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#' Perform MCMC algorithm to generate the posterior samples for longitudinal ordinal data
#'
#' This function is used to generate the posterior samples using MCMC algorithm from the
#' cumulative probit model with the hypersphere decomposition applied to model the correlation structure
#' in the serial dependence of repeated responses.
#'
#' @param fixed a two-sided linear formula object to describe fixed-effects with the response on the left of
#' a \samp{~} operator and the terms separated by \samp{+} or \samp{*} operators, on the right.
#' The specification \code{first*second} indicates the cross of \code{first} and \code{second}.
#' This is the same as \code{first + second + first:second}.
#' @param data an optional data frame containing the variables named in \samp{fixed} and \samp{random}.
#' It requires an ``integer'' variable named by \samp{id} to denote the identifications of subjects.
#' @param random a one-sided linear formula object to describe random-effects with the terms separated by
#' \samp{+} or \samp{*} operators on the right of a \samp{~} operator.
#' @param Robustness logical. If 'TRUE' the distribution of random effects is assumed to be \cr
#' t-distribution; otherwise normal distribution.
#' @param na.action a function that indicates what should happen when the data contain NA’s.
#' The default action (\samp{na.omit}, inherited from the \samp{factory fresh} value of \cr
#' \samp{getOption("na.action")}) strips any observations with any missing values in any variables.
#' @param subset an optional expression indicating the subset of the rows of \samp{data} that should be used in the fit.
#' This can be a logical vector, or a numeric vector indicating which observation numbers are to be included,
#' or a character vector of the row names to be included. All observations are included by default.
#' @param HS.model a specification of the correlation structure in HSD model:
#' \itemize{
#' \item \code{HS.model = ~0} denotes independence, that is, \eqn{R_i} is an identity matrix,
#' \item \code{HS.model = ~IndTime+}\eqn{\cdots}\code{+IndTimer} denotes AR(r) correlation structure,
#' \item \code{HS.model = ~DiffTime1+}\eqn{\cdots}\code{+DiffTimer} denotes correlation structure related to \eqn{r}th order
#' of time difference.
#' }
#' @param hyper.params specify the values in hyperparameters in priors.
#' @param num.of.iter an integer to specify the total number of iterations; default is 20000.
#' @param Interactive logical. If 'TRUE' when the program is being run interactively for progress bar and 'FALSE' otherwise.
#' @return a list of posterior samples, parameters estimates, AIC, BIC, CIC, DIC, MPL, RJR, predicted values,
#' and the acceptance rates in MH are returned.
#'
#' @note Only a model either HSD (\samp{HS.model}) or ARMA (\samp{arma.order}) model should be specified in the function.
#' We'll provide the reference for details of the model and the algorithm for performing
#' model estimation whenever the manuscript is accepted.
#'
#' @author Kuo-Jung Lee <kuojunglee@ncku.edu.tw>
#' @references{
#' \insertRef{Lee:etal:2021}{BayesRGMM}
#'
#' \insertRef{Lee:etal:2020}{BayesRGMM}
#'
#'}
#'
#' @examples
#' \dontrun{
#' library(BayesRGMM)
#' rm(list=ls(all=TRUE))
#'
#' Fixed.Effs = c(-0.1, 0.1, -0.1) #c(-0.8, -0.3, 1.8, -0.4)
#' P = length(Fixed.Effs)
#' q = 1 #number of random effects
#' T = 7 #time points
#' N = 100 #number of subjects
#' Num.of.Cats = 3 #in KBLEE simulation studies, please fix it.
#' num.of.iter = 1000 #number of iterations
#'
#' HSD.para = c(-0.9, -0.6) #the parameters in HSD model
#' a = length(HSD.para)
#' w = array(runif(T*T*a), c(T, T, a)) #design matrix in HSD model
#'
#' for(time.diff in 1:a)
#' w[, , time.diff] = 1*(as.matrix(dist(1:T, 1:T, method="manhattan")) ==time.diff)
#'
#' x = array(0, c(T, P, N))
#' for(i in 1:N){
#' #x[,, i] = t(rmvnorm(P, rep(0, T), AR1.cor(T, Cor.in.DesignMat)))
#' x[, 1, i] = 1:T
#' x[, 2, i] = rbinom(1, 1, 0.5)
#' x[, 3, i] = x[, 1, i]*x[, 2, i]
#' }
#'
#' DesignMat = x
#'
#' #Generate a data with HSD model
#'
#'
#' #MAR
#' CPREM.sim.data = SimulatedDataGenerator.CumulativeProbit(
#' Num.of.Obs = N, Num.of.TimePoints = T, Num.of.Cats = Num.of.Cats,
#' Fixed.Effs = Fixed.Effs, Random.Effs = list(Sigma = 0.5*diag(1), df=3),
#' DesignMat = DesignMat, Missing = list(Missing.Mechanism = 2,
#' MissingRegCoefs=c(-0.7, -0.2, -0.1)),
#' HSD.DesignMat.para = list(HSD.para = HSD.para, DesignMat = w))
#'
#' print(table(CPREM.sim.data$sim.data$y))
#' print(CPREM.sim.data$classes)
#'
#' BCP.output = BayesCumulativeProbitHSD(
#' fixed = as.formula(paste("y~", paste0("x", 1:P, collapse="+"))),
#' data=CPREM.sim.data$sim.data, random = ~ 1, Robustness = TRUE,
#' subset = NULL, na.action='na.exclude', HS.model = ~IndTime1+IndTime2,
#' hyper.params=NULL, num.of.iter=num.of.iter, Interactive=0)
#'
#' BCP.Est.output = BayesRobustProbitSummary(BCP.output)
#' }
BayesCumulativeProbitHSD = function(fixed, data, random, Robustness, subset, na.action, HS.model, hyper.params, num.of.iter, Interactive)
{
# process data: reponse, fixed and random effects matrices.
cl <- match.call()
mf <- match.call(expand.dots = FALSE)
m <- match(c("fixed", "data", "subset", "na.action"), names(mf), 0L)
mf <- mf[c(1L, m)]
mf$drop.unused.levels <- TRUE
mf[[1L]] <- quote(model.frame)
names(mf)[2] = "formula"
fixed.eff = all.vars.character(fixed[-2])$m[[2]]
#fixed.eff.intercept.included = !any(grepl("-1", fixed.eff))
random.eff = all.vars.character(random)$m[[2]]
HS.model.cov = all.vars.character(HS.model)$m[[2]]
#cat("HS.model.cov = ", HS.model.cov, "\n")
TimeOrder = sort(gsub("IndTime", "", HS.model.cov[HS.model.cov %in% paste0("IndTime", 1:10)]))
#cat("TimeOrder = ", TimeOrder, "\n")
DiffTime = sort(gsub("DiffTime", "", HS.model.cov[HS.model.cov %in% paste0("DiffTime", 1:10)]))
#cat("DiffTime = ", DiffTime, "\n")
#cat("HS.model = \n")
#print(as.formula(HS.model))
interaction.terms = attr(terms.formula(as.formula(HS.model)), "term.labels")
#cat("HS.model = ", interaction.terms, "\n")
mf2 = eval(mf, parent.frame())
Terms = attr(mf2, "terms")
fixed.eff = colnames(model.matrix(Terms, mf2))
fixed.eff = fixed.eff[-1]
mf[[2L]] = update(fixed, as.formula(paste("~.+", paste(random.eff, collapse="+") )))
mf[[2L]] = update(mf[[2L]], ~.+id)
mf <- eval(mf, parent.frame())
m.design.mat <- attr(mf, "terms")
#cat("mfixed.design.mat = \n")
#print(mfixed.design.mat)
yy <- model.response(mf, "numeric") #model.response(mf, "numeric")
xx <- model.matrix(m.design.mat, mf)
#fixed.eff = attr(terms.formula(fixed), "term.labels")
#if(fixed.eff.intercept.included)
# fixed.eff = c("(Intercept)", fixed.eff)
#cat("fixed.eff = ", fixed.eff, "\n")
random.eff[random.eff=="1"] = "(Intercept)"
#cat("random.eff = ", random.eff, "\n")
x.fixed = xx[, colnames(xx)%in%fixed.eff, drop=FALSE]
z.random = xx[, colnames(xx)%in%random.eff, drop=FALSE]
id = xx[, colnames(xx)%in%"id"]
p = dim(x.fixed)[2]
q = dim(z.random)[2]
N = length(table(id))
T = range(table(id))[2]
a = length(interaction.terms)
#cat("a = ", a, "\n")
u = NULL
delta.num = 0
# for HSD model
TimeOrder = unique(TimeOrder)
DiffTime = unique(DiffTime)
if(a>0){
u = array(0, c(T, T, N, a))
if(length(TimeOrder)>0){
for(t in 1:length(TimeOrder))
u[,,1:N, t][as.matrix(dist(1:T, method="euclidean", diag = TRUE, upper = TRUE))==t] = 1
delta.num = delta.num + length(TimeOrder)
}
#cat("length(DiffTime) = ", length(DiffTime)>0, "\n")
if(length(DiffTime)>0){
for(t in 1:length(DiffTime))
u[,,1:N, delta.num+t] = (as.matrix(dist(1:T, method="euclidean", diag = TRUE, upper = TRUE)))^t
delta.num = delta.num + length(DiffTime)
}
main.terms = interaction.terms[-grep("Time",interaction.terms)]
int.terms = interaction.terms[grep(":",interaction.terms)]
#cat("main = ", main.terms, "\n")
#cat("int.terms = ", int.terms, "\n")
if(length(main.terms)>0){
#cat("=============== 1 ============\n")
for(t in 1:length(main.terms)){
uu = data[, names(data)%in%c("id", main.terms[t]), drop=FALSE]
HSD.cov = unique(uu)[, , drop=FALSE]
HSD.cov = HSD.cov[HSD.cov$id%in%id, ]
HSD.cov = as.matrix(HSD.cov[, -1])
for(sub in 1:N)
u[,,sub, delta.num+t] = matrix(HSD.cov[sub], T, T)
}
delta.num = delta.num + length(main.terms)
}
if(length(int.terms)>0){
#cat("=============== 2 ============\n")
for(t in 1:length(int.terms)){
#cat("int.terms[t]= ", int.terms[t], "\n")
int.terms.tmp = strsplit(int.terms[t], ":")[[1]]
#cat("int.terms.tmp = ", int.terms.tmp, "\n")
int.terms.tmp.IndTime.TF = int.terms.tmp %in% paste0("IndTime", 1:10)
int.terms.tmp.DiffTime.TF = int.terms.tmp %in% paste0("DiffTime", 1:10)
int.terms.tmp.IndTime = int.terms.tmp[int.terms.tmp %in% paste0("IndTime", 1:10)]
int.terms.tmp.DiffTime = int.terms.tmp[int.terms.tmp %in% paste0("DiffTime", 1:10)]
#cat("any(int.terms.tmp.IndTime) = ", length(int.terms.tmp.IndTime), "\n")
#cat("any(int.terms.tmp.DiffTime) = ", length(int.terms.tmp.DiffTime), "\n")
if(length(int.terms.tmp.IndTime)>0){
#cat("=============== 3 ============\n")
#cat("int.terms.tmp.IndTime = ", int.terms.tmp.IndTime, "\n")
IndTime.tmp = as.numeric(gsub("\\D", "", int.terms.tmp.IndTime))
#as.numeric(int.terms.tmp.IndTime) #as.numeric(gsub("IndTime", "", int.terms.tmp.IndTime))
#cat("IndTime.tmp = ", IndTime.tmp, "\n")
HSD.cov.tmp = int.terms.tmp[!int.terms.tmp.IndTime.TF]
#cat("HSD.cov.tmp = ", HSD.cov.tmp, "\n")
uu.tmp = matrix(0, T, T)
uu.tmp[as.matrix(dist(1:T, method="euclidean", diag = TRUE, upper = TRUE))==IndTime.tmp] = 1
uu = data[, names(data)%in%c("id", HSD.cov.tmp), drop=FALSE]
uu = uu[complete.cases(uu), ]
#cat("uu = \n")
#print(uu)
HSD.cov = unique(uu)[, , drop=FALSE]
#cat("HSD.cov 1= \n")
#print(HSD.cov)
HSD.cov = HSD.cov[HSD.cov$id%in%id, ]
HSD.cov = as.matrix(HSD.cov[, -1])
#cat("HSD.cov 2= \n")
#print(HSD.cov)
for(sub in 1:N){
u[,, sub, delta.num+1] = uu.tmp*matrix(HSD.cov[sub], T, T)
}
delta.num = delta.num + 1
}
else if(length(int.terms.tmp.DiffTime)>0){
#cat("=============== 4 ============\n")
DiffTime.tmp = as.numeric(gsub("DiffTime", "", int.terms.tmp.DiffTime))
HSD.cov.tmp = int.terms.tmp[!int.terms.tmp.DiffTime.TF]
uu.tmp = matrix(0, T, T)
uu.tmp[as.matrix(dist(1:T, method="euclidean", diag = TRUE, upper = TRUE))==DiffTime.tmp] = 1
uu = data[, names(data)%in%c("id", HSD.cov.tmp), drop=FALSE]
HSD.cov = unique(uu)[, , drop=FALSE]
HSD.cov = HSD.cov[HSD.cov$id%in%id, ]
HSD.cov = as.matrix(HSD.cov[, -1])
for(sub in 1:N)
u[,, sub, delta.num + 1] = uu.tmp*matrix(HSD.cov[sub], T, T)
delta.num = delta.num + 1
}
else{
#cat("=============== 5 ============\n")
#cat("int.terms.tmp = ", int.terms.tmp, "\n")
uu = data[, names(data)%in%c("id", int.terms.tmp), drop=FALSE]
HSD.cov1 = unique(uu)[, 1, drop=FALSE]
#print(head(HSD.cov1))
HSD.cov2 = unique(uu)[, 2, drop=FALSE]
#print(head(HSD.cov2))
HSD.cov = cbind(HSD.cov1, HSD.cov2)
HSD.cov = HSD.cov[HSD.cov$id%in%id, ]
#print(head(HSD.cov))
HSD.cov = as.matrix(HSD.cov[, -1])
for(sub in 1:N)
u[,, sub ,delta.num + 1] = matrix(prod(HSD.cov[sub, ]), T, T)
}
}
}
}
#print(dim(u))
#cat("delta.num = ", delta.num, "\n")
#cat("=============== 6 ============\n")
if(a != delta.num)
stop("Something wrong to assing the design matrix in HSD model.\n")
if(any(HS.model.cov==1)){
a = a+1
u = abind(array(1, c(T, T, N)), u, along=4)
}
uu = u
#cat("dim(u) = ", dim(u), "\n")
#cat("u = \n")
if(a>0)
dim(u) = c(T, T, N*a)
TimePointsAvailable = as.vector(table(id))
y = matrix(NA, T, N)
x = array(0, c(T, p, N)) #intercept,
z = array(0, c(T, length(random.eff), N)) #intercept,
id.index = unique(id)
for(i in 1:N){
y[1:TimePointsAvailable[i], i] = yy[id==id.index[i]]
x[1:TimePointsAvailable[i], , i] = as.matrix(x.fixed[id==id.index[i], ])
z[1:TimePointsAvailable[i], , i] = as.matrix(z.random[id==id.index[i], ], drop=FALSE)
}
#Defult values for hyperparameters
#sigma2.alpha = 0.1
#sigma2.beta = 1
#sigma2.delta = 1
#v.gamma = 5
#InvWishart.df = 5
#InvWishart.Lambda = diag(q)
sigma2.alpha = ifelse(is.null(hyper.params$sigma2.alpha), 0.1, hyper.params$sigma2.alpha)
sigma2.beta = ifelse(is.null(hyper.params$sigma2.beta), 1, hyper.params$sigma2.beta)
sigma2.delta = ifelse(is.null(hyper.params$sigma2.delta), 1, hyper.params$sigma2.delta)
v.gamma = ifelse(is.null(hyper.params$v.gamma), 5, hyper.params$v.gamma)
InvWishart.df = ifelse(is.null(hyper.params$InvWishart.df), 5, hyper.params$InvWishart.df)
InvWishart.Lambda = if(is.null(hyper.params$InvWishart.Lambda)) diag(q) else hyper.params$InvWishart.Lambda
UpdateYstar = TRUE
UpdateAlpha = TRUE
UpdateRandomEffect = TRUE
UpdateBeta = TRUE
UpdateSigma = TRUE
UpdateNu = TRUE
UpdateDelta = ifelse(is.null(u), FALSE, TRUE)
Num.of.Cats = length(unique(na.omit(c(y))))
y.star.ini = matrix(0, T, N)
alpha.ini = c(-Inf, seq(-5, 5, length = Num.of.Cats-1), Inf)
#print(alpha.ini)
y = y-min(y, na.rm=TRUE)+1 # to make the categorical variable begin with 1
#y[is.na(y)] = 1000 # missing values specified by 1000
y[!is.finite(y)] = 1000
#y[is.nan(y)] = 1000
#print(head(is.finite(y)))
#print(head(y))
for(i in 1:Num.of.Cats)
y.star.ini[y%in%i] = rtnorm(sum(y%in%i), lower=alpha.ini[i], upper=alpha.ini[i+1])
#print(head(y.star.ini))
b.ini = NULL
Sigma = diag(q)
for(i in 1:N)
b.ini = cbind(b.ini, t(rmvnorm(1, rep(0, q), Sigma)))
nu.ini = rgamma(N, 5, 5)
beta.ini = matrix(rep(0, p), ncol=1)
Sigma.ini = as.matrix(rWishart(1,q,diag(q))[,,1])
delta.ini = rep(0, a)#runif(a, -1, 1)
Data = list(Y = y, X = x, Z=z, U = u, TimePointsAvailable = TimePointsAvailable)
InitialValues = list(y.star = y.star.ini, alpha = alpha.ini, b = b.ini, nu = nu.ini, beta = beta.ini , Sigma = Sigma.ini, delta = delta.ini)
HyperPara = list(sigma2.beta = sigma2.beta, sigma2.delta=sigma2.delta, sigma2.alpha=sigma2.alpha, v.gamma = v.gamma, InvWishart.df = InvWishart.df, InvWishart.Lambda=InvWishart.Lambda)
UpdatePara = list(UpdateYstar = UpdateYstar, UpdateAlpha = UpdateAlpha, UpdateRandomEffect = UpdateRandomEffect, UpdateNu = UpdateNu,
UpdateBeta = UpdateBeta, UpdateSigma = UpdateSigma, UpdateDelta = UpdateDelta)
TuningPara = list(TuningDelta = 0.01)
if(1){
start.time <- Sys.time()
PosteriorSamplesCP = CumulativeProbitMCMC(num.of.iter, Data, Robustness, InitialValues, HyperPara, UpdatePara, TuningPara, Interactive)
end.time <- Sys.time()
#cat("\nCall:\n", printCall(cl), "\n\n", sep = "")
cat("\nData Descriptives:\n")
cat("Longitudinal Data Information:")
cat("\nNumber of Observations: ", sum(TimePointsAvailable), "\tNumber of Covariates: ", p-1)
cat("\nNumber of subjects:", N, "\n\n")
out <- list(Posterior.Samples = PosteriorSamplesCP, Fixed.Effects.Names = fixed.eff,
Random.Effects.Names = random.eff,
Response = y, Fixed.Effects.Mat = x, Random.Effects.Mat = z,
HS.model.Mat = uu, call = cl, Num.of.Iter = num.of.iter)
#class(out)
out
}
}
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