#### Bayesian variable selection for (binomial) logit regression model
#### with random intercept using Dirac spike and Student-t/ or normal slab
#### (invisible function)
#### last change: 2016/03/24
## -------------------------
select_logit <- function(y, N, W, H=NULL, compmix.bin=NULL, model, prior, mcmc, param, imc){
# linear predictior in binomial logit model
muL <- W%*%param$alpha
if (model$ri == 1){
ranEff <- param$atilde[model$Zl]
linp <- muL + ranEff*param$theta
} else linp <- muL
#### Step A --- data augmentation for the binomial logit model (binomial dRUM)
## DATA AUGMENTATION (part I) for binomial logit models taken from "binomlogit"
if (model$family == "pogit"){
compmix.bin <- dataug_binom_dRUM1(y, N)
#compmix.bin <- do.call(dataug_binom_dRUM1, list(y, N))
}
## DATA AUGMENTATION (part II) for binomial logit models taken from "binomlogit"
augBinom <- dataug_binom_dRUM2(y, N, linp, compmix.bin)
z <- augBinom$ystar
invSig <- augBinom$invSig
yS <- z*invSig # =sqrt(Sigma^-1)*z
if (model$ri == 0){
Wall <- W*kronecker(matrix(1, 1, model$d + 1), invSig)
} else {
Wall <- cbind(W,ranEff)*kronecker(matrix(1, 1, model$d + model$ri + 1), invSig)
}
# inverse prior variance of regression effects (updated)
invA0 <- diag(c(prior$invM0, 1/param$psi), nrow = model$d + model$ri + 1)
#### Step B --- starts Bayesian variable selection
if (imc > mcmc$startsel && sum(sum(!model$deltafix) + sum(!model$gammafix))>0){
## (1) update mixture weights
incfix <- sum(param$delta == 1)
omega <- rbeta(1, prior$w['wa0'] + incfix, prior$w['wb0'] + model$d - incfix)
if (model$ri==1){
incran <- sum(param$gamma == 1)
pi <- rbeta(1, prior$pi['pa0'] + incran, prior$pi['pb0'] + model$ri - incran)
} else {
pi <- NULL
}
## (2) sample the indicators, the regression coefficients and the scale parameters
## --- (i) sample the indicators delta_{alpha,j}, gamma_{alpha} for the slab component
indic <- draw_indicators(yS, Wall, param$delta, param$gamma, omega, pi, model, prior, invA0)
delta <- indic$deltanew
pdelta <- indic$pdeltanew
gamma <- indic$gammanew
pgamma <- indic$pgammanew
} else {
delta <- param$delta
pdelta <- param$pdelta
gamma <- param$gamma
pgamma <- param$pgamma
omega <- param$omega
pi <- param$pi
}
## --- (ii_A) sample the (selected) regression effects
if (model$ri == 0){
index <- c(1, which(delta == 1) + 1)
} else {
index <- c(1, which(c(delta, gamma) == 1) + 1)
}
Zsel <- Wall[, index, drop=FALSE] # Z*=[1, W^delta,atilde]*sqrt(Sigma^-1)
dsel <- length(index)
invA0_sel <- invA0[index, index, drop=FALSE]
a0_sel <- prior$a0[index,,drop=FALSE]
AL <- solve(invA0_sel + t(Zsel)%*%Zsel) # A = (A0^-1 + (Z*)'Sigma^-1 Z*)
aL <- AL%*%(invA0_sel%*%a0_sel + t(Zsel)%*%yS) # a = A(A0^-1*a0 + (Z*)'Sigma^-1*y
zetaL <- t(chol(AL))%*%matrix(rnorm(dsel), dsel, 1) + aL
v1 <- matrix(0, 1, model$d + model$ri + 1)
v1[index] <- t(zetaL)
mu_alpha <- v1[1]
if (model$d > 0){
alpha <- v1[2:(model$d + 1)]
} else {
alpha <- matrix(0, 1, model$d)
}
muL <- W%*%c(mu_alpha, alpha)
## --- (ii_B) sample the random intercepts
if (model$ri == 1){
theta <- v1[((model$d + 1) + 1):(model$d + model$ri + 1)]
yh <- (z-muL)*invSig
Xh <- (H*theta)*kronecker(matrix(1, 1, max(model$Zl)), invSig)
BL <- solve(prior$invB0 + t(Xh)%*%Xh) # B = (theta^2*H'*Sigma^-1 + B0)
bL <- BL%*%t(Xh)%*%yh
atilde <- t(chol(BL))%*%matrix(rnorm(max(model$Zl)), max(model$Zl),1) + bL
# perform a random sign-switch (sign-switching step)
sswitch <- sign(runif(1) - 0.5)
theta <- theta*sswitch
atilde <- atilde*sswitch
} else {
atilde <- NULL
theta <- NULL
}
## --- (iii) sample the variance parameter of Student-t/ or normal slab
ind <- c(delta, gamma)
if (model$ri == 1){
effAlpha <- c(t(alpha), t(theta))
} else {
effAlpha <- t(alpha)
}
psi <- draw_psi(effAlpha, ind, prior)
# returns updated par-list with parameters used for subsequent step
return(list(alpha = c(mu_alpha, alpha), delta = delta, pdelta = pdelta,
omega = omega, psi = psi, atilde = atilde, theta = theta,
gamma = gamma, pgamma = pgamma, pi = pi))
}
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