Nothing
R/MCMCordfactanal.R
In MCMCpack: Markov Chain Monte Carlo (MCMC) Package
###########################################################################
## sample from the posterior distribution of a factor analysis model
## model in R using linked C++ code in Scythe.
##
## The model is:
##
## x*_i = \Lambda \phi_i + \epsilon_i, \epsilon_i \sim N(0, I)
##
## \lambda_{ij} \sim N(l0_{ij}, L0^{-1}_{ij})
## \phi_i \sim N(0,I)
##
## and x*_i is the latent variable formed from the observed ordinal
## variable in the usual (Albert and Chib, 1993) way.
##
## Andrew D. Martin
## Washington University
##
## Kevin M. Quinn
## Harvard University
##
## This software is distributed under the terms of the GNU GENERAL
## PUBLIC LICENSE Version 2, June 1991. See the package LICENSE
## file for more information.
##
## May 12, 2003
## Revised to accommodate new spec 7/13/2004
##
## Copyright (C) 2003-2007 Andrew D. Martin and Kevin M. Quinn
## Copyright (C) 2007-present Andrew D. Martin, Kevin M. Quinn,
## and Jong Hee Park
##########################################################################
#' Markov Chain Monte Carlo for Ordinal Data Factor Analysis Model
#'
#' This function generates a sample from the posterior distribution of an
#' ordinal data factor analysis model. Normal priors are assumed on the factor
#' loadings and factor scores while improper uniform priors are assumed on the
#' cutpoints. The user supplies data and parameters for the prior
#' distributions, and a sample from the posterior distribution is returned as
#' an mcmc object, which can be subsequently analyzed with functions provided
#' in the coda package.
#'
#' The model takes the following form:
#'
#' Let \eqn{i=1,\ldots,N} index observations and \eqn{j=1,\ldots,K}
#' index response variables within an observation. The typical
#' observed variable \eqn{x_{ij}} is ordinal with a total of \eqn{C_j}
#' categories. The distribution of \eqn{X} is governed by a \eqn{N
#' \times K} matrix of latent variables \eqn{X^*} and a series of
#' cutpoints \eqn{\gamma}. \eqn{X^*} is assumed to be generated
#' according to:
#'
#' \deqn{x^*_i = \Lambda \phi_i + \epsilon_i}
#'
#' \deqn{\epsilon_i \sim \mathcal{N}(0,I)}
#'
#' where \eqn{x^*_i} is the \eqn{k}-vector of latent variables
#' specific to observation \eqn{i}, \eqn{\Lambda} is the \eqn{k \times
#' d} matrix of factor loadings, and \eqn{\phi_i} is the
#' \eqn{d}-vector of latent factor scores. It is assumed that the
#' first element of \eqn{\phi_i} is equal to 1 for all \eqn{i}.
#'
#' The probability that the \eqn{j}th variable in observation \eqn{i}
#' takes the value \eqn{c} is:
#'
#' \deqn{\pi_{ijc} = \Phi(\gamma_{jc} - \Lambda'_j\phi_i) - \Phi(\gamma_{j(c-1)} - \Lambda'_j\phi_i) }
#'
#' The implementation used here assumes independent conjugate priors
#' for each element of \eqn{\Lambda} and each \eqn{\phi_i}. More
#' specifically we assume:
#'
#' \deqn{\Lambda_{ij} \sim \mathcal{N}(l_{0_{ij}}, L_{0_{ij}}^{-1}),
#' i=1,\ldots,k, j=1,\ldots,d}
#'
#' \deqn{\phi_{i(2:d)} \sim \mathcal{N}(0, I), i=1,\dots,n}
#'
#' The standard two-parameter item response theory model with probit link is a
#' special case of the model sketched above.
#'
#' \code{MCMCordfactanal} simulates from the posterior distribution using a
#' Metropolis-Hastings within Gibbs sampling algorithm. The algorithm employed
#' is based on work by Cowles (1996). Note that the first element of
#' \eqn{\phi_i} is a 1. As a result, the first column of
#' \eqn{\Lambda} can be interpretated as item difficulty parameters.
#' Further, the first element \eqn{\gamma_1} is normalized to zero,
#' and thus not returned in the mcmc object. The simulation proper is done in
#' compiled C++ code to maximize efficiency. Please consult the coda
#' documentation for a comprehensive list of functions that can be used to
#' analyze the posterior sample.
#'
#' As is the case with all measurement models, make sure that you have plenty
#' of free memory, especially when storing the scores.
#'
#' @param x Either a formula or a numeric matrix containing the manifest
#' variables.
#'
#' @param factors The number of factors to be fitted.
#'
#' @param lambda.constraints List of lists specifying possible equality or
#' simple inequality constraints on the factor loadings. A typical entry in the
#' list has one of three forms: \code{varname=list(d,c)} which will constrain
#' the dth loading for the variable named varname to be equal to c,
#' \code{varname=list(d,"+")} which will constrain the dth loading for the
#' variable named varname to be positive, and \code{varname=list(d, "-")} which
#' will constrain the dth loading for the variable named varname to be
#' negative. If x is a matrix without column names defaults names of ``V1",
#' ``V2", ... , etc will be used. Note that, unlike \code{MCMCfactanal}, the
#' \eqn{\Lambda} matrix used here has \code{factors}+1 columns. The
#' first column of \eqn{\Lambda} corresponds to negative item
#' difficulty parameters and should generally not be constrained.
#'
#' @param data A data frame.
#'
#' @param burnin The number of burn-in iterations for the sampler.
#'
#' @param mcmc The number of iterations for the sampler.
#'
#' @param thin The thinning interval used in the simulation. The number of
#' iterations must be divisible by this value.
#'
#' @param tune The tuning parameter for the Metropolis-Hastings sampling. Can
#' be either a scalar or a \eqn{k}-vector. Must be strictly positive.
#'
#' @param verbose A switch which determines whether or not the progress of the
#' sampler is printed to the screen. If \code{verbose} is greater than 0 the
#' iteration number and the Metropolis-Hastings acceptance rate are printed to
#' the screen every \code{verbose}th iteration.
#'
#' @param seed The seed for the random number generator. If NA, the Mersenne
#' Twister generator is used with default seed 12345; if an integer is passed
#' it is used to seed the Mersenne twister. The user can also pass a list of
#' length two to use the L'Ecuyer random number generator, which is suitable
#' for parallel computation. The first element of the list is the L'Ecuyer
#' seed, which is a vector of length six or NA (if NA a default seed of
#' \code{rep(12345,6)} is used). The second element of list is a positive
#' substream number. See the MCMCpack specification for more details.
#'
#' @param lambda.start Starting values for the factor loading matrix Lambda. If
#' \code{lambda.start} is set to a scalar the starting value for all
#' unconstrained loadings will be set to that scalar. If \code{lambda.start} is
#' a matrix of the same dimensions as Lambda then the \code{lambda.start}
#' matrix is used as the starting values (except for equality-constrained
#' elements). If \code{lambda.start} is set to \code{NA} (the default) then
#' starting values for unconstrained elements in the first column of Lambda are
#' based on the observed response pattern, the remaining unconstrained elements
#' of Lambda are set to , and starting values for inequality constrained
#' elements are set to either 1.0 or -1.0 depending on the nature of the
#' constraints.
#'
#' @param l0 The means of the independent Normal prior on the factor loadings.
#' Can be either a scalar or a matrix with the same dimensions as
#' \code{Lambda}.
#'
#' @param L0 The precisions (inverse variances) of the independent Normal prior
#' on the factor loadings. Can be either a scalar or a matrix with the same
#' dimensions as \code{Lambda}.
#'
#' @param store.lambda A switch that determines whether or not to store the
#' factor loadings for posterior analysis. By default, the factor loadings are
#' all stored.
#'
#' @param store.scores A switch that determines whether or not to store the
#' factor scores for posterior analysis. \emph{NOTE: This takes an enormous
#' amount of memory, so should only be used if the chain is thinned heavily, or
#' for applications with a small number of observations}. By default, the
#' factor scores are not stored.
#'
#' @param drop.constantvars A switch that determines whether or not manifest
#' variables that have no variation should be deleted before fitting the model.
#' Default = TRUE.
#'
#' @param ... further arguments to be passed
#'
#' @return An mcmc object that contains the posterior sample. This object can
#' be summarized by functions provided by the coda package.
#'
#' @export
#'
#' @seealso \code{\link[coda]{plot.mcmc}}, \code{\link[coda]{summary.mcmc}},
#' \code{\link[stats]{factanal}}, \code{\link[MCMCpack]{MCMCfactanal}},
#' \code{\link[MCMCpack]{MCMCirt1d}}, \code{\link[MCMCpack]{MCMCirtKd}}
#'
#' @references Shawn Treier and Simon Jackman. 2008. ``Democracy as a Latent
#' Variable." \emph{American Journal of Political Science}. 52: 201-217.
#'
#' Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park. 2011. ``MCMCpack:
#' Markov Chain Monte Carlo in R.'', \emph{Journal of Statistical Software}.
#' 42(9): 1-21. \doi{10.18637/jss.v042.i09}.
#'
#' M. K. Cowles. 1996. ``Accelerating Monte Carlo Markov Chain Convergence for
#' Cumulative-link Generalized Linear Models." \emph{Statistics and Computing.}
#' 6: 101-110.
#'
#' Valen E. Johnson and James H. Albert. 1999. ``Ordinal Data Modeling."
#' Springer: New York.
#'
#' Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. \emph{Scythe
#' Statistical Library 1.0.} \url{http://scythe.wustl.edu.s3-website-us-east-1.amazonaws.com/}.
#'
#' Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2006. ``Output
#' Analysis and Diagnostics for MCMC (CODA)'', \emph{R News}. 6(1): 7-11.
#' \url{https://CRAN.R-project.org/doc/Rnews/Rnews_2006-1.pdf}.
#'
#' @keywords models
#'
#' @examples
#'
#' \dontrun{
#' data(painters)
#' new.painters <- painters[,1:4]
#' cuts <- apply(new.painters, 2, quantile, c(.25, .50, .75))
#' for (i in 1:4){
#' new.painters[new.painters[,i]<cuts[1,i],i] <- 100
#' new.painters[new.painters[,i]<cuts[2,i],i] <- 200
#' new.painters[new.painters[,i]<cuts[3,i],i] <- 300
#' new.painters[new.painters[,i]<100,i] <- 400
#' }
#'
#' posterior <- MCMCordfactanal(~Composition+Drawing+Colour+Expression,
#' data=new.painters, factors=1,
#' lambda.constraints=list(Drawing=list(2,"+")),
#' burnin=5000, mcmc=500000, thin=200, verbose=500,
#' L0=0.5, store.lambda=TRUE,
#' store.scores=TRUE, tune=1.2)
#' plot(posterior)
#' summary(posterior)
#' }
#'
"MCMCordfactanal" <-
function(x, factors, lambda.constraints=list(),
data=parent.frame(), burnin = 1000, mcmc = 20000,
thin=1, tune=NA, verbose = 0, seed = NA,
lambda.start = NA, l0=0, L0=0,
store.lambda=TRUE, store.scores=FALSE,
drop.constantvars=TRUE, ... ) {
## check for MCMCirtKd special case, this is used to tell the R
## and C++ code what to echo (1 if regular, 2 if MCMCirtKd)
## the test is based on the existence of model="MCMCirtKd"
## passed through ...
args <- list(...)
if (length(args$model) > 0){ # if model arg is passed
if (args$model=="MCMCirtKd"){
case.switch <- 2
echo.name <- "MCMCirtKd"
}
## could allow for other possibities here but not clear what they
## would be
}
else { # if model arg not passed then assume MCMCordfactanal
case.switch <- 1
echo.name <- "MCMCordfactanal"
}
# extract X and variable names from the model formula and frame
if (is.matrix(x)){
if (drop.constantvars==TRUE){
x.col.var <- apply(x, 2, var, na.rm=TRUE)
keep.inds <- x.col.var>0
keep.inds[is.na(keep.inds)] <- FALSE
x <- x[,keep.inds]
}
X <- as.data.frame(x)
xvars <- dimnames(X)[[2]]
xobs <- dimnames(X)[[1]]
N <- nrow(X) # number of observations
K <- ncol(X) # number of manifest variables
ncat <- matrix(NA, K, 1) # vector of number of categ. in each man. var.
for (i in 1:K){
X[,i] <- factor(X[,i], ordered=TRUE)
ncat[i] <- nlevels(X[,i])
X[,i] <- as.integer(X[,i])
X[is.na(X[,i]), i] <- -999
}
X <- as.matrix(X)
}
else {
call <- match.call()
mt <- terms(x, data=data)
if (attr(mt, "response") > 0)
stop("Response not allowed in formula in ", echo.name, "().\n")
if(missing(data)) data <- sys.frame(sys.parent())
mf <- match.call(expand.dots = FALSE)
mf$factors <- mf$lambda.constraints <- mf$burnin <- mf$mcmc <- NULL
mf$thin <- mf$tune <- mf$verbose <- mf$seed <- NULL
mf$lambda.start <- mf$l0 <- mf$L0 <- NULL
mf$store.lambda <- mf$store.scores <- mf$drop.constantvars <- NULL
mf$... <- NULL
mf$drop.unused.levels <- TRUE
mf[[1]] <- as.name("model.frame")
mf$na.action <- 'na.pass'
mf <- eval(mf, sys.frame(sys.parent()))
attributes(mt)$intercept <- 0
Xterm.length <- length(attr(mt, "variables"))
X <- subset(mf,
select=as.character(attr(mt, "variables"))[2:Xterm.length])
if (drop.constantvars==TRUE){
x.col.var <- apply(X, 2, var, na.rm=TRUE)
X <- X[,x.col.var!=0]
}
N <- nrow(X) # number of observations
K <- ncol(X) # number of manifest variables
ncat <- matrix(NA, K, 1) # vector of number of categ. in each man. var.
for (i in 1:K){
X[,i] <- factor(X[,i], ordered=TRUE)
ncat[i] <- nlevels(X[,i])
X[,i] <- as.integer(X[,i])
X[is.na(X[,i]), i] <- -999
}
X <- as.matrix(X)
xvars <- dimnames(X)[[2]] # X variable names
xobs <- dimnames(X)[[1]] # observation names
}
## take care of the case where X has no row names
if (is.null(xobs)){
xobs <- 1:N
}
check.offset(list(...))
check.mcmc.parameters(burnin, mcmc, thin)
## setup constraints on Lambda
holder <- build.factor.constraints(lambda.constraints, X, K, factors+1)
Lambda.eq.constraints <- holder[[1]]
Lambda.ineq.constraints <- holder[[2]]
X.names <- holder[[3]]
## setup prior on Lambda
holder <- form.factload.norm.prior(l0, L0, K, factors+1, X.names)
Lambda.prior.mean <- holder[[1]]
Lambda.prior.prec <- holder[[2]]
if (case.switch==2){# if MCMCirtKD make it a prior on the diff. param.
Lambda.prior.mean[,1] <- Lambda.prior.mean[,1] * -1
}
# seeds
seeds <- form.seeds(seed)
lecuyer <- seeds[[1]]
seed.array <- seeds[[2]]
lecuyer.stream <- seeds[[3]]
## Starting values for Lambda
Lambda <- matrix(0, K, factors+1)
if (is.na(lambda.start)){# sets Lambda to equality constraints & 0s
for (i in 1:K){
for (j in 1:(factors+1)){
if (Lambda.eq.constraints[i,j]==-999){
if(Lambda.ineq.constraints[i,j]==0){
if (j==1){
if (ncat[i] == 2){
probit.out <- glm(as.factor(X[X[,i]!=-999,i])~1,
family=binomial(link="probit"))
probit.beta <- coef(probit.out)
Lambda[i,j] <- probit.beta[1]
}
if (ncat[i] > 2){
polr.out <- polr(ordered(X[X[,i]!=-999,i])~1)
Lambda[i,j] <- -polr.out$zeta[1]*.588
}
}
}
if(Lambda.ineq.constraints[i,j]>0){
Lambda[i,j] <- 1.0
}
if(Lambda.ineq.constraints[i,j]<0){
Lambda[i,j] <- -1.0
}
}
else Lambda[i,j] <- Lambda.eq.constraints[i,j]
}
}
}
else if (is.matrix(lambda.start)){
if (nrow(lambda.start)==K && ncol(lambda.start)==(factors+1))
Lambda <- lambda.start
else {
cat("Starting values not of correct size for model specification.\n")
stop("Please respecify and call ", echo.name, "() again\n")
}
}
else if (length(lambda.start)==1 && is.numeric(lambda.start)){
Lambda <- matrix(lambda.start, K, factors+1)
for (i in 1:K){
for (j in 1:(factors+1)){
if (Lambda.eq.constraints[i,j] != -999)
Lambda[i,j] <- Lambda.eq.constraints[i,j]
}
}
}
else {
cat("Starting values neither NA, matrix, nor scalar.\n")
stop("Please respecify and call ", echo.name, "() again\n")
}
## check MH tuning parameter
if (is.na(tune)){
tune <- matrix(NA, K, 1)
for (i in 1:K){
tune[i] <- 0.05/ncat[i]
}
}
else if (is.double(tune)){
tune <- matrix(tune/ncat, K, 1)
}
if(min(tune) < 0) {
cat("Tuning parameter is negative.\n")
stop("Please respecify and call ", echo.name, "() again\n")
}
## starting values for gamma (note: not changeable by user)
gamma <- matrix(0, max(ncat)+1, K)
for (i in 1:K){
if (ncat[i]<=2){
gamma[1,i] <- -300
gamma[2,i] <- 0
gamma[3,i] <- 300
}
if(ncat[i] > 2) {
polr.out <- polr(ordered(X[X[,i]!=-999,i])~1)
gamma[1,i] <- -300
gamma[2,i] <- 0
gamma[3:ncat[i],i] <- (polr.out$zeta[2:(ncat[i]-1)] -
polr.out$zeta[1])*.588
gamma[ncat[i]+1,i] <- 300
}
}
## define holder for posterior sample
if (store.scores == FALSE && store.lambda == FALSE){
sample <- matrix(data=0, mcmc/thin, length(gamma))
}
else if (store.scores == TRUE && store.lambda == FALSE){
sample <- matrix(data=0, mcmc/thin, (factors+1)*N + length(gamma))
}
else if(store.scores == FALSE && store.lambda == TRUE) {
sample <- matrix(data=0, mcmc/thin, K*(factors+1)+length(gamma))
}
else { # store.scores==TRUE && store.lambda==TRUE
sample <- matrix(data=0, mcmc/thin, K*(factors+1)+(factors+1)*N +
length(gamma))
}
accepts <- matrix(0, K, 1)
## Call the C++ code to do the real work
posterior <- .C("ordfactanalpost",
samdata = as.double(sample),
samrow = as.integer(nrow(sample)),
samcol = as.integer(ncol(sample)),
X = as.integer(X),
Xrow = as.integer(nrow(X)),
Xcol = as.integer(ncol(X)),
burnin = as.integer(burnin),
mcmc = as.integer(mcmc),
thin = as.integer(thin),
tune = as.double(tune),
lecuyer = as.integer(lecuyer),
seedarray = as.integer(seed.array),
lecuyerstream = as.integer(lecuyer.stream),
verbose = as.integer(verbose),
Lambda = as.double(Lambda),
Lambdarow = as.integer(nrow(Lambda)),
Lambdacol = as.integer(ncol(Lambda)),
gamma = as.double(gamma),
gammarow = as.integer(nrow(gamma)),
gammacol = as.integer(ncol(gamma)),
ncat = as.integer(ncat),
ncatrow = as.integer(nrow(ncat)),
ncatcol = as.integer(ncol(ncat)),
Lameq = as.double(Lambda.eq.constraints),
Lameqrow = as.integer(nrow(Lambda.eq.constraints)),
Lameqcol = as.integer(ncol(Lambda.ineq.constraints)),
Lamineq = as.double(Lambda.ineq.constraints),
Lamineqrow = as.integer(nrow(Lambda.ineq.constraints)),
Lamineqcol = as.integer(ncol(Lambda.ineq.constraints)),
Lampmean = as.double(Lambda.prior.mean),
Lampmeanrow = as.integer(nrow(Lambda.prior.mean)),
Lampmeancol = as.integer(ncol(Lambda.prior.prec)),
Lampprec = as.double(Lambda.prior.prec),
Lampprecrow = as.integer(nrow(Lambda.prior.prec)),
Lamppreccol = as.integer(ncol(Lambda.prior.prec)),
storelambda = as.integer(store.lambda),
storescores = as.integer(store.scores),
accepts = as.integer(accepts),
acceptsrow = as.integer(nrow(accepts)),
acceptscol = as.integer(ncol(accepts)),
outswitch = as.integer(case.switch),
PACKAGE="MCMCpack"
)
if(case.switch==1) {
accepts <- matrix(posterior$accepts, posterior$acceptsrow,
posterior$acceptscol, byrow=FALSE)
rownames(accepts) <- X.names
colnames(accepts) <- ""
cat("\n\nAcceptance rates:\n")
print(t(accepts) / (posterior$burnin+posterior$mcmc), digits=2)
}
# put together matrix and build MCMC object to return
sample <- matrix(posterior$samdata, posterior$samrow, posterior$samcol,
byrow=FALSE)
output <- mcmc(data=sample,start=1, end=mcmc, thin=thin)
par.names <- NULL
if (store.lambda==TRUE){
if(case.switch==1) {
Lambda.names <- paste(paste("Lambda",
rep(X.names,
each=(factors+1)), sep=""),
rep(1:(factors+1),K), sep=".")
}
if(case.switch==2) {
alpha.hold <- paste("alpha", X.names, sep=".")
beta.hold <- paste("beta", X.names, sep = ".")
beta.hold <- rep(beta.hold, factors, each=factors)
beta.hold <- paste(beta.hold, 1:factors, sep=".")
Lambda.names <- t(cbind(matrix(alpha.hold, K, 1),
matrix(beta.hold,K,factors,byrow=TRUE)))
dim(Lambda.names) <- NULL
}
par.names <- c(par.names, Lambda.names)
}
gamma.names <- paste(paste("gamma",
rep(0:(nrow(gamma)-1),
each=K), sep=""),
rep(X.names, nrow(gamma)), sep=".")
par.names <- c(par.names, gamma.names)
if (store.scores==TRUE){
if(case.switch==1) {
phi.names <- paste(paste("phi",
rep(xobs, each=(factors+1)), sep="."),
rep(1:(factors+1),(factors+1)), sep=".")
par.names <- c(par.names, phi.names)
}
if(case.switch==2) {
phi.names <- paste(paste("theta",
rep(xobs, each=(factors+1)), sep="."),
rep(0:factors,(factors+1)), sep=".")
par.names <- c(par.names, phi.names)
}
}
varnames(output) <- par.names
# get rid of columns for constrained parameters
output.df <- as.data.frame(as.matrix(output))
output.var <- diag(var(output.df))
output.df <- output.df[,output.var != 0]
output <- mcmc(as.matrix(output.df), start=burnin+1, end=burnin+mcmc,
thin=thin)
# add constraint info so this isn't lost
attr(output, "constraints") <- lambda.constraints
attr(output, "n.manifest") <- K
attr(output, "n.factors") <- factors
attr(output, "accept.rates") <- t(accepts) / (posterior$burnin+posterior$mcmc)
if(case.switch==1) {
attr(output,"title") <-
"MCMCpack Ordinal Data Factor Analysis Posterior Sample"
}
if(case.switch==2) {
attr(output,"title") <-
"MCMCpack K-Dimensional Item Response Theory Model Posterior Sample"
alphas <- grep("^alpha\\.", varnames(output))
output[,alphas] <- -1 * output[,alphas]
}
return(output)
}
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###########################################################################
## sample from the posterior distribution of a factor analysis model
## model in R using linked C++ code in Scythe.
##
## The model is:
##
## x*_i = \Lambda \phi_i + \epsilon_i, \epsilon_i \sim N(0, I)
##
## \lambda_{ij} \sim N(l0_{ij}, L0^{-1}_{ij})
## \phi_i \sim N(0,I)
##
## and x*_i is the latent variable formed from the observed ordinal
## variable in the usual (Albert and Chib, 1993) way.
##
## Andrew D. Martin
## Washington University
##
## Kevin M. Quinn
## Harvard University
##
## This software is distributed under the terms of the GNU GENERAL
## PUBLIC LICENSE Version 2, June 1991. See the package LICENSE
## file for more information.
##
## May 12, 2003
## Revised to accommodate new spec 7/13/2004
##
## Copyright (C) 2003-2007 Andrew D. Martin and Kevin M. Quinn
## Copyright (C) 2007-present Andrew D. Martin, Kevin M. Quinn,
## and Jong Hee Park
##########################################################################
#' Markov Chain Monte Carlo for Ordinal Data Factor Analysis Model
#'
#' This function generates a sample from the posterior distribution of an
#' ordinal data factor analysis model. Normal priors are assumed on the factor
#' loadings and factor scores while improper uniform priors are assumed on the
#' cutpoints. The user supplies data and parameters for the prior
#' distributions, and a sample from the posterior distribution is returned as
#' an mcmc object, which can be subsequently analyzed with functions provided
#' in the coda package.
#'
#' The model takes the following form:
#'
#' Let \eqn{i=1,\ldots,N} index observations and \eqn{j=1,\ldots,K}
#' index response variables within an observation. The typical
#' observed variable \eqn{x_{ij}} is ordinal with a total of \eqn{C_j}
#' categories. The distribution of \eqn{X} is governed by a \eqn{N
#' \times K} matrix of latent variables \eqn{X^*} and a series of
#' cutpoints \eqn{\gamma}. \eqn{X^*} is assumed to be generated
#' according to:
#'
#' \deqn{x^*_i = \Lambda \phi_i + \epsilon_i}
#'
#' \deqn{\epsilon_i \sim \mathcal{N}(0,I)}
#'
#' where \eqn{x^*_i} is the \eqn{k}-vector of latent variables
#' specific to observation \eqn{i}, \eqn{\Lambda} is the \eqn{k \times
#' d} matrix of factor loadings, and \eqn{\phi_i} is the
#' \eqn{d}-vector of latent factor scores. It is assumed that the
#' first element of \eqn{\phi_i} is equal to 1 for all \eqn{i}.
#'
#' The probability that the \eqn{j}th variable in observation \eqn{i}
#' takes the value \eqn{c} is:
#'
#' \deqn{\pi_{ijc} = \Phi(\gamma_{jc} - \Lambda'_j\phi_i) - \Phi(\gamma_{j(c-1)} - \Lambda'_j\phi_i) }
#'
#' The implementation used here assumes independent conjugate priors
#' for each element of \eqn{\Lambda} and each \eqn{\phi_i}. More
#' specifically we assume:
#'
#' \deqn{\Lambda_{ij} \sim \mathcal{N}(l_{0_{ij}}, L_{0_{ij}}^{-1}),
#' i=1,\ldots,k, j=1,\ldots,d}
#'
#' \deqn{\phi_{i(2:d)} \sim \mathcal{N}(0, I), i=1,\dots,n}
#'
#' The standard two-parameter item response theory model with probit link is a
#' special case of the model sketched above.
#'
#' \code{MCMCordfactanal} simulates from the posterior distribution using a
#' Metropolis-Hastings within Gibbs sampling algorithm. The algorithm employed
#' is based on work by Cowles (1996). Note that the first element of
#' \eqn{\phi_i} is a 1. As a result, the first column of
#' \eqn{\Lambda} can be interpretated as item difficulty parameters.
#' Further, the first element \eqn{\gamma_1} is normalized to zero,
#' and thus not returned in the mcmc object. The simulation proper is done in
#' compiled C++ code to maximize efficiency. Please consult the coda
#' documentation for a comprehensive list of functions that can be used to
#' analyze the posterior sample.
#'
#' As is the case with all measurement models, make sure that you have plenty
#' of free memory, especially when storing the scores.
#'
#' @param x Either a formula or a numeric matrix containing the manifest
#' variables.
#'
#' @param factors The number of factors to be fitted.
#'
#' @param lambda.constraints List of lists specifying possible equality or
#' simple inequality constraints on the factor loadings. A typical entry in the
#' list has one of three forms: \code{varname=list(d,c)} which will constrain
#' the dth loading for the variable named varname to be equal to c,
#' \code{varname=list(d,"+")} which will constrain the dth loading for the
#' variable named varname to be positive, and \code{varname=list(d, "-")} which
#' will constrain the dth loading for the variable named varname to be
#' negative. If x is a matrix without column names defaults names of ``V1",
#' ``V2", ... , etc will be used. Note that, unlike \code{MCMCfactanal}, the
#' \eqn{\Lambda} matrix used here has \code{factors}+1 columns. The
#' first column of \eqn{\Lambda} corresponds to negative item
#' difficulty parameters and should generally not be constrained.
#'
#' @param data A data frame.
#'
#' @param burnin The number of burn-in iterations for the sampler.
#'
#' @param mcmc The number of iterations for the sampler.
#'
#' @param thin The thinning interval used in the simulation. The number of
#' iterations must be divisible by this value.
#'
#' @param tune The tuning parameter for the Metropolis-Hastings sampling. Can
#' be either a scalar or a \eqn{k}-vector. Must be strictly positive.
#'
#' @param verbose A switch which determines whether or not the progress of the
#' sampler is printed to the screen. If \code{verbose} is greater than 0 the
#' iteration number and the Metropolis-Hastings acceptance rate are printed to
#' the screen every \code{verbose}th iteration.
#'
#' @param seed The seed for the random number generator. If NA, the Mersenne
#' Twister generator is used with default seed 12345; if an integer is passed
#' it is used to seed the Mersenne twister. The user can also pass a list of
#' length two to use the L'Ecuyer random number generator, which is suitable
#' for parallel computation. The first element of the list is the L'Ecuyer
#' seed, which is a vector of length six or NA (if NA a default seed of
#' \code{rep(12345,6)} is used). The second element of list is a positive
#' substream number. See the MCMCpack specification for more details.
#'
#' @param lambda.start Starting values for the factor loading matrix Lambda. If
#' \code{lambda.start} is set to a scalar the starting value for all
#' unconstrained loadings will be set to that scalar. If \code{lambda.start} is
#' a matrix of the same dimensions as Lambda then the \code{lambda.start}
#' matrix is used as the starting values (except for equality-constrained
#' elements). If \code{lambda.start} is set to \code{NA} (the default) then
#' starting values for unconstrained elements in the first column of Lambda are
#' based on the observed response pattern, the remaining unconstrained elements
#' of Lambda are set to , and starting values for inequality constrained
#' elements are set to either 1.0 or -1.0 depending on the nature of the
#' constraints.
#'
#' @param l0 The means of the independent Normal prior on the factor loadings.
#' Can be either a scalar or a matrix with the same dimensions as
#' \code{Lambda}.
#'
#' @param L0 The precisions (inverse variances) of the independent Normal prior
#' on the factor loadings. Can be either a scalar or a matrix with the same
#' dimensions as \code{Lambda}.
#'
#' @param store.lambda A switch that determines whether or not to store the
#' factor loadings for posterior analysis. By default, the factor loadings are
#' all stored.
#'
#' @param store.scores A switch that determines whether or not to store the
#' factor scores for posterior analysis. \emph{NOTE: This takes an enormous
#' amount of memory, so should only be used if the chain is thinned heavily, or
#' for applications with a small number of observations}. By default, the
#' factor scores are not stored.
#'
#' @param drop.constantvars A switch that determines whether or not manifest
#' variables that have no variation should be deleted before fitting the model.
#' Default = TRUE.
#'
#' @param ... further arguments to be passed
#'
#' @return An mcmc object that contains the posterior sample. This object can
#' be summarized by functions provided by the coda package.
#'
#' @export
#'
#' @seealso \code{\link[coda]{plot.mcmc}}, \code{\link[coda]{summary.mcmc}},
#' \code{\link[stats]{factanal}}, \code{\link[MCMCpack]{MCMCfactanal}},
#' \code{\link[MCMCpack]{MCMCirt1d}}, \code{\link[MCMCpack]{MCMCirtKd}}
#'
#' @references Shawn Treier and Simon Jackman. 2008. ``Democracy as a Latent
#' Variable." \emph{American Journal of Political Science}. 52: 201-217.
#'
#' Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park. 2011. ``MCMCpack:
#' Markov Chain Monte Carlo in R.'', \emph{Journal of Statistical Software}.
#' 42(9): 1-21. \doi{10.18637/jss.v042.i09}.
#'
#' M. K. Cowles. 1996. ``Accelerating Monte Carlo Markov Chain Convergence for
#' Cumulative-link Generalized Linear Models." \emph{Statistics and Computing.}
#' 6: 101-110.
#'
#' Valen E. Johnson and James H. Albert. 1999. ``Ordinal Data Modeling."
#' Springer: New York.
#'
#' Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. \emph{Scythe
#' Statistical Library 1.0.} \url{http://scythe.wustl.edu.s3-website-us-east-1.amazonaws.com/}.
#'
#' Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2006. ``Output
#' Analysis and Diagnostics for MCMC (CODA)'', \emph{R News}. 6(1): 7-11.
#' \url{https://CRAN.R-project.org/doc/Rnews/Rnews_2006-1.pdf}.
#'
#' @keywords models
#'
#' @examples
#'
#' \dontrun{
#' data(painters)
#' new.painters <- painters[,1:4]
#' cuts <- apply(new.painters, 2, quantile, c(.25, .50, .75))
#' for (i in 1:4){
#' new.painters[new.painters[,i]<cuts[1,i],i] <- 100
#' new.painters[new.painters[,i]<cuts[2,i],i] <- 200
#' new.painters[new.painters[,i]<cuts[3,i],i] <- 300
#' new.painters[new.painters[,i]<100,i] <- 400
#' }
#'
#' posterior <- MCMCordfactanal(~Composition+Drawing+Colour+Expression,
#' data=new.painters, factors=1,
#' lambda.constraints=list(Drawing=list(2,"+")),
#' burnin=5000, mcmc=500000, thin=200, verbose=500,
#' L0=0.5, store.lambda=TRUE,
#' store.scores=TRUE, tune=1.2)
#' plot(posterior)
#' summary(posterior)
#' }
#'
"MCMCordfactanal" <-
function(x, factors, lambda.constraints=list(),
data=parent.frame(), burnin = 1000, mcmc = 20000,
thin=1, tune=NA, verbose = 0, seed = NA,
lambda.start = NA, l0=0, L0=0,
store.lambda=TRUE, store.scores=FALSE,
drop.constantvars=TRUE, ... ) {
## check for MCMCirtKd special case, this is used to tell the R
## and C++ code what to echo (1 if regular, 2 if MCMCirtKd)
## the test is based on the existence of model="MCMCirtKd"
## passed through ...
args <- list(...)
if (length(args$model) > 0){ # if model arg is passed
if (args$model=="MCMCirtKd"){
case.switch <- 2
echo.name <- "MCMCirtKd"
}
## could allow for other possibities here but not clear what they
## would be
}
else { # if model arg not passed then assume MCMCordfactanal
case.switch <- 1
echo.name <- "MCMCordfactanal"
}
# extract X and variable names from the model formula and frame
if (is.matrix(x)){
if (drop.constantvars==TRUE){
x.col.var <- apply(x, 2, var, na.rm=TRUE)
keep.inds <- x.col.var>0
keep.inds[is.na(keep.inds)] <- FALSE
x <- x[,keep.inds]
}
X <- as.data.frame(x)
xvars <- dimnames(X)[[2]]
xobs <- dimnames(X)[[1]]
N <- nrow(X) # number of observations
K <- ncol(X) # number of manifest variables
ncat <- matrix(NA, K, 1) # vector of number of categ. in each man. var.
for (i in 1:K){
X[,i] <- factor(X[,i], ordered=TRUE)
ncat[i] <- nlevels(X[,i])
X[,i] <- as.integer(X[,i])
X[is.na(X[,i]), i] <- -999
}
X <- as.matrix(X)
}
else {
call <- match.call()
mt <- terms(x, data=data)
if (attr(mt, "response") > 0)
stop("Response not allowed in formula in ", echo.name, "().\n")
if(missing(data)) data <- sys.frame(sys.parent())
mf <- match.call(expand.dots = FALSE)
mf$factors <- mf$lambda.constraints <- mf$burnin <- mf$mcmc <- NULL
mf$thin <- mf$tune <- mf$verbose <- mf$seed <- NULL
mf$lambda.start <- mf$l0 <- mf$L0 <- NULL
mf$store.lambda <- mf$store.scores <- mf$drop.constantvars <- NULL
mf$... <- NULL
mf$drop.unused.levels <- TRUE
mf[[1]] <- as.name("model.frame")
mf$na.action <- 'na.pass'
mf <- eval(mf, sys.frame(sys.parent()))
attributes(mt)$intercept <- 0
Xterm.length <- length(attr(mt, "variables"))
X <- subset(mf,
select=as.character(attr(mt, "variables"))[2:Xterm.length])
if (drop.constantvars==TRUE){
x.col.var <- apply(X, 2, var, na.rm=TRUE)
X <- X[,x.col.var!=0]
}
N <- nrow(X) # number of observations
K <- ncol(X) # number of manifest variables
ncat <- matrix(NA, K, 1) # vector of number of categ. in each man. var.
for (i in 1:K){
X[,i] <- factor(X[,i], ordered=TRUE)
ncat[i] <- nlevels(X[,i])
X[,i] <- as.integer(X[,i])
X[is.na(X[,i]), i] <- -999
}
X <- as.matrix(X)
xvars <- dimnames(X)[[2]] # X variable names
xobs <- dimnames(X)[[1]] # observation names
}
## take care of the case where X has no row names
if (is.null(xobs)){
xobs <- 1:N
}
check.offset(list(...))
check.mcmc.parameters(burnin, mcmc, thin)
## setup constraints on Lambda
holder <- build.factor.constraints(lambda.constraints, X, K, factors+1)
Lambda.eq.constraints <- holder[[1]]
Lambda.ineq.constraints <- holder[[2]]
X.names <- holder[[3]]
## setup prior on Lambda
holder <- form.factload.norm.prior(l0, L0, K, factors+1, X.names)
Lambda.prior.mean <- holder[[1]]
Lambda.prior.prec <- holder[[2]]
if (case.switch==2){# if MCMCirtKD make it a prior on the diff. param.
Lambda.prior.mean[,1] <- Lambda.prior.mean[,1] * -1
}
# seeds
seeds <- form.seeds(seed)
lecuyer <- seeds[[1]]
seed.array <- seeds[[2]]
lecuyer.stream <- seeds[[3]]
## Starting values for Lambda
Lambda <- matrix(0, K, factors+1)
if (is.na(lambda.start)){# sets Lambda to equality constraints & 0s
for (i in 1:K){
for (j in 1:(factors+1)){
if (Lambda.eq.constraints[i,j]==-999){
if(Lambda.ineq.constraints[i,j]==0){
if (j==1){
if (ncat[i] == 2){
probit.out <- glm(as.factor(X[X[,i]!=-999,i])~1,
family=binomial(link="probit"))
probit.beta <- coef(probit.out)
Lambda[i,j] <- probit.beta[1]
}
if (ncat[i] > 2){
polr.out <- polr(ordered(X[X[,i]!=-999,i])~1)
Lambda[i,j] <- -polr.out$zeta[1]*.588
}
}
}
if(Lambda.ineq.constraints[i,j]>0){
Lambda[i,j] <- 1.0
}
if(Lambda.ineq.constraints[i,j]<0){
Lambda[i,j] <- -1.0
}
}
else Lambda[i,j] <- Lambda.eq.constraints[i,j]
}
}
}
else if (is.matrix(lambda.start)){
if (nrow(lambda.start)==K && ncol(lambda.start)==(factors+1))
Lambda <- lambda.start
else {
cat("Starting values not of correct size for model specification.\n")
stop("Please respecify and call ", echo.name, "() again\n")
}
}
else if (length(lambda.start)==1 && is.numeric(lambda.start)){
Lambda <- matrix(lambda.start, K, factors+1)
for (i in 1:K){
for (j in 1:(factors+1)){
if (Lambda.eq.constraints[i,j] != -999)
Lambda[i,j] <- Lambda.eq.constraints[i,j]
}
}
}
else {
cat("Starting values neither NA, matrix, nor scalar.\n")
stop("Please respecify and call ", echo.name, "() again\n")
}
## check MH tuning parameter
if (is.na(tune)){
tune <- matrix(NA, K, 1)
for (i in 1:K){
tune[i] <- 0.05/ncat[i]
}
}
else if (is.double(tune)){
tune <- matrix(tune/ncat, K, 1)
}
if(min(tune) < 0) {
cat("Tuning parameter is negative.\n")
stop("Please respecify and call ", echo.name, "() again\n")
}
## starting values for gamma (note: not changeable by user)
gamma <- matrix(0, max(ncat)+1, K)
for (i in 1:K){
if (ncat[i]<=2){
gamma[1,i] <- -300
gamma[2,i] <- 0
gamma[3,i] <- 300
}
if(ncat[i] > 2) {
polr.out <- polr(ordered(X[X[,i]!=-999,i])~1)
gamma[1,i] <- -300
gamma[2,i] <- 0
gamma[3:ncat[i],i] <- (polr.out$zeta[2:(ncat[i]-1)] -
polr.out$zeta[1])*.588
gamma[ncat[i]+1,i] <- 300
}
}
## define holder for posterior sample
if (store.scores == FALSE && store.lambda == FALSE){
sample <- matrix(data=0, mcmc/thin, length(gamma))
}
else if (store.scores == TRUE && store.lambda == FALSE){
sample <- matrix(data=0, mcmc/thin, (factors+1)*N + length(gamma))
}
else if(store.scores == FALSE && store.lambda == TRUE) {
sample <- matrix(data=0, mcmc/thin, K*(factors+1)+length(gamma))
}
else { # store.scores==TRUE && store.lambda==TRUE
sample <- matrix(data=0, mcmc/thin, K*(factors+1)+(factors+1)*N +
length(gamma))
}
accepts <- matrix(0, K, 1)
## Call the C++ code to do the real work
posterior <- .C("ordfactanalpost",
samdata = as.double(sample),
samrow = as.integer(nrow(sample)),
samcol = as.integer(ncol(sample)),
X = as.integer(X),
Xrow = as.integer(nrow(X)),
Xcol = as.integer(ncol(X)),
burnin = as.integer(burnin),
mcmc = as.integer(mcmc),
thin = as.integer(thin),
tune = as.double(tune),
lecuyer = as.integer(lecuyer),
seedarray = as.integer(seed.array),
lecuyerstream = as.integer(lecuyer.stream),
verbose = as.integer(verbose),
Lambda = as.double(Lambda),
Lambdarow = as.integer(nrow(Lambda)),
Lambdacol = as.integer(ncol(Lambda)),
gamma = as.double(gamma),
gammarow = as.integer(nrow(gamma)),
gammacol = as.integer(ncol(gamma)),
ncat = as.integer(ncat),
ncatrow = as.integer(nrow(ncat)),
ncatcol = as.integer(ncol(ncat)),
Lameq = as.double(Lambda.eq.constraints),
Lameqrow = as.integer(nrow(Lambda.eq.constraints)),
Lameqcol = as.integer(ncol(Lambda.ineq.constraints)),
Lamineq = as.double(Lambda.ineq.constraints),
Lamineqrow = as.integer(nrow(Lambda.ineq.constraints)),
Lamineqcol = as.integer(ncol(Lambda.ineq.constraints)),
Lampmean = as.double(Lambda.prior.mean),
Lampmeanrow = as.integer(nrow(Lambda.prior.mean)),
Lampmeancol = as.integer(ncol(Lambda.prior.prec)),
Lampprec = as.double(Lambda.prior.prec),
Lampprecrow = as.integer(nrow(Lambda.prior.prec)),
Lamppreccol = as.integer(ncol(Lambda.prior.prec)),
storelambda = as.integer(store.lambda),
storescores = as.integer(store.scores),
accepts = as.integer(accepts),
acceptsrow = as.integer(nrow(accepts)),
acceptscol = as.integer(ncol(accepts)),
outswitch = as.integer(case.switch),
PACKAGE="MCMCpack"
)
if(case.switch==1) {
accepts <- matrix(posterior$accepts, posterior$acceptsrow,
posterior$acceptscol, byrow=FALSE)
rownames(accepts) <- X.names
colnames(accepts) <- ""
cat("\n\nAcceptance rates:\n")
print(t(accepts) / (posterior$burnin+posterior$mcmc), digits=2)
}
# put together matrix and build MCMC object to return
sample <- matrix(posterior$samdata, posterior$samrow, posterior$samcol,
byrow=FALSE)
output <- mcmc(data=sample,start=1, end=mcmc, thin=thin)
par.names <- NULL
if (store.lambda==TRUE){
if(case.switch==1) {
Lambda.names <- paste(paste("Lambda",
rep(X.names,
each=(factors+1)), sep=""),
rep(1:(factors+1),K), sep=".")
}
if(case.switch==2) {
alpha.hold <- paste("alpha", X.names, sep=".")
beta.hold <- paste("beta", X.names, sep = ".")
beta.hold <- rep(beta.hold, factors, each=factors)
beta.hold <- paste(beta.hold, 1:factors, sep=".")
Lambda.names <- t(cbind(matrix(alpha.hold, K, 1),
matrix(beta.hold,K,factors,byrow=TRUE)))
dim(Lambda.names) <- NULL
}
par.names <- c(par.names, Lambda.names)
}
gamma.names <- paste(paste("gamma",
rep(0:(nrow(gamma)-1),
each=K), sep=""),
rep(X.names, nrow(gamma)), sep=".")
par.names <- c(par.names, gamma.names)
if (store.scores==TRUE){
if(case.switch==1) {
phi.names <- paste(paste("phi",
rep(xobs, each=(factors+1)), sep="."),
rep(1:(factors+1),(factors+1)), sep=".")
par.names <- c(par.names, phi.names)
}
if(case.switch==2) {
phi.names <- paste(paste("theta",
rep(xobs, each=(factors+1)), sep="."),
rep(0:factors,(factors+1)), sep=".")
par.names <- c(par.names, phi.names)
}
}
varnames(output) <- par.names
# get rid of columns for constrained parameters
output.df <- as.data.frame(as.matrix(output))
output.var <- diag(var(output.df))
output.df <- output.df[,output.var != 0]
output <- mcmc(as.matrix(output.df), start=burnin+1, end=burnin+mcmc,
thin=thin)
# add constraint info so this isn't lost
attr(output, "constraints") <- lambda.constraints
attr(output, "n.manifest") <- K
attr(output, "n.factors") <- factors
attr(output, "accept.rates") <- t(accepts) / (posterior$burnin+posterior$mcmc)
if(case.switch==1) {
attr(output,"title") <-
"MCMCpack Ordinal Data Factor Analysis Posterior Sample"
}
if(case.switch==2) {
attr(output,"title") <-
"MCMCpack K-Dimensional Item Response Theory Model Posterior Sample"
alphas <- grep("^alpha\\.", varnames(output))
output[,alphas] <- -1 * output[,alphas]
}
return(output)
}
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