# calculate hessian matrix without component of bet (bet is given)
hessian.LRT <- function(par, rdata, edata, par.pos, bet){
if(any(is.na(par))){
return(NULL)
}
ne <- nrow(edata$data)
nr <- nrow(rdata$data)
a <- par['a']
if('b' %in% names(par)){
b <- par['b']
est.b <- TRUE
}else{
b <- 0
est.b <- FALSE
}
c <- par['c']
alp.0 <- par['alp.0']
if('alp.x' %in% par.pos$par.name){
alp.x <- par[par.pos['alp.x', 'start']:par.pos['alp.x', 'end']]
nx <- length(alp.x)
}else{
alp.x <- NA
nx <- 0
}
if('alp.h' %in% par.pos$par.name){
alp.h <- par[par.pos['alp.h', 'start']:par.pos['alp.h', 'end']]
nh <- length(alp.h)
}else{
alp.h <- NA
nh <- 0
}
alp.g <- par[par.pos['alp.g', 'start']:par.pos['alp.g', 'end']]
ng <- length(alp.g)
if('bet.y' %in% par.pos$par.name){
bet.y <- par[par.pos['bet.y', 'start']:par.pos['bet.y', 'end']]
ny <- length(bet.y)
}else{
bet.y <- NA
ny <- 0
}
if('bet.x' %in% par.pos$par.name){
bet.x <- par[par.pos['bet.x', 'start']:par.pos['bet.x', 'end']]
}else{
bet.x <- NA
}
bet.z <- bet
##################
rg <- as.matrix(rdata$data[, rdata$vg, drop = FALSE])
eg <- as.matrix(edata$data[, edata$vg, drop = FALSE])
eta <- as.vector(rg %*% alp.g)
if(nx > 0){
rx <- as.matrix(rdata$data[, rdata$vx, drop = FALSE])
ex <- as.matrix(edata$data[, edata$vx, drop = FALSE])
#eta.x <- as.vector(rx %*% alp.x)
}
if(nh > 0){
eh <- as.matrix(edata$data[, edata$vh, drop = FALSE])
}
if(ny > 0){
ry <- as.matrix(rdata$data[, rdata$vy, drop = FALSE])
}
rd <- as.vector(rdata$data[, rdata$vd])
n1 <- sum(rd)
n0 <- nr - n1
ed <- as.vector(edata$data[, edata$vd])
ez <- as.vector(edata$data[, edata$vz])
lin <- a + rg %*% alp.g * bet.z
if(nx > 0){
lin <- lin + rx %*% bet.x
}
if(ny > 0){
lin <- lin + ry %*% bet.y
}
lin <- as.vector(lin)
delta <- exp(lin)
p <- 1/ n0 / (1 + n1/n0 * delta) # 1 / (n0 + n1 * delta)
Delta <- -1 + 1 / (1 + n1/n0 * delta) # -n1 p delta
xi <- Delta * (1 + Delta)
r <- ez - alp.0 - eg %*% alp.g - (bet.z * c + b) * ed
if(nx > 0){
r <- r - ex %*% alp.x
}
if(nh > 0){
r <- r - eh %*% alp.h
}
r <- as.vector(r)
one <- rep(1, ne)
name.alp.g <- paste0('alp.', edata$vg)
name.alp.x <- paste0('alp.', edata$vx)
name.alp.h <- paste0('alp.', edata$vh)
name.bet.x <- paste0('bet.', rdata$vx)
name.bet.y <- paste0('bet.', rdata$vy)
###########################
## calculate Hess matrix ##
###########################
hess <- matrix(0, nrow = length(par), ncol = length(par))
rownames(hess) <- names(par)
colnames(hess) <- names(par)
hess['c', 'c'] <- ne/c^2/2 - 1/c^3 * sum(r^2) -
2 * bet.z/c^2 * sum(ed * r) - bet.z^2/c * sum(ed)
hess['c', 'alp.0'] <- -1/c^2 * sum(r) -
bet.z/c * sum(ed)
if(nx > 0){
hess['c', name.alp.x] <- -1/c^2 * t(t(ex) %*% r) -
bet.z/c * t(t(ex) %*% ed)
}
if(nh > 0){
hess['c', name.alp.h] <- -1/c^2 * t(t(eh) %*% r) -
bet.z/c * t(t(eh) %*% ed)
}
hess['c', name.alp.g] <- -1/c^2 * t(t(eg) %*% r) -
bet.z/c * t(t(eg) %*% ed)
if(est.b){
hess['c', 'b'] <- -bet.z/c * sum(ed) - 1/c^2 * sum(ed * r)
}
############
hess['alp.0', 'alp.0'] <- -ne/c
if(nx > 0){
hess['alp.0', name.alp.x] <- -1/c * t(t(ex) %*% one)
}
if(nh > 0){
hess['alp.0', name.alp.h] <- -1/c * t(t(eh) %*% one)
}
hess['alp.0', name.alp.g] <- -1/c * t(t(eg) %*% one)
if(est.b){
hess['alp.0', 'b'] <- -1/c * sum(ed)
}
############
if(nx > 0){
hess[name.alp.x, name.alp.x] <- -1/c * (t(ex) %*% ex)
if(nh > 0){
hess[name.alp.x, name.alp.h] <- -1/c * (t(ex) %*% eh)
}
hess[name.alp.x, name.alp.g] <- -1/c * (t(ex) %*% eg)
if(est.b){
hess[name.alp.x, 'b'] <- -1/c * (t(ex) %*% ed)
}
}
############
if(nh > 0){
hess[name.alp.h, name.alp.h] <- -1/c * (t(eh) %*% eh)
hess[name.alp.h, name.alp.g] <- -1/c * (t(eh) %*% eg)
if(est.b){
hess[name.alp.h, 'b'] <- -1/c * (t(eh) %*% ed)
}
}
############
hess[name.alp.g, name.alp.g] <- bet.z^2 * (t(rg) %*% (xi * rg)) - 1/c * (t(eg) %*% eg)
if(est.b){
hess[name.alp.g, 'b'] <- -1/c * (t(eg) %*% ed)
}
hess[name.alp.g, 'a'] <- bet.z * (t(rg) %*% xi)
if(nx > 0){
hess[name.alp.g, name.bet.x] <- bet.z * (t(rg) %*% (xi * rx))
}
if(ny > 0){
hess[name.alp.g, name.bet.y] <- bet.z * (t(rg) %*% (xi * ry))
}
############
if(est.b){
hess['b', 'b'] <- -1/c * sum(ed)
}
hess['a', 'a'] <- sum(xi)
if(nx > 0){
hess['a', name.bet.x] <- t(t(rx) %*% xi)
}
if(ny > 0){
hess['a', name.bet.y] <- t(t(ry) %*% xi)
}
if(nx > 0){
hess[name.bet.x, name.bet.x] <- t(rx) %*% (xi * rx)
if(ny > 0){
hess[name.bet.x, name.bet.y] <- t(rx) %*% (xi * ry)
}
}
############
if(ny > 0){
hess[name.bet.y, name.bet.y] <- t(ry) %*% (xi * ry)
}
############
for(i in 1:nrow(hess)){
for(j in 1:ncol(hess)){
if(hess[i, j] == 0){
hess[i, j] <- hess[j, i]
}
}
}
hess <- (hess + t(hess))/2
#N <- length(unique(c(rownames(rdata$data)), rownames(edata$data)))
#hess/N
hess
}
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