>= data(validity) @ \subsection{Plotting} {\codefont <<>>= print(validity) @ } It would be a wise precaution to examine the $p$--values graphically or otherwise before subjecting them to further analysis. Two functions are provided for this purpose: \func{plotp} and \func{schweder}. \subsubsection{Plotting using \func{plotp}} The \func{plotp} provides a Q--Q plot of the $p$--values to detect departure from the uniform distribution. An example is shown in Figure \ref{plotp}. The standard line through through the quartiles is superimposed. This is the function which is called when the \func{plot} method is used on an object returned by any of the meta--analysis functions. <>= plotp(validity) @ \mygraph{metap-plotp}{Q--Q plot from \func{plotp}}{plotp} \subsubsection{Plotting using \func{schweder}} A function \func{schweder} provides plots with a variety of informative lines superimposed. It plots the ordered $p$--values, $p_{[i]}$, against $i$. Although the original motivation for the plot is \citet{schweder82} the function uses a different choice of axes due to \citet{benjamini00}. We will use an example dataset on the validity of student ratings quoted in \citet{becker94}. Figure \ref{simple} shows the plot from \func{schweder} which is the same as from \func{plotp} but without the line. <>= schweder(validity) @ \func{schweder} also offers the possibility of drawing one of a number of straight line summaries. The three possible straight line summaries are shown in Figure \ref{withlines} and are: \begin{itemize} \item the lowest slope line of Benjaimin and Hochberg which is drawn by default as solid, \item a least squares line drawn passing through the point $k+1, 1$ and using a specified fraction of the points which is drawn by default as dotted, \item a line with user specified intercept and slope which is drawn by default as dashed. \end{itemize} <>= schweder(validity, drawline = c("bh", "ls", "ab"), ls.control = list(frac = 0.5), ab.control = list(a = 0, b = 0.01)) @ \twograph{metap-simple}{Simple graph}{simple}{metap-withlines}{With lines}{withlines}{Output from schweder}{schweder} \subsection{Reporting problems in the primary studies} Another issue is what to do with studies which have simply reported on whether a conventional level of significance like 0.05 was achieved or not. If the exact associated $p$ cannot be derived from the statistics quoted in the primary source then the value of the level achieved, in this case 0.05, can be used although this may be conservative. Studies which simply report not significant could be included as having $p=1$ (or $p=0.5$ if it is known that the direction was right) although this is very conservative. The theory of handling $p$--values which have been truncated like this has been developed by \citet{zaykin02} and \func{truncated} provides a convenience wrapper for two methods available in other CRAN packages. \section{Using the methods} All the methods in the package take as their first argument the vector of $p$--values. To use Fisher's method as an example: <<>>= sumlog(validity) @ A few require extra information. Those which rely on inverse transformations often need a vector of degrees of freedom. Currently this applies to \func{invchisq} and \func{invt}. Stouffer's method in \func{sumz} optionally uses weights if a vector of weights is provided. \section{Miscellanea} \begin{description} \item[Extractor functions] The standard \code{print} and \code{plot} methods are provided. \item[Reading] An annotated bibliography is provided by \citet{cousins08} \end{description} \bibliography{metap} \bibliographystyle{plainnat} \end{document} metap/inst/doc/compare.pdf0000644000176200001440000114426513611774146015252 0ustar liggesusers%PDF-1.5 % 3 0 obj << /Length 1934 /Filter /FlateDecode >> stream xڵK4T&~ !C%{Ldtvy㲋C'ro[Z5~SEcjv-^V_-u5+[,cEjp ruE崍v4xƨ5:oۥ ۥf@ uc+]X9|'䏎ߊp 2|'a;\U+nд#b80a051o]蠌ш7U:YY/WZ42lr.ՅNp)dvs,U6 YGxOE}g0y`7pΒGo4dbR #$t/'Mw=Bx'S};f&AmΫn&+(ë6-#zHe%^kīבaׯjPE{rZ"v@q8D$vOXFu Ydy3 L`cjus5(kO4Îɢ\2(0/DDi.UhgJrc[i1aR{tQMrw#cғ1rGvec;!_V:)>LN㖿H"QI41Oз$b#z\1rtOG|ЁPWr˶Yn3~F獰竾ƩoKtɛdz`BO6́\M;Yw(qk Er( @,@zys u(Ee!eP.W r(,@3g<P&TGOcD+~D/P7[?I(yFiڦqRQ*n, a`}l:z/R^ҡV!_maz){L.u5?4pWq@yUTnJa>S]2!yF K95`mږ(DVK0n$z)cSnĶIy]Pe:ZA]PlBJ}=^kbk# S߮o s"##4m⧨0/gćK0m&t:.ATCFМ1' i+Av3@D'nyOsClY0i}5mtQϯ;\wPwc@_B'ǻ=kU3 %rzFSÄ픵]j[p mœN8g]&,sSu]2߬_f endstream endobj 17 0 obj << /Length 1790 /Filter /FlateDecode >> stream xڵXKs6WVj&Bޚ6C݃"ѱ&䐒]'ƒ4-)@ $v|9֪s3_P5*5d\L+:µ=Y~(:ݲ[l:L=g+dkWtf a2U Y`T ;*ޡע}Ju9I-CCtG%q5iSd2k8H˩)6MQ]$l)kܧ'&8GM)@tݠw=·/O˙@HHY^~_Թn>;2# g1(M+JA8mЅ 4e^'?' \ Cb wV]uW+r=|GTgu]'n i;󁝟Od}wҥSM,)iw19;("=&xur'חxѬC(}zuu#\_Ěۇož3 n{tGn4}v?XR:-n(`ދxy?Ngo@$ h@ىXCW޾AP`^\-[/qTuj$UULY+.^UD)3XRڛBRbbAdl:HrbӞY=ulLrrtDT\E No%ʶPY yyݰCR(1>}ɢܒGWܴ"6qZid?WvBL8 h8ƚC7ÚX}Iɞ6ZER|rP*އqƽ8½g]p ln؝pm: rͽ!1幸,Y-U^~ 4:E,͆b1zDgU)ᇟ"ʣ3UŽ5MDCKCq(;J}} 1|+hrh\$0W[NW:1s^A#ʀ$ /K)$3運L3fV cV SU>b g 2t&@^bY|vuATZ"|LN[AL=@zlLYC&ЖzWxn.śv>p>,n -ٺw  o*;֤+on墹V\)އNk U4=m ذNfDAWA.VtӢ*] &+_Yo(SHUeUߠat9:Nd ~mw;t?XZ7B,C~hY>rLi#_gFz ǙpZG{G1N75_њB6Bw 3Ș82-{񈪫 P]\L1(w,|3@@DχٱFgD#nZ+!ZY+cw^͟V endstream endobj 21 0 obj << /Length 2821 /Filter /FlateDecode >> stream xZo#~_G o@. 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################################################### ### code chunk number 1: metap.Rnw:119-120 ################################################### library(metap) ################################################### ### code chunk number 2: metap.Rnw:137-141 ################################################### pvals <- c(0.1, 0.1, 0.9, 0.9, 0.9, 0.9) istwo <- c(TRUE, FALSE, TRUE, FALSE, TRUE, FALSE) toinvert <- c(FALSE, TRUE, FALSE, FALSE, TRUE, TRUE) two2one(pvals, two = istwo, invert = toinvert) ################################################### ### code chunk number 3: metap.Rnw:147-148 ################################################### data(validity) ################################################### ### code chunk number 4: metap.Rnw:153-154 ################################################### print(validity) ################################################### ### code chunk number 5: plotp ################################################### plotp(validity) ################################################### ### code chunk number 6: simple ################################################### schweder(validity) ################################################### ### code chunk number 7: withlines ################################################### schweder(validity, drawline = c("bh", "ls", "ab"), ls.control = list(frac = 0.5), ab.control = list(a = 0, b = 0.01)) ################################################### ### code chunk number 8: metap.Rnw:253-254 ################################################### sumlog(validity) metap/inst/doc/compare.R0000644000176200001440000001237713611774146014677 0ustar liggesusers### R code from vignette source 'compare.Rnw' ################################################### ### code chunk number 1: compare.Rnw:178-179 ################################################### cancel <- c(0.001, 0.001, 0.999, 0.999) ################################################### ### code chunk number 2: compare.Rnw:183-201 ################################################### library(metap) data(validity) genvec <- function(pvals, kvals, fun, name) { ps <- length(pvals) ks <- length(kvals) temp <- matrix(-1, nrow = ps, ncol = ks) for(i in 1:ps) for(j in 1:ks) { temp[i, j] <- fun(rep(pvals[i], kvals[j]))$p } temp2 <- as.vector(temp) res <- data.frame(method = rep(name, length(temp2)), p = rep(pvals, ks), k = rep(kvals, each = ps), g = temp2 ) res } ################################################### ### code chunk number 3: compare.Rnw:228-236 ################################################### kvals <- c(4, 5, 6, 8, 10, 15, 20) pvals <- c(0.2, 0.3, 0.3679, 0.4, 0.5, 0.6) dat <- rbind( genvec(pvals, kvals, logitp, "logitp"), genvec(pvals, kvals, meanz, "meanz"), genvec(pvals, kvals, sumlog, "sumlog"), genvec(pvals, kvals, sumz, "sumz") ) ################################################### ### code chunk number 4: transeqp ################################################### lattice::xyplot(g ~ k | method, groups = p, type = "l", data = dat, auto.key = list(space = "left", lines = TRUE, title = "p"), ylab = "g(p)" ) ################################################### ### code chunk number 5: compare.Rnw:258-281 ################################################### set.seed(18122019) temp <- matrix(runif(10000), nrow = 100) fisher <- apply(temp, 1, function(x) sumlog(x)$p) lanc4 <- apply(temp, 1, function(x) invchisq(x, 4)$p) lanc16 <- apply(temp, 1, function(x) invchisq(x, 16)$p) lanc256 <- apply(temp, 1, function(x) invchisq(x, 256)$p) banda <- function(x, y) { res <- data.frame(sum = x + y, diff = (x - y)) res } dat <- data.frame(rbind(banda(fisher, lanc4), banda(fisher, lanc16), banda(fisher, lanc256), banda(lanc4, lanc16), banda(lanc4, lanc256), banda(lanc16, lanc256) ), name = factor(c(rep("FL4", 100), rep("FL16", 100), rep("FL256", 100), rep("L4L16", 100), rep("L4L256", 100), rep("L16L256", 100)), levels = c("FL4", "FL16", "FL256", "L4L16", "L4L256", "L16L256") ) ) ################################################### ### code chunk number 6: fishlanc ################################################### lattice::xyplot(diff ~ sum | name, data = dat, panel = function(x, y, ...) { lattice::panel.xyplot(x, y, ...) lattice::panel.abline(h = mean(y), lty = 2) lattice::panel.abline(h = mean(y) + 1.96 * sd(y), lty = 3) lattice::panel.abline(h = mean(y) - 1.96 * sd(y), lty = 3) } ) ################################################### ### code chunk number 7: compare.Rnw:295-316 ################################################### stouff <- apply(temp, 1, function(x) sumz(x)$p) invt4 <- apply(temp, 1, function(x) invt(x, 4)$p) invt16 <- apply(temp, 1, function(x) invt(x, 16)$p) invt256 <- apply(temp, 1, function(x) invt(x, 256)$p) banda <- function(x, y) { res <- data.frame(sum = x + y, diff = (x - y)) res } dat <- data.frame(rbind(banda(stouff, invt4), banda(stouff, invt16), banda(stouff, invt256), banda(invt4, invt16), banda(invt4, invt256), banda(invt16, invt256) ), name = factor(c(rep("St4", 100), rep("St16", 100), rep("St256", 100), rep("t4t16", 100), rep("t4t256", 100), rep("t16t256", 100)), levels = c("St4", "St16", "St256", "t4t16", "t4t256", "t16t256") ) ) ################################################### ### code chunk number 8: stouffinvt ################################################### lattice::xyplot(diff ~ sum | name, data = dat, panel = function(x, y, ...) { lattice::panel.xyplot(x, y, ...) lattice::panel.abline(h = mean(y), lty = 2) lattice::panel.abline(h = mean(y) + 1.96 * sd(y), lty = 3) lattice::panel.abline(h = mean(y) - 1.96 * sd(y), lty = 3) } ) ################################################### ### code chunk number 9: compare.Rnw:399-401 ################################################### meanz(c(0.3, 0.31))$p meanz(c(0.1, 0.2))$p ################################################### ### code chunk number 10: compare.Rnw:481-490 ################################################### kvals <- c(4, 5, 6, 8, 10, 15, 20) pvals <- c(0.2, 0.3, 0.3679, 0.4, 0.5, 0.6) dat <- rbind( genvec(pvals, kvals, meanp, "meanp"), genvec(pvals, kvals, maximump, "maximump"), genvec(pvals, kvals, minimump, "minimump"), genvec(pvals, kvals, sump, "sump"), genvec(pvals, kvals, votep, "votep") ) ################################################### ### code chunk number 11: untranseqp ################################################### lattice::xyplot(g ~ k | method, groups = p, type = "l", data = dat, auto.key = list(space = "left", lines = TRUE, title = "p"), ylab = "g(p)" ) metap/inst/doc/strength.gv0000644000176200001440000000622213576717176015324 0ustar liggesusersdigraph test { rankdir = "LR" fontname = "Helvetica" node[fontname = "Helvetica"] edge[fontname = "Helvetica"] {rank = same; eqklt10; eqany; someklt10; someany; majority; modst; anypower; strongtot; modtot; weaktot;} evidence [label = "Location\nof\nevidence", shape = diamond] eqinall [label = "N\nof\nstudies", shape = diamond] evidence -> eqinall [label = "Equally in all"] eqklt10 [label = <
sump (Edgington)
maximump
>] eqany [label = <
sumz (Stouffer)
logitp
>] eqinall -> eqklt10 [label = "< 10"] eqinall -> eqany [label = "any"] someinall [label = "N\nof\nstudies", shape = diamond] evidence -> someinall [label = "Some in all"] someklt10 [label = <
sump (Edgington)
maximump
>] someany [label = <
sumz (Stouffer)
logitp
>] someinall -> someklt10 [label = "< 10"] someinall -> someany [label = "any"] majority [label = <
sumz (Stouffer)
logitp
>] evidence -> majority [label = "In the majority"] minority [label = "Strength\nof\nevidence", shape = diamond] evidence -> minority [label = "In the minority"] modst [label = <
sumlog (Fisher)
>] anypower [label = <
sumz (Stouffer)
logitp
>] minority -> modst [label = "Moderate or strong"] minority -> anypower [label = "Any power"] onetest [label = "Strength\nof\ntotal evidence", shape = diamond] evidence -> onetest [label = "In one test only"] strongtot [label = <
minimump (Tippett)
>] modtot [label = <
sumlog (Fisher)
>] weaktot [label = <
sumz (Stouffer)
logitp
>] onetest -> strongtot [label = "Strong total evidence"] onetest -> modtot [label = "Moderate total evidence"] onetest -> weaktot [label = "Weak total evidence"] } metap/inst/CITATION0000644000176200001440000000103412457431154013476 0ustar liggesuserscitHeader("To cite package 'metap' in publications use:") year <- sub(".*(2[[:digit:]]{3})-.*", "\\1", meta$Date) vers <- paste("R package version", meta$Version) citEntry(entry="Manual", title = "{metap}: meta-analysis of significance values", author = personList(as.person("Michael Dewey")), year = year, note = vers, textVersion = paste("Michael Dewey (", year, "). metap: meta-analysis of significance values. ", vers, ".", sep=""))