Nothing
R/MCMCpaircompare2d.R
In MCMCpack: Markov Chain Monte Carlo (MCMC) Package
##########################################################################
## sample from the posterior distribution of a 2-dimensional pairwise
## comparisons model in R using linked C++ code in Scythe.
##
## Model is:
##
## i = 1,...,n.resp (resondents)
## j = 1,...,n.cand (candidates)
##
## Y_{ijj'} = 1 if i chooses j over j'
## Y_{ijj'} = 0 if i chooses j' over j
## Y_{ijj'} = NA if i chooses neither;
##
## Pr(Y_{ijj'} = 1) = \Phi( z_{i}^{T} [\theta_{j} - \theta_{ j'} ] )
## z_{i}=[cos(gamma_{i}) sin(gamma_{i})]^{T}
##
## gamma_i \overset{iid}{\sim} Unif(0, \pi/2)
## theta_j \overset{ind}{\sim} N_{2}(0, I_{2})
## (some theta_js truncated above or below 0, or fixed to constants)
##
##
## candidate IDs in columns 2 to 4 need to begin with a letter
##
## 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.
##
## Based on 1-d code from KQ 3/17/2015
## Initial 2-d code QY 10/5/2019
## Various improvements and modifications KQ and QY 2019 to 2021
## Added to MCMCpack KQ 6/26/2021
##
## 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 the Two-Dimensional Pairwise Comparisons
#' Model in Yu and Quinn (2021)
#'
#' This function generates a sample from the posterior distribution of a
#' model for pairwise comparisons data with a probit link. Unlike standard
#' models for pairwise comparisons data, in this model the latent attribute
#' of each item being compared is a vector in two-dimensional Euclidean space.
#'
#' \code{MCMCpaircompare2d} uses the data augmentation approach of Albert and
#' Chib (1993) in conjunction with Gibbs and Metropolis-within-Gibbs steps
#' to fit the model. The user supplies data and a sample from the
#' posterior is returned as an \code{mcmc} object, which can be subsequently
#' analyzed in the \code{coda} package.
#'
#' The simulation is done in compiled C++ code to maximize efficiency.
#'
#' Please consult the \code{coda} package documentation for a comprehensive
#' list of functions that can be used to analyze the posterior sample.
#'
#' The model takes the following form:
#'
#' \deqn{i = 1,...,I \ \ \ \ (raters) }
#' \deqn{j = 1,...,J \ \ \ \ (items) }
#' \deqn{Y_{ijj'} = 1 \ \ if \ \ i \ \ chooses \ \ j \ \ over \ \ j'}
#' \deqn{Y_{ijj'} = 0 \ \ if \ \ i \ \ chooses \ \ j' \ \ over \ \ j}
#' \deqn{Y_{ijj'} = NA \ \ if \ \ i \ \ chooses \ \ neither}
#'
#' \deqn{\Pr(Y_{ijj'} = 1) = \Phi( \mathbf{z}_{i}' [\boldsymbol{\theta}_{j} -
#' \boldsymbol{\theta}_{ j'} ])}
#' \deqn{\mathbf{z}_{i}=[\cos(\gamma_{i}), \ \sin(\gamma_{i})]' }
#'
#' The following priors are assumed:
#' \deqn{\gamma_i \sim \mathcal{U}nif(0, \ \pi/2)}
#' \deqn{\boldsymbol{\theta}_j \sim
#' \mathcal{N}_{2}(\mathbf{0}, \mathbf{I}_{2})}
#' For identification, some \eqn{\boldsymbol{\theta}_j}s are truncated
#' above or below 0, or fixed to constants.
#'
#'
#' @param pwc.data A data.frame containing the pairwise comparisons data.
#' Each row of \code{pwc.data} corresponds to a single pairwise comparison.
#' \code{pwc.data} needs to have exactly four columns. The first column
#' contains a unique identifier for the rater. Column two contains the unique
#' identifier for the first item being compared. Column three contains the
#' unique identifier for the second item being compared. Column four contains
#' the unique identifier of the item selected from the two items being
#' compared. If a tie occurred, the entry in the fourth column should be NA.
#' \strong{The identifiers in columns 2 through 4 must start with a letter.
#' Examples are provided below.}
#'
#' @param theta.constraints A list specifying possible simple equality or
#' inequality constraints on the item parameters. A
#' typical entry in the list has one of three forms:
#' \code{itemname=list(d,c)} which will constrain the dth dimension of
#' theta for the item named \code{itemname} to be equal to c,
#' \code{itemname=list(d,"+")} which will constrain the dth dimension of
#' theta for the item named \code{itemname} to be positive, and
#' \code{itemname=list(d, "-")} which will constrain the dth dimension of
#' theta for the item named \code{itemname} to be negative.
#'
#' @param burnin The number of burn-in iterations for the sampler.
#'
#' @param mcmc The number of Gibbs iterations for the sampler.
#'
#' @param thin The thinning interval used in the simulation. The number of
#' Gibbs iterations must be divisible by this value.
#'
#' @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
#' output is 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 gamma.start The starting value for the gamma vector. This
#' can either be a scalar or a column vector with dimension equal to the number
#' of raters. If this takes a scalar value, then that value will serve as the
#' starting value for all of the gammas. The default value of NA will set the
#' starting value of each gamma parameter to \eqn{\pi/4}.
#'
#'
#' @param theta.start Starting values for the theta. Can be either a numeric
#' scalar, a J by 2 matrix (where J is the number of items compared), or NA.
#' If a scalar, all theta values are set to that value (except elements already
#' specified via theta.contraints. If NA, then non constrained elements of
#' theta are set equal to 0, elements constrained to be positive are set equal
#' to 0.5, elements constrained to be negative are set equal to -0.5 and
#' elements with equality constraints are set to satisfy those constraints.
#'
#' @param store.theta Should the theta draws be returned? Default is TRUE.
#'
#' @param store.gamma Should the gamma draws be returned? Default is TRUE.
#'
#' @param tune Tuning parameter for the random walk Metropolis proposal for
#' each gamma_i. \code{tune} is the width of the uniform proposal centered at
#' the current value of gamma_i. Must be a positive scalar.
#'
#' @param procrustes Should the theta and gamma draws be post-processed with
#' a Procrustes transformation? Default is FALSE. The Procrustes target matrix
#' is derived from the constrained elements of theta. Each row of theta that
#' has both theta values constrained is part of the of the target matrix.
#' Elements with equality constraints are set to those values. Elements
#' constrained to be positive are set to 1. Elements constrained to be negative
#' are set to -1. If \code{procrustes} is set to \code{TRUE} theta.constraints
#' must be set so that there are at least three rows of theta that have both
#' elements of theta constrained.
#'
#' @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.
#'
#' @author Qiushi Yu <yuqiushi@umich.edu> and
#' Kevin M. Quinn <kmq@umich.edu>
#'
#' @seealso \code{\link[coda]{plot.mcmc}},\code{\link[coda]{summary.mcmc}},
#' \code{\link[MCMCpack]{MCMCpaircompare}},
#' \code{\link[MCMCpack]{MCMCpaircompare2dDP}}
#'
#' @references Albert, J. H. and S. Chib. 1993. ``Bayesian Analysis of Binary
#' and Polychotomous Response Data.'' \emph{J. Amer. Statist. Assoc.} 88,
#' 669-679
#'
#' Yu, Qiushi and Kevin M. Quinn. 2021. ``A Multidimensional Pairwise
#' Comparison Model for Heterogeneous Perceptions with an Application to
#' Modeling the Perceived Truthfulness of Public Statements on COVID-19.''
#' University of Michigan Working Paper.
#'
#' 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}.
#'
#' Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. \emph{Scythe
#' Statistical Library 1.0.} \url{http://scythe.lsa.umich.edu}.
#'
#' 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{
#' ## a synthetic data example
#' set.seed(123)
#'
#' I <- 65 ## number of raters
#' J <- 50 ## number of items to be compared
#'
#'
#' ## raters 1 to 5 put most weight on dimension 1
#' ## raters 6 to 10 put most weight on dimension 2
#' ## raters 11 to I put substantial weight on both dimensions
#' gamma.true <- c(runif(5, 0, 0.1),
#' runif(5, 1.47, 1.57),
#' runif(I-10, 0.58, 0.98) )
#' theta1.true <- rnorm(J, m=0, s=1)
#' theta2.true <- rnorm(J, m=0, s=1)
#' theta1.true[1] <- 2
#' theta2.true[1] <- 2
#' theta1.true[2] <- -2
#' theta2.true[2] <- -2
#' theta1.true[3] <- 2
#' theta2.true[3] <- -2
#'
#'
#'
#' n.comparisons <- 125 ## number of pairwise comparisons for each rater
#'
#' ## generate synthetic data according to the assumed model
#' rater.id <- NULL
#' item.1.id <- NULL
#' item.2.id <- NULL
#' choice.id <- NULL
#' for (i in 1:I){
#' for (c in 1:n.comparisons){
#' rater.id <- c(rater.id, i+100)
#' item.numbers <- sample(1:J, size=2, replace=FALSE)
#' item.1 <- item.numbers[1]
#' item.2 <- item.numbers[2]
#' item.1.id <- c(item.1.id, item.1)
#' item.2.id <- c(item.2.id, item.2)
#' z <- c(cos(gamma.true[i]), sin(gamma.true[i]))
#' eta <- z[1] * (theta1.true[item.1] - theta1.true[item.2]) +
#' z[2] * (theta2.true[item.1] - theta2.true[item.2])
#' prob.item.1.chosen <- pnorm(eta)
#' u <- runif(1)
#' if (u <= prob.item.1.chosen){
#' choice.id <- c(choice.id, item.1)
#' }
#' else{
#' choice.id <- c(choice.id, item.2)
#' }
#' }
#' }
#' item.1.id <- paste("item", item.1.id+100, sep=".")
#' item.2.id <- paste("item", item.2.id+100, sep=".")
#' choice.id <- paste("item", choice.id+100, sep=".")
#'
#' sim.data <- data.frame(rater.id, item.1.id, item.2.id, choice.id)
#'
#'
#' ## fit the model
#' posterior <- MCMCpaircompare2d(pwc.data=sim.data,
#' theta.constraints=list(item.101=list(1,2),
#' item.101=list(2,2),
#' item.102=list(1,-2),
#' item.102=list(2,-2),
#' item.103=list(1,"+"),
#' item.103=list(2,"-")),
#' verbose=1000,
#' burnin=500, mcmc=20000, thin=10,
#' store.theta=TRUE, store.gamma=TRUE, tune=0.5)
#'
#'
#'
#'
#'
#' theta1.draws <- posterior[, grep("theta1", colnames(posterior))]
#' theta2.draws <- posterior[, grep("theta2", colnames(posterior))]
#' gamma.draws <- posterior[, grep("gamma", colnames(posterior))]
#'
#' theta1.post.med <- apply(theta1.draws, 2, median)
#' theta2.post.med <- apply(theta2.draws, 2, median)
#' gamma.post.med <- apply(gamma.draws, 2, median)
#'
#' theta1.post.025 <- apply(theta1.draws, 2, quantile, prob=0.025)
#' theta1.post.975 <- apply(theta1.draws, 2, quantile, prob=0.975)
#' theta2.post.025 <- apply(theta2.draws, 2, quantile, prob=0.025)
#' theta2.post.975 <- apply(theta2.draws, 2, quantile, prob=0.975)
#' gamma.post.025 <- apply(gamma.draws, 2, quantile, prob=0.025)
#' gamma.post.975 <- apply(gamma.draws, 2, quantile, prob=0.975)
#'
#'
#'
#' ## compare estimates to truth
#' par(mfrow=c(2,2))
#' plot(theta1.true, theta1.post.med, xlim=c(-2.5, 2.5), ylim=c(-2.5, 2.5),
#' col=rgb(0,0,0,0.3))
#' segments(x0=theta1.true, x1=theta1.true,
#' y0=theta1.post.025, y1=theta1.post.975,
#' col=rgb(0,0,0,0.3))
#' abline(0, 1, col=rgb(1,0,0,0.5))
#'
#' plot(theta2.true, theta2.post.med, xlim=c(-2.5, 2.5), ylim=c(-2.5, 2.5),
#' col=rgb(0,0,0,0.3))
#' segments(x0=theta2.true, x1=theta2.true,
#' y0=theta2.post.025, y1=theta2.post.975,
#' col=rgb(0,0,0,0.3))
#' abline(0, 1, col=rgb(1,0,0,0.5))
#'
#' plot(gamma.true, gamma.post.med, xlim=c(0, 1.6), ylim=c(0, 1.6),
#' col=rgb(0,0,0,0.3))
#' segments(x0=gamma.true, x1=gamma.true,
#' y0=gamma.post.025, y1=gamma.post.975,
#' col=rgb(0,0,0,0.3))
#' abline(0, 1, col=rgb(1,0,0,0.5))
#'
#'
#' ## plot point estimates
#' plot(theta1.post.med, theta2.post.med,
#' xlim=c(-2.5, 2.5), ylim=c(-2.5, 2.5),
#' col=rgb(0,0,0,0.3))
#' for (i in 1:length(gamma.post.med)){
#' arrows(x0=0, y0=0,
#' x1=cos(gamma.post.med[i]),
#' y1=sin(gamma.post.med[i]),
#' col=rgb(1,0,0,0.2), len=0.05, lwd=0.5)
#' }
#'}
#' @export
"MCMCpaircompare2d" <- function(pwc.data, theta.constraints=list(),
burnin=1000, mcmc=20000, thin=1,
verbose=0, seed=NA,
gamma.start=NA,
theta.start=NA,
store.theta=TRUE,
store.gamma=TRUE,
tune=0.3, procrustes=FALSE,
...){
## checks
check.offset(list(...))
check.mcmc.parameters(burnin, mcmc, thin)
if (!is.numeric(tune)){
cat("tune must be numeric.\n")
stop("Please check function arguments and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
if (length(tune) > 1){
cat("tune must be a scalar.\n")
stop("Please check function arguments and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
if (tune <= 0){
cat("tune must be a positive.\n")
stop("Please check function arguments and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
if (!(procrustes %in% c(TRUE, FALSE))){
cat("procrustes cannot take a value other than TRUE or FALSE.\n")
stop("Please check function arguments and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
## convert all columns to character data
pwc.data[,1] <- as.character(pwc.data[,1])
pwc.data[,2] <- as.character(pwc.data[,2])
pwc.data[,3] <- as.character(pwc.data[,3])
pwc.data[,4] <- as.character(pwc.data[,4])
## check input data
if (ncol(pwc.data) != 4){
cat("pwc.data must have 4 columns. The specified pwc.data does not have 4 columns.\n")
stop("Please check data and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
for (i in 1:nrow(pwc.data)){
if (!(pwc.data[i,4] %in% c(NA, pwc.data[i,2], pwc.data[i,3]))){
cat("pwc.data[", i, ",4] is not in {NA, pwc.data[", i, ",2:3]}.\n", sep="")
stop("Please check data and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
}
## extract key constants from pwc.data
n <- nrow(pwc.data)
n.resp <- length(unique(pwc.data[,1]))
n.cand <- length(unique( c(pwc.data[,2], pwc.data[,3])))
## convert pwc.data into purely numeric matrix
resp.codes <- sort(unique(pwc.data[,1]))
cand.codes <- sort(unique( c(pwc.data[,2], pwc.data[,3]) ))
pwc.data.numeric <- matrix(-999, nrow(pwc.data), 4)
for (p in 1:n){
if (pwc.data[p,1] > 0){
pwc.data.numeric[p,1] <- which(pwc.data[p,1] == resp.codes)
}
if (pwc.data[p,2] > 0){
pwc.data.numeric[p,2] <- which(pwc.data[p,2] == cand.codes)
}
if (pwc.data[p,3] > 0){
pwc.data.numeric[p,3] <- which(pwc.data[p,3] == cand.codes)
}
if (pwc.data[p,4] > 0){
pwc.data.numeric[p,4] <- which(pwc.data[p,4] == cand.codes)
}
}
## set up constraints on theta
holder <- build.pairwise.theta.constraints(theta.constraints,
cand.codes, n.cand, 2)
theta.eq.constraints <- holder[[1]]
theta.ineq.constraints <- holder[[2]]
## starting values for theta
theta <- pairwise.theta.start(theta.start, n.cand, 2,
theta.eq.constraints,
theta.ineq.constraints)
## starting values for gamma
gamma <- gamma.start
if (all(is.na(gamma.start))){
gamma <- gamma.start <- rep(pi/4, n.resp)
}
if (length(gamma.start) < n.resp){
gamma <- rep(gamma.start, length.out=n.resp)
}
if (!is.numeric(gamma.start)){
cat("gamma.start is non-numeric in MCMCpaircompare2d().\n")
stop("Please check specification and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
if (min(gamma) < 0){
cat("gamma.start takes a value < 0 in MCMCpaircompare2d().\n")
stop("Please check specification and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
if (max(gamma) > pi/2){
cat("gamma.start takes a value > pi/2 in MCMCpaircompare2d().\n")
stop("Please check specification and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
## define holder for posterior sample
if(store.gamma == FALSE & store.theta == TRUE) {
sample <- matrix(data=0, mcmc/thin, 2*n.cand)
}
else if (store.gamma == TRUE & store.theta == FALSE){
sample <- matrix(data=0, mcmc/thin, n.resp)
}
else if (store.gamma == TRUE & store.theta == TRUE){
sample <- matrix(data=0, mcmc/thin, 2*n.cand + n.resp)
}
else{
cat("Error: store.gamma == FALSE & store.theta == FALSE.\n")
stop("Please respecify and call MCMCpaircompare() again.\n",
call.=FALSE)
}
## define holder for the acceptance rate of each gamma
gamma_accept_rate<-rep(0,n.resp)
## seeds
seeds <- form.seeds(seed)
lecuyer <- seeds[[1]]
seed.array <- seeds[[2]]
lecuyer.stream <- seeds[[3]]
## create theta.sub.target and associated row indicator
theta.eq.const.holder <- theta.eq.constraints
theta.ineq.const.holder <- theta.ineq.constraints
theta.eq.const.holder[theta.eq.const.holder[,1] == -999 &
theta.ineq.const.holder[,1] != 0, 1] <-
theta.ineq.const.holder[theta.eq.const.holder[,1] == -999 &
theta.ineq.const.holder[,1] != 0, 1]
theta.eq.const.holder[theta.eq.const.holder[,2] == -999 &
theta.ineq.const.holder[,2] != 0, 2] <-
theta.ineq.const.holder[theta.eq.const.holder[,2] == -999 &
theta.ineq.const.holder[,2] != 0, 2]
theta.sub.target.indic <- ((theta.eq.const.holder[,1] != -999) &
(theta.eq.const.holder[,2] != -999))
theta.sub.target <- theta.eq.const.holder[theta.sub.target.indic,]
if (procrustes == TRUE){
if (nrow(theta.sub.target) < 3){
cat("Error: procrustes == TRUE but theta.constraints has < 3 rows.\n")
stop("Please respecify and call MCMCpaircompare() again.\n",
call.=FALSE)
}
else{
theta.eq.constraints <- 0 * theta.eq.constraints - 999
theta.ineq.constraints <- 0 * theta.ineq.constraints
}
}
## call C++ code to draw sample
posterior <- .C("cMCMCpaircompare2d",
sampledata = as.double(sample),
samplerow = as.integer(nrow(sample)),
samplecol = as.integer(ncol(sample)),
pwc.datanumericdata = as.integer(pwc.data.numeric-1),#minus one because respondents and items are indexed from zero in C++
pwc.datanumericrow = as.integer(nrow(pwc.data.numeric)),
pwc.datanumericcol = as.integer(ncol(pwc.data.numeric)),
burnin = as.integer(burnin),
mcmc = as.integer(mcmc),
thin = as.integer(thin),
lecuyer = as.integer(lecuyer),
seedarray = as.integer(seed.array),
lecuyerstream = as.integer(lecuyer.stream),
verbose = as.integer(verbose),
thetastartdata = as.double(theta),
thetastartrow = as.integer(nrow(theta)),
thetastartcol = as.integer(ncol(theta)),
gammastartdata = as.double(gamma),
gammastartrow = as.integer(length(gamma)),
gammastartcol = as.integer(1),
tunevalue=as.double(tune),
thetaeqdata = as.double(theta.eq.constraints),
thetaeqrow = as.integer(nrow(theta.eq.constraints)),
thetaeqcol = as.integer(ncol(theta.eq.constraints)),
thetaineqdata = as.double(theta.ineq.constraints),
thetaineqrow = as.integer(nrow(theta.ineq.constraints)),
thetaineqcol = as.integer(ncol(theta.ineq.constraints)),
storegamma = as.integer(store.gamma),
storetheta = as.integer(store.theta),
gammaacceptrate=as.double(gamma_accept_rate),
PACKAGE="MCMCpack"
)
## undo the C++ indexing by 0
posterior$pwc.datanumericdata <- posterior$pwc.datanumericdata + 1
theta.names <- c(paste("theta1.", cand.codes, sep = ""), paste("theta2.", cand.codes, sep = ""))
## I store theta's in the following way:
## if we have 1,2,3,4,5 cand.codes,
## then theta.names is "theta1.1" "theta1.2" "theta1.3" "theta1.4" "theta1.5"
## "theta2.1" "theta2.2" "theta2.3" "theta2.4" "theta2.5"
gamma.names <- paste("gamma.", resp.codes, sep = "")
## put together matrix and build MCMC object to return
sample <- matrix(posterior$sampledata, posterior$samplerow,
posterior$samplecol,
byrow=FALSE)
output <- mcmc(data=sample, start=burnin+1, end=burnin+mcmc, thin=thin)
names <- NULL
if(store.theta == TRUE) {
names <- c(names, theta.names)
}
if (store.gamma == TRUE){
names <- c(names, gamma.names)
}
varnames(output) <- names
if (procrustes){
gammas <- output[, grep("gamma", colnames(output))]
thetas <- output[, grep("theta", colnames(output))]
thetas1 <- thetas[,1:n.cand]
thetas2 <- thetas[,(n.cand+1):ncol(thetas)]
for (iter in 1:nrow(thetas)){
theta.sub <- cbind(thetas1[iter, theta.sub.target.indic],
thetas2[iter, theta.sub.target.indic])
procrust.out <- procrustes(theta.sub, theta.sub.target,
translation=FALSE, dilation=FALSE)
R <- procrust.out$R
theta.mat <- cbind(thetas1[iter,], thetas2[iter,])
theta.mat <- theta.mat %*% R
thetas1[iter, ] <- theta.mat[,1]
thetas2[iter, ] <- theta.mat[,2]
z <- cbind(cos(gammas[iter,]), sin(gammas[iter,]))
z <- t(R) %*% t(z)
gammas[iter,] <- atan2(z[2,], z[1,])
}
output <- mcmc(data=cbind(thetas1, thetas2, gammas),
start=burnin+1, end=burnin+mcmc, thin=thin)
}
attr(output,"title") <-
"MCMCpaircompare2d Posterior Sample"
gamma_accept_rate <- posterior$gammaacceptrate
attr(output, "gamma_accept_rate") <- gamma_accept_rate
attr(output, "procrustes") <- procrustes
return(output)
} ## end MCMCpaircompare
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##########################################################################
## sample from the posterior distribution of a 2-dimensional pairwise
## comparisons model in R using linked C++ code in Scythe.
##
## Model is:
##
## i = 1,...,n.resp (resondents)
## j = 1,...,n.cand (candidates)
##
## Y_{ijj'} = 1 if i chooses j over j'
## Y_{ijj'} = 0 if i chooses j' over j
## Y_{ijj'} = NA if i chooses neither;
##
## Pr(Y_{ijj'} = 1) = \Phi( z_{i}^{T} [\theta_{j} - \theta_{ j'} ] )
## z_{i}=[cos(gamma_{i}) sin(gamma_{i})]^{T}
##
## gamma_i \overset{iid}{\sim} Unif(0, \pi/2)
## theta_j \overset{ind}{\sim} N_{2}(0, I_{2})
## (some theta_js truncated above or below 0, or fixed to constants)
##
##
## candidate IDs in columns 2 to 4 need to begin with a letter
##
## 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.
##
## Based on 1-d code from KQ 3/17/2015
## Initial 2-d code QY 10/5/2019
## Various improvements and modifications KQ and QY 2019 to 2021
## Added to MCMCpack KQ 6/26/2021
##
## 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 the Two-Dimensional Pairwise Comparisons
#' Model in Yu and Quinn (2021)
#'
#' This function generates a sample from the posterior distribution of a
#' model for pairwise comparisons data with a probit link. Unlike standard
#' models for pairwise comparisons data, in this model the latent attribute
#' of each item being compared is a vector in two-dimensional Euclidean space.
#'
#' \code{MCMCpaircompare2d} uses the data augmentation approach of Albert and
#' Chib (1993) in conjunction with Gibbs and Metropolis-within-Gibbs steps
#' to fit the model. The user supplies data and a sample from the
#' posterior is returned as an \code{mcmc} object, which can be subsequently
#' analyzed in the \code{coda} package.
#'
#' The simulation is done in compiled C++ code to maximize efficiency.
#'
#' Please consult the \code{coda} package documentation for a comprehensive
#' list of functions that can be used to analyze the posterior sample.
#'
#' The model takes the following form:
#'
#' \deqn{i = 1,...,I \ \ \ \ (raters) }
#' \deqn{j = 1,...,J \ \ \ \ (items) }
#' \deqn{Y_{ijj'} = 1 \ \ if \ \ i \ \ chooses \ \ j \ \ over \ \ j'}
#' \deqn{Y_{ijj'} = 0 \ \ if \ \ i \ \ chooses \ \ j' \ \ over \ \ j}
#' \deqn{Y_{ijj'} = NA \ \ if \ \ i \ \ chooses \ \ neither}
#'
#' \deqn{\Pr(Y_{ijj'} = 1) = \Phi( \mathbf{z}_{i}' [\boldsymbol{\theta}_{j} -
#' \boldsymbol{\theta}_{ j'} ])}
#' \deqn{\mathbf{z}_{i}=[\cos(\gamma_{i}), \ \sin(\gamma_{i})]' }
#'
#' The following priors are assumed:
#' \deqn{\gamma_i \sim \mathcal{U}nif(0, \ \pi/2)}
#' \deqn{\boldsymbol{\theta}_j \sim
#' \mathcal{N}_{2}(\mathbf{0}, \mathbf{I}_{2})}
#' For identification, some \eqn{\boldsymbol{\theta}_j}s are truncated
#' above or below 0, or fixed to constants.
#'
#'
#' @param pwc.data A data.frame containing the pairwise comparisons data.
#' Each row of \code{pwc.data} corresponds to a single pairwise comparison.
#' \code{pwc.data} needs to have exactly four columns. The first column
#' contains a unique identifier for the rater. Column two contains the unique
#' identifier for the first item being compared. Column three contains the
#' unique identifier for the second item being compared. Column four contains
#' the unique identifier of the item selected from the two items being
#' compared. If a tie occurred, the entry in the fourth column should be NA.
#' \strong{The identifiers in columns 2 through 4 must start with a letter.
#' Examples are provided below.}
#'
#' @param theta.constraints A list specifying possible simple equality or
#' inequality constraints on the item parameters. A
#' typical entry in the list has one of three forms:
#' \code{itemname=list(d,c)} which will constrain the dth dimension of
#' theta for the item named \code{itemname} to be equal to c,
#' \code{itemname=list(d,"+")} which will constrain the dth dimension of
#' theta for the item named \code{itemname} to be positive, and
#' \code{itemname=list(d, "-")} which will constrain the dth dimension of
#' theta for the item named \code{itemname} to be negative.
#'
#' @param burnin The number of burn-in iterations for the sampler.
#'
#' @param mcmc The number of Gibbs iterations for the sampler.
#'
#' @param thin The thinning interval used in the simulation. The number of
#' Gibbs iterations must be divisible by this value.
#'
#' @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
#' output is 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 gamma.start The starting value for the gamma vector. This
#' can either be a scalar or a column vector with dimension equal to the number
#' of raters. If this takes a scalar value, then that value will serve as the
#' starting value for all of the gammas. The default value of NA will set the
#' starting value of each gamma parameter to \eqn{\pi/4}.
#'
#'
#' @param theta.start Starting values for the theta. Can be either a numeric
#' scalar, a J by 2 matrix (where J is the number of items compared), or NA.
#' If a scalar, all theta values are set to that value (except elements already
#' specified via theta.contraints. If NA, then non constrained elements of
#' theta are set equal to 0, elements constrained to be positive are set equal
#' to 0.5, elements constrained to be negative are set equal to -0.5 and
#' elements with equality constraints are set to satisfy those constraints.
#'
#' @param store.theta Should the theta draws be returned? Default is TRUE.
#'
#' @param store.gamma Should the gamma draws be returned? Default is TRUE.
#'
#' @param tune Tuning parameter for the random walk Metropolis proposal for
#' each gamma_i. \code{tune} is the width of the uniform proposal centered at
#' the current value of gamma_i. Must be a positive scalar.
#'
#' @param procrustes Should the theta and gamma draws be post-processed with
#' a Procrustes transformation? Default is FALSE. The Procrustes target matrix
#' is derived from the constrained elements of theta. Each row of theta that
#' has both theta values constrained is part of the of the target matrix.
#' Elements with equality constraints are set to those values. Elements
#' constrained to be positive are set to 1. Elements constrained to be negative
#' are set to -1. If \code{procrustes} is set to \code{TRUE} theta.constraints
#' must be set so that there are at least three rows of theta that have both
#' elements of theta constrained.
#'
#' @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.
#'
#' @author Qiushi Yu <yuqiushi@umich.edu> and
#' Kevin M. Quinn <kmq@umich.edu>
#'
#' @seealso \code{\link[coda]{plot.mcmc}},\code{\link[coda]{summary.mcmc}},
#' \code{\link[MCMCpack]{MCMCpaircompare}},
#' \code{\link[MCMCpack]{MCMCpaircompare2dDP}}
#'
#' @references Albert, J. H. and S. Chib. 1993. ``Bayesian Analysis of Binary
#' and Polychotomous Response Data.'' \emph{J. Amer. Statist. Assoc.} 88,
#' 669-679
#'
#' Yu, Qiushi and Kevin M. Quinn. 2021. ``A Multidimensional Pairwise
#' Comparison Model for Heterogeneous Perceptions with an Application to
#' Modeling the Perceived Truthfulness of Public Statements on COVID-19.''
#' University of Michigan Working Paper.
#'
#' 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}.
#'
#' Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. \emph{Scythe
#' Statistical Library 1.0.} \url{http://scythe.lsa.umich.edu}.
#'
#' 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{
#' ## a synthetic data example
#' set.seed(123)
#'
#' I <- 65 ## number of raters
#' J <- 50 ## number of items to be compared
#'
#'
#' ## raters 1 to 5 put most weight on dimension 1
#' ## raters 6 to 10 put most weight on dimension 2
#' ## raters 11 to I put substantial weight on both dimensions
#' gamma.true <- c(runif(5, 0, 0.1),
#' runif(5, 1.47, 1.57),
#' runif(I-10, 0.58, 0.98) )
#' theta1.true <- rnorm(J, m=0, s=1)
#' theta2.true <- rnorm(J, m=0, s=1)
#' theta1.true[1] <- 2
#' theta2.true[1] <- 2
#' theta1.true[2] <- -2
#' theta2.true[2] <- -2
#' theta1.true[3] <- 2
#' theta2.true[3] <- -2
#'
#'
#'
#' n.comparisons <- 125 ## number of pairwise comparisons for each rater
#'
#' ## generate synthetic data according to the assumed model
#' rater.id <- NULL
#' item.1.id <- NULL
#' item.2.id <- NULL
#' choice.id <- NULL
#' for (i in 1:I){
#' for (c in 1:n.comparisons){
#' rater.id <- c(rater.id, i+100)
#' item.numbers <- sample(1:J, size=2, replace=FALSE)
#' item.1 <- item.numbers[1]
#' item.2 <- item.numbers[2]
#' item.1.id <- c(item.1.id, item.1)
#' item.2.id <- c(item.2.id, item.2)
#' z <- c(cos(gamma.true[i]), sin(gamma.true[i]))
#' eta <- z[1] * (theta1.true[item.1] - theta1.true[item.2]) +
#' z[2] * (theta2.true[item.1] - theta2.true[item.2])
#' prob.item.1.chosen <- pnorm(eta)
#' u <- runif(1)
#' if (u <= prob.item.1.chosen){
#' choice.id <- c(choice.id, item.1)
#' }
#' else{
#' choice.id <- c(choice.id, item.2)
#' }
#' }
#' }
#' item.1.id <- paste("item", item.1.id+100, sep=".")
#' item.2.id <- paste("item", item.2.id+100, sep=".")
#' choice.id <- paste("item", choice.id+100, sep=".")
#'
#' sim.data <- data.frame(rater.id, item.1.id, item.2.id, choice.id)
#'
#'
#' ## fit the model
#' posterior <- MCMCpaircompare2d(pwc.data=sim.data,
#' theta.constraints=list(item.101=list(1,2),
#' item.101=list(2,2),
#' item.102=list(1,-2),
#' item.102=list(2,-2),
#' item.103=list(1,"+"),
#' item.103=list(2,"-")),
#' verbose=1000,
#' burnin=500, mcmc=20000, thin=10,
#' store.theta=TRUE, store.gamma=TRUE, tune=0.5)
#'
#'
#'
#'
#'
#' theta1.draws <- posterior[, grep("theta1", colnames(posterior))]
#' theta2.draws <- posterior[, grep("theta2", colnames(posterior))]
#' gamma.draws <- posterior[, grep("gamma", colnames(posterior))]
#'
#' theta1.post.med <- apply(theta1.draws, 2, median)
#' theta2.post.med <- apply(theta2.draws, 2, median)
#' gamma.post.med <- apply(gamma.draws, 2, median)
#'
#' theta1.post.025 <- apply(theta1.draws, 2, quantile, prob=0.025)
#' theta1.post.975 <- apply(theta1.draws, 2, quantile, prob=0.975)
#' theta2.post.025 <- apply(theta2.draws, 2, quantile, prob=0.025)
#' theta2.post.975 <- apply(theta2.draws, 2, quantile, prob=0.975)
#' gamma.post.025 <- apply(gamma.draws, 2, quantile, prob=0.025)
#' gamma.post.975 <- apply(gamma.draws, 2, quantile, prob=0.975)
#'
#'
#'
#' ## compare estimates to truth
#' par(mfrow=c(2,2))
#' plot(theta1.true, theta1.post.med, xlim=c(-2.5, 2.5), ylim=c(-2.5, 2.5),
#' col=rgb(0,0,0,0.3))
#' segments(x0=theta1.true, x1=theta1.true,
#' y0=theta1.post.025, y1=theta1.post.975,
#' col=rgb(0,0,0,0.3))
#' abline(0, 1, col=rgb(1,0,0,0.5))
#'
#' plot(theta2.true, theta2.post.med, xlim=c(-2.5, 2.5), ylim=c(-2.5, 2.5),
#' col=rgb(0,0,0,0.3))
#' segments(x0=theta2.true, x1=theta2.true,
#' y0=theta2.post.025, y1=theta2.post.975,
#' col=rgb(0,0,0,0.3))
#' abline(0, 1, col=rgb(1,0,0,0.5))
#'
#' plot(gamma.true, gamma.post.med, xlim=c(0, 1.6), ylim=c(0, 1.6),
#' col=rgb(0,0,0,0.3))
#' segments(x0=gamma.true, x1=gamma.true,
#' y0=gamma.post.025, y1=gamma.post.975,
#' col=rgb(0,0,0,0.3))
#' abline(0, 1, col=rgb(1,0,0,0.5))
#'
#'
#' ## plot point estimates
#' plot(theta1.post.med, theta2.post.med,
#' xlim=c(-2.5, 2.5), ylim=c(-2.5, 2.5),
#' col=rgb(0,0,0,0.3))
#' for (i in 1:length(gamma.post.med)){
#' arrows(x0=0, y0=0,
#' x1=cos(gamma.post.med[i]),
#' y1=sin(gamma.post.med[i]),
#' col=rgb(1,0,0,0.2), len=0.05, lwd=0.5)
#' }
#'}
#' @export
"MCMCpaircompare2d" <- function(pwc.data, theta.constraints=list(),
burnin=1000, mcmc=20000, thin=1,
verbose=0, seed=NA,
gamma.start=NA,
theta.start=NA,
store.theta=TRUE,
store.gamma=TRUE,
tune=0.3, procrustes=FALSE,
...){
## checks
check.offset(list(...))
check.mcmc.parameters(burnin, mcmc, thin)
if (!is.numeric(tune)){
cat("tune must be numeric.\n")
stop("Please check function arguments and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
if (length(tune) > 1){
cat("tune must be a scalar.\n")
stop("Please check function arguments and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
if (tune <= 0){
cat("tune must be a positive.\n")
stop("Please check function arguments and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
if (!(procrustes %in% c(TRUE, FALSE))){
cat("procrustes cannot take a value other than TRUE or FALSE.\n")
stop("Please check function arguments and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
## convert all columns to character data
pwc.data[,1] <- as.character(pwc.data[,1])
pwc.data[,2] <- as.character(pwc.data[,2])
pwc.data[,3] <- as.character(pwc.data[,3])
pwc.data[,4] <- as.character(pwc.data[,4])
## check input data
if (ncol(pwc.data) != 4){
cat("pwc.data must have 4 columns. The specified pwc.data does not have 4 columns.\n")
stop("Please check data and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
for (i in 1:nrow(pwc.data)){
if (!(pwc.data[i,4] %in% c(NA, pwc.data[i,2], pwc.data[i,3]))){
cat("pwc.data[", i, ",4] is not in {NA, pwc.data[", i, ",2:3]}.\n", sep="")
stop("Please check data and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
}
## extract key constants from pwc.data
n <- nrow(pwc.data)
n.resp <- length(unique(pwc.data[,1]))
n.cand <- length(unique( c(pwc.data[,2], pwc.data[,3])))
## convert pwc.data into purely numeric matrix
resp.codes <- sort(unique(pwc.data[,1]))
cand.codes <- sort(unique( c(pwc.data[,2], pwc.data[,3]) ))
pwc.data.numeric <- matrix(-999, nrow(pwc.data), 4)
for (p in 1:n){
if (pwc.data[p,1] > 0){
pwc.data.numeric[p,1] <- which(pwc.data[p,1] == resp.codes)
}
if (pwc.data[p,2] > 0){
pwc.data.numeric[p,2] <- which(pwc.data[p,2] == cand.codes)
}
if (pwc.data[p,3] > 0){
pwc.data.numeric[p,3] <- which(pwc.data[p,3] == cand.codes)
}
if (pwc.data[p,4] > 0){
pwc.data.numeric[p,4] <- which(pwc.data[p,4] == cand.codes)
}
}
## set up constraints on theta
holder <- build.pairwise.theta.constraints(theta.constraints,
cand.codes, n.cand, 2)
theta.eq.constraints <- holder[[1]]
theta.ineq.constraints <- holder[[2]]
## starting values for theta
theta <- pairwise.theta.start(theta.start, n.cand, 2,
theta.eq.constraints,
theta.ineq.constraints)
## starting values for gamma
gamma <- gamma.start
if (all(is.na(gamma.start))){
gamma <- gamma.start <- rep(pi/4, n.resp)
}
if (length(gamma.start) < n.resp){
gamma <- rep(gamma.start, length.out=n.resp)
}
if (!is.numeric(gamma.start)){
cat("gamma.start is non-numeric in MCMCpaircompare2d().\n")
stop("Please check specification and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
if (min(gamma) < 0){
cat("gamma.start takes a value < 0 in MCMCpaircompare2d().\n")
stop("Please check specification and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
if (max(gamma) > pi/2){
cat("gamma.start takes a value > pi/2 in MCMCpaircompare2d().\n")
stop("Please check specification and try MCMCpaircompare2d() again.\n",
call.=FALSE)
}
## define holder for posterior sample
if(store.gamma == FALSE & store.theta == TRUE) {
sample <- matrix(data=0, mcmc/thin, 2*n.cand)
}
else if (store.gamma == TRUE & store.theta == FALSE){
sample <- matrix(data=0, mcmc/thin, n.resp)
}
else if (store.gamma == TRUE & store.theta == TRUE){
sample <- matrix(data=0, mcmc/thin, 2*n.cand + n.resp)
}
else{
cat("Error: store.gamma == FALSE & store.theta == FALSE.\n")
stop("Please respecify and call MCMCpaircompare() again.\n",
call.=FALSE)
}
## define holder for the acceptance rate of each gamma
gamma_accept_rate<-rep(0,n.resp)
## seeds
seeds <- form.seeds(seed)
lecuyer <- seeds[[1]]
seed.array <- seeds[[2]]
lecuyer.stream <- seeds[[3]]
## create theta.sub.target and associated row indicator
theta.eq.const.holder <- theta.eq.constraints
theta.ineq.const.holder <- theta.ineq.constraints
theta.eq.const.holder[theta.eq.const.holder[,1] == -999 &
theta.ineq.const.holder[,1] != 0, 1] <-
theta.ineq.const.holder[theta.eq.const.holder[,1] == -999 &
theta.ineq.const.holder[,1] != 0, 1]
theta.eq.const.holder[theta.eq.const.holder[,2] == -999 &
theta.ineq.const.holder[,2] != 0, 2] <-
theta.ineq.const.holder[theta.eq.const.holder[,2] == -999 &
theta.ineq.const.holder[,2] != 0, 2]
theta.sub.target.indic <- ((theta.eq.const.holder[,1] != -999) &
(theta.eq.const.holder[,2] != -999))
theta.sub.target <- theta.eq.const.holder[theta.sub.target.indic,]
if (procrustes == TRUE){
if (nrow(theta.sub.target) < 3){
cat("Error: procrustes == TRUE but theta.constraints has < 3 rows.\n")
stop("Please respecify and call MCMCpaircompare() again.\n",
call.=FALSE)
}
else{
theta.eq.constraints <- 0 * theta.eq.constraints - 999
theta.ineq.constraints <- 0 * theta.ineq.constraints
}
}
## call C++ code to draw sample
posterior <- .C("cMCMCpaircompare2d",
sampledata = as.double(sample),
samplerow = as.integer(nrow(sample)),
samplecol = as.integer(ncol(sample)),
pwc.datanumericdata = as.integer(pwc.data.numeric-1),#minus one because respondents and items are indexed from zero in C++
pwc.datanumericrow = as.integer(nrow(pwc.data.numeric)),
pwc.datanumericcol = as.integer(ncol(pwc.data.numeric)),
burnin = as.integer(burnin),
mcmc = as.integer(mcmc),
thin = as.integer(thin),
lecuyer = as.integer(lecuyer),
seedarray = as.integer(seed.array),
lecuyerstream = as.integer(lecuyer.stream),
verbose = as.integer(verbose),
thetastartdata = as.double(theta),
thetastartrow = as.integer(nrow(theta)),
thetastartcol = as.integer(ncol(theta)),
gammastartdata = as.double(gamma),
gammastartrow = as.integer(length(gamma)),
gammastartcol = as.integer(1),
tunevalue=as.double(tune),
thetaeqdata = as.double(theta.eq.constraints),
thetaeqrow = as.integer(nrow(theta.eq.constraints)),
thetaeqcol = as.integer(ncol(theta.eq.constraints)),
thetaineqdata = as.double(theta.ineq.constraints),
thetaineqrow = as.integer(nrow(theta.ineq.constraints)),
thetaineqcol = as.integer(ncol(theta.ineq.constraints)),
storegamma = as.integer(store.gamma),
storetheta = as.integer(store.theta),
gammaacceptrate=as.double(gamma_accept_rate),
PACKAGE="MCMCpack"
)
## undo the C++ indexing by 0
posterior$pwc.datanumericdata <- posterior$pwc.datanumericdata + 1
theta.names <- c(paste("theta1.", cand.codes, sep = ""), paste("theta2.", cand.codes, sep = ""))
## I store theta's in the following way:
## if we have 1,2,3,4,5 cand.codes,
## then theta.names is "theta1.1" "theta1.2" "theta1.3" "theta1.4" "theta1.5"
## "theta2.1" "theta2.2" "theta2.3" "theta2.4" "theta2.5"
gamma.names <- paste("gamma.", resp.codes, sep = "")
## put together matrix and build MCMC object to return
sample <- matrix(posterior$sampledata, posterior$samplerow,
posterior$samplecol,
byrow=FALSE)
output <- mcmc(data=sample, start=burnin+1, end=burnin+mcmc, thin=thin)
names <- NULL
if(store.theta == TRUE) {
names <- c(names, theta.names)
}
if (store.gamma == TRUE){
names <- c(names, gamma.names)
}
varnames(output) <- names
if (procrustes){
gammas <- output[, grep("gamma", colnames(output))]
thetas <- output[, grep("theta", colnames(output))]
thetas1 <- thetas[,1:n.cand]
thetas2 <- thetas[,(n.cand+1):ncol(thetas)]
for (iter in 1:nrow(thetas)){
theta.sub <- cbind(thetas1[iter, theta.sub.target.indic],
thetas2[iter, theta.sub.target.indic])
procrust.out <- procrustes(theta.sub, theta.sub.target,
translation=FALSE, dilation=FALSE)
R <- procrust.out$R
theta.mat <- cbind(thetas1[iter,], thetas2[iter,])
theta.mat <- theta.mat %*% R
thetas1[iter, ] <- theta.mat[,1]
thetas2[iter, ] <- theta.mat[,2]
z <- cbind(cos(gammas[iter,]), sin(gammas[iter,]))
z <- t(R) %*% t(z)
gammas[iter,] <- atan2(z[2,], z[1,])
}
output <- mcmc(data=cbind(thetas1, thetas2, gammas),
start=burnin+1, end=burnin+mcmc, thin=thin)
}
attr(output,"title") <-
"MCMCpaircompare2d Posterior Sample"
gamma_accept_rate <- posterior$gammaacceptrate
attr(output, "gamma_accept_rate") <- gamma_accept_rate
attr(output, "procrustes") <- procrustes
return(output)
} ## end MCMCpaircompare
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