binomial.MVlerouxCAR <- function(formula, data=NULL, trials, W, burnin, n.sample, thin=1, prior.mean.beta=NULL, prior.var.beta=NULL, prior.Sigma.df=NULL, prior.Sigma.scale=NULL, rho=NULL, MALA=FALSE, verbose=TRUE)
{
##############################################
#### Format the arguments and check for errors
##############################################
#### Verbose
a <- common.verbose(verbose)
#### Frame object
frame.results <- common.frame(formula, data, "binomial")
K <- frame.results$n
p <- frame.results$p
X <- frame.results$X
X.standardised <- frame.results$X.standardised
X.sd <- frame.results$X.sd
X.mean <- frame.results$X.mean
X.indicator <- frame.results$X.indicator
offset <- frame.results$offset
Y <- frame.results$Y
which.miss <- frame.results$which.miss
n.miss <- frame.results$n.miss
J <- ncol(Y)
N.all <- K * J
Y.DA <- Y
#### Create a missing list
if(n.miss>0)
{
miss.locator <- array(NA, c(n.miss, 2))
colnames(miss.locator) <- c("row", "column")
locations <- which(t(which.miss)==0)
miss.locator[ ,1] <- ceiling(locations/J)
miss.locator[ ,2] <- locations - (miss.locator[ ,1]-1) * J
}else
{}
#### Check on MALA argument
if(length(MALA)!=1) stop("MALA is not length 1.", call.=FALSE)
if(!is.logical(MALA)) stop("MALA is not logical.", call.=FALSE)
#### Check and format the trials argument
if(ncol(trials)!=J) stop("trials has the wrong number of columns.", call.=FALSE)
if(nrow(trials)!=K) stop("trials has the wrong number of rows.", call.=FALSE)
if(sum(is.na(trials))>0) stop("the numbers of trials has missing 'NA' values.", call.=FALSE)
if(!is.numeric(trials)) stop("the numbers of trials has non-numeric values.", call.=FALSE)
int.check <- N.all-sum(ceiling(trials)==floor(trials))
if(int.check > 0) stop("the numbers of trials has non-integer values.", call.=FALSE)
if(min(trials)<=0) stop("the numbers of trials has zero or negative values.", call.=FALSE)
failures <- trials - Y
failures.DA <- trials - Y.DA
if(sum(Y>trials, na.rm=TRUE)>0) stop("the response variable has larger values that the numbers of trials.", call.=FALSE)
#### W matrix
if(!is.matrix(W)) stop("W is not a matrix.", call.=FALSE)
if(ceiling(N.all/K)!= floor(N.all/K)) stop("The number of data points divided by the number of rows in W is not a whole number.", call.=FALSE)
#### rho
if(is.null(rho))
{
rho <- runif(1)
fix.rho <- FALSE
}else
{
fix.rho <- TRUE
}
if(!is.numeric(rho) ) stop("rho is fixed but is not numeric.", call.=FALSE)
if(rho<0 ) stop("rho is outside the range [0, 1].", call.=FALSE)
if(rho>1 ) stop("rho is outside the range [0, 1].", call.=FALSE)
#### Priors
if(is.null(prior.mean.beta)) prior.mean.beta <- rep(0, p)
if(is.null(prior.var.beta)) prior.var.beta <- rep(100000, p)
if(is.null(prior.Sigma.df)) prior.Sigma.df <- J+1
if(is.null(prior.Sigma.scale)) prior.Sigma.scale <- diag(rep(1,J)) / 1000
common.prior.beta.check(prior.mean.beta, prior.var.beta, p)
common.prior.varmat.check(prior.Sigma.scale, J)
#### Compute the blocking structure for beta
block.temp <- common.betablock(p)
beta.beg <- block.temp[[1]]
beta.fin <- block.temp[[2]]
n.beta.block <- block.temp[[3]]
list.block <- as.list(rep(NA, n.beta.block*2))
for(r in 1:n.beta.block)
{
list.block[[r]] <- beta.beg[r]:beta.fin[r]-1
list.block[[r+n.beta.block]] <- length(list.block[[r]])
}
#### MCMC quantities - burnin, n.sample, thin
common.burnin.nsample.thin.check(burnin, n.sample, thin)
#############################
#### Initial parameter values
#############################
beta <- array(NA, c(p, J))
for(i in 1:J)
{
mod.glm <- glm(cbind(Y[ ,i], failures[ ,i])~X.standardised-1, offset=offset[ ,i], family="quasibinomial")
beta.mean <- mod.glm$coefficients
beta.sd <- sqrt(diag(summary(mod.glm)$cov.scaled))
beta[ ,i] <- rnorm(n=p, mean=beta.mean, sd=beta.sd)
}
theta.hat <- Y / trials
theta.hat[theta.hat==0] <- 0.01
theta.hat[theta.hat==1] <- 0.99
res.temp <- log(theta.hat / (1 - theta.hat)) - X.standardised %*% beta - offset
res.sd <- sd(res.temp, na.rm=TRUE)/5
phi.vec <- rnorm(n=N.all, mean=0, sd=res.sd)
phi <- matrix(phi.vec, nrow=K, byrow=TRUE)
Sigma <- cov(phi)
Sigma.inv <- solve(Sigma)
regression <- X.standardised %*% beta
lp <- regression + phi + offset
prob <- exp(lp) / (1 + exp(lp))
###############################
#### Set up the MCMC quantities
###############################
#### Matrices to store samples
n.keep <- floor((n.sample - burnin)/thin)
samples.beta <- array(NA, c(n.keep, J*p))
samples.phi <- array(NA, c(n.keep, N.all))
samples.Sigma <- array(NA, c(n.keep, J, J))
if(!fix.rho) samples.rho <- array(NA, c(n.keep, 1))
samples.loglike <- array(NA, c(n.keep, N.all))
samples.fitted <- array(NA, c(n.keep, N.all))
if(n.miss>0) samples.Y <- array(NA, c(n.keep, n.miss))
#### Metropolis quantities
accept <- rep(0,4)
accept.beta <- rep(0,2*J)
proposal.sd.beta <- rep(0.01, J)
proposal.sd.phi <- 0.1
proposal.sd.rho <- 0.02
Sigma.post.df <- prior.Sigma.df + K
##################################
#### Set up the spatial quantities
##################################
#### CAR quantities
W.quants <- common.Wcheckformat(W)
W <- W.quants$W
W.triplet <- W.quants$W.triplet
n.triplet <- W.quants$n.triplet
W.triplet.sum <- W.quants$W.triplet.sum
n.neighbours <- W.quants$n.neighbours
W.begfin <- W.quants$W.begfin
Wstar <- diag(apply(W,1,sum)) - W
Q <- rho * Wstar + diag(rep(1-rho,K))
#### Create the determinant
if(!fix.rho)
{
Wstar.eigen <- eigen(Wstar)
Wstar.val <- Wstar.eigen$values
det.Q <- sum(log((rho * Wstar.val + (1-rho))))
}else
{}
#### Check for islands
W.list<- mat2listw(W)
W.nb <- W.list$neighbours
W.islands <- n.comp.nb(W.nb)
islands <- W.islands$comp.id
islands.all <- rep(islands,J)
n.islands <- max(W.islands$nc)
if(rho==1) Sigma.post.df <- prior.Sigma.df + K - n.islands
#### Make matrix versions of the variables
Y.vec <- as.numeric(t(Y))
trials.vec <- as.numeric(t(trials))
###########################
#### Run the Bayesian model
###########################
#### Start timer
if(verbose)
{
cat("Generating", n.keep, "post burnin and thinned (if requested) samples.\n", sep = " ")
progressBar <- txtProgressBar(style = 3)
percentage.points<-round((1:100/100)*n.sample)
}else
{
percentage.points<-round((1:100/100)*n.sample)
}
#### Create the MCMC samples
for(j in 1:n.sample)
{
####################################
## Sample from Y - data augmentation
####################################
if(n.miss>0)
{
Y.DA[miss.locator] <- rbinom(n=n.miss, size=trials[miss.locator], prob=prob[miss.locator])
failures.DA <- trials - Y.DA
}else
{}
###################
## Sample from beta
###################
offset.temp <- phi + offset
for(r in 1:J)
{
if(MALA)
{
temp <- binomialbetaupdateMALA(X.standardised, K, p, beta[ ,r], offset.temp[ ,r], Y.DA[ ,r], failures.DA[ ,r], trials[ ,r], prior.mean.beta, prior.var.beta, n.beta.block, proposal.sd.beta[r], list.block)
}else
{
temp <- binomialbetaupdateRW(X.standardised, K, p, beta[ ,r], offset.temp[ ,r], Y.DA[ ,r], failures.DA[ ,r], prior.mean.beta, prior.var.beta, n.beta.block, proposal.sd.beta[r], list.block)
}
beta[ ,r] <- temp[[1]]
accept.beta[r] <- accept.beta[r] + temp[[2]]
accept.beta[(r+J)] <- accept.beta[(r+J)] + n.beta.block
}
regression <- X.standardised %*% beta
##################
## Sample from phi
##################
den.offset <- rho * W.triplet.sum + 1 - rho
phi.offset <- regression + offset
if(MALA)
{
temp1 <- binomialmcarupdateMALA(W.triplet, W.begfin, K, J, phi, Y.DA, failures.DA, trials, phi.offset, den.offset, Sigma.inv, rho, proposal.sd.phi)
}else
{
temp1 <- binomialmcarupdateRW(W.triplet, W.begfin, K, J, phi, Y.DA, failures.DA, phi.offset, den.offset, Sigma.inv, rho, proposal.sd.phi)
}
phi <- temp1[[1]]
for(r in 1:J)
{
phi[ ,r] <- phi[ ,r] - mean(phi[ ,r])
}
accept[1] <- accept[1] + temp1[[2]]
accept[2] <- accept[2] + K
####################
## Sample from Sigma
####################
Sigma.post.scale <- t(phi) %*% Q %*% phi + prior.Sigma.scale
Sigma <- riwish(Sigma.post.df, Sigma.post.scale)
Sigma.inv <- solve(Sigma)
##################
## Sample from rho
##################
if(!fix.rho)
{
## Propose a new value
proposal.rho <- rtruncnorm(n=1, a=0, b=1, mean=rho, sd=proposal.sd.rho)
Q.prop <- proposal.rho * Wstar + diag(rep(1-proposal.rho), K)
det.Q.prop <- sum(log((proposal.rho * Wstar.val + (1-proposal.rho))))
## Compute the acceptance rate
logprob.current <- 0.5 * J * det.Q - 0.5 * sum(diag(t(phi) %*% Q %*% phi %*% Sigma.inv))
logprob.proposal <- 0.5 * J * det.Q.prop - 0.5 * sum(diag(t(phi) %*% Q.prop %*% phi %*% Sigma.inv))
hastings <- log(dtruncnorm(x=rho, a=0, b=1, mean=proposal.rho, sd=proposal.sd.rho)) - log(dtruncnorm(x=proposal.rho, a=0, b=1, mean=rho, sd=proposal.sd.rho))
prob <- exp(logprob.proposal - logprob.current + hastings)
if(prob > runif(1))
{
rho <- proposal.rho
det.Q <- det.Q.prop
Q <- Q.prop
accept[3] <- accept[3] + 1
}else
{
}
accept[4] <- accept[4] + 1
}else
{}
#########################
## Calculate the deviance
#########################
lp <- regression + phi + offset
prob <- exp(lp) / (1 + exp(lp))
fitted <- trials * prob
loglike <- dbinom(x=as.numeric(t(Y)), size=as.numeric(t(trials)), prob=as.numeric(t(prob)), log=TRUE)
###################
## Save the results
###################
if(j > burnin & (j-burnin)%%thin==0)
{
ele <- (j - burnin) / thin
samples.beta[ele, ] <- as.numeric(beta)
samples.phi[ele, ] <- as.numeric(t(phi))
samples.Sigma[ele, , ] <- Sigma
if(!fix.rho) samples.rho[ele, ] <- rho
samples.loglike[ele, ] <- loglike
samples.fitted[ele, ] <- as.numeric(t(fitted))
if(n.miss>0) samples.Y[ele, ] <- Y.DA[miss.locator]
}else
{}
########################################
## Self tune the acceptance probabilties
########################################
if(ceiling(j/100)==floor(j/100) & j < burnin)
{
#### Update the proposal sds
for(r in 1:J)
{
if(p>2)
{
proposal.sd.beta[r] <- common.accceptrates1(accept.beta[c(r, (r+J))], proposal.sd.beta[r], 40, 50)
}else
{
proposal.sd.beta[r] <- common.accceptrates1(accept.beta[c(r, (r+J))], proposal.sd.beta[r], 30, 40)
}
}
proposal.sd.phi <- common.accceptrates1(accept[1:2], proposal.sd.phi, 40, 50)
if(!fix.rho)
{
proposal.sd.rho <- common.accceptrates2(accept[3:4], proposal.sd.rho, 40, 50, 0.5)
}
accept <- c(0,0,0,0)
accept.beta <- rep(0,2*J)
}else
{}
################################
## print progress to the console
################################
if(j %in% percentage.points & verbose)
{
setTxtProgressBar(progressBar, j/n.sample)
}
}
##### end timer
if(verbose)
{
cat("\nSummarising results.")
close(progressBar)
}else
{}
###################################
#### Summarise and save the results
###################################
#### Compute the acceptance rates
accept.beta <- 100 * sum(accept.beta[1:J]) / sum(accept.beta[(J+1):(2*J)])
accept.phi <- 100 * accept[1] / accept[2]
if(!fix.rho)
{
accept.rho <- 100 * accept[3] / accept[4]
}else
{
accept.rho <- NA
}
accept.Sigma <- 100
accept.final <- c(accept.beta, accept.phi, accept.rho, accept.Sigma)
names(accept.final) <- c("beta", "phi", "rho", "Sigma")
#### Compute the fitted deviance
mean.beta <- matrix(apply(samples.beta, 2, mean), nrow=p, ncol=J, byrow=F)
mean.phi <- matrix(apply(samples.phi, 2, mean), nrow=K, ncol=J, byrow=T)
mean.logit <- X.standardised %*% mean.beta + mean.phi + offset
mean.prob <- exp(mean.logit) / (1 + exp(mean.logit))
fitted.mean <- trials * mean.prob
deviance.fitted <- -2 * sum(dbinom(x=as.numeric(t(Y)), size=as.numeric(t(trials)), prob=as.numeric(t(mean.prob)), log=TRUE), na.rm=TRUE)
#### Model fit criteria
modelfit <- common.modelfit(samples.loglike, deviance.fitted)
#### transform the parameters back to the origianl covariate scale.
samples.beta.orig <- samples.beta
for(r in 1:J)
{
samples.beta.orig[ ,((r-1)*p+1):(r*p)] <- common.betatransform(samples.beta[ ,((r-1)*p+1):(r*p)], X.indicator, X.mean, X.sd, p, FALSE)
}
#### Create a summary object
samples.beta.orig <- mcmc(samples.beta.orig)
summary.beta <- t(apply(samples.beta.orig, 2, quantile, c(0.5, 0.025, 0.975)))
summary.beta <- cbind(summary.beta, rep(n.keep, p), rep(accept.beta,J*p), effectiveSize(samples.beta.orig), geweke.diag(samples.beta.orig)$z)
col.name <- rep(NA, p*(J-1))
if(is.null(colnames(Y)))
{
for(r in 1:J)
{
col.name[((r-1)*p+1):(r*p)] <- paste("Variable ", r, " - ", colnames(X), sep="")
}
}else
{
for(r in 1:J)
{
col.name[((r-1)*p+1):(r*p)] <- paste(colnames(Y)[r], " - ", colnames(X), sep="")
}
}
rownames(summary.beta) <- col.name
colnames(summary.beta) <- c("Median", "2.5%", "97.5%", "n.sample", "% accept", "n.effective", "Geweke.diag")
summary.hyper <- array(NA, c((J+1) ,7))
summary.hyper[1:J, 1] <- diag(apply(samples.Sigma, c(2,3), quantile, c(0.5)))
summary.hyper[1:J, 2] <- diag(apply(samples.Sigma, c(2,3), quantile, c(0.025)))
summary.hyper[1:J, 3] <- diag(apply(samples.Sigma, c(2,3), quantile, c(0.975)))
summary.hyper[1:J, 4] <- n.keep
summary.hyper[1:J, 5] <- accept.Sigma
summary.hyper[1:J, 6] <- diag(apply(samples.Sigma, c(2,3), effectiveSize))
for(r in 1:J)
{
summary.hyper[r, 7] <- geweke.diag(samples.Sigma[ ,r,r])$z
}
if(!fix.rho)
{
summary.hyper[(J+1), 1:3] <- quantile(samples.rho, c(0.5, 0.025, 0.975))
summary.hyper[(J+1), 4:7] <- c(n.keep, accept.rho, effectiveSize(samples.rho), geweke.diag(samples.rho)$z)
}else
{
summary.hyper[(J+1), 1:3] <- c(rho, rho, rho)
summary.hyper[(J+1), 4:7] <- rep(NA, 4)
}
summary.results <- rbind(summary.beta, summary.hyper)
rownames(summary.results)[((J*p)+1): nrow(summary.results)] <- c(paste(rep("Sigma",J), 1:J, 1:J, sep=""), "rho")
summary.results[ , 1:3] <- round(summary.results[ , 1:3], 4)
summary.results[ , 4:7] <- round(summary.results[ , 4:7], 1)
#### Create the fitted values and residuals
fitted.values <- matrix(apply(samples.fitted, 2, mean), nrow=K, ncol=J, byrow=T)
response.residuals <- Y - fitted.values
pearson.residuals <- response.residuals /sqrt(fitted.values * (1 - mean.prob))
residuals <- list(response=response.residuals, pearson=pearson.residuals)
#### Compile and return the results
model.string <- c("Likelihood model - Binomial (logit link function)", "\nRandom effects model - Leroux MCAR\n")
if(fix.rho) samples.rho=NA
if(n.miss==0) samples.Y = NA
samples <- list(beta=samples.beta.orig, phi=mcmc(samples.phi), Sigma=samples.Sigma, rho=mcmc(samples.rho), fitted=mcmc(samples.fitted), Y=mcmc(samples.Y))
results <- list(summary.results=summary.results, samples=samples, fitted.values=fitted.values, residuals=residuals, modelfit=modelfit, accept=accept.final, localised.structure=NULL, formula=formula, model=model.string, X=X)
class(results) <- "CARBayes"
#### Finish by stating the time taken
if(verbose)
{
b<-proc.time()
cat("Finished in ", round(b[3]-a[3], 1), "seconds.\n")
}else
{}
return(results)
}
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