pan/0000755000176200001440000000000014470620226011033 5ustar liggesuserspan/NAMESPACE0000644000176200001440000000006713374503040012251 0ustar liggesusersuseDynLib(pan) export(ecme, pan, pan.bd) pan/ChangeLog0000644000176200001440000000465114470211164012610 0ustar liggesusersVersion 1.8 2023-08-19 Fix on remaining dfloat Version 1.8 2023-06-25 Replace 13 dfloat with dble in pan.f as suggested by Prof Brian Ripley from CRAN Revise CITATION according to check --as-cran Version 1.7 2022-12-03 Refine pan.bd.Rd 2020-11-09 Refine vignette nhyp=nhyp,hyp=hyp specification for pan() Apply -fc-prototypes-external for C prototypes Add marijuana to INDEX Version 1.6 2018-06-29 Add reference to DESCRIPTION/Description: Version 1.5 2018-06-24 Change email address 2018-02-04 Remove a4wide.sty 2018-01-25 Employ tools::package_native_routine_registration_skeleton("pan") Version 1.4 2016-02-10 Minor change to pan example as suggested by 이찬영 Revise URL in pan-tr.Rnw and remove URL from DESCRIPTION Revise License to GPL-3 according to CRAN policy 2015-02-24 Add YWC.data example for multivariate imputation Version 1.0-1.3 2015-02-10 Add reference to the book chapter in New Methods for the Analysis of Change To comply with title case and revise URL for TR-9704 as in vignette/Rd Change rngs to real function due to Simon Grund Revise DESCRIPTION, remove .Rinstignore and regenerate INDEX 2013-12-10 Modify file LICENSE as lmm Version 0.9 2013--09-27 Move pan-tr.Rnw as with tr9704.pdf into vignettes Version 0.8 2013-09-27 Change require(lme4) to if(require(lme4)) Add MIT to file LICENEE Version 0.7 2013-04-16 Fix error in the bivariate example, i.e., pred <- cbind() than c() Notified by Prof Brian Ripley and revealed via Makevars/PKG_FFLAGS=-fbounds-check 2013-01-10 Suggests mitools and lme4 Add comparison with lmer in ecme.Rd Add code for combining multiple imputation results 2013-01-07 Revise bivariate examples in pan.Rd Version 0.6 2013-01-07 Add bivariate example from Maria Valaste following earlier communications with Turkiewicz, Aleksandra 2012-08-31 Remove reference to panex.R in pan.Rd Version 0.5 2012-08-06 Correct minor typos according to Speidel Matthias Add CITATION 2012-04-06 Add marijuana data from pannex Remove inst/tests and remove dontrun{} in ecme.Rd and pan.Rd To enable LazyData and LazyLoad Version 0.4 2012-03-16 Add this file, .Rinstignore, and pan-tr.Rnw Add pointers to ecmeex.R and panex.R in the documentation Version 0.3 2011-04-28 Add references Version 0.1 2003-11-15 Made available from CRAN pan/data/0000755000176200001440000000000013374503040011740 5ustar liggesuserspan/data/datalist0000644000176200001440000000003213374503040013463 0ustar liggesusersbitest marijuana YWC.data pan/data/marijuana.rda0000644000176200001440000000057313374503040014404 0ustar liggesusers r0b```b`@& `d`aҜEYy  b3 `@,@"A"-@"-B"-A"-C"@"D7 FT\J[a`y;@7A`4]셫7Cs =6Py4s1UoFh?;@Uǀ4zGKV=(=4Z Fzh˾QTSotoT=uc77: UO(sּb C)&eAٌPs~r2WeVPx(<)0AQ~E@$ 90gRKҊ [;pan/data/YWC.data.rda0000644000176200001440000013036613374503040014013 0ustar liggesusersMٹ߇UuN;Cbx"O@E.`X?$$II=HM$ ቁ܁܁ 2c-F h^奛dʮ_ϩ*27N^>YZ{k4M;nO?/ O9Mwoo>xiέjyF9Qάߙז+Y\Y0Z`Y^̡vVQ^3ֆ\~6ʹ;7(fn 1*m2_}ͨ㏭ڹնVϷ44]vn5oipU[ jf.68ri4\ ȥVse4c7ouvflͷ~敬lV}1;瞹_xiF&G6VWͬV6iӕi̕/FQpNN빍ljIze~j-+tVͬM?6lFe-kqֹq|zk9nNy:37_i̵omVz=fVo.[}kcmmұ6~mum<6am6`m\6-vXdegl5w==s-:k{kk5;[iIڤcmұ1>}872?7p~f?7 }6@+&5Ysʜwn<nsMf?7=isz~e̩dv̢}mc5o93٬,׌.ϟon|k0,¹Iهsswf7ٴ(k_3*[K_#k6Sɤ$ocz~c~tƼ3]lLơ1s6gc\1.߬qqƼm6Y6oi0ژ11{c\111d~F ӭC6n6/fE-j]YdtefRlmٚ.mMn-5bkzwkݭ[նkl-:ۚt3]0 ^%,.>3ۚ}}`1}k}k\Hq{n˷'&[Ӧ[Wkm6rnW<*_=Z nMؚܚ55^,ݙ3 ؙx2+3<ϰ3+Y^cgyI$gg}g}gr[y9wܙ ؙ ؙ<ٙEؙEؙ<ߙ Y3ؙ,,,,΢ههf0dgzwr19ZyKɦŢK*^3L:v.0< #:0.̒>#ydKW^Y^xeyW+,wrkK˭kk(Fs, =}͠,#f{52vn5(ož&vZf_ؘ>K,=o̺elN}؎2 Ky,|brjbk6V%^Llviي |f5َG ;eg|L[1Hۙ\ZķԦ->ZZ|gi;K--Zz 2[;1IY_'׌̪r,GU٘^:62QrO4v_1lf_ـҤcik7?<+!^Jgf^]L6sc|m;ˍvbtN777Fyd6}_1M:6#l>lL:ly_Q[7o?myer2i\Ɏ}Ǵ:ks.}3/my_3J]}(a,tʴʵ͹ΪL3L\7l.`ksmzi|̭y:?mf3lM&S5-;bfVvf]yjsn5t4$` L޵\ ΍wm'{i;4/mx5=}!5ٴʴf_>|>jkYTnYԍy;l'{_,΢rۑ^ڮf_,=}mg5smc_}}F1k3 ؙ7c;gұ3egV/mziKۭl>˾,X&üv5l<^{ڙ3hg΢K#p ,i{yL{/swo[N_ݿ;5o[{ߓ7-}[ſ?7 7 ŷOg~K[[[;xkOGo翃 K~K o-+ooo߷߷o7|W_}WWտa_麯}|*ZyʯN+k?li+oW'D/0 ˯=s8rў?@?#=?_.\jۀW׉_phq V>>O3K Np OE~䧓u|&~6x:Ӽԯ߾;#vnq{.7OZuҺ6?QO|GbOt:j'_= WtݴX?>yZ]ӮڸsimϣyEOGit%s՟96Oj=: P~j'z>{Exh{y{N{N}@ WWr[nCt~䟇xj'9}O|#z?OۯWe'zCtU]{S+}/s[/q%_C?D_Iy<˒">#_G]OԺ _?O;..wПv[|~[+h/XC|:/xvVp<Ӟ+Q\'͞?ۭ^x? 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This was just what I intended. Of course you can include the example in the documentation. It's very good idea. Best regards, Maria Valaste Lainaus "Jing Hua Zhao" : > Dear Maria, > > I just had time to try out your data/code; is this close to what you intend? > ... > > If you don't mind I could include your example in the documentation to > let others share your experiment. > > Best wishes, > > > Jing Hua > From: Maria Valaste [mailto:maria.valaste@helsinki.fi] Sent: 08 December 2012 19:09 To: Jing Hua Zhao Subject: Help for using R package PAN Dear Jing hua Zhao, I'm trying to use R package PAN. For one Y variable I have managed succesfully to impute using package PAN. When I try to impute two Y variables, I have encountered a problem that I can't solve. Could please help me? I'm not expert on R program and I'm really desperate. I have R version 2.15.2 (Platform: x86_64-w64-mingw32/x64 (64-bit)). Below is code that I have tried to run for two Y variables. There also the error message (Error: subscript out of bounds). I also attach the test data below after the R code. Best regards, Maria Valaste University of Helsinki, Finland } } \usage{data(bitest)} \format{A data frame} \source{Maria Valaste } \keyword{datasets} pan/man/ecme.Rd0000644000176200001440000002021713374503040013004 0ustar liggesusers\name{ecme} \alias{ecme} \title{ ECME algorithm for general linear mixed model } \description{ Performs maximum-likelihood estimation for generalized linear mixed models. The model, which is typically applied to longitudinal or clustered responses, is yi = Xi\%*\%beta + Zi\%*\%bi + ei , i=1,\dots,m, where yi = (ni x 1) response vector for subject or cluster i; Xi = (ni x p) matrix of covariates; Zi = (ni x q) matrix of covariates; beta = (p x 1) vector of coefficients common to the population (fixed effects); bi = (q x 1) vector of coefficients specific to subject or cluster i (random effects); and ei = (ni x 1) vector of residual errors. The vector bi is assumed to be normally distributed with mean zero and unstructured covariance matrix psi, bi \eqn{\sim}{~} N(0,psi) independently for i=1,\dots,m. The residual vector ei is assumed to be ei \eqn{\sim}{~} N(0,sigma2*Vi) where Vi is a known (ni x ni) matrix. In most applications, Vi is the identity matrix. } \usage{ ecme(y, subj, occ, pred, xcol, zcol=NULL, vmax, start, maxits=1000, eps=0.0001, random.effects=F) } \arguments{ \item{y}{ vector of responses. This is simply the individual yi vectors stacked upon one another. Each element of y represents the observed response for a particular subject-occasion, or for a particular unit within a cluster. } \item{subj}{ vector of same length as y, giving the subject (or cluster) indicators i for the elements of y. For example, suppose that y is in fact c(y1,y2,y3,y4) where length(y1)=2, length(y2)=3, length(y3)=2, and length(y4)=7. Then subj should be c(1,1,2,2,2,3,3,4,4,4,4,4,4,4). } \item{occ}{ vector of same length as y indicating the "occasions" for the elements of y. In a longitudinal dataset where each individual is measured on at most nmax distinct occasions, each element of y corresponds to one subject-occasion, and the elements of of occ should be coded as 1,2,\dots,nmax to indicate these occasion labels. (You should label the occasions as 1,2,\dots,nmax even if they are not equally spaced in time; the actual times of measurement will be incorporated into the matrix "pred" below.) In a clustered dataset, the elements of occ label the units within each cluster i, using the labels 1,2,\dots,ni. } \item{pred}{ matrix of covariates used to predict y. The number of rows should be length(y). The first column will typically be constant (one), and the remaining columns correspond to other variables appearing in Xi and Zi. } \item{xcol}{ vector of integers indicating which columns of pred will be used in Xi. That is, pred[,xcol] is the Xi matrices (stacked upon one another. } \item{zcol}{ vector of integers indicating which columns of pred will be used in Zi. That is, pred[,zcol] is the Zi matrices (stacked upon one another). If zcol=NULL then the model is assumed to have no random effects; in that case the parameters are estimated noniteratively by generalized least squares. } \item{vmax}{ optional matrix of dimension c(max(occ),max(occ)) from which the Vi matrices will be extracted. In a longitudinal dataset, vmax would represent the Vi matrix for an individual with responses at all possible occasions 1,2,\dots,nmax=max(occ); for individuals with responses at only a subset of these occasions, the Vi will be obtained by extracting the rows and columns of vmax for those occasions. If no vmax is specified by the user, an identity matrix is used. In most applications of this model one will want to have Vi = identity, so most of the time this argument can be omitted. } \item{start}{ optional starting values of the parameters. If this argument is not given then ecme() chooses its own starting values. This argument should be a list of three elements named "beta", "psi", and "sigma2". Note that "beta" should be a vector of the same length as "xcol", "psi" should be a matrix of dimension c(length(zcol),length(zcol)), and "sigma2" should be a scalar. This argument has no effect if zcol=NULL. } \item{maxits}{ maximum number of cycles of ECME to be performed. The algorithm runs to convergence or until "maxits" iterations, whichever comes first. } \item{eps}{ convergence criterion. The algorithm is considered to have converged if the relative differences in all parameters from one iteration to the next are less than eps--that is, if all(abs(new-old) 1 response variables. If we had # r response variables rather than one, the only difference would be # that the vector y below would become a matrix with r columns, one # for each response variable. The dimensions of Sigma (the residual # covariance matrix for the response) and Psi (the covariance matrix # for the random effects) would also change to (r x r) and (r*q x r*q), # respectively, where q is the number of random coefficients in the # model (in this case q=1 because we have only random intercepts). The # new dimensions for Sigma and Psi will be reflected in the prior # distribution, as Dinv and Binv become (r x r) and (r*q x r*q). # # The pred matrix has the same number of rows as y, the number of # subject-occasions. Each column of Xi and Zi must be represented in # pred. Because Zi is merely the first column of Xi, we do not need to # enter that column twice. So pred is simply the matrix Xi, stacked # upon itself nine times. # data(marijuana) attach(marijuana) pred <- with(marijuana,cbind(int,dummy1,dummy2,dummy3,dummy4,dummy5)) # # Now we must tell pan that all six columns of pred are to be used in # Xi, but only the first column of pred appears in Zi. # xcol <- 1:6 zcol <- 1 ######################################################################## # The model specification is now complete. The only task that remains # is to specify the prior distributions for the covariance matrices # Sigma and Psi. # # Recall that the dimension of Sigma is (r x r) where r # is the number of response variables (in this case, r=1). The prior # distribution for Sigma is inverted Wishart with hyperparameters a # (scalar) and Binv (r x r), where a is the imaginary degrees of freedom # and Binv/a is the prior guesstimate of Sigma. The value of a must be # greater than or equal to r. The "least informative" prior possible # would have a=r, so here we will take a=1. As a prior guesstimate of # Sigma we will use the (r x r) identity matrix, so Binv = 1*1 = 1. # # By similar reasoning we choose the prior distribution for Psi. The # dimension of Psi is (r*q x r*q) where q is the number of random # effects in the model (i.e. the length of zcol, which in this case is # one). The hyperparameters for Psi are c and Dinv, where c is the # imaginary degrees of freedom (which must be greater than or equal to # r*q) and Dinv/c is the prior guesstimate of Psi. We will take c=1 # and Dinv=1*1 = 1. # # The prior is specified as a list with four components named a, Binv, # c, and Dinv, respectively. # prior <- list(a=1,Binv=1,c=1,Dinv=1) ######################################################################## # Now we are ready to run pan(). Let's assume that the pan function # and the object code have already been loaded into R. First we # do a preliminary run of 1000 iterations. # result <- pan(y,subj,pred,xcol,zcol,prior,seed=13579,iter=1000) # # Check the convergence behavior by making time-series plots and acfs # for the model parameters. Variances will be plotted on a log # scale. We'll assume that a graphics device has already been opened. # plot(1:1000,log(result$sigma[1,1,]),type="l") acf(log(result$sigma[1,1,])) plot(1:1000,log(result$psi[1,1,]),type="l") acf(log(result$psi[1,1,])) par(mfrow=c(3,2)) for(i in 1:6) plot(1:1000,result$beta[i,1,],type="l") for(i in 1:6) acf(result$beta[i,1,]) # # This example appears to converge very rapidly; the only appreciable # autocorrelations are found in Psi, and even those die down by lag # 10. With a sample this small we can afford to be cautious, so let's # impute the missing data m=10 times taking 100 steps between # imputations. We'll use the current simulated value of y as the first # imputation, then restart the chain where we left off to produce # the second through the tenth. # y1 <- result$y result <- pan(y,subj,pred,xcol,zcol,prior,seed=9565,iter=100,start=result$last) y2 <- result$y result <- pan(y,subj,pred,xcol,zcol,prior,seed=6047,iter=100,start=result$last) y3 <- result$y result <- pan(y,subj,pred,xcol,zcol,prior,seed=3955,iter=100,start=result$last) y4 <- result$y result <- pan(y,subj,pred,xcol,zcol,prior,seed=4761,iter=100,start=result$last) y5 <- result$y result <- pan(y,subj,pred,xcol,zcol,prior,seed=9188,iter=100,start=result$last) y6 <- result$y result <- pan(y,subj,pred,xcol,zcol,prior,seed=9029,iter=100,start=result$last) y7 <- result$y result <- pan(y,subj,pred,xcol,zcol,prior,seed=4343,iter=100,start=result$last) y8 <- result$y result <- pan(y,subj,pred,xcol,zcol,prior,seed=2372,iter=100,start=result$last) y9 <- result$y result <- pan(y,subj,pred,xcol,zcol,prior,seed=7081,iter=100,start=result$last) y10 <- result$y ######################################################################## # Now we combine the imputation results according to mitools ######################################################################## # First, we build data frames from above, d1 <- data.frame(y=y1,subj,pred) d2 <- data.frame(y=y2,subj,pred) d3 <- data.frame(y=y3,subj,pred) d4 <- data.frame(y=y4,subj,pred) d5 <- data.frame(y=y5,subj,pred) d6 <- data.frame(y=y6,subj,pred) d7 <- data.frame(y=y7,subj,pred) d8 <- data.frame(y=y8,subj,pred) d9 <- data.frame(y=y9,subj,pred) d10 <- data.frame(y=y10,subj,pred) # Second, we establish a S3 object as needed for the function MIcombine # nevertheless we start with an ordinary least squares regression require(mitools) d <- imputationList(list(d1,d2,d3,d4,d5,d6,d7,d8,d9,d10)) w <- with(d,lm(y~-1+pred)) MIcombine(w) # Now, we can turn to lmer as in lme4 package but in this case it is the # same. if(require(lme4)) { w2 <- with(d,lmer(y~-1+pred+(1|subj))) b <- MIextract(w2,fun=fixef) Var <- function(obj) unlist(lapply(diag(vcov(obj)),function(m) m)) v <- MIextract(w2,fun=Var) MIcombine(b,v) detach(marijuana) } ### bivariate example data(bitest) attach(bitest) y <- with(bitest,cbind(y1,y2)) subj <- c(clusterid) pred <- cbind (int, x1, x2, x3) xcol <- 1:4 zcol <- 1 a <- 2 c <- 2 id2 <- matrix(c(1,0,0,1),ncol=2,nrow=2) Binv <- a*id2 Dinv <- c*id2 prior <- list(a=a, Binv=Binv, c=c, Dinv=Dinv) result <- pan(y, subj, pred, xcol, zcol, prior, seed=12345, iter=1000) } \keyword{models} % Converted by Sd2Rd version 1.21. pan/man/marijuana.Rd0000644000176200001440000000141613752203466014053 0ustar liggesusers\name{marijuana} \docType{data} \alias{marijuana} \title{A pilot study of the clinical and psychological effects of maarijuana} \description{ Nine male subjects were given three treatments in the form of low-dose, high-dose, and placebo cigarettes. The order of treatments within subjects was balanced in a replicated 3 x 3 Latin square, but because the order for each subject was not reported in the article, (I shall proceed as if) the order effects are negligible. Changes in heart rate were recorded 15 and 90 minutes after marijuana use, and five of the 54 data values are missing. } \usage{data(marijuana)} \format{A data frame} \source{Wei AT, Zinberg NE, Nelson JM. Clinical and psychological effects of marijuana in man. Science 1968; 162:1234-1242} \keyword{datasets} pan/man/YWC.data.Rd0000644000176200001440000000166313752012346013455 0ustar liggesusers\name{YWC.data} \docType{data} \alias{YWC.data} \title{A data frame for multivariate imputation} \description{ The data serves as an worked example for PAN as follows, \preformatted{ head(YWC.data) Yi.Matrix <- as.matrix(YWC.data[,4:39]) Subj.vector <- as.vector(as.matrix(YWC.data[,2])) YWC.data$intercept <- 1. Pred.Matrix <- as.matrix(YWC.data[,c(40,3)]) xcol <- 1:2 zcol <- 1 a.parameter = ncol(Yi.Matrix) Binv.parameter = diag(a.parameter*a.parameter) c.parameter = (zcol*ncol(Yi.Matrix)) Dinv.parameter = diag(c.parameter*c.parameter) Prior.Distribution <- list(a = a.parameter, Binv = Binv.parameter, c = c.parameter, Dinv = Dinv.parameter) Imputation.Result <- pan(Yi.Matrix, Subj.vector, Pred.Matrix, xcol, zcol, Prior.Distribution, seed=44896, iter=1000) } } \usage{data(YWC.data)} \format{A data frame} \source{Steven J. Pierce} \keyword{datasets} pan/DESCRIPTION0000644000176200001440000000147714470620225012551 0ustar liggesusersPackage: pan Version: 1.9 Date: 2023-8-19 Title: Multiple Imputation for Multivariate Panel or Clustered Data Author: Original by Joseph L. Schafer Maintainer: Jing hua Zhao Suggests: mitools, lme4 LazyData: Yes LazyLoad: Yes Description: It provides functions and examples for maximum likelihood estimation for generalized linear mixed models and Gibbs sampler for multivariate linear mixed models with incomplete data, as described in Schafer JL (1997) "Imputation of missing covariates under a multivariate linear mixed model". Technical report 97-04, Dept. of Statistics, The Pennsylvania State University. License: GPL-3 NeedsCompilation: yes Packaged: 2023-08-19 19:05:44 UTC; jhz22 Repository: CRAN Date/Publication: 2023-08-21 08:30:13 UTC pan/build/0000755000176200001440000000000014470211210012117 5ustar liggesuserspan/build/vignette.rds0000644000176200001440000000036014470211210014455 0ustar liggesusersuOj0lB ?„^!!#q{'vhv`F={ǂ1tnҼ=tTS+i T\lEMFK}€FVh du3ng9{p*í^&uw7aTIa Xh! @8Gpan/src/0000755000176200001440000000000014470211210011607 5ustar liggesuserspan/src/pan.f0000644000176200001440000020160014470211044012540 0ustar liggesusersC*********************************************************************** C The following subroutines are used by both pan() and ecme(). C*********************************************************************** subroutine istfin(ntot,subj,m,ist,ifin) C creates vectors of starting and finishing positions for subjects integer ntot,subj(ntot),m,ist(m),ifin(m),scur scur=-999 icur=0 do 100 i=1,ntot if(subj(i).ne.scur) then icur=icur+1 ist(icur)=i scur=subj(i) endif 100 continue do 200 i=1,m-1 ifin(i)=ist(i+1)-1 200 continue ifin(m)=ntot return end C*********************************************************************** subroutine chfc(p,pw,s) C overwrites s (upper tri) with cholesky factor integer p,pw double precision s(p,p),sum do 50 i=1,pw sum=dble(0.) do 10 k=1,i-1 sum=sum+s(k,i)**2 10 continue s(i,i)=dsqrt(s(i,i)-sum) do 40 j=i+1,pw sum=dble(0.) do 30 k=1,i-1 sum=sum+s(k,i)*s(k,j) 30 continue s(i,j)=(s(i,j)-sum)/s(i,i) 40 continue 50 continue return end C*********************************************************************** subroutine chl(p,pw,m,s,l) C overwrites lth layer of s (upper tri) with cholesky factor integer p,pw,m,l double precision s(p,p,m),sum do 50 i=1,pw sum=dble(0.) do 10 k=1,i-1 sum=sum+s(k,i,l)**2 10 continue s(i,i,l)=dsqrt(s(i,i,l)-sum) do 40 j=i+1,pw sum=dble(0.) do 30 k=1,i-1 sum=sum+s(k,i,l)*s(k,j,l) 30 continue s(i,j,l)=(s(i,j,l)-sum)/s(i,i,l) 40 continue 50 continue return end C*********************************************************************** subroutine bkslv(p,pw,s) C inverts an upper triangular matrix integer p,pw double precision s(p,p),sum s(1,1)=dble(1.)/s(1,1) do 10 k=2,pw s(k,k)=dble(1.)/s(k,k) do 5 j=1,k-1 sum=dble(0.) do 3 i=j,k-1 sum=sum+s(j,i)*s(i,k) 3 continue s(j,k)=-sum*s(k,k) 5 continue 10 continue return end C*********************************************************************** subroutine mm(p,pw,wm,cm) C calculates upper tri part of cm = wm%*%t(wm), where wm is upper tri integer p,pw double precision wm(p,p),cm(p,p),sum do 10 i=1,pw do 5 j=i,pw sum=dble(0.) do 2 k=max(i,j),pw sum=sum+wm(i,k)*wm(j,k) 2 continue cm(i,j)=sum 5 continue 10 continue return end C*********************************************************************** subroutine bkslvl(p,pw,m,s,l) C inverts an upper triangular matrix in layer l of s integer p,pw,m,l double precision s(p,p,m),sum s(1,1,l)=dble(1.)/s(1,1,l) do 10 k=2,pw s(k,k,l)=dble(1.)/s(k,k,l) do 5 j=1,k-1 sum=dble(0.) do 3 i=j,k-1 sum=sum+s(j,i,l)*s(i,k,l) 3 continue s(j,k,l)=-sum*s(k,k,l) 5 continue 10 continue return end C*********************************************************************** subroutine mmul(p,pw,m,wm,l,cm) C calculates upper tri part of cm = wm%*%t(wm), where wm is upper tri C(works in layer l) integer p,pw,m,l double precision wm(p,p,m),cm(p,p),sum do 10 i=1,pw do 5 j=i,pw sum=dble(0.) do 2 k=max(i,j),pw sum=sum+wm(i,k,l)*wm(j,k,l) 2 continue cm(i,j)=sum 5 continue 10 continue return end C*********************************************************************** C The following are used only by pan(). C*********************************************************************** subroutine prelimm(ntot,subj,m,ist,ifin,pcol,pred,q, / zcol,ztz,patt,nstar,p,xcol,xtxinv,wkpp) C Preliminary manipulations for pan. After execution, C ist = vector of starting positions for subject i, i=1,...,m C ifin = vector of finishing positions for subject i, i=1,...,m C ztz = t(z_i)%*%z, i=1,...,m C nstar = total number of rows in y containing data C xtxinv = inv( sum of t(X_i)%*%X_i ) integer ntot,subj(ntot),m,ist(m),ifin(m),pcol, / q,zcol(q),nstar,s,patt(ntot),p,xcol(p) double precision pred(ntot,pcol),ztz(q,q,m),sum,xtxinv(p,p), / wkpp(p,p) call istfin(ntot,subj,m,ist,ifin) nstar=0 do 10 i=1,ntot if(patt(i).ne.0) nstar=nstar+1 10 continue do 100 s=1,m do 90 i=1,q do 80 j=i,q sum=dble(0.) do 60 k=ist(s),ifin(s) if(patt(k).ne.0) then sum=sum+pred(k,zcol(i))*pred(k,zcol(j)) endif 60 continue ztz(i,j,s)=sum if(i.ne.j) ztz(j,i,s)=sum 80 continue 90 continue 100 continue do 150 i=1,p do 140 j=i,p sum=dble(0.) do 130 k=1,ntot if(patt(k).ne.0) then sum=sum+pred(k,xcol(i))*pred(k,xcol(j)) endif 130 continue wkpp(i,j)=sum 140 continue 150 continue call chfc(p,p,wkpp) call bkslv(p,p,wkpp) call mm(p,p,wkpp,xtxinv) do 160 i=1,p do 155 j=i,p xtxinv(j,i)=xtxinv(i,j) 155 continue 160 continue return end C*********************************************************************** subroutine mgibbsbd(ntot,subj,m,ist,ifin,pcol,pred,q,zcol,ztz, / patt,nstar,r,y,p,xcol,npatt,rmat,sflag,beta,sigma,psi,b, / xtxinv,wkpp,wkpr,eps,wkrqrq1,wkrqrq2,sig,wkrr1,wkrr2,iter, / wkqr1,wkqr2,wkqrv,nhyp,hyp,delta,iposn,pstfin,betas, / sigmas,psis,wkqq1,wkqq2) C conventional Gibbs sampler for the block-diagonal version of pan integer ntot,subj(ntot),m,ist(m),ifin(m),pcol,sflag, / q,zcol(q),nstar,patt(ntot),r,p,xcol(p),npatt,rmat(npatt,r), / iter,nhyp,iposn(ntot),pstfin(npatt,2),it integer oc(100),mc(100) double precision pred(ntot,pcol),ztz(q,q,m),y(ntot,r), / beta(p,r),sigma(r,r),psi(q,q,r),b(q,r,m),xtxinv(p,p), / wkpp(p,p),wkpr(p,r),eps(ntot,r),wkrqrq1(r*q,r*q), / wkrqrq2(r*q,r*q),sig(r*q,r*q,m),wkrr1(r,r),wkrr2(r,r), / wkqr1(q,r),wkqr2(q,r),wkqrv(q*r),hyp(nhyp), / delta(ntot,r),betas(p,r,iter),sigmas(r,r,iter), / psis(q,q,r,iter),wkqq1(q,q),wkqq2(q,q) double precision wkr(100) real junk junk=gauss() call prelimm(ntot,subj,m,ist,ifin,pcol,pred,q,zcol,ztz,patt, / nstar,p,xcol,xtxinv,wkpp) if(sflag.ne.1) then C get starting values of parameters call mimpy(ntot,r,y,patt,npatt,rmat) call mkxty(ntot,r,y,pcol,pred,p,xcol,patt,wkpr) call mkbeta(p,r,xtxinv,wkpr,beta) call mkeps1(ntot,r,y,pcol,pred,p,xcol,beta,eps,patt) call mksigma(ntot,r,eps,nstar,sigma,patt) call mksigbd(r,q,m,psi,sigma,ztz,sig,wkrr1,wkrr2, / wkrqrq2,1,nhyp,hyp,wkqq1,wkqq2) call mkpsi0bd(r,q,m,psi,sig,wkrqrq1) else C set starting value of y to eps do 50 i=1,ntot do 40 j=1,r y(i,j)=eps(i,j) 40 continue 50 continue endif do 100 it=1,iter call mkeps1(ntot,r,y,pcol,pred,p,xcol,beta,eps,patt) call mksigbd(r,q,m,psi,sigma,ztz,sig,wkrr1,wkrr2, / wkrqrq2,0,nhyp,hyp,wkqq1,wkqq2) call drb(r,q,m,b,wkrr2,eps,pcol,pred,zcol,wkqr1, / wkqr2,ist,ifin,patt,ntot,sig,wkrqrq2,wkqrv) call drpsibd(r,q,m,psi,wkqq1,wkqq2,wkqrv,nhyp,hyp,b) call mkeps2(ntot,m,r,y,pcol,pred,q,zcol,b,eps,patt,ist,ifin) call mkxty(ntot,r,eps,pcol,pred,p,xcol,patt,wkpr) call mkbeta(p,r,xtxinv,wkpr,beta) call mkeps1(ntot,r,eps,pcol,pred,p,xcol,beta,delta,patt) call drsigma(ntot,r,delta,nstar,sigma,patt,nhyp,hyp,wkrr1, / wkrr2,p) call drbeta(r,sigma,xtxinv,p,beta,wkrr1,wkpp,wkpr) call mkeps1(ntot,r,eps,pcol,pred,p,xcol,beta,delta,patt) call dreps(100,100,oc,mc,100,wkr,ntot,iposn,npatt,pstfin, / r,rmat,delta,sigma,wkrr1,wkrr2) call mky(ntot,r,pcol,pred,delta,y,p,xcol,q,zcol,beta,m,b, / ist,ifin,npatt,rmat,patt) call storebd(iter,it,p,r,beta,q,psi,sigma,betas,sigmas,psis) 100 continue return end C*********************************************************************** subroutine mgibbs(ntot,subj,m,ist,ifin,pcol,pred,q,zcol,ztz,patt, / nstar,r,y,p,xcol,npatt,rmat,sflag,beta,sigma,psi,b, / xtxinv,wkpp,wkpr,eps,wkrqrq1,wkrqrq2,sig,wkrr1,wkrr2,iter, / wkqr1,wkqr2,wkqrv,nhyp,hyp,delta,iposn,pstfin,betas, / sigmas,psis) C conventional Gibbs sampler. integer ntot,subj(ntot),m,ist(m),ifin(m),pcol,sflag, / q,zcol(q),nstar,patt(ntot),r,p,xcol(p),npatt,rmat(npatt,r), / iter,nhyp,iposn(ntot),pstfin(npatt,2),it integer oc(100),mc(100) double precision pred(ntot,pcol),ztz(q,q,m),y(ntot,r), / beta(p,r),sigma(r,r),psi(r*q,r*q),b(q,r,m),xtxinv(p,p), / wkpp(p,p),wkpr(p,r),eps(ntot,r),wkrqrq1(r*q,r*q), / wkrqrq2(r*q,r*q),sig(r*q,r*q,m),wkrr1(r,r),wkrr2(r,r), / wkqr1(q,r),wkqr2(q,r),wkqrv(q*r),hyp(nhyp), / delta(ntot,r),betas(p,r,iter),sigmas(r,r,iter), / psis(r*q,r*q,iter) double precision wkr(100) real junk junk=gauss() call prelimm(ntot,subj,m,ist,ifin,pcol,pred,q,zcol,ztz,patt, / nstar,p,xcol,xtxinv,wkpp) if(sflag.ne.1) then C get starting values of parameters call mimpy(ntot,r,y,patt,npatt,rmat) call mkxty(ntot,r,y,pcol,pred,p,xcol,patt,wkpr) call mkbeta(p,r,xtxinv,wkpr,beta) call mkeps1(ntot,r,y,pcol,pred,p,xcol,beta,eps,patt) call mksigma(ntot,r,eps,nstar,sigma,patt) call mksig(r,q,m,psi,sigma,ztz,sig,wkrr1,wkrr2,wkrqrq1, / wkrqrq2,1,nhyp,hyp) call mkpsi0(r,q,m,psi,sig,wkrqrq1) else C set starting value of y to eps do 50 i=1,ntot do 40 j=1,r y(i,j)=eps(i,j) 40 continue 50 continue endif do 100 it=1,iter call mkeps1(ntot,r,y,pcol,pred,p,xcol,beta,eps,patt) call mksig(r,q,m,psi,sigma,ztz,sig,wkrr1,wkrr2,wkrqrq1, / wkrqrq2,0,nhyp,hyp) call drb(r,q,m,b,wkrr2,eps,pcol,pred,zcol,wkqr1, / wkqr2,ist,ifin,patt,ntot,sig,wkrqrq2,wkqrv) call drpsi(r,q,m,psi,wkrqrq1,wkrqrq2,wkqrv,nhyp,hyp,b) call mkeps2(ntot,m,r,y,pcol,pred,q,zcol,b,eps,patt,ist,ifin) call mkxty(ntot,r,eps,pcol,pred,p,xcol,patt,wkpr) call mkbeta(p,r,xtxinv,wkpr,beta) call mkeps1(ntot,r,eps,pcol,pred,p,xcol,beta,delta,patt) call drsigma(ntot,r,delta,nstar,sigma,patt,nhyp,hyp,wkrr1, / wkrr2,p) call drbeta(r,sigma,xtxinv,p,beta,wkrr1,wkpp,wkpr) call mkeps1(ntot,r,eps,pcol,pred,p,xcol,beta,delta,patt) call dreps(100,100,oc,mc,100,wkr,ntot,iposn,npatt,pstfin, / r,rmat,delta,sigma,wkrr1,wkrr2) call mky(ntot,r,pcol,pred,delta,y,p,xcol,q,zcol,beta,m,b, / ist,ifin,npatt,rmat,patt) call store(iter,it,p,r,beta,q,psi,sigma,betas,sigmas,psis) 100 continue return end C*********************************************************************** subroutine storebd(iter,it,p,r,beta,q,psi,sigma,betas,sigmas,psis) integer iter,it,p,r,q double precision beta(p,r),betas(p,r,iter),sigma(r,r), / sigmas(r,r,iter),psi(q,q,r),psis(q,q,r,iter) do 10 j=1,r do 5 i=1,p betas(i,j,it)=beta(i,j) 5 continue 10 continue do 21 l=1,r do 20 j=1,q do 15 i=1,q psis(i,j,l,it)=psi(i,j,l) 15 continue 20 continue 21 continue do 30 j=1,r do 25 i=1,r sigmas(i,j,it)=sigma(i,j) 25 continue 30 continue return end C*********************************************************************** subroutine store(iter,it,p,r,beta,q,psi,sigma,betas,sigmas,psis) integer iter,it,p,r,q double precision beta(p,r),betas(p,r,iter),sigma(r,r), / sigmas(r,r,iter),psi(r*q,r*q),psis(r*q,r*q,iter) do 10 j=1,r do 5 i=1,p betas(i,j,it)=beta(i,j) 5 continue 10 continue do 20 j=1,r*q do 15 i=1,r*q psis(i,j,it)=psi(i,j) 15 continue 20 continue do 30 j=1,r do 25 i=1,r sigmas(i,j,it)=sigma(i,j) 25 continue 30 continue return end C*********************************************************************** subroutine mky(ntot,r,pcol,pred,delta,y,p,xcol,q,zcol,beta,m,b, / ist,ifin,npatt,rmat,patt) C calculates y = delta + X%*%beta + Z%*%b for missing observations integer ntot,r,pcol,p,xcol(p),q,zcol(q),m,ist(m),ifin(m),npatt, / rmat(npatt,r),patt(ntot),s,st,fin double precision pred(ntot,pcol),delta(ntot,r),y(ntot,r), / beta(p,r),b(q,r,m),sum do 100 s=1,m st=ist(s) fin=ifin(s) do 90 i=st,fin if(patt(i).eq.0) then do 20 j=1,r sum=dble(0.) do 10 k=1,p sum=sum+pred(i,xcol(k))*beta(k,j) 10 continue do 15 k=1,q sum=sum+pred(i,zcol(k))*b(k,j,s) 15 continue y(i,j)=sum+delta(i,j) 20 continue else do 85 j=1,r if(rmat(patt(i),j).eq.0) then sum=dble(0.) do 40 k=1,p sum=sum+pred(i,xcol(k))*beta(k,j) 40 continue do 50 k=1,q sum=sum+pred(i,zcol(k))*b(k,j,s) 50 continue y(i,j)=sum+delta(i,j) endif 85 continue endif 90 continue 100 continue return end C*********************************************************************** subroutine dreps(loc,lmc,oc,mc,lwkr,wkr,ntot,iposn,npatt,pstfin, / r,rmat,delta,sigma,wkrr1,wkrr2) C draws residuals given parameters and random effects integer loc,lmc,oc(loc),mc(lmc),lwkr,nmc,noc,ntot,iposn(ntot), / npatt,pstfin(npatt,2),r,rmat(npatt,r),patt(ntot),pt double precision wkr(lwkr),delta(ntot,r),sigma(r,r),wkrr1(r,r), / wkrr2(r,r),sum C take care of completely missing cases if(pstfin(1,1).ne.1) then do 5 i=1,r do 4 j=i,r wkrr1(i,j)=sigma(i,j) 4 continue 5 continue call chfc(r,r,wkrr1) do 100 i=1,(pstfin(1,1)-1) do 10 j=1,r wkr(j)=dble(gauss()) 10 continue do 50 j=r,1,-1 sum=dble(0.) do 40 k=1,j sum=sum+wkrr1(k,j)*wkr(k) 40 continue wkr(j)=sum 50 continue do 70 j=1,r delta(iposn(i),j)=wkr(j) 70 continue 100 continue endif C now take care of partially observed cases do 110 i=1,r do 105 j=i,r wkrr1(i,j)=sigma(i,j) 105 continue 110 continue do 500 pt=1,npatt call swpobs(r,wkrr1,npatt,rmat,pt) do 120 i=1,r do 119 j=i+1,r wkrr1(j,i)=wkrr1(i,j) 119 continue 120 continue call getmc(r,npatt,rmat,pt,lmc,mc,nmc) call getoc(r,npatt,rmat,pt,loc,oc,noc) call chsub(r,wkrr1,lmc,mc,nmc,wkrr2) do 450 i=pstfin(pt,1),pstfin(pt,2) do 150 j=1,nmc wkr(j)=dble(gauss()) 150 continue do 200 j=nmc,1,-1 sum=dble(0.) do 190 k=1,j sum=sum+wkrr2(k,j)*wkr(k) 190 continue wkr(j)=sum 200 continue do 300 j=1,nmc sum=dble(0.) do 210 k=1,noc sum=sum+wkrr1(oc(k),mc(j))*delta(iposn(i),oc(k)) 210 continue delta(iposn(i),mc(j))=sum+wkr(j) 300 continue 450 continue 500 continue return end C*********************************************************************** subroutine chsub(r,sigma,lmc,mc,nmc,wkrr) C takes cholesky factor of submatrix of sigma corresponding to mc, C storing it in the upper-left (nmc x nmc) corner of wkrr integer r,lmc,mc(lmc),nmc double precision sigma(r,r),wkrr(r,r) do 10 i=1,nmc do 5 j=i,nmc wkrr(i,j)=sigma(mc(i),mc(j)) 5 continue 10 continue call chfc(r,nmc,wkrr) return end C*********************************************************************** subroutine getmc(r,npatt,rmat,pt,lmc,mc,nmc) integer r,npatt,rmat(npatt,r),pt,lmc,mc(lmc),nmc,posn nmc=0 posn=0 do 10 j=1,r if(rmat(pt,j).eq.0) then nmc=nmc+1 posn=posn+1 mc(posn)=j endif 10 continue return end C*********************************************************************** subroutine getoc(r,npatt,rmat,pt,loc,oc,noc) integer r,npatt,rmat(npatt,r),pt,loc,oc(loc),noc,posn noc=0 posn=0 do 10 j=1,r if(rmat(pt,j).eq.1) then noc=noc+1 posn=posn+1 oc(posn)=j endif 10 continue return end C*********************************************************************** subroutine swpobs(r,sigma,npatt,rmat,pt) C sweeps sigma on positions of rmat(pt,*) containing 1's integer r,npatt,rmat(npatt,r),pt,col double precision sigma(r,r) do 100 col=1,r if((rmat(pt,col).eq.1).and.(sigma(col,col).gt.dble(0.))) / call swp(r,sigma,col) if((rmat(pt,col).eq.0).and.(sigma(col,col).le.dble(0.))) / call rsw(r,sigma,col) 100 continue return end C*********************************************************************** subroutine swp(n,mat,p) C sweep upper-triangular portion of mat on position p integer n,p double precision mat(n,n) mat(p,p)=-dble(1.)/mat(p,p) do 1 i=1,p-1 mat(i,p)=-mat(i,p)*mat(p,p) 1 continue do 2 j=p+1,n mat(p,j)=-mat(p,j)*mat(p,p) 2 continue do 6 i=1,p-1 do 4 j=i,p-1 mat(i,j)=mat(i,j)+mat(i,p)*mat(j,p)/mat(p,p) 4 continue do 5 j=p+1,n mat(i,j)=mat(i,j)+mat(i,p)*mat(p,j)/mat(p,p) 5 continue 6 continue do 8 i=p+1,n do 7 j=i,n mat(i,j)=mat(i,j)+mat(p,i)*mat(p,j)/mat(p,p) 7 continue 8 continue return end C*********************************************************************** subroutine rsw(n,mat,p) C reverse-sweep upper-triangular portion of mat on position p integer n,p double precision mat(n,n) mat(p,p)=-dble(1.)/mat(p,p) do 1 i=1,p-1 mat(i,p)=mat(i,p)*mat(p,p) 1 continue do 2 j=p+1,n mat(p,j)=mat(p,j)*mat(p,p) 2 continue do 6 i=1,p-1 do 4 j=i,p-1 mat(i,j)=mat(i,j)+mat(i,p)*mat(j,p)/mat(p,p) 4 continue do 5 j=p+1,n mat(i,j)=mat(i,j)+mat(i,p)*mat(p,j)/mat(p,p) 5 continue 6 continue do 8 i=p+1,n do 7 j=i,n mat(i,j)=mat(i,j)+mat(p,i)*mat(p,j)/mat(p,p) 7 continue 8 continue return end C*********************************************************************** subroutine drbeta(r,sigma,xtxinv,p,beta,wkrr1,wkpp,wkpr) C draws adds normal(0, sigma Otimes xtxinv) noise to beta integer r,p double precision sigma(r,r),xtxinv(p,p),beta(p,r),wkrr1(r,r), / wkpp(p,p),wkpr(p,r),sum do 2 i=1,r do 1 j=i,r wkrr1(i,j)=sigma(i,j) 1 continue 2 continue do 5 i=1,p do 4 j=i,p wkpp(i,j)=xtxinv(i,j) 4 continue 5 continue call chfc(r,r,wkrr1) call chfc(p,p,wkpp) do 10 i=1,p do 9 j=1,r wkpr(i,j)=dble(gauss()) 9 continue 10 continue C multiply each column of wkpr by t(wkpp) do 100 j=1,r do 80 i=p,1,-1 sum=dble(0.) do 70 k=1,i sum=sum+wkpp(k,i)*wkpr(k,j) 70 continue wkpr(i,j)=sum 80 continue 100 continue C add noise do 200 j=1,r do 180 i=1,j do 170 k=1,p beta(k,j)=beta(k,j)+wkrr1(i,j)*wkpr(k,i) 170 continue 180 continue 200 continue return end C*********************************************************************** subroutine drsigma(ntot,r,delta,nstar,sigma,patt,nhyp,hyp,wkrr1, / wkrr2,p) C draws sigma integer ntot,r,patt(ntot),nstar,nhyp,p double precision delta(ntot,r),sigma(r,r),hyp(nhyp),a, / wkrr1(r,r),wkrr2(r,r) C initialize wkrr1 to Binv k=1 a=hyp(k) do 5 i=1,r do 4 j=i,r k=k+1 wkrr1(i,j)=hyp(k) 4 continue 5 continue C accumulate sum_i t(delta_i)%*%delta_i do 100 i=1,ntot if(patt(i).ne.0) then do 80 j=1,r do 70 k=j,r wkrr1(j,k)=wkrr1(j,k)+delta(i,j)*delta(i,k) 70 continue 80 continue endif 100 continue C draw inverse Wishart call chfc(r,r,wkrr1) call bfac(r,sngl(a)+float(nstar-p),sigma) call bkslv(r,r,sigma) do 200 i=1,r do 190 j=1,r sum=dble(0.) do 180 k=1,min(i,j) sum=sum+wkrr1(k,i)*sigma(k,j) 180 continue wkrr2(i,j)=sum 190 continue 200 continue do 300 i=1,r do 280 j=i,r sum=dble(0.) do 250 k=1,r sum=sum+wkrr2(i,k)*wkrr2(j,k) 250 continue sigma(i,j)=sum if(i.ne.j) sigma(j,i)=sigma(i,j) 280 continue 300 continue return end C*********************************************************************** subroutine mkeps2(ntot,m,r,y,pcol,pred,q,zcol,b,eps,patt,ist,ifin) C calculates eps = y - Z%*% b integer ntot,r,patt(ntot),pcol,q,zcol(q),ist(m),ifin(m),s, / st,fin,m double precision y(ntot,r),pred(ntot,pcol),b(q,r,m),eps(ntot,r), / sum do 100 s=1,m st=ist(s) fin=ifin(s) do 90 i=st,fin if(patt(i).ne.0) then do 80 j=1,r sum=dble(0.) do 70 k=1,q sum=sum+pred(i,zcol(k))*b(k,j,s) 70 continue eps(i,j)=y(i,j)-sum 80 continue endif 90 continue 100 continue return end C*********************************************************************** subroutine drpsibd(r,q,m,psi,wkqq1,wkqq2,wkqrv,nhyp,hyp,b) C Version of drpsi for block-diagonal pan integer r,q,m,s,nhyp double precision psi(q,q,r),sig(r*q,r*q,m),wkqq1(q,q), / hyp(nhyp),wkqrv(q*r),c,b(q,r,m),sum,wkqq2(q,q) do 400 l=1,r C initialize wkqq1 to appropriate portion of Dinv k=1 + r*(r+1)/2 + l c=hyp(k) k=1 + r*(r+1)/2 + r + (l-1)*q*(q+1)/2 do 5 i=1,q do 4 j=i,q k=k+1 wkqq1(i,j)=hyp(k) 4 continue 5 continue C accumulate BtB into wkrr1 do 100 s=1,m do 15 i=1,q wkqrv(i)=b(i,l,s) 15 continue do 40 i=1,q do 30 j=i,q wkqq1(i,j)=wkqq1(i,j)+wkqrv(i)*wkqrv(j) 30 continue 40 continue 100 continue C draw inverse Wishart call chfc(q,q,wkqq1) call bfac(q,sngl(c)+float(m),wkqq2) call bkslv(q,q,wkqq2) do 200 i=1,q do 190 j=1,q sum=dble(0.) do 180 k=1,min(i,j) sum=sum+wkqq1(k,i)*wkqq2(k,j) 180 continue psi(i,j,l)=sum 190 continue 200 continue do 205 i=1,q do 204 j=1,q wkqq2(i,j)=psi(i,j,l) 204 continue 205 continue do 300 i=1,q do 280 j=i,q sum=dble(0.) do 250 k=1,q sum=sum+wkqq2(i,k)*wkqq2(j,k) 250 continue psi(i,j,l)=sum if(i.ne.j) psi(j,i,l)=psi(i,j,l) 280 continue 300 continue 400 continue return end C*********************************************************************** subroutine drpsi(r,q,m,psi,wkrqrq1,wkrqrq2,wkqrv,nhyp,hyp,b) C draws new value of psi integer r,q,m,s,nhyp double precision psi(r*q,r*q),sig(r*q,r*q,m),wkrqrq1(r*q,r*q), / hyp(nhyp),wkqrv(q*r),c,b(q,r,m),sum,wkrqrq2(r*q,r*q) C initialize wkrqrq1 to Dinv k=r*(r+1)/2 + 2 c=hyp(k) do 5 i=1,r*q do 4 j=i,r*q k=k+1 wkrqrq1(i,j)=hyp(k) 4 continue 5 continue C put vec(b_i) into wkqrv do 100 s=1,m ia=0 do 20 j=1,r do 15 i=1,q ia=ia+1 wkqrv(ia)=b(i,j,s) 15 continue 20 continue do 40 i=1,r*q do 30 j=i,r*q wkrqrq1(i,j)=wkrqrq1(i,j)+wkqrv(i)*wkqrv(j) 30 continue 40 continue 100 continue C draw inverse Wishart call chfc(r*q,r*q,wkrqrq1) call bfac(r*q,sngl(c)+float(m),psi) call bkslv(r*q,r*q,psi) do 200 i=1,r*q do 190 j=1,r*q sum=dble(0.) do 180 k=1,min(i,j) sum=sum+wkrqrq1(k,i)*psi(k,j) 180 continue wkrqrq2(i,j)=sum 190 continue 200 continue do 300 i=1,r*q do 280 j=i,r*q sum=dble(0.) do 250 k=1,r*q sum=sum+wkrqrq2(i,k)*wkrqrq2(j,k) 250 continue psi(i,j)=sum if(i.ne.j) psi(j,i)=psi(i,j) 280 continue 300 continue return end C*********************************************************************** subroutine drb(r,q,m,b,sigmainv,eps,pcol,pred,zcol,wkqr1, / wkqr2,ist,ifin,patt,ntot,sig,wkrqrq2,wkqrv) C draws b_i, i=1,...,m. integer r,q,m,s,pcol,zcol(q),ist(m),ifin(m),st,fin,patt(ntot), / ntot double precision sig(r*q,r*q,m),eps(ntot,r),pred(ntot,pcol), / sigmainv(r,r),b(q,r,m),wkqr1(q,r),wkqr2(q,r), / wkrqrq1(r*q,r*q),wkrqrq2(r*q,r*q),sum,wkqrv(q*r) do 500 s=1,m st=ist(s) fin=ifin(s) C put t(Z_i)%*%(jth column of eps_i) into jth column of wkqr1, j=1..r do 100 j=1,r do 90 i=1,q sum=dble(0.) do 80 k=st,fin if(patt(k).ne.0) sum=sum+pred(k,zcol(i))*eps(k,j) 80 continue wkqr1(i,j)=sum 90 continue 100 continue C put sum_k=1^r sigmainv(j,k) *t(Z_i)%*% (col k of eps_i) into wkqr2 do 200 j=1,r do 190 i=1,q sum=dble(0.) do 180 k=1,r sum=sum+sigmainv(j,k)*wkqr1(i,k) 180 continue wkqr2(i,j)=sum 190 continue 200 continue C put sig_i into wkrqrq2 and calculate b_i call mmul(r*q,r*q,m,sig,s,wkrqrq2) do 220 i=1,r*q do 210 j=i+1,r*q wkrqrq2(j,i)=wkrqrq2(i,j) 210 continue 220 continue ia=0 do 400 i=1,r do 380 ii=1,q ia=ia+1 ja=0 sum=dble(0.) do 370 j=1,r do 360 jj=1,q ja=ja+1 sum=sum+wkrqrq2(ia,ja)*wkqr2(jj,j) 360 continue 370 continue b(ii,i,s)=sum 380 continue 400 continue do 410 i=1,r*q wkqrv(i)=dble(gauss()) 410 continue ia=0 do 450 i=1,r do 440 ii=1,q ia=ia+1 sum=dble(0.) do 430 ja=ia,r*q sum=sum+sig(ia,ja,s)*wkqrv(ja) 430 continue b(ii,i,s)=b(ii,i,s)+sum 440 continue 450 continue 500 continue return end C*********************************************************************** subroutine mkpsi0bd(r,q,m,psi,sig,wkrqrq1) C calculates initial estimate of psi given sig. integer r,q,m,s double precision psi(q,q,r),sig(r*q,r*q,m),wkrqrq1(r*q,r*q) do 6 l=1,r do 5 i=1,q do 4 j=i,q psi(i,j,l)=dble(0.) 4 continue 5 continue 6 continue do 100 s=1,m call mmul(r*q,r*q,m,sig,s,wkrqrq1) do 30 l=1,r do 20 i=1,q do 10 j=i,q ii=(l-1)*q + i jj=(l-1)*q + j psi(i,j,l)=psi(i,j,l)+wkrqrq1(ii,jj) 10 continue 20 continue 30 continue 100 continue do 120 l=1,r do 110 i=1,q do 105 j=i,q psi(i,j,l)=psi(i,j,l)/dble(m) if(i.ne.j) psi(j,i,l)=psi(i,j,l) 105 continue 110 continue 120 continue return end C*********************************************************************** subroutine mkpsi0(r,q,m,psi,sig,wkrqrq1) C calculates initial estimate of psi given sig. integer r,q,m,s double precision psi(r*q,r*q),sig(r*q,r*q,m),wkrqrq1(r*q,r*q) do 5 i=1,r*q do 4 j=i,r*q psi(i,j)=dble(0.) 4 continue 5 continue do 100 s=1,m call mmul(r*q,r*q,m,sig,s,wkrqrq1) do 20 i=1,r*q do 10 j=i,r*q psi(i,j)=psi(i,j)+wkrqrq1(i,j) 10 continue 20 continue 100 continue do 110 i=1,r*q do 105 j=i,r*q psi(i,j)=psi(i,j)/dble(m) if(i.ne.j) psi(j,i)=psi(i,j) 105 continue 110 continue return end C*********************************************************************** subroutine mksigbd(r,q,m,psi,sigma,ztz,sig,wkrr1,wkrr2, / wkrqrq2,zflag,nhyp,hyp,wkqq1,wkqq2) C Version of mksig for the block-diagonal pan. C calculates the square root of C sig = inv( inv(psi) + ( inv(sigma) Otimes ztz ) ), i=1,..m C Use zflag = 1 to set psi equal to its prior guess. C When finished, wkrr2 contains inv(sigma). integer r,q,m,s,zflag,nhyp double precision psi(q,q,r),sigma(r,r),ztz(q,q,m), / sig(r*q,r*q,m),wkrr1(r,r),wkrr2(r,r),wkrqrq1(r*q,r*q), / wkrqrq2(r*q,r*q),hyp(nhyp),c,wkqq1(q,q),wkqq2(q,q) if(zflag.eq.1) then k1= 1 + r*(r+1)/2 k2= 1 + r*(r+1)/2 + r do 5 l=1,r k1=k1+1 c=hyp(k1) do 4 i=1,q do 3 j=i,q k2=k2+1 psi(i,j,l)=hyp(k2)/c 3 continue 4 continue 5 continue endif C invert psi and put into wkrqrq2, using the fact that it's block-diagonal do 102 i=1,r*q do 101 j=i,r*q wkrqrq2(i,j)=dble(0.) 101 continue 102 continue do 110 l=1,r do 105 i=1,q do 104 j=i,q wkqq1(i,j)=psi(i,j,l) 104 continue 105 continue call chfc(q,q,wkqq1) call bkslv(q,q,wkqq1) call mm(q,q,wkqq1,wkqq2) do 108 i=1,q do 107 j=i,q ii=(l-1)*q + i jj=(l-1)*q + j wkrqrq2(ii,jj)=wkqq2(i,j) 107 continue 108 continue 110 continue C invert sigma and put into wkrr2 do 7 i=1,r do 6 j=i,r wkrr1(i,j)=sigma(i,j) 6 continue 7 continue call chfc(r,r,wkrr1) call bkslv(r,r,wkrr1) call mm(r,r,wkrr1,wkrr2) do 9 i=1,r do 8 j=i+1,r wkrr2(j,i)=wkrr2(i,j) 8 continue 9 continue do 100 s=1,m C initialize sig(,,s) to inv(sigma) Otimes t(Z_i)%*%Z_i do 30 i=1,r do 20 j=i,r do 15 ii=1,q do 10 jj=1,q ia=(i-1)*q+ii ja=(j-1)*q+jj sig(ia,ja,s)=wkrr2(i,j)*ztz(ii,jj,s) 10 continue 15 continue 20 continue 30 continue C add in inv(psi) and take inverse square-root do 40 i=1,r*q do 35 j=i,r*q sig(i,j,s)=sig(i,j,s)+wkrqrq2(i,j) 35 continue 40 continue call chl(r*q,r*q,m,sig,s) call bkslvl(r*q,r*q,m,sig,s) 100 continue return end C*********************************************************************** subroutine mksig(r,q,m,psi,sigma,ztz,sig,wkrr1,wkrr2,wkrqrq1, / wkrqrq2,zflag,nhyp,hyp) C calculates the square root of C sig = inv( inv(psi) + ( inv(sigma) Otimes ztz ) ), i=1,..m C Use zflag = 1 to set psi equal to its prior guess. C When finished, wkrr2 contains inv(sigma). integer r,q,m,s,zflag,nhyp double precision psi(r*q,r*q),sigma(r,r),ztz(q,q,m), / sig(r*q,r*q,m),wkrr1(r,r),wkrr2(r,r),wkrqrq1(r*q,r*q), / wkrqrq2(r*q,r*q),hyp(nhyp),c if(zflag.eq.1) then k=r*(r+1)/2 + 2 c=hyp(k) do 4 i=1,r*q do 3 j=i,r*q k=k+1 psi(i,j)=hyp(k)/c 3 continue 4 continue endif C invert psi and put into wkrqrq2 do 2 i=1,r*q do 1 j=i,r*q wkrqrq1(i,j)=psi(i,j) 1 continue 2 continue call chfc(r*q,r*q,wkrqrq1) call bkslv(r*q,r*q,wkrqrq1) call mm(r*q,r*q,wkrqrq1,wkrqrq2) C invert sigma and put into wkrr2 do 6 i=1,r do 5 j=i,r wkrr1(i,j)=sigma(i,j) 5 continue 6 continue call chfc(r,r,wkrr1) call bkslv(r,r,wkrr1) call mm(r,r,wkrr1,wkrr2) do 9 i=1,r do 8 j=i+1,r wkrr2(j,i)=wkrr2(i,j) 8 continue 9 continue do 100 s=1,m C initialize sig(,,s) to inv(sigma) Otimes t(Z_i)%*%Z_i do 30 i=1,r do 20 j=i,r do 15 ii=1,q do 10 jj=1,q ia=(i-1)*q+ii ja=(j-1)*q+jj sig(ia,ja,s)=wkrr2(i,j)*ztz(ii,jj,s) 10 continue 15 continue 20 continue 30 continue C add in inv(psi) and take inverse square-root do 40 i=1,r*q do 35 j=i,r*q sig(i,j,s)=sig(i,j,s)+wkrqrq2(i,j) 35 continue 40 continue call chl(r*q,r*q,m,sig,s) call bkslvl(r*q,r*q,m,sig,s) 100 continue return end C*********************************************************************** subroutine mksigma(ntot,r,eps,nstar,sigma,patt) C calculates (1/nstar)*sum of t(eps_i)%*%eps_i integer ntot,r,patt(ntot),nstar double precision eps(ntot,r),sigma(r,r) do 2 i=1,r do 1 j=i,r sigma(i,j)=dble(0.) 1 continue 2 continue do 100 i=1,ntot if(patt(i).ne.0) then do 90 j=1,r do 80 k=j,r sigma(j,k)=sigma(j,k)+eps(i,j)*eps(i,k) 80 continue 90 continue endif 100 continue do 110 i=1,r do 108 j=i,r sigma(i,j)=sigma(i,j)/dble(nstar) if(i.ne.j) sigma(j,i)=sigma(i,j) 108 continue 110 continue return end C*********************************************************************** subroutine mkeps1(ntot,r,y,pcol,pred,p,xcol,beta,eps,patt) C calculates eps = y - X%*% beta integer ntot,r,patt(ntot),pcol,p,xcol(p) double precision y(ntot,r),pred(ntot,pcol),beta(p,r),eps(ntot,r), / sum do 100 i=1,ntot if(patt(i).ne.0) then do 90 j=1,r sum=dble(0.) do 80 k=1,p sum=sum+pred(i,xcol(k))*beta(k,j) 80 continue eps(i,j)=y(i,j)-sum 90 continue endif 100 continue continue return end C*********************************************************************** subroutine mkbeta(p,r,xtxinv,xty,beta) C calculates betahat integer p,r double precision xtxinv(p,p),xty(p,r),beta(p,r),sum do 100 i=1,p do 90 j=1,r sum=dble(0.) do 50 k=1,p sum=sum+xtxinv(i,k)*xty(k,j) 50 continue beta(i,j)=sum 90 continue 100 continue return end C*********************************************************************** subroutine mkxty(ntot,r,y,pcol,pred,p,xcol,patt,xty) integer ntot,r,pcol,p,xcol(p),patt(ntot) double precision y(ntot,r),pred(ntot,pcol),xty(p,r),sum do 100 i=1,p do 80 j=1,r sum=dble(0.) do 50 k=1,ntot if(patt(k).ne.0) then sum=sum+pred(k,xcol(i))*y(k,j) endif 50 continue xty(i,j)=sum 80 continue 100 continue return end C*********************************************************************** subroutine mimpy(ntot,r,y,patt,npatt,rmat) C unconditional mean imputation for variables in y integer ntot,r,patt(ntot),npatt,rmat(npatt,r),denom,rij double precision y(ntot,r),sum,mean do 200 j=1,r sum=dble(0.) denom=0 do 100 i=1,ntot if(patt(i).ne.0) then rij=rmat(patt(i),j) sum=sum+dble(rij)*y(i,j) denom=denom+rij endif 100 continue mean=sum/dble(denom) do 150 i=1,ntot if(patt(i).ne.0) then if(rmat(patt(i),j).eq.0) y(i,j)=mean endif 150 continue 200 continue return end C*********************************************************************** real function rangen(init) integer a,p,ix,b15,b16,xhi,xalo,leftflo,fhi,k,init save ix data a/16807/,b15/32768/,b16/65536/,p/2147483647/ if(init.ne.0) ix=init xhi=ix/b16 xalo=(ix-xhi*b16)*a leftflo=xalo/b16 fhi=xhi*a+leftflo k=fhi/b15 ix=(((xalo-leftflo*b16)-p)+(fhi-k*b15)*b16)+k if (ix.lt.0)ix=ix+p rangen=float(ix)*4.656612875E-10 return end C*********************************************************************** real function rngs(seed) C initializes rangen with seed integer seed rngs=rangen(seed) return end C*********************************************************************** function gamm(a) C Generates a random gamma(a) deviate. If a>=1, uses the method of C Fishman (1976); if 0 #include // for NULL #include /* FIXME: Check these declarations against the C/Fortran source code. */ /* .Fortran calls */ extern void F77_NAME(bdiag)(int *q, int *m, double *sig); extern void F77_NAME(ecme3)(int *ntot, int *subj, int *m, int *ist, int *ifin, int *occ, int *nmax, double *vi, double *vh, int *pcol, double *pred, int *q, int *zcol, double *ztv, double *sig0, int *iflag, double *sig, double *psi, double *sigma2, int *p, int *xcol, double *beta, double *wkq1, double *wkq2, double *wkq3, double *y, double *delta, double *b, double *wk, double *w, double *xtw, double *xtwx, double *xtwy, double *xtwxinv, double *llk, double *vmax, int *sflag, double *eps, double *obeta, double *opsi, int *maxits, int *iter, int *cvgd); extern void F77_NAME(mgibbs)(int *ntot, int *subj, int *m, int *ist, int *ifin, int *pcol, double *pred, int *q, int *zcol, double *ztz, int *patt, int *nstar, int *r, double *y, int *p, int *xcol, int *npatt, int *rmat, int *sflag, double *beta, double *sigma, double *psi, double *b, double *xtxinv, double *wkpp, double *wkpr, double *eps, double *wkrqrq1, double *wkrqrq2, double *sig, double *wkrr1, double *wkrr2, int *iter, double *wkqr1, double *wkqr2, double *wkqrv, int *nhyp, double *hyp, double *delta, int *iposn, int *pstfin, double *betas, double *sigmas, double *psis); extern void F77_NAME(mgibbsbd)(int *ntot, int *subj, int *m, int *ist, int *ifin, int *pcol, double *pred, int *q, int *zcol, double *ztz, int *patt, int *nstar, int *r, double *y, int *p, int *xcol, int *npatt, int *rmat, int *sflag, double *beta, double *sigma, double *psi, double *b, double *xtxinv, double *wkpp, double *wkpr, double *eps, double *wkrqrq1, double *wkrqrq2, double *sig, double *wkrr1, double *wkrr2, int *iter, double *wkqr1, double *wkqr2, double *wkqrv, int *nhyp, double *hyp, double *delta, int *iposn, int *pstfin, double *betas, double *sigmas, double *psis, double *wkqq1, double *wkqq2); extern void F77_NAME(nopsi)(int *ntot, int *m, int *ist, int *ifin, int *occ, int *nmax, double *vi, double *vh, int *pcol, double *pred, int *q, int *zcol, double *ztv, double *sig0, int *iflag, double *sig, double *psi, double *sigma2, int *p, int *xcol, double *beta, double *wkq1, double *wkq2, double *wkq3, double *y, double *delta, double *b, double *wk, double *w, double *xtw, double *xtwx, double *xtwy, double *xtwxinv, double *ll); extern void F77_NAME(prelim)(int *ntot, int *subj, int *m, int *ist, int *ifin, int *occ, int *nmax, double *vmax, double *vh, double *vi, int *pcol, double *pred, int *q, int *zcol, double *ztv, double *sig0, int *iflag); extern float F77_NAME(rngs)(int *seed); static const R_FortranMethodDef FortranEntries[] = { {"bdiag", (DL_FUNC) &F77_NAME(bdiag), 3}, {"ecme3", (DL_FUNC) &F77_NAME(ecme3), 43}, {"mgibbs", (DL_FUNC) &F77_NAME(mgibbs), 44}, {"mgibbsbd", (DL_FUNC) &F77_NAME(mgibbsbd), 46}, {"nopsi", (DL_FUNC) &F77_NAME(nopsi), 34}, {"prelim", (DL_FUNC) &F77_NAME(prelim), 17}, {"rngs", (DL_FUNC) &F77_NAME(rngs), 1}, {NULL, NULL, 0} }; void R_init_pan(DllInfo *dll) { R_registerRoutines(dll, NULL, NULL, FortranEntries, NULL); R_useDynamicSymbols(dll, FALSE); } pan/vignettes/0000755000176200001440000000000014470211210013030 5ustar liggesuserspan/vignettes/schafer01.pdf0000755000176200001440000131433213752014434015324 0ustar liggesusers%PDF-1.5 % 30 0 obj << /BitsPerComponent 1 /ColorSpace /DeviceGray /Height 1350 /Intent /RelativeColormetric /Subtype /Image /Type /XObject /Width 863 /Filter /JBIG2Decode /Length 8902 >> stream 0_F  Qk'㎔Q) +'߇I$CSף䚊O /zn?Ʒ-=XS\p"as!zl);RquF GӜuaf0"1@%3c+~oRȞf8P}8/H'|(fA "'OPAdK5ҁpYq /-UoTVMg9)XC|# _YR { Zv1Ҁ([$ 90N۔Hm{Xlvgxڙ@P&e]I[z(d} yK?1FZ +@|~5_A!R T "͕>ws\;-g8x+#8`ۈE ::s+zmKg2ӭEAy5I&:CB,H˛Z5<Ҙz3Gvp/wrn!r~FibDzÑh-ǻHj̎{δs&f"N`wD-$OeNq7>^v3S\MEΙ8:@Q`ӡ0S3gUJ#'*MՇE@ũ`d5K*x֫=$HG6Qف*/0mU16mc4ӽ -?Hx;FGTRY؜< Bä SڅvflaknjY//*tw=gtC_j3_V͗;w3c̵-.x~k`HЊ 3*UȦ 9l#Vm5rs\iFـI"HR/  T:e DljHy1(U;#)@5'}YqF7X cg@O8; 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 /TGIG;oVZB} |i37?) endstream endobj startxref 365855 %%EOF pan/vignettes/pan-tr.Rnw0000644000176200001440000000530113752203466014740 0ustar liggesusers%% \VignetteIndexEntry{Imputation of missing covariates under a multivariate linear mixed model} %% \Vignettekeywords{multiple imputation, multivariate linear mixed model} %% \VignettePackage{pan} \documentclass[12pt]{article} \usepackage{Sweave,hyperref,pdfpages} \usepackage{amsmath,epsfig} \usepackage{graphicx} \usepackage[authoryear,round]{natbib} \bibliographystyle{plainnat} \parindent 0in \begin{document} \title{Imputation of missing covariates under a multivariate linear mixed model} \author{Joseph L. Schafer} \date{} \maketitle \tableofcontents \thispagestyle{empty} \newpage \thispagestyle{empty} \section{Maintainer's note} The reference should read\footnote{Note that the technical report is now available from \textit{http://sites.stat.psu.edu/reports/1997/tr9704.pdf} and briefly described at \textit{http://stat.psu.edu/research-old/technical-reports/archived-technical-reports}}.. \vskip 0.25cm Schafer, J.L. (1997) Imputation of missing covariates under a multivariate linear mixed model. Technical report 97-04, Dept. of Statistics, The Pennsylvania State University. \vskip 0.25cm You can also refer to the following paper. \vskip 0.25cm Schafer J L, Yucel RM (2002). Computational strategies for multivariate linear mixed-effects models with missing values. Journal of Computational and Graphical Statistics. 11:437-457 \vskip 0.25cm The marijuana data in the package is reproduced here, \begin{table}[h] \centering \caption{Change in heart rate recorded 15 and 90 minutes after marijuana use, measured in beats per minute above baseline} \begin{tabular}{crrr|rrr} \\ \hline & \multicolumn{3}{c}{15 minutes} & \multicolumn{3}{c}{90 minutes} \\ \cline{2-4} \cline{5-7} Subject & Placebo & Low & High & Placebo & Low & High\\ \hline 1 & 16 & 20 & 16 & 20 & -6 & -4\\ 2 & 12 & 24 & 12 & -6 & 4 & -8\\ 3 & 8 & 8 & 26 & -4 & 4 & 8\\ 4 & 20 & 8 & - & - & 20 & -4\\ 5 & 8 & 4 & -8 & - & 22 & -8\\ 6 & 10 & 20 & 28 & -20 & -4 & -4\\ 7 & 4 & 28 & 24 & 12 & 8 & 18\\ 8 & -8 & 20 & 24 & -3 & 8 & -24\\ 9 & - & 20 & 24 & 8 & 12 & -\\ \hline \\ \end{tabular} \end{table} \vskip 0.15cm One can use {\bf help(ecme,package=``pan'')} to see the example code. \section{Technical Report} The technical report starts from next page. \includepdf[pages=-,width=8.5in]{tr9704.pdf}\label{tr9704} \includepdf[pages=-,width=8.5in]{schafer01.pdf}\label{schafer01} \includepdf[pages=-,width=8.5in]{schafer02.pdf}\label{schafer02} \end{document} pan/vignettes/tr9704.pdf0000644000176200001440000066416013374503040014520 0ustar liggesusers%PDF-1.2 %쏢 5 0 obj <> stream xZYsFNվq#(U\MGNe;ʃCr$b4;ʯߞ3 v"eCDwO_KQ$~A"e/_-ߥ5WtISlYwdW?-ޞ$Pߏ$?Oj"ΰ}ѶEphQ22xS I4,`Sޠ<(Oڪf˕Z&T#n uO3@'!Q:l%<eQ)i9|#cQb__!b޽YxsESl)P[naw)l2 zb!FhNh,\YDɸtBr"Apc&YLL/.KMrYuqppJx$bgb$;z,8դ!rSDSJ52x +OڹdT+gp TCmCblNS3#ϩӒhHS2+sxjJ>֢LkctҦu쬾=W sC #Y- רOnLl$e]=]-@sH3z'ޔ}V:kgHenZ%$ڥv`l6RsEP}wg$`Z$<ϞNT${}ca}mMCx:͸ZӜusXN3;6hH ]pRV_w5FY`<PWj mLv﬙ˡZ)>џܻvifԖǽ#uDD8κT{`D>dGIxT8n2 gf]1G0v#wʦ+6DV(dv+SMU;PVΣxH{6 H4Jy/܀7A~`,L7.뢺{\d&uYr— {&BfT]9<~S|z5BBmI=IPD h<Or0υipG:oo ۞;tSZ a D)d ,F /CrRgL (.F+9+<VTթu1LO_P׋C_dx"|vRefE]i{նus6` ^v$uUʉ ~m Cx;!91}Ȅ7}2TUXz9 l@Y}>Y:uAMA&-:%B4N9DzΕ͟D1Ss 0׼EM-M2V #Qb:Y(aO6Pd~z$e%걇RyC1bpWp3nӢ#,1#$x)xh*s5属=&/?$1+f!L_ ` s$ !p}!ea{7ig%n7 5k7y4Іq< 7ϻtE=՛^iw> stream x[KoFrYc EIB<\v= qóCILr"'ȏO?dH1LwWWU_U>gc~ȫ*|Ϋo/WzGD}zE59C/:#e:'V]~8ua3/bxbkS*Ekm,/>4Eo^,aJ/>BA=mOT.$ S/Al x{!2cL1Z(>Ñ{MŶ_MR2v)զbTYߖnMZ%JcL x⾻hXbd7bS|;S(jPCzƩ\$BGz0$=?Ez\ =վP3rȜK`K#7ϒ{PUmy+mf.,I1^(۴yYT YWiQxQC^ =^y#K׏FOz跱cYߌҰ$fN"Y,)W:\[T[]Wy\^EL5D:@F1sp GYɺse$3K:ʆuףBfbT&; s- ۲V8vP4uM—? +n{ \H^}@",{J22+-59@ֳ=ߗβߋ?&D_ bA3gN"Jٗ7ßʥ6Jt[U>~`<ϹyI]Fx}8Bƾ?* ~H_r6;4mJiTՔ Zv\P5muf2dtpXaP-bX }掣t6zDCTuN7B+m6,CC&g&+[u*@Ӊ'c\*؜4.: fJ$T+s)v?3J} ʸW]ZH"-ߩM oHcȼ̛)S^2۬F48vMYoU+sʩ.Z5S1IbLa}>|3K#l6w]_Vw6:&{~oϋ-?*dXEpGڶVz, ~,Ѣ3SY> stream xZYoO44&9F/U';omq~ʆ˪O"7Nv:Ue e],0wmz Ωmy/in o{'?IFòɫq_aJ;۬ɪ+=okghc5_Գ t[Wf8-8Uaܗܱf zar%GJ؁]{:9L xvxn%qbd[}<}ߙ̿)Z@6rmӅ7v+~슽q=Msz)W*AzR)NXiBɆm ׾nǷ2&g)mc!Yp^l$J8h/e>]/8j+"_E!C" `5G޵}^ Dw Vݽsl2,d4ܯmyh څvڿ4܃Sn (mx|.ID+6 "!T==o7?tj,"t:}! 0N]eā641MрOeUIW:ҐO2yU,9 q@ܩv'xv(4ky<mw$ceBå! *#t|bf XNqr Yr;8/ {f<{ci-$zcΆ̗k&!ZiZc6Fy&؂D&FpfqS e>VYg@!iBDٳ}&,7mUs=dBQ4~A6۹$Y M>z!u~rW`%ΰO ΦHXOM';?ج9r' ()u= ` xE{册nwБ)ϻ4Xaz[j;  Q -8 LA1{!,^ `*1)$ZÜ^ReMюƮ>2#SYbK) N[m7v= sPܛis̺,"sz&8B{P+m[!?)ABS(̊ײekpDI6d0YC=Vl}D5P\W:cJ}/Mk9Aw F' kj2!YBHLdAAU@f͝ aR@ș4t$uAE{)li#y̳RoO%G4)9^ Be>3DW 2@D NL>eVsIe2oBx!e@2a}'UmHb/t$&w,KΥB%vh)>fPXPaP8YRmV-3V>GL>Yb,"XMՅ^@ZHG3v⣳|SL!(lZh$L5\8oS,Jxo^]NhFWh%@_6{LT%ڗha4t}g*a /LR0UjL&U@,-z!6LIE>ţ>ӎ"և,~jeT@x<Ԭ&[Qo F",'GW[ƆihHէ a"<'B̠tzԨuUpq' +4Ww>Zf?Mvptc 6@1dF`0%g/>Nt1ڙX*qVreP5W G p8dLL䜒Y}eNJT!gAhS< ]kǙ _%.5fˡ>`  ph^TeZ~Hߎ{oVHtpc,(G\xr9p g dzʊ`<4C66@+jfe|*kѶ2MiĎݠʜ+"5ǭa:R֪%Y ^}LbڤXUQ.C;]7 l\~}kGuq;a'KS 0;ƱJ\ ЅJy(q8R}S8,0s#yG/&cȫA?Mk8/!O1o/R)4aW!*ӟ7(8}endstream endobj 230 0 obj 2824 endobj 238 0 obj <> stream xZKs 0v%AQ{f8r^+iʹ-t7y].f7 BCY«Շ.H?sY_x}>"{c]m چ ʯnպ_W6SxZߘ ߋ\ 񊮟ơtkaޭza>[%YS;VdtUp P4n.Sz=%r޲2p00F:U 'Ew?_mskx%Ptzc.m4 UP)fjUci}l;{m] !Ynݏכ̘WU֎i"͗6Z!I]OV7v8gBZQ;Ab#aO k3sԪA_+#^wSa?}Q;Oy3~3 c$1~ )/zߪ1dۙM9D(=ؗ|οCg><RZRNe_!m8OG>Wsܯk4؞[t@Gb2W4:Dt9xUciWrc.NдtQcUǀ8Z7bsQ'i 62.}d "(qN֟5~Nfv3Af%K ȅD ]DA"!1x4bnv4ڢu$`^_w% s2x;N`Y 閩ޘs9a|.pk]l%'Gym?O=SIҺ}Ed^ԙ(fC`.?+#y6g5ѣchMLv(vGH k7\sJݔxe8x6%xׄM|y.N 1YsV!&1޹c A\t heV,Y&2~ӗ%N9],s  I:e;*v8]8EaH@VC9މp={.|W\ZSEbNM $FFV&⭩W@/ 0 m}úE /"+T YJ:c@{IȞJPH Y!y$yfN*lzoQF E Y&$<t2"W.\3ZE9>DU($`%[7&+cy(4J)_ս.vREߠpbc&JѧeDPWo:`ݜ vTK#2r?FӴMShh[ B*Zg2{F280lAaO%d;N>Mi^UNq>lO/dJ8X_o/~% Z+%]ss7>u0YT4j 5Pr4o,> oD ~kr[dwrU?a&xW@ۆ,_ᬛ'Hl%~*fE5C1Ή\I պMn}FrIyyppils]._FB#o>D*\2Hىh]h4cm?8&H`d?3bD$$=~c$V\'Lt@GWX%k N\XmJ%Kr;ӅZBȏ7hGZ[##)1apD?Ej?-E=Hճ0\9L QU8~9ܤKw",f1W<Mj݌yn~VL#.c‰X~jJi@Jwl:UT܂Y&D0ͣjLљv=ǖ-o @#hH -Yhkecd1' O{heniŹM/L#9D((wJfwqcځ٣+oXtnKuRD95n~goT5A=kVAyDo0ʒQ& a`/^p)=zg( ^9f*\N ?&endstream endobj 239 0 obj 2988 endobj 250 0 obj <> stream xZKoF-ahh>N _,! @-M'3X=Y,ُZN OC-ŧE$}_N\˛Ed_,UdpDnqVϪUSw,4X.YS'Q<)<I}hUq ^շvx͙$$v0ymʱIΖ%xkΗI^H h!;A+om;>r>npÑH zynSա؎S6m+䉌c8eki)~׹Yl4\.΋h~2cBWbU4ֽg̍p/;4HI엿rgGu"r{]d'IB1+l pfn1^kLqxkc%23Hͺ=e?\.~u%1KuYhu9rࡌ]ECj܈L \䲉;NvQBleW:;iw.YV5މn|:SGˑ7v !WYE q=^ 6 gnN.#8 ?<l 0 K?JEktP*~KYU;B঱lrrTq9k*?.Qk@dQdjS]ŃN=8'pa$>.WﻍCpxmL(; l:87j XQZ25= `u'1eu()&Y|QvE6p5~S9@0*~ > e ,KmjIDw>:' |t0N :!|wW14֙8Cdop975" AnP6vwȩ8y˺C1O^ޫ;@sm2䣆zHNʨ,:'& @E7jR8ue>cui;jt>L?5VMX(MPf#,-^+a$ E˶.9˴X!.hT5ctS#|FTeLtӂĎo8.t. 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Also, the prior # hyperparameters for psi, c and Dinv, have a new form; c is now # a vector of length r, and Dinv is now a (q x q x r) array. pan.bd <- function(y,subj,pred,xcol,zcol,prior,seed,iter=1,start){ if(any(is.na(pred))) stop("missing values in pred not allowed") # change y and pred to matrices, if necessary if(is.vector(y)) y <- matrix(y,ncol=1) if(is.vector(pred)) pred <- matrix(pred,ncol=1) # change prior$Binv and Dinv to matrices if necessary if(length(prior$Binv)==1) prior$Binv <- matrix(prior$Binv,1,1) if(length(prior$Dinv)==1) prior$Dinv <- matrix(prior$Binv,1,1) m <- as.integer(length(table(subj))) ntot <- as.integer(nrow(y)) r <- as.integer(ncol(y)) p <- length(xcol) q <- length(zcol) zcol <- as.integer(zcol) xcol <- as.integer(xcol) pcol <- as.integer(ncol(pred)) # {if(missing(start)){ beta <- matrix(0,p,r) sigma <- matrix(0,r,r) psi <- array(0,c(q,q,r)) eps <- matrix(0,ntot,r) sflag <- as.integer(0)} else{ beta <- start$beta sigma <- start$sigma psi <- start$psi eps <- start$y sflag <- as.integer(1) storage.mode(eps) <- "double" storage.mode(beta) <- "double" storage.mode(sigma) <- "double" storage.mode(psi) <- "double"}} # This part packs the hyperparameters a, Binv, c, Dinv into a vector # of length 1 + r*(r+1)/2 + r + r*q*(q+1)/2 nhyp <- as.integer(1 + r*(r+1)/2 + r + r*q*(q+1)/2) hyp <- rep(0,nhyp) hyp[1] <- prior$a k <- 1 for(i in 1:r){ for(j in i:r){ k <- k+1 hyp[k] <- prior$Binv[i,j]}} for(i in 1:r){ k <- k+1 hyp[k] <- prior$c[i]} for(l in 1:r){ for(i in 1:q){ for(j in i:q){ k <- k+1 hyp[k] <- prior$Dinv[i,j,l]}}} storage.mode(hyp) <- "double" # create rmat, npatt and patt to keep track of missingness patterns rmat <- 1-1*is.na(y) storage.mode(rmat) <- "integer" revcpatt <- rep("",ntot) for(i in 1:r) revcpatt <- paste(as.character(rmat[,i]),revcpatt,sep="") nulpat0 <- "" nulpat2 <- "" for(i in 1:r){ nulpat0 <- paste(nulpat0,"0",sep="") nulpat2 <- paste(nulpat2,"2",sep="")} revcpatt[revcpatt==nulpat0] <- nulpat2 tmp <- rev(table(revcpatt)) npatt <- length(tmp) if(any(revcpatt==nulpat2)) npatt <- npatt-1 w <- !duplicated(revcpatt) upatt <- revcpatt[w] rmat <- rmat[w,] if(r==1) rmat <- matrix(rmat,ncol=1) w <- rev(order(upatt)) upatt <- upatt[w] rmat <- rmat[w,] if(r==1) rmat <- matrix(rmat,ncol=1) if(any(upatt==nulpat2)){ rmat <- rmat[-1,] upatt <- upatt[-1]} patt <- integer(ntot) patt[revcpatt==nulpat2] <- 0 for(i in 1:npatt) patt[revcpatt==upatt[i]] <- i storage.mode(npatt) <- "integer" storage.mode(rmat) <- "integer" storage.mode(patt) <- "integer" iposn <- as.integer(1:ntot) w <- order(patt) iposn <- iposn[w] pstfin <- matrix(0,npatt,2) {if(any(patt==0)){ st <- tmp[1]+1 for(i in 2:(npatt+1)){ pstfin[i-1,1] <- st pstfin[i-1,2] <- st+tmp[i]-1 st <- st+tmp[i]}} else{ st <- 1 for(i in 1:npatt){ pstfin[i,1] <- st pstfin[i,2] <- st+tmp[i]-1 st <- st+tmp[i]}}} storage.mode(pstfin) <- "integer" # storage.mode(y) <- "double" y[is.na(y)] <- -999.99 storage.mode(pred) <- "double" # .Fortran("rngs",as.integer(seed),PACKAGE="pan") # tmp <- .Fortran("mgibbsbd",ntot,as.integer(subj),m,ist=integer(m), ifin=integer(m),pcol,pred, q,zcol,ztz=array(0,c(q,q,m)),patt,nstar=integer(1), r,y=y,p,xcol,npatt,rmat,sflag, beta=beta,sigma=sigma,psi=psi, b=array(0,c(q,r,m)),xtxinv=matrix(0,p,p),wkpp=matrix(0,p,p), wkpr=matrix(0,p,r),eps=eps, wkrqrq1=matrix(0,r*q,r*q),wkrqrq2=matrix(0,r*q,r*q), sig=array(0,c(r*q,r*q,m)),wkrr1=matrix(0,r,r), wkrr2=matrix(0,r,r), iter=as.integer(iter),wkqr1=matrix(0,q,r),wkqr2=matrix(0,q,r), wkqrv=numeric(q*r),nhyp=nhyp,hyp=hyp,delta=matrix(0,ntot,r), iposn=iposn,pstfin=pstfin, betas=array(0,c(p,r,iter)),sigmas=array(0,c(r,r,iter)), psis=array(0,c(q,q,r,iter)), wkqq1=matrix(0,q,q),wkqq2=matrix(0,q,q),PACKAGE="pan") # restore y to being a vector if necessary if(r==1) tmp$y <- as.vector(tmp$y) # list(beta=tmp$betas,sigma=tmp$sigmas,psi=tmp$psis, y=tmp$y,last=list(beta=tmp$beta,sigma=tmp$sigma, psi=tmp$psi,y=tmp$y))} ############################################################################### pan <- function(y,subj,pred,xcol,zcol,prior,seed,iter=1,start){ if(any(is.na(pred))) stop("missing values in pred not allowed") # change y and pred to matrices, if necessary if(is.vector(y)) y <- matrix(y,ncol=1) if(is.vector(pred)) pred <- matrix(pred,ncol=1) # change prior$Binv and Dinv to matrices if necessary if(length(prior$Binv)==1) prior$Binv <- matrix(prior$Binv,1,1) if(length(prior$Dinv)==1) prior$Dinv <- matrix(prior$Binv,1,1) m <- as.integer(length(table(subj))) ntot <- as.integer(nrow(y)) r <- as.integer(ncol(y)) p <- length(xcol) q <- length(zcol) zcol <- as.integer(zcol) xcol <- as.integer(xcol) pcol <- as.integer(ncol(pred)) # {if(missing(start)){ beta <- matrix(0,p,r) sigma <- matrix(0,r,r) psi <- matrix(0,q*r,q*r) eps <- matrix(0,ntot,r) sflag <- as.integer(0)} else{ beta <- start$beta sigma <- start$sigma psi <- start$psi eps <- start$y sflag <- as.integer(1) storage.mode(eps) <- "double" storage.mode(beta) <- "double" storage.mode(sigma) <- "double" storage.mode(psi) <- "double"}} # nhyp <- as.integer(2+(r*(r+1)/2)+(r*q*(r*q+1)/2)) hyp <- rep(0,nhyp) hyp[1] <- prior$a k <- 1 for(i in 1:r){ for(j in i:r){ k <- k+1 hyp[k] <- prior$Binv[i,j]}} k <- k+1 hyp[k] <- prior$c for(i in 1:(r*q)){ for(j in i:(r*q)){ k <- k+1 hyp[k] <- prior$Dinv[i,j]}} storage.mode(hyp) <- "double" # create rmat, npatt and patt to keep track of missingness patterns rmat <- 1-1*is.na(y) storage.mode(rmat) <- "integer" revcpatt <- rep("",ntot) for(i in 1:r) revcpatt <- paste(as.character(rmat[,i]),revcpatt,sep="") nulpat0 <- "" nulpat2 <- "" for(i in 1:r){ nulpat0 <- paste(nulpat0,"0",sep="") nulpat2 <- paste(nulpat2,"2",sep="")} revcpatt[revcpatt==nulpat0] <- nulpat2 tmp <- rev(table(revcpatt)) npatt <- length(tmp) if(any(revcpatt==nulpat2)) npatt <- npatt-1 w <- !duplicated(revcpatt) upatt <- revcpatt[w] rmat <- rmat[w,] if(r==1) rmat <- matrix(rmat,ncol=1) w <- rev(order(upatt)) upatt <- upatt[w] rmat <- rmat[w,] if(r==1) rmat <- matrix(rmat,ncol=1) if(any(upatt==nulpat2)){ rmat <- rmat[-1,] upatt <- upatt[-1]} patt <- integer(ntot) patt[revcpatt==nulpat2] <- 0 for(i in 1:npatt) patt[revcpatt==upatt[i]] <- i storage.mode(npatt) <- "integer" storage.mode(rmat) <- "integer" storage.mode(patt) <- "integer" iposn <- as.integer(1:ntot) w <- order(patt) iposn <- iposn[w] pstfin <- matrix(0,npatt,2) {if(any(patt==0)){ st <- tmp[1]+1 for(i in 2:(npatt+1)){ pstfin[i-1,1] <- st pstfin[i-1,2] <- st+tmp[i]-1 st <- st+tmp[i]}} else{ st <- 1 for(i in 1:npatt){ pstfin[i,1] <- st pstfin[i,2] <- st+tmp[i]-1 st <- st+tmp[i]}}} storage.mode(pstfin) <- "integer" # storage.mode(y) <- "double" y[is.na(y)] <- -999.99 storage.mode(pred) <- "double" # .Fortran("rngs",as.integer(seed),PACKAGE="pan") # tmp <- .Fortran("mgibbs",ntot,as.integer(subj),m,ist=integer(m), ifin=integer(m),pcol,pred, q,zcol,ztz=array(0,c(q,q,m)),patt,nstar=integer(1), r,y=y,p,xcol,npatt,rmat,sflag, beta=beta,sigma=sigma,psi=psi, b=array(0,c(q,r,m)),xtxinv=matrix(0,p,p),wkpp=matrix(0,p,p), wkpr=matrix(0,p,r),eps=eps, wkrqrq1=matrix(0,r*q,r*q),wkrqrq2=matrix(0,r*q,r*q), sig=array(0,c(r*q,r*q,m)),wkrr1=matrix(0,r,r), wkrr2=matrix(0,r,r), iter=as.integer(iter),wkqr1=matrix(0,q,r),wkqr2=matrix(0,q,r), wkqrv=numeric(q*r),nhyp=nhyp,hyp=hyp,delta=matrix(0,ntot,r), iposn=iposn,pstfin=pstfin, betas=array(0,c(p,r,iter)),sigmas=array(0,c(r,r,iter)), psis=array(0,c(r*q,r*q,iter)),PACKAGE="pan") # restore y to being a vector if necessary if(r==1) tmp$y <- as.vector(tmp$y) # list(beta=tmp$betas,sigma=tmp$sigmas,psi=tmp$psis, y=tmp$y,last=list(beta=tmp$beta,sigma=tmp$sigma, psi=tmp$psi,y=tmp$y))} ############################################################################### ecme <- function(y,subj,occ,pred,xcol,zcol=NULL,vmax,start,maxits=1000, eps=.0001,random.effects=F){ # Calculates ML estimates for the univariate linear mixed model # using the ECME algorithm described by Schafer (1997). m <- length(table(subj)) ntot <- length(y) nmax <- as.integer(max(occ)) pcol <- ncol(pred) q <- length(zcol) p <- length(xcol) # {if(missing(vmax)){ vmax <- diag(rep(1,nmax)) iflag <- as.integer(1)} else iflag <- as.integer(0)} storage.mode(vmax) <- "double" # if(is.null(zcol)){ q <- as.integer(1) tmp <- .Fortran("prelim",ntot,as.integer(subj),m,ist=integer(m), ifin=integer(m),as.integer(occ),nmax,as.double(vmax), vh=array(0,c(nmax,nmax,m)),vi=array(0,c(nmax,nmax,m)),pcol, as.double(pred),q, integer(1),ztv=array(0,c(q,nmax,m)), sig=array(0,c(q,q,m)),iflag,PACKAGE="pan") ist <- tmp$ist ifin <- tmp$ifin vh <- tmp$vh vi <- tmp$vi ztv <- tmp$ztv sig0 <- tmp$sig tmp <- .Fortran("nopsi",ntot,m,ist,ifin,as.integer(occ),nmax, vi,vh,pcol,as.double(pred),q,integer(1),ztv, sig0,iflag,sig=array(0,c(q,q,m)),psi=matrix(0,q,q), sigma2=numeric(1), p,as.integer(xcol),beta=numeric(p),matrix(0,q,q), matrix(0,q,q), matrix(0,q,q),as.double(y),delta=numeric(ntot), b=matrix(0,q,m), wk=array(0,c(q,nmax,m)),w=array(0,c(nmax,nmax,m)), xtw=matrix(0,p,nmax), xtwx=matrix(0,p,p),xtwy=numeric(p),xtwxinv=matrix(0,p,p), ll=numeric(1),PACKAGE="pan") beta <- tmp$beta sigma2 <- tmp$sigma2 #llvec <- tmp$ll I'm not sure that this loglikelihood is right, # so I'll omit it for now. llvec <- NULL cov.beta <- tmp$xtwxinv converged <- T iter <- as.integer(1) psi <- NULL bhat <- NULL cov.b <- NULL } # else{ {if(!missing(start)){ beta <- start$beta sigma2 <- start$sigma2 psi <- start$psi storage.mode(beta) <- "double" storage.mode(psi) <- "double" storage.mode(sigma2) <- "double" sflag <- as.integer(1)} else{ beta <- numeric(p) psi <- matrix(0,q,q) sigma2 <- 0 sflag <- as.integer(0)}} llvec <- numeric(as.integer(maxits)) cat("Performing ECME...") tmp <- .Fortran("ecme3",ntot,as.integer(subj),m,ist=integer(m), ifin=integer(m),as.integer(occ),nmax, vi=array(0,c(nmax,nmax,m)), vh=array(0,c(nmax,nmax,m)), pcol,as.double(pred),q,as.integer(zcol), ztv=array(0,c(q,nmax,m)), sig0=array(0,c(q,q,m)),iflag=iflag, sig=array(0,c(q,q,m)),psi=psi, sigma2=sigma2, p,as.integer(xcol),beta=beta, matrix(0,q,q),matrix(0,q,q), matrix(0,q,q),as.double(y),delta=rep(0,ntot), b=matrix(0,q,m),wk=array(0,c(q,nmax,m)), w=array(0,c(nmax,nmax,m)),xtw=matrix(0,p,nmax), xtwx=matrix(0,p,p),xtwy=numeric(p), xtwxinv=matrix(0,p,p),llvec=llvec,vmax, sflag=sflag,eps=as.double(eps), obeta=rep(0,p),opsi=matrix(0,q,q), maxits=as.integer(maxits), iter=integer(1),cvgd=integer(1),PACKAGE="pan") cat("\n") iter <- tmp$iter llvec <- tmp$llvec[1:iter] converged <- tmp$cvgd==as.integer(1) cov.beta <- tmp$xtwxinv*tmp$sigma2 cov.beta <- cov.beta+t(cov.beta)-diag(cov.beta) {if(random.effects){ bhat <- tmp$b cov.b <- tmp$sig cov.b <- .Fortran("bdiag",q,m,cov.b=cov.b,PACKAGE="pan")$cov.b} else{ bhat <- NULL cov.b <- NULL}} } # list(beta=tmp$beta,sigma2=tmp$sigma2,psi=tmp$psi, converged=converged,iter=iter, loglik=llvec,cov.beta=cov.beta,bhat=bhat,cov.b=cov.b)} pan/R/zzz.R0000644000176200001440000000010213374503040012201 0ustar liggesusers# .First.lib <- function(lib, pkg) 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[RR'ol'+ۺ]l26vvm%>FIqaN^hd2VXq@8;3g]yw1M㹸Y:.͓h`()طƀArw\=9U:Ut@l-I%OBBkPvO[$-mNBG$(h'"X7ޡ]/ 1.1幦w\\sF&yX8rKUqWԣHu6M:ϵ@H~M 1Dj⩆`b w3CcxJ9Z"*X! 9 㭙Ʉ53`\-6QLr ^l,YEenAߐnf):pUz ",σHZ 247)mie4-< +L$CIib@\Y-?lxU"8C| 3Rz]l- @fz&כf>lKg3N碟Pl^>a!d.z W")Kpm2\ xapjU;=T2ۈ0gUhXwWW X>N},wZy P;\H~Jz͑~\(n{%spp !] Р)-SL!lJBr5^< .51'4ݾ kc]Lc8)Vc ;qFlj @g“: ם{ʫt4=xcW C@u'%Zw__#چ+ɮ?u/,J1l鬋ey#J03$Q&fOOԬa<'YKy1J7:x<63"T씳,nF𷸞͛P& ^rV)ӈ¸v!VڟMl/nX]=~ Ý9?䔂CE;^zS֊k~KyIIb|Sڞ6e{ԡ>ĭ/jkUH8"| y.| ^\V?2ZCC ?%v`h8NnIݬ; Q1HNɨLwD:1MNNK uƌ@,ka `Oi1%b.hRMTo.D޽PS endstream endobj 956 0 obj << /Length 74 /Filter /FlateDecode >> stream x3T0BC]=eUedA z@` #1!FB=sMM\@!(  endstream endobj 953 0 obj << /Type /XObject /Subtype /Form /FormType 1 /PTEX.FileName (./schafer02.pdf) /PTEX.PageNumber 7 /PTEX.InfoDict 870 0 R /BBox [0 0 612 792] /Resources << /Font << /F15 872 0 R /F14 871 0 R /F2 894 0 R /F5 903 0 R /F3 897 0 R /F4 900 0 R /F6 906 0 R /F9 938 0 R /F8 935 0 R >> /ExtGState << /GS1 877 0 R >>>> /Length 4944 /Filter /FlateDecode >> stream x\Y7r~hK/N-,"")Aj7 ]3v=ҿ~UwDpTH$2߾`<^ > g0Ģ(RL͋_•wcQ0ܵ۷Oo}{ PB gPʶ}_ -%ҭWcCK[([ulhU6շ>62KVͻ>DZ.J_FWLNݟ_@hT=cQYW^R26(^péj*=nf~_@"SR Qn/e0KF$im)ԓ(4fhVvb,8c"qZ_w38?ܰa, r<&7%p=]7g,wUHk]zC|(+]J/ !.\7V%0FDHf9E"?Kb \akkM,ў!g]s\$Y_Y mT *D8ۦS is^!qW4p Àv5_7m,k)JL{=ݭ4sWx `g+(46J^A 2hօ5]e!$'{z"muT7[QZo#Q ~p#1op/FeLXӡc;N*Ѣ/,3 vWe %]gN'?5ՠ)^Y9vwg 0 bȿ-nU\d{J *Z4~1ˍx}i8Z]e+Jt4Qޚ~DOl裤Ӷ)lXr%ϼ+$p=hDB/Q*0G%K$ ̶ wZR}KBYǗg;v#bwQkEnRZHqV~ե_gMWW:`gn F:XJ,qbd;]Vă}UR6*Ѽ}$:s BXXydB;cGtBy S:+ <-N/RwiME{h6 谄 :t(&MR Ae o|L[pXUGB)۠LwqUZ}fނa!G,@ h,=:Gz"p Tiʋu&A](5҉+v0tB:7q SCIH^4):B1i_dekc}NjO36dOC$uA6Jna9>V!#/W>fp9ybh|b,VxZc4(~ǿQ",T;JSlY\+L|ǖ% QY2 rܨ0dm]>?P2Rbv`Btey)4-D0Z2.LQ`Ǣi:˭ BܐcĐ6FwE2Yu¥`< EIa{;Ee(%V uJ?#9@1ZO0wGmZa!=(ugG74o(Cy_I F7 .ZsXGf;j:Q(h;?Y-k #A`'93Ƞ"x94 |UaX[\r h0!B&\L+-w fC@[*OgJQɕAzkFVrB~mgo/:zDNp kLq2 {CGBaU~.H2!x{@W1BeԶCxЎbplao|FJ#iZhGR6a0Fb@;S)mv*gPI :6cdKH0PAvp}΍ȸ7 %:+5JP | xXYtڢVse,5.jDbLma0IcOCI]ŦtZ}; t ӳ)잏N|ȶBp(];S+oU\-|}};Mcq"'RIy8w@̜rB%|dp5}I%kħ'CdGx'm_rU1o/vzʊ#+T%kGLwEO\T'rZݗTzzn?=x OSNOrXf~= kY$7b1S2a^(ӔSi8MH+;)\c_;p2 եM:um1j6n9tբmYX(mKpuǁ>{ .}uh endstream endobj 961 0 obj << /Length 74 /Filter /FlateDecode >> stream x3T0BC]=eUedA z@` #1!FB=sM\@!(6 endstream endobj 958 0 obj << /Type /XObject /Subtype /Form /FormType 1 /PTEX.FileName (./schafer02.pdf) /PTEX.PageNumber 8 /PTEX.InfoDict 870 0 R /BBox [0 0 612 792] /Resources << /Font << /F14 871 0 R /F15 872 0 R /F2 894 0 R /F3 897 0 R /F5 903 0 R /F6 906 0 R /F4 900 0 R >> /ExtGState << /GS1 877 0 R >>>> /Length 4608 /Filter /FlateDecode >> stream x;iȕ+k 8퉃$,@Inf(RKRq:~QU[k-|GֱL'>ދhc}bwILGo_$$M31:?Dxj'z&YX[t}b͇Ͱ̈́3展/aHd^lUjRSd~ x1?9weו#7MۗMqy@’W@gH"'0RſE3zOCU O?5N8wgYXna%%훺{<"=>z cwriggp @&KY6K^-;"\>Hke\L|Q\) QbBmተoz1SE) q僇VOe:eVmMX-7Em DI6ב'||V-0Mfob gJӤp@]HxY KjмvK[Ed}e=4RS$ Z@|W#Dy  MP|LKj Nlp!x/1@pH7򍶨dv4qp"`POW_|-UO;%adzƯۀSQ{xlⱨ7\l+elI?SkQ(p"üo*+a.Zj'L(Y(R ίVP4h1_Ĩpt}і7FN-;G륊TКj( 0>,%V}{E\K{V- Wأz,h$)؜R X$#ɉbg, Px;_]yJ;C=ַWatng~yu(ߡD*u%b^5qN݉!nJr/HP6WS76|nWar?9yoo$ԾxN|(O,9Ie7yWMM^cV{0+.Rs? <,˂)N_$j )XP%>cVBY3DE-7?xj=U9yش)+i!Тk`h]Te1Q[_i2wqE~Kr9<'#HX,cydBA*6*jȹ Rkl/c^+]J Ȥ/_4@,j5¼p}ýk\'3؞L,C EzweJ$ Pfsp0Mc O8gYD_f7 _8wezS}qAx]IQ+?ͦis ?H -5 6`Zny{عzc$6Zt׫z!hb;s9ӛ0pQ(qJ+<=5j[{tty5{"܁&AOa@ ҁH+%KglOzgh]~T؋ހvLo"C=kDpmJCh߾hhĶk0N]h~x-?ow D9p C?oXnv5u"/EW=BfYuyj,0^RFfn΍wD<彇A]͖ d9ϼ&"eIJn `uu,&P x$8pf[/@fWI`ݕ[T%^eP13 .1P}l2;>ι͆a"!8OŊ8YGZ,i 5d&!;Y wBx֞͡4}9GF0`nY{;ۼG5nA>!QĨtI5%b[YсI% .6`sx4@V!)*R2h  # &4 5Bew}<јњ4kK9>r 6Go8 l oZMbHFKF"5qM> ^(aPeq$ x=*`y+ '^x!PߙM3c*2q?`dh7Q#q HyM@Q`q<'}0_M|'@vtj/T 7?/oq47= r_7C0tn4hxiL"7+Ogj tPL.fn |xf.(>rmd%y_'wC}ƾ7qv'o)S F<6BK5m$uhHZGsamȲ͓U>aWE2XAtiH"i#"< 0GC;>N 1 "-eDxjl{̓v``r u`/7 rW{QK0O+Ai'6.&t65ҿ.S>u[lX2Q%/0-AKr}gzƕ󮮆kŽ/sAG6)G[ $gTe%A<Ӣ(}r$I!z֍3ESxp$\8)4/@I<c$IBXi"dQ%~{3<ŵ_>fpN!":_7+)o[wܦ!>B*6έvc[7@8 7MD C8;7əY"t 252SDEVK|!}}$_y$|WCH-!~OD_6o*tu=XQ6&cpd6ٹ"WՆ8ڹOl,x*!=o+tRQx>S)C˄j6}{(=m7 `)L9]q\nYxxTC}2 r,0 CL4b Ϛ_&hîFk0Y{y1_gb M 8~yW[퐿a8IGiM(##P'HF#.p /!cpVHbxqgn&+)/v蚇zլXL硵+Qv5~?t`}b >0nZnqR!6t8QgE-};mCu VP. 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W.@hd޸V5 %*a؟B4Z# _+CSԩNSB7}@C7㧁AоEJe}5HR  g3^,HHK~ ,]2&kW3V RaW}=wb}1@jmc \EkHg9H*+) ۯxړl1՞Ueja! eWZQnU>rT# %@sG@W=Km;n|N.7>V\ز~=K8RU>uv(Z3sbϗ.urlA"X`n@fEWI@VIBnϡ Xae& ^]XcÉB sPXU/z`}1gJ8N8({qk=%hT1"-:фaq(F{ҿ=Iy'b|ʽB D#]&{%>,ɰ$/bIƞh2.eIsYj4X&^R*-)(})H8ϔpOg1tKDNXR;ޓMs4`֤}:P#\7ȽI+O-LXk8>(t9q8(B#yp8MR2%'p߭Eшͤ>34-@<[i5>>-_x0(Hۋh=Z?0!h*T#oPGMM~Kh`grdxx[,ASjl3bF[J91$ڇĊYCD}VTIm=nnCFUX0weǏ41<> J\1HR|nاhoTIh6]֑VKo_CWJV XNsQ{o7|&ZTn^:wo+p .I hffDzF6gfDV6/L@ !2o]|Z x&&ovL?(AGӍ؍Vo./XG1VLʥeKbrv nXM˔F N3 Chͥa ~^Dr.7HR}K㭩Rvnnp|sK8[bzݷ5S'CĎuW\,YSȢ< Lu- _4O9|f*'>2>>w3(@H_1RI_!KNŞ3. ?lhK:눖Pku3\-k*PCdI/y /}˺ߨMfBߪK_P!*0h/ sTUq˙ѓ{,Qc|Ao60a31at'>u"HwwPH*,@h/%*فlGB mm \ endstream endobj 982 0 obj << /Length 74 /Filter /FlateDecode >> stream x3T0BC]=eUedA z@` #1!FB=sM-\@!( endstream endobj 979 0 obj << /Type /XObject /Subtype /Form /FormType 1 /PTEX.FileName (./schafer02.pdf) /PTEX.PageNumber 11 /PTEX.InfoDict 870 0 R /BBox [0 0 612 792] /Resources << /Font << /F15 872 0 R /F14 871 0 R /F2 894 0 R /F3 897 0 R /F5 903 0 R /F9 938 0 R /F8 935 0 R /F4 900 0 R /F6 906 0 R >> /ExtGState << /GS1 877 0 R >>>> /Length 4562 /Filter /FlateDecode >> stream x\[o~ׯX+7b~\v.Ӈʒ~gCr%KR,.wsܾsᳳ#xw|n{^#_y#̂_}) cUŤ]}8Z~{8瑩.V߽:{ӿ|zvFf~ NWJ/~|ђ)ӔꜸq"LJWa (@xYa l,?O^cqv8{J֚2&-&. }>b7Fɜ7`ڕMa$)š I'V3Y`H}#WSyh*C(S1 HS0ZʼnD,yT/@1d>f{B۟CN>M; UEYJ+ZeI82὘UTQ Dkf=r`ð5pK=X!Eщ?JH t<ͩ:`A"GG @"dKQB^FtS0W!|3ꐑXch ד~تћW.)/ k--|EOf5 wM[]^/ޭכOWv>liYo#'CdI^pq4(!C**(ƾ#mčON5q=M}8 HY! FT'!/spt1y/!%l !h":(bH1C8l8QH[%q[paH@2F_B+xbLVޞesQM>)њHN09BVyRQ(EBtH, X ͓m3O+"ȅF (,慟jl*4D&[Qra'@ ^MAp]do~yCBm-=pXA:f;#aBJ-g; s w(d e z$40WAx{K팾/(Fi{Lg% mo)I2b=13MťX(%CA5:maғ$ChqҶ h(xO$Oɧц73L0kls#HŗG^!UFmbmWLVbQ^\jaQă4K)/U\>lss5*%D A[Svn M}iU4QhfU &mʔ a*b <կ]YjҴ󢱔_JVZg4pom/[Ȱ,kpbc x}{YMC֖a0ԨXmP.Dţm BNDqbgoQhO}ŘjLd29OX!IA03 q`f@ &g2t$zaVp3Vb=—noZ0*fR32ע#*r=QF{3)4$>Irߨ$g0^1 )fJXC=WJ0t4CBNM3 D8YA1oo;7 4P'vuBXU›.w+ks:Ns+7/13ޒ$^{霸q"|?g shc:hUct` :ouTp4Uw+NF[FP;,l@7:?AMu_io3DVߙo!3l:$}^ wu'pt,pK\#rSNX<=p,qA8{gHotpwZdM+ۥפ.81Iu8$zHM)#T;݂!¥&>DIDžtNCENu=TcčGGQ#&d3Lj;Gb]ŭ?(z\E<Šɚ{R EN( \P@|]̓m3(eʊQӑ*,uw(4lpK'r B槺K:@a-yQ:X<&KGG%qi 781"{?ͫHy Ar /`NCQl}xr 6\onۓ撚o>]}]׀7 5u,NZY~ܽ/Pלhjޭ7 WԲ[ViIM/I:E=d}HԁD1ej M~wq(u5]x5eަ]^n5ޥ[W&};@ 9{@ 2yMGT>mv&(:k|tapNbO@&(@NHÈԝv85 277WW]x'N.nu7]swh绛Ͷ_1F@KɂPf=}բ=E/"ˠS*61sTbO bʋ̄-71!bp0YH賩_% (컄Ո: Nq6&sczeQÊ-9Jxk"q4Vs*w,"G&HԹO I l4M=&V81U(u DLV;W}Q'XN$(m0iؑnږދh׳(zYc vW+1FjhPq?AրM}ͧ 'T,<FEԂLp $ꆢJ n~ht6#J7ΕQTHGI<Hblg^OJBӉ!֦T;xb^~E<;@[^ iRs-PG[%/`.B¸S[Hbp "g9N(;=,!NvDΔNORGǜޝx"*!»jb$0bH ^ )qvvoHw0Oj⽋YY/*يӳg*=LCgGV6MZTLfr*-ڒ S@iPɫ6-곖bjwvmZޑs]Ӳ qUi1ɨx% ļҙmS녥P6N#wBTMQ1צL0y2r,(<;r~3GNeYʹ)jޟ)-O/-J+T(x8k,܃={ |k endstream endobj 987 0 obj << /Length 74 /Filter /FlateDecode >> stream x3T0BC]=eUedA z@` #1!FB=s \@!(` endstream endobj 984 0 obj << /Type /XObject /Subtype /Form /FormType 1 /PTEX.FileName (./schafer02.pdf) /PTEX.PageNumber 12 /PTEX.InfoDict 870 0 R /BBox [0 0 612 792] /Resources << /Font << /F14 871 0 R /F15 872 0 R /F2 894 0 R /F5 903 0 R /F8 935 0 R /F3 897 0 R /F6 906 0 R /F9 938 0 R /F4 900 0 R >> /ExtGState << /GS1 877 0 R >>>> /Length 4732 /Filter /FlateDecode >> stream x\m䶑bp};."N Lp8MfFqODR{=_xldCHŪ͛+| WMXUȤF&0W7oVRf뫛SXȵ\ cY߇՟5W]NB$.o_έx:d ̽ﷁ7ܳa0Il|;PŇRn/MRl`FVW2]6LDY7H͞HZf wZ]a=L ;R3i_."yUD"M^;*ocܯeiԪoE˄ ίDlf(U3d K̹g r4wg42NeRL'2YS3?ղZdA1qM&9ǜDǜɓsъ%t2p;ЉKH/כ˝Vvv 0cOÜe;wFCmsbK$DExqSBSf"Va s^*|_X;4Eܗ1 ղ<vz՝]z#oS:Q vvlF/XY3_RE(mFQtQt lI+sKǴԟVɪg$+q0GQ옋]oѺak_hu+X :G/JАUK M)bھW?_<)w1&S(ch}{=y$Xs@[ HO6oHR iʆV1o8ફd )ʾ sGrٓx &I4d0pZ4 Д S=f'"R/d3A!o}KxKH_GdBZ Q8A N'r躑vJ9ul&MB{fKn~)R =4pH u 8W@H )Pbp[X}(аzD~ }5a89 eOV>ǖƅ#\Z[bSԵg(@_,Rs[qLO.rmPnb~Z ?7il 8ZȬ@c|i(>ePӲt!#cr!ٰZ(SJ6y$z .'ߖ=.i+ b=6HHe{OO.cP5{.U՟ɹK b pI 0~nji `_  vCi3ޅJ; 0cW7.A%J6?<9T?)ׁþJNQ0Z?.>Uv8=Gz" y,% ՗[LO(8ܐ?r^Jf PѹBdk ptn|>w? v;ƯW(9AqUߜ7?AL˞%y )ڴ]ڏB dC?o 2 Dnm'FN̰-Omq0 Wh 6&O]>ԉ1j@ ]KUaK0SKyߢX\UB4 ٹ{;7U[zX/eq 0 WXF@~!GxSwN ǞpP;ĤV }[gIʆK'Ksf`t- w%Y{BѮ,\AST%r{1Fበ-x#^ЉT/@3ǙgI1TZwy}նY:_L?Odxt0ف~\trɰ ,#< [Mf{7MR~,*SZ}+\Xo} :{O)*5:۔) nq='R| x )5RU1,T\g.E&_w߿lJlHӑ=I рZDi4k4:_\-DHM?XC:63ImY~z!-.%fNr8;#aSg[ e vfq3նt>4kka4 `2FԵu^H V)9hDDi5g*hj$g> stream x3T0BC]=eUedA z@` #1!FB=s \@!( endstream endobj 989 0 obj << /Type /XObject /Subtype /Form /FormType 1 /PTEX.FileName (./schafer02.pdf) /PTEX.PageNumber 13 /PTEX.InfoDict 870 0 R /BBox [0 0 612 792] /Resources << /Font << /F15 872 0 R /F14 871 0 R /F2 894 0 R /F6 906 0 R /F3 897 0 R /F5 903 0 R /F4 900 0 R /F8 935 0 R /F10 971 0 R /F9 938 0 R /F7 909 0 R >> /ExtGState << /GS1 877 0 R >>>> /Length 5424 /Filter /FlateDecode >> stream x\Ys7~`l6gL cp<<ڰg#V#BއY"{1o&P@DQU@"ǗD}wuFo~[v~{8kՇ3Gə%Fsm$œ_=~83M?[797D3=yqʿ50D"R7{rkA%t-LGގu-,eukie:c=kFWeDJy7~s9G2ҹ8? ,Ah,tnJtQac(hnz]j%tI}u>>荇P$qus e߆t'H*%1FhPQ%[P9 (-luloz>nw:Dg: ̹j'8Oh!>xWfu$é(ْI"Q斑-#ƙ쏛t]7u D^_2"BW00XM?'9/f;S@PmJgZaE \}\bf%Wqs<9L6'N%1@+AfdAXY ]:RXPv43.%%K\RVp pG{~7LnߕIAmL ӦnL :IO溊Wǻij?U}aLJTT`5=g3:,/ |%&hP-]=B-^kCas8Vx]DڇPvsCF4QMIv6u=h1==x}CFZR2? >݃zoy)* 9Ƶ+N0WB8mD1Ѓ0cXE LzRv_5KRNfS.Sm׷ὴRb񖣹!`% PY::%@N-dNGSkyԗ_xvƩ|$ձ7YlsGFݷ }mQxF"cL2D1QspE/ TC|⻩E oڻ~_j7)Ab.yA vK0>z .1Adyv > S7L~Z0)܀aFM 7p:7>st{0niWY&u;0,B F䣽KPN*XSe Lԯj{sH ]e]80AͯTn0&{)s@)_RXnc*K Zeс,[fB,Z2;,qșirEJ;v/ԩ52nF!/0@ғ*67Cu~q%N|;ڳ0I<8H1'"h9136"&CG3AECN]̃-bT s(fIdԛciװXm٭wpN@`PUvfZ`Ё;1S]` -z Xev}w&'{ ƈh)rx wѩ͑/(рp?t_ j0-u/Vb6HzJaKWJ7/5X|^.R3!-ÉF7;n{{Z{IckhgY)"]3(v +yB!p *Y97@h}Z5ZHU'e˘ H>&~20 N}rto 1.=Cp U9`i%Z@@Thx TV$j•Y s8%j=/!8 y +Lk3T0mjKU$mf:?uZ$M.,-8(R q`aE3Iv$J8]*\z8ԌLuEI-H++0 j![L|KP eSs"Y -RY"fPA )i_밙xiC1Ð-_Z'I]};D([oǪbWtAȘS$ePP)`q$/)QSNkSU4d1|"?X XYEɏ5 YV}}k!*pOUQC0{Yoe0A:>/ a7@`b>Bu w^\FIiFcu(/-jL{,1S3fD<%0ŅI+ V^";u )`Ƅ8SP:Q,D -:#/: *-&Lnذ`MC#*Lַr9XxnhVnY`ah%omw@rǦYn2 !fO[_ <;{XvRI(i}KJց`)Sψh% e$npB,VlM8 -'ñzW]j>RǦV̟w`(R.6Orn.c C\ȑr Xs9ᗻuz WA}a}cy.8uS'u{[u+Ÿ? :ic94^oԿ|s' ?nc(t{_w[1)MaIJEL[hj_b[mLgM7lot@Luٗ (>Wv`R +H}C˟Mtx#qwb[i`EbXsc\be6#A\  Rn}g2β8 [Z'"X;T)λͫãKgҥcFq4='p/ J (r 86|tde,Q>T^N~&{\>l>1\uЛB?vp5۳ڢK0>p@>Rs}\LcXtF|}IeTc {txhO=;kyB Bd,Z%jv$qMZRW 8wv$F!ao|r-<_d-n4OJ4s ~!6 '#Rc]T  W9bA-X:+EW(I :?XBFXUߎ[=pćg%k%Mε> 'oڂ=Z/hѲsLv<|^&1^b U}o"&~[H&]"(l,|ps%|dk;$3,^LU`qk*7[3hϱ 0^YkCœ(mˎنK4Wu#@S9nmʆJvLX+3l?~qw:\LN|M?FezUfMbԃ9P&'q7o)a&įqoޜ|s,'n'ᵚ;*aD^2t*nCJ1@}߉yC==5X t@XD!2?CMGS U$8ϔ{ZSb+TmЂH٤9Vc]]NeNNR!LȌaQ .p2r@;eaSXqB_x4f+։_F4P[Kǜ^έs9Xͺ)0~@9ANk @t`pjb'ٺH崸_rFԬ&k;0UΊm48@x9եŤf՝o\:,b:MdGÂH}z3uOFd7[Z&,x0u0 $A:Ə3d v1`z2ۤ8X܈>XեʃgUNCfcaٸV`-/ Q9}^U(Q歌CPo!DߟjR Nt-u0}yx3C5t8KoP?t`| endstream endobj 997 0 obj << /Length 74 /Filter /FlateDecode >> stream x3T0BC]=eUedA z@` #1!FB=s͌\@!( endstream endobj 994 0 obj << /Type /XObject /Subtype /Form /FormType 1 /PTEX.FileName (./schafer02.pdf) /PTEX.PageNumber 14 /PTEX.InfoDict 870 0 R /BBox [0 0 612 792] /Resources << /Font << /F14 871 0 R /F15 872 0 R /F5 903 0 R /F2 894 0 R /F3 897 0 R /F9 938 0 R /F8 935 0 R /F4 900 0 R /F6 906 0 R >> /ExtGState << /GS1 877 0 R >>>> /Length 3798 /Filter /FlateDecode >> stream xk }(P81 dcps¾(|Nl(R!8fg_|;9.x%㙜]]dQLqދdH~إzuDFtTQ 5],e2_1_r8^Fh>T"Jg"RqLk=37DXI5[n[D6bO, 3'.'5Q"{`3?1(\Q"5J+DnfPpA>De:!~|,2U&٭r:JGqA{hU>Ivo0S%?QަI"8 xNGzȤC&0* wH,p|X&Iv[N@ 0x?MUOi<(spmSɘ' Zf_e^I?'̩ ^'y$o `@i?Gm*u|G"" G.$ p~(D??Ve԰S2Q&29p5y,dd`"Y+^WEWE_olg=mP1 dK?C߭ގh};O):ễ۔MYg9%]4+(p~$[ 绶I+{ eCu]ٮ+ڏ7L}YQ@VMK 4+޷ N '兘 m`Co'fǂ,hS["=uM EU} w 7BA)viuS >c ,Q*ʅu^()Sy 9~Hejmu)pI,V(ErmmpkW/ͨO(&]770+7[Oڐ5.=XwϠ۲n枮.v]zl+o#7ss85 VnͲǏ~`[k:݂6!n!e4L4Ra%(Tx`-0ʆ(3p~ #dJPr0arB T7'6-J {2V ⾩㶲[0i@e`uHz6q4p<52FQ W p"̠ jvL;'J"w`N+,O1s\Mߥ?lJt pt mlƣv tMůvXed^ mz;^s! pP>OvF{y6C `0xհU3ڴq[PDf#u ut`*~-I~6E|lw_*==X%]G1gx>8#$Mrz_|{[RS(=|24xyed~0v4X0)MKrb%.KɈmȭy_RXint_}XU.b¿^ټtFx1‡<]SU ~^v49 = ւ>mYV0IKC=^W\L`Ks9ETMd` 3L^= p$O2W8HSAs!BNN R#* d8%v-u?YZP#b fR3 4$xhOzC-A t+nM9JQHLF!JI^\@r\}H~Q 0ڰa(/+2}O49P$Οy .;1!!Ky 4/cuy~#n9kiH?N:7D)GCZjV?_o)?jF.`: ޅ &t?Pd%k*Hz<|a.$=1OhUElUn8.Naf^ݦر$t3J٠riVJYEPY4 @""t9{SN~;Xj;-k=ºqKC0<ɕ*gia!*jp Ũ2%ÜpЭ8Hu⁦n4тih2תܖ5KdzR<ʈD.')Yq]v=dړ0fO"Υ m㟱 o"s1MD0=m Wr&G:~">: Dp+@ԕ?'B`eZ>^D:28e'! M\Q&5/9 (Q%$ qa46z[}2vNb SFUpE {T 夗sRCKG5K]f~bR,c lȐdU271]I<DDt9f tA5omEJ>eZY0dJkrU)dQz_U]U`zW#lu=8ylB4n&TMsF#;1hh46 IXJ6a't ߖ= D~8-W uBT&n5.:5nSgλ_njSv]޺B>}%:@'q5OЫ3FN)gp>#wz h=;lLC䷝LA {E8!A!چ#q)o2,\ae5wqNo]iУ v$/wx8'?4'1\;ץGY0&8:!<2-'yxus?>kN:zҙgH)u4ףnJf0'4 G'=)z:GOvMٮZ3 vmeG O.YDWp endstream endobj 1002 0 obj << /Length 74 /Filter /FlateDecode >> stream x3T0BC]=eUedA z@` #1!FB=s͌\@!( endstream endobj 999 0 obj << /Type /XObject /Subtype /Form /FormType 1 /PTEX.FileName (./schafer02.pdf) /PTEX.PageNumber 15 /PTEX.InfoDict 870 0 R /BBox [0 0 612 792] /Resources << /Font << /F15 872 0 R /F14 871 0 R /F2 894 0 R /F5 903 0 R /F4 900 0 R /F3 897 0 R /F9 938 0 R /F8 935 0 R >> /ExtGState << /GS1 877 0 R >>>> /Length 3615 /Filter /FlateDecode >> stream xrF_P[>xR831e+Gx(H$e[>@av}+Wѷ?XR?dtH8Q@dz͑w99(X%4րR~$؟.f?ޜh֞WhgVUPGCߝ]"_ 1!:}8D9~*1~uw~{{~4c=@}zF$@:4Ӏ:I}EС89l2ՁmmiU^MVɺlk}oKESe^,Unv^8wHSu7mùEk|"Bsy(ɿg _E'2s'LtL*~uxEkpy6CuvMVT(7m襵Lz[QMhPс^bwzɲ"z2ae-^x]]Id붮N̳eF2vvYYi dqhA b) 1!nmaxb3%# k9ݤ]z/k@b6_1 ne;.sLlz/2X`LH.pe6 ,*g5K򦩛a!k2آܠyvo/͚/(D{My>z"C vɫ.w'n.&5e08A g?-)OA kSY'wHͶ 1mgb h;.4t;[ADcQ F#k1!l&DA_;n@ji]5ny3&g=Ǣ"eCQ"4śbF8%h`Mb1Jk놖o^^v 4FѾ m9= U. ,E1Tm$'ȩܚ%GR~MR|~?>.ȭͽ<ߒ9ĪQH.z;ԭ)}C3~ v~r%8!s*855~ceVD$Q |jGN`zC10c1P1ph HAFIPR g{!J _`_aӭ҃eJ&ilUeY(N 8{986bXA&GFVZcB(A*NKpXQg- /i j/1R'/kqvOKp㇑ irxO'Ux Ulʣۣf;ʖ$~I⅕8Ibe\4 ^#S4# QM*NwY"ii2f,!86ޙw&n{B?IdDehY' H=vYi^_iWZE~(Ȯ4dx])JGc]mA w]a2 :Ty8;' %cDc7$ BBc/׈? &n;\FdN㱁bPѷqlYHRmCQ_ڮMłƛ共gqdPy+ghQcu[WpTruFѺ fP-|ځ|ysvC%0(H̅_^@' fnjOB@ $Io @1Wg^K(}&n:2^J 6j=i>XW,apy_;S(Wp5^039r.5Odn(D޶(gv_'#F#zH^ÏݞPtوr SS<_dPDAK%Ek,S1N1o_T=(* ]@tTnK<Hn.!Oar*u+HzUO ۤ$Ӷv8/O J^y@hg$ !#u"B %9H(g 04x5F8Yf`C wNj3c`!y"60Y|u8QSa01x0{w}qvjF[F t8?%/!wTx{0q0!> Vޜ __$P&W t9#XD6:Bl Bnxle|WM͈e]":z]|lw)lMT5wzqv799|I6jUt%^ڶvr:Z؆7ha3~GKjW?,db*ukS~%ިrd^cRy}Ȫ87:X3z@f'HbuQ.~&_ԺدЎ^}ɳoVMG% A%mscu=5uEM2IЈ fy( Jؾ2Rl_l;%q@}k{!,aGVȿf14Tg SBb"1FWz] * 橡M5ݜ_#PJ5vNm 3˭JP6o9/ڍk(@`m (b?$w=2vz蚃k+VU λ()~3h^i9Gt1:[MpQ.av  y- ReeEf9|DXC}Grz},iӼ +Bݺޮ֝Ck s 5s̍{(,`܄W|a1As. 3|"-3NnUc>2~0[mݬMݓCqYk9r )dwZ0-PfD$* xy-8`#!"KΩ2! yc(V 9O heLB&0Bݞl@(]Bwp9ikM\!>v 7;elԸ\M55_TʧPW A80}f8k,B;6 :r&_CF`*K—lPBM X3—dӅh\vwCP8h5?7GrU1`K 5Ri|j!KJ迲Cf endstream endobj 1007 0 obj << /Length 74 /Filter /FlateDecode >> stream x3T0BC]=eUedA z@` #1!FB=sL\@!(  endstream endobj 1004 0 obj << /Type /XObject /Subtype /Form /FormType 1 /PTEX.FileName (./schafer02.pdf) /PTEX.PageNumber 16 /PTEX.InfoDict 870 0 R /BBox [0 0 612 792] /Resources << /Font << /F14 871 0 R /F15 872 0 R /F17 1010 0 R/F18 1011 0 R/F2 894 0 R /F5 903 0 R /F3 897 0 R /F4 900 0 R /F1 875 0 R >> /ExtGState << /GS1 877 0 R >>>> /Length 3060 /Filter /FlateDecode >> stream xZs6_n, 2i'^t:7=m*HJVAK |vۏbv]|F虘-.rVXg)Ypn^dl_PJ})-gB~|! ~ -p*ށa&?7W eׯnǽ??ӭa_)Zb~1nf(R ɳrγ9 *PL 3ÔꮫwㇲyfN$\늬|{*L9e1^.5&I6a;NNjWKok~V P5Rrm=9 ,9s 0'-WOUZp+`|a͊TAr44ChV$($?뷠JHLpJ&IƖ8m>[H`Iﯱ ]ݘDaSiTkR˿˜WoiL]t>iϏ4=vqQ2,fs9y\~ ffWEܕ/]]FA[uUٮG]_oB\92H76m3&{duhsq}[UO^$dW<9SVђ!ĜS?i3]\ݵ6T_y =/7 \NF3g xIkdlnqduf€r ^PܯH;̎u$R"2vߴ=D?|˶& Mwa n #C *)!#Oٛ$&-#mY?wMK5҂"vOYoU[ؠ.Re$b6?c՝Z]جMdnUyӑ|F :j,at7x@<41.aU^Z+//|TP P} h)eV-43(D0N7Z$o ?Ǹ p 0 %}ܸԾmVЩ]{|sXG$ʐ?2LgݾZ7\YT1#׷]Hxl %5U|naIF]f_p j]U>}K&UJy #<^pn᧞yb%9 XTٖ} ﺀ#{(Ou]+77uX8/|2_xҶZTsˇp(::/ejCDr܃6 ГH i1u^ t?` :@'37sF쯊`W! K IjmI|:aK{AV(hq׮?̭A >P=~i&uB=Hxd$zGY$Ij+xd6uԻ l#!X5\U"_efs\0 8 VmyG_D/.Rދzbz:O 9,pOx7265Kf Oj(|| DB7D65auYMuU`aD']|Amj7\W I>iˤ22]) $zv00@r˴eǪH4YIqzxJzϏR[8s82sg^%tq)|Bg$)aߩ@MIef7٪igY' @H`r.7TuЛE07B)Z$sBao&4?\^Mz{,ϥG8(׹UuD> stream xW XTacvSP Dn*$J59QJAMҬ&KyJ"(o7Y39<=go}oXƩò{dddTdϤ2ADǐ,YEz;5WWpӢIth]'uips s-XRπА _XCkgLτ¼̬<Ϙٹ9ٹi{zFBgnf^f<,"K))sgy58ӱ Z&!?Fiqt|8L5jԽ0BDXU0C7@ԉ>xG 8j bLhwX +31^a$ F _YUd4ܕ0*; DÇ'0JzOi;-bD,D`x>*4 F~KĬLV>CTYR'vq1&`7~Z7ՄHw._&^w*x~3)i(M|wݓO&LTEo][!@foU"$@2{Ͷi\3DD*i h!&ЄE2>f(;)ݝށ;_ .A}ard;#Dك8eQrR)wP YQF%=GjwQ:'rYw}/āl؟2e{Uc;N ([K<0{<Ο0 2R5k-#͸菣Ru\́Ȭ k9\® ܙm~P+>8L~OFuo.Xc&||EO̝A31Fި9j 1GoI?;[(3|tbV):tPXU כ_ _R2<E i5udoStp/೐>> 1|1Y:0<`4x\}=vs;mlk;PzaaA *XǷHC6ЗP`<>_nXSL-JH?1&[tWCĊz8&iɋpV|~cgiI.ak0!H'sCPÕ^8 GÞm|*J"}Oz*}7k]]]g $ਰGϓpq!>}9H1l*7V;] RLmH~q:v1G3|"`W\ڇs>BTm)B\M,|خrk{+l?ڡIH[%)9O8dx,{D]ib#ҤJ+g`a:tmg侫/SpP!'A•hzƺ-7tՕoU.eo+,`gТ4BѽTf,TBc?wj屍2U;߯.25FTQΉE!_~T;'L8 GfFnfsTbѳt86IG!uOˆZPnWA?ej&&)z28h. ĭ|Y >[_*~a©~8Toz[w˛@ VazZoasE|BB!D:yr{*DÔHpEM)2} "zʥBs+AqIj5U$IMU;nZfl־rJ]Ьْ$IPcN1&S,,B? ڏR 2I}˅)T+o{C5nNB|I:4ͽTKj3/ߝ\].Ӑxn]u݋fXgH0dоr/\ /nڝ/cYPi  ,-9lPg@/pxt^-iрf7zDG/Cڔ4>ZR7⍣XK9pMy(YXP\cwFP{ oÊQf$iJq@(?9P- Aoi_/ujy$n+m:p"0?dFvU9^l`$.^kaK Ľs|hGC0xwԸ#_|upa~1d ֶU/ !V|x'.6weh6\QCm?ՑU+Sty+bN ֖V}XZ-M !"Rs`Ԙ0,^Al7 M⦝ƁJEEpIh"{#|cq]<+TN}P6쁢:(SQh,2MlUW/YuXin<7C&I^ىW9?& F 'l9gVR2;a)2/9'^nA{PZfnJbDs28HGݪN)2=V f x\82bې,4]`G/՗úk'͑w3\7L/ 0koBsO K|Sz}TJ-="zx [+N,n[n[v43QW7=Hu xk#*[nVî"R@10YO6KY{ 5qCxo>;6I D&{E7E7a> stream HU PWfneɸL{:~@ Q~*"(!3` / E姠h~F $N!76V*U[U];sϻ>pt H2+mfMY=a^Қԏ25}G$x  C g6R?@^șKrBJN5\1%XK]*C96y۽N}CM(szeXZVթ34)5kJFe3WjeNY֭VjW(禦i3r5YJuZ$N*3)j]FgjBE…$8*K " B%$h]QZBB_k2~!J8'XU5x5$Bb /; XGlٙK0PhY &g& EOY`TcLC1*A~&xPسV< $L{05wyVZAMb"p*AEh #?tW,:|ldãn7l|<<>~P I~c=Kf7M-R<'3ύc6 ^Ԭz i H%; K`+aFM-t~V k͐n&d7PG#I%C ;gAS'!iDà *yu^AQ=^~] 6 8#Vpx+^A0RXYu5qTQ`rN@J݉:6hcZvԍ[9#v\cȐ=rB'VY{Pwϻç 8B/Ov>'U\^~MdO鵎'd/me@Z䧬s:/`k]HyVGH:0zN?^\:_ug$N~پup.4s.'oGM4(n5f}Km8e K;q=g@]d,OZ"j^*|7Y߻sX].`慫U [(&7fa?{K`a80{)X"- h1M#0I^kZT+E|D3vÁ0`.,1<(HU?vl 'fmdqbŶD-؇bf޲]{7ośO6oY~E^J9T QmL)._3mPY/{b<GgZE̲>fHgX,ǿm.]7O m> stream x3T0BC]=eUedA z@` #1!FB=sL\@!(7 endstream endobj 1016 0 obj << /Type /XObject /Subtype /Form /FormType 1 /PTEX.FileName (./schafer02.pdf) /PTEX.PageNumber 17 /PTEX.InfoDict 870 0 R /BBox [0 0 612 792] /Resources << /Font << /F15 872 0 R /F14 871 0 R /F20 1021 0 R/F16 876 0 R /F5 903 0 R /F2 894 0 R /F6 906 0 R /F4 900 0 R >> /ExtGState << /GS1 877 0 R >>>> /Length 3287 /Filter /FlateDecode >> stream xZ[s~#{x<4Lv"&TDBjPPH[u_Vf9ջk1ݴWoE2$gfǃRd,5rjR͸Jd~>]gbXzPʔh~y?\}Nܐ/!Kو_ߞ9ZqՋVB1HVB>wNY&̘Νle2Sl뫿;wxT5ә5,WcK8vNhN,gEЉ$R/5%(5B3ʦz?vec;Jfts m[6z7mVuG\R p>M;DKZ &a*˧ۉ7 izLIfDLQ$i !MD\0nH1F˜erO4&cJ1~wm'דo'+rᒩkݛ/Tb8frJfpLXv=#e* rk>u,-&Z\[f)Kӈ$9G$L Ej@=Ne>( 'oJ>. s%%l!2\3h3--XJd)SNqצY"B?¶?J!1'+=ZIU>.G cx>S=5B0Kdj6H`$1M> sc ~5&L(<# SFb$\!;b@=b 4GDI gf_Dtx1 FJ5X/W0Q2?SG##<&pi61BL4v =`2IpCnLƙ MP"J;H' Ln45JZeC5j&Fc"Pf4A@R2Jw8cR!7iIDo{ɑKˠ^Iws"LZS#rط]sXu 7þ:Û=lroup,zWb}s^0[wꜲkzQݕ-aoE8@]dVm\)3IZKK >++Ox[Uu~ R fwnkzUcTE<(WoKGwky7~wy ;-op nܧ\c-WKQ1,=%uD-$ϑ6ѭ̠Cz~K]ѕGBa+i.6 ue}>'nB/nI ]j']m&ƙ2ۺH*O[H&XȊ+9jj޺Ot1.J-!M9j^qava&pUك9ف P;3_= F һ,0{x19BGQ[)H?-l1!4&gYq`NKrA_[ˣB(%JXI[oe"aTpO[>/,( ^QCA5]wu˶ aMTHVUP87no۰h֦ABY $XM:艢>lq\(E'U2cs6(C>R֤'$-Wš{0F&Zk(-/uOjGxz*د]XwĹket}Wնd3 k[=V#z*Zjd`_Wc>LqEh)ʵDoGn ê| 8@h 5|)|ꢈ'3{"ŪGχ/^3S>\Z3χߐ9Uƪ@-ߣT2~ȋ:#a1/9nuPI^0)kAӢXZp[Ќ Q0c NHυ>hjv, y7 T/?$Odա }ѱ.)UTZ^g#2sh]cQr&*ptzگ—Hbسƙ Zl [@RbIx'MMFIDc3L]GcJVs<9=d>}!':-Čz"(~tdJLRFe! ŀ';ݟj{uFuONrof8,֧8=ϓ_4ybSQTx6ԩ?kjI$#S?/ifAQOhC}#%/2}F%QTNx*mko; Kk_i,|>#-*Ur&YQy9.\:`hCW\j mztK K"L&caAS\m,mWJ>*?eJk7('*e1Ar:!-脴|(k>[q SYVjvV=gLd?w]EZwT>XHWݪEW| qn׌[mH[蜥SwBÀQ`52ڮvuӛF`1+VGLLՄ^MDg?ǩ&&{zж*Mv AvE ʅ \ed'LI32O#aE1{Y:G` Wt.R0L,kd`m taK ,Ž jS)xm#Jf*4)P赤 Ժ)}fT Na*45}T87Gʲ3)FhQD: WoٮJҘ[D1VcV ̙f3,]w$|R p9Um|X>cy6 {JGWR7Cm'vvnqu~GC_I endstream endobj 1026 0 obj << /Length 74 /Filter /FlateDecode >> stream x3T0BC]=eUedA z@` #1!FB=s\@!(b endstream endobj 1023 0 obj << /Type /XObject /Subtype /Form /FormType 1 /PTEX.FileName (./schafer02.pdf) /PTEX.PageNumber 18 /PTEX.InfoDict 870 0 R /BBox [0 0 612 792] /Resources << /Font << /F14 871 0 R /F15 872 0 R /F20 1021 0 R /F17 1010 0 R /F18 1011 0 R /F1 875 0 R /F11 1030 0 R /F12 1033 0 R /F5 903 0 R /F4 900 0 R /F2 894 0 R >> /ExtGState << /GS1 877 0 R >>>> /Length 3110 /Filter /FlateDecode >> stream x]8="/8ײ-Evڛδd{pb%ciE0@CRoR7~y$v;|%nE܃?0GqzA<_fN(|>#L Џߧ?fA 'ҕ?=>/]߹}|K/?;߻;LKPϟ6NyH7fx*[,9k<$L&i$Y"p'*wmWN | @7BL2En>G^S=WgSkB:7˪>="!*`#tttXnog{eOk^xz$CFh&)?$DTC76AO[TX 眏O+ڲ?ٿW닉Pç?V{ckW4UŹ ڈ&{o?t?.0 S2*8djR;n7LzQ e&.Y1> lBU쳒7$KWnզkb+s[7)' HHa= 0cYﲦ8V|,Es> z6+qA/.Jۄ@#;!/X.2%cҴ8Ŕ (c`5ZW'$)_>2NU9/-3ƕ0c˾em8!f .$(zRI, )3CLxQbxSۑ- G6{3cx]T~V`_>:~A( TgUgL,q崀N ΋Xfq/#ȟ X{ B)}DB/+vy1Pd8hM-H{+k@N`f*j3ܘ"/BV-B'j1)/uiϺvbM{zX@jtʳ _ ىbb:ѢL#}\֐a4QW2j 8=iWAɵQ+ՂxY#8c5Q=Zʨz7u6!u-Rj~Zm~XC= /[ʢO:i!0dیƾkoFGwD(*F}GtȇXtMAu0 -q!9[}Lj-vOj\[5 7n'dBWcݸg=]"TD֏]k 2t]ճsw3.%TY]єpQ [PT` ԫ3 *xlA&b >X_iai5&W"蚠L@ ogot5V[ Ƴ]O(Iq賯T@a5U+ `a.Zv[74 !\D3_A?UAf؏;Hx@P-8;KF_zB]CEcrl1ix!}_jC$SmX+tCMCCrQMZi,YhzSC(K5RB"BSV :Es&7!ôT-q%-Q"`Nv5&'!eBmfr G1Ե u '-QyǜK! } Ng-i#ZazƆp{I˅UolQbr  نPƂ05=Ppm-O eg!^T?"pCHA?#/1AASj.g 4| > stream HTP= {I]d臚)8 BU*d??avl~ Nadn ᆣ%j0V5+''q#N k",o~6o*8!E %oʟԄƠhDhD%;6<ӵ$+D%YߜO%d\̦,X!{0 mL endstream endobj 1036 0 obj << /Length 215 /Filter /FlateDecode >> stream HTP=0 +4t(i E63ߟ~Qӓ$Б  V,AY:ܳ> stream x3T0BC]=eUedA z@` #1!FB=s\@!( endstream endobj 1038 0 obj << /Type /XObject /Subtype /Form /FormType 1 /PTEX.FileName (./schafer02.pdf) /PTEX.PageNumber 19 /PTEX.InfoDict 870 0 R /BBox [0 0 612 792] /Resources << /Font << /F15 872 0 R /F14 871 0 R /F2 894 0 R /F3 897 0 R /F4 900 0 R /F5 903 0 R >> /ExtGState << /GS1 877 0 R >>>> /Length 3762 /Filter /FlateDecode >> stream xk۶M!ăi&3}IN|I:%NL)R%_} )Q<v Exz{w2^\JsHHR#"./oV_$"OuFsT*O//pG_&"ί!1d?^_XZGW-&L}w֙0RNziiL&SziD\1F"ͦ/-"5Sraq))/EFch2X:1M M`6QaTH2o˭m/խѲyqE2imwJR?ʦ]inӏYѭXOJTJ!u'+osphq+^ncۺ\W8=q,,X<\h3-4fAbuGW?-1t%xx4~tlYTOM[})[<1㥭mg=u֩=TM C]oaVe4,٢ᓁ@(IQuQ=/ <E[[0)`-wS >,ܲev anFF{苲Fsgx\^<*ڕ̖Oݍ'8|+fc6 nmi5Y9e0,l<𡬪/lt҇? Pn\IDAJ$73+{Xݾ]8`aaw̼aXW_T<}nPϣI1jXOD|؅7~l;R\#h$uD>6,J+jm{} SٮiA.!AÉz˱`[Xfv=Cl$ ](`=8`y\˲P׷EaZnko` cGRv}G/!_Vajq1lKTT属 ?8w$XF4#7[<5 A`hfOӆT@8Nٗ;'rbp.;!i(i!ye!E#;qEփH zUO[8ԐX9xʁ>]lJQJ4%GvAж- Aw{CgrB{3|!<r.˰3G:F)Q3,> Piǩגh\xzgsEuzlW ##yF{WP\փ؃9|Z!L :-?P j?c? &% ;qb 9ύ>iң%&^zx\:TŞr(nFХuYQDxsqSKMg&I|8 4F lx !!{U /Cܘ YM!Ѿ?mE!?ruG, xP E&U<:n@ m[ #ʐ3Ԃ+5{M4>Ь9HPk'RDq:2e`;v0q\N$ ӄ ]aZ~e_Z$rC5 d4nƘF'e4F7+)ܶd~L:ztG(B~А U%!f!raˤu"Jub3;`ش#Rra}cJ)nTLob* nxgwPjh&GOĭ=M@pk.Ԉ0L9pySelFg {).!bG󎋮<+]] 6D+[3K]E)UtE|E<h~o 5;%_ܜD.TahRj#F[u[,@Vn=iHBkӟe CyƆDfr5S<1ZF6:zj gI$jpбǜE CHFZBgТDD2l'd,`r]xDeq5HCd Bq+ê!a<tu ބ A]v'3( 0\i2zv bJL_e"#?J p42!2Wu` |,6a ]2a-hq@"8m\I+"4F2ϯQ.R21BEiX5rwVyC`W6eCЙyyR h s*#!?t%t䡡E4JL׵j350j]8G8ɏu*բ!䡶bh}/l DM̱5DG{(³ ߌς+Z6†zƘuq.\ܻ;*|#pFRϕ]s Ǒ:S]2,+8sG Zct<1<z5JJ)榉@߳26Yϲbo]> ЧDC,"({SRIȹڙz~ [Ȉ'tF]L^f$˜/I2P@6& KcbոshNq _CBiLx#=-ա{8 =c-2aU˩ >c\72<@ؕ(9E1zʴHl~BBO);<ז.eI~aqUy֫4P?ab3#ԍucO.t2*2 R,!Ľ#J?Vх;'D~YdTڪpm $M4Lo<ڮsr'7QWg<) VOrHO.*f,9!jگR^x32cV\sh"oȆKH(:܁T})WFaal$S1pC=` endstream endobj 1046 0 obj << /Length 74 /Filter /FlateDecode >> stream x3T0BC]=eUedA z@` #1!FB=s,\@!( endstream endobj 1043 0 obj << /Type /XObject /Subtype /Form /FormType 1 /PTEX.FileName (./schafer02.pdf) /PTEX.PageNumber 20 /PTEX.InfoDict 870 0 R /BBox [0 0 612 792] /Resources << /Font << /F14 871 0 R /F15 872 0 R /F5 903 0 R /F2 894 0 R /F3 897 0 R /F4 900 0 R /F7 909 0 R /F16 876 0 R >> /ExtGState << /GS1 877 0 R >>>> /Length 3758 /Filter /FlateDecode >> stream xZYs~ׯ`^Rdc08].WIcEE%q0]O3yHbUY=w7 t>?q'o/grܜ(̟N"7pGA Gdh0&n@m2yx#F o$q"jE7ۭWg/F;b]t-yiz+r5VJi4am~1Ā佣>iu*lD*r,@H/q,h@{GVϿA8".t+L$LݞP:9">/pv;WՃW[ujfw} MOp!DIE8ŅVP<uѧОJN'vW5D4>yV߅+TAN@BN)K 2K='NM~ :Gi #1, ̷uI:uC@;t:N0g2@RbW֊|"/-mް*mQאfSdƀZ\;r xrݬP9 N@?_}`]_*ϡq))B >6ΰ OmޱHDq@LFc)ےH"+[H…-VuIB46P3.[06:_lq# @Hǂ(JU+^ًsd|RbEbf:8m]ږGha벛9 ]5Ľh *jܗdcKV5I[(Ax;ǨqA4 aGϿ;i{M-# [giQ2cK8nl6ЍE8\ei-5>hW̻eǕ=W@Uc F ;%+ȶ+{|"/#(l}ҋEJ/L<7F]K8pOzx-Uxappg;,x ui;ׄL{މ; gÅ;Bga'H8pH k%2?DBH~p"I s7A]jfd̾r^AzұlOwurzq7ǚY0cGRwZN P4i+1-=OHwWr\'ek7> |w?NF3<5%eR^0#tS-K#_2Y٩ӎ88!n_8Fă%9 (3G|˓0!%+(CfW4&cu,>d#2%.=0ݛQ輏ڔ`ݍ*ͫB|?kgZL)la?9Cנp\Tox+?"8\زcw,2A!OL˛lRt( %<xg#&MB=Mû2W0hU5\ShP\nu΃%,^:GMnlݒ0τf U^x|0Q-8Ղ=ͽfPj߂~HYH*Gaz .R?LN*2:/HwG._[݂F/?; 2ȟ4kԠR|]Cc㤆 H],"l!Ó:NF mNgS~0] 8L9 C6VS 3,j<肇#剮qn ;4f&/ ȲAOHyNЖワh_2cg57_K(iN iajM{0,Q@ ʣ.HzOrHIxEc١*I bD龬+~ jS+Qpv(6p 4d|Dw V.G[2д4Oٵm|P #$P£lᄫ|)Ah*$.%^PR*q*Š]'I]&]* DX:4QidV73k%XxջI50GzHb )Bkd' X:M_9Ί\ 7(6 @D'PR:LTWjX$Do ;p&,i'['YؙL,)DI5z DcEsS) rָb9rCd¶"NN6_1t=,X:=_xOofz\B,tSX.R퀑 #ߊK;/BS֚eyǛ wE3t ]b^UCo7(k%`W`#ǑbWId&pg GFi ~wXǚb%gh6@&|yP 2aji J&)CpiCd7gHIaw/.{I>W'!dIWe;uAƘW&EN/6o`5sʭp1*D]=_m'l#|$n]BXѓDP ܃I1MVGIVXȑY,6FlIʤqld OcR~u,i[tϚ֋<vtfS:~dߤMl7A6jD"_V/Kⅶ`'NZܻ#?[ݽ+TB]1їp5\!?uT(YK"o#Vg6{Z^<+z$s!?CnEꗏqvDLrw !B݄n5cLtmkGnmd6]VVC^6Z- ulL yD=wJ9x{^4$*q|i;J \$mpӋɿs endstream endobj 1051 0 obj << /Length 74 /Filter /FlateDecode >> stream x3T0BC]=eUedA z@` #1!FB=s,\@!( endstream endobj 1048 0 obj << /Type /XObject /Subtype /Form /FormType 1 /PTEX.FileName (./schafer02.pdf) /PTEX.PageNumber 21 /PTEX.InfoDict 870 0 R /BBox [0 0 612 792] /Resources << /Font << /F15 872 0 R /F14 871 0 R /F16 876 0 R /F11 1030 0 R /F13 1055 0 R >> /ExtGState << /GS1 877 0 R >>>> /Length 3968 /Filter /FlateDecode >> stream xZo8_ 4(3( 8Alwo{ʒOΐdMwE2~o>9OGo/g:z{c~< D~=,m'$P&gjҋb:#x~hie 3ywx9 ӺYfx↜;f1ie!w2K;rԌeaN$_63(/'Yְ;cW9o7 x_o8TGanp9_ZM(S~v;b)ʮ]DpzUqR q9̉c3 ߌOaqWuZokƴL/2g4WRf?}Hs<ף%(#10V`uREIRzE44K'3Q[b0P]L߇(j}Vןfr EH#ZSxVc$Q i-RgB%H.AGM"7#GC+ix:1({*JzFZS.Ӈ4fz+(4II0a W Kj }3 w~ZøKB%ܢ]<&CsWy-b4,-ֲ.5E!8>8wB{"udKީэ[!n 0>zY/_v?4>H̘I^It@YE\ R=,Í,TTnm :1At&SX%yZ+IsC%-VE%j˭]Ds:k=SOM"-퀉'aOG6Pː"=ݦgY)3:稭{Z>ָ2Ɍ.V/ Ҫ ܞ^v5dSVFW9~WWլxzxrOnud sRj3ܦ˞ݝ-[AlփRmZ3gPEXX{ish1*8!0bQ(B/yVnXC>W }"qϡS[QwrWڬGʤږ2m%/·p+Q0CQ|p[Rwu;6:e5u5f\srY ndUɬ,;``~o3o`a%z. um)Tx/\3R"wrlkiJ@*!A}(%橒!fŶlVbVk^R˴Zl+{a{hH ?8|)3?ũK8fWJ8:, Wn=O QD j.jcpƒNƅoJ4ˌ7@V1_پ|x,HSYGQ}b}t[U"% 5E=cu뻜bY?˃06a2|_@6gS(zq2{ޮ ZiY |i8Y|KChxjH+S1DKy_{.LŌ_cAo3u@(m툌-F7NSp#K8 bC*TQI]5A5s5ےH7_ ,bG|R'߷GZ9'ٕZ έBgiq0ځYZټlINZ#ŎXa [ (;*`zEj\2".eAXTW('FH> Eլme@u/5w@KϹFe/^9B(Rjl! 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