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Liu_orm_code.r
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################# code for sims and applications in orm paper ################
library(rms)
################################### FUNCTIONS ################
#### function to estimate conditional mean and its standard error for orm model
mean.orm <- function(mod, new.data, se=TRUE){
if(is.null(mod$yunique)) {
stop("Need to set x=TRUE and y=TRUE for orm")
} else{
order.y <- mod$yunique
n.alpha <- length(order.y)-1
xb <- as.matrix(new.data)%*%matrix(coef(mod)[colnames(new.data)])
m.alpha <- mod$coef[1:n.alpha]
lb <- t(outer(m.alpha, xb, "+")[,,1])
m.s <- mod$trans$cumprob(lb)
m.f <- t(apply(m.s, 1, FUN=function(x) c(1,x[1:n.alpha]) - c(x[1:n.alpha], 0)))
m.mean <- apply(m.f, 1, FUN=function(x) sum(x*order.y))
if(se){
if(mod$family=="logistic") mod$trans$deriv <- function(x) exp(-x)/(1+exp(-x))^2
dmean.dalpha <- t(apply(mod$trans$deriv(lb), 1, FUN=function(x) x*(order.y[2:length(order.y)] - order.y[1:n.alpha])))
dmean.dbeta <- apply(dmean.dalpha, 1, sum)*as.matrix(new.data)
dmean.dtheta <- cbind(dmean.dalpha, dmean.dbeta)
mean.var <-diag(dmean.dtheta%*%solve(mod$info.matrix)%*%t(dmean.dtheta))
mean.se <- sqrt(mean.var)
result <- cbind(m.mean, mean.se)
ci <- t(apply(result, 1, FUN=function(x) c(x[1]- qnorm(0.975)*x[2], x[1]+ qnorm(0.975)*x[2])))
result <- cbind(result, ci)
colnames(result) <- c("est", "se", "lb", "ub")
} else{
result <- matrix(m.mean)
colnames(result) <- c("est")
}
return(result)
}
}
#### function to estimate conditional quantiles and confidence intervals for orm model
quantile.orm <- function(mod, new.data, probs=0.5, se=TRUE){
quantile <- matrix(NA, nrow=dim(new.data)[1], ncol=length(probs))
order.y <- mod$yunique
#n.alpha <- length(order.y)-1
xb <- as.matrix(new.data)%*%matrix(coef(mod)[colnames(new.data)])
alpha <- mod$coef[1:(length(unique(order.y))-1)]
lb <- t(outer(alpha, xb, "+")[,,1])
m.cdf <- 1- mod$trans$cumprob(lb)
m.cdf <- cbind(0, m.cdf, 1)
for(i in 1: length(probs)){
try({
index.1 <- apply(m.cdf, 1, FUN=function(x){ max(which(x<=probs[i]))[1]} )
index.2 <- apply(m.cdf, 1, FUN=function(x){ min(which(x>=probs[i]))[1]} )
index.y1 <- ifelse(index.1>length(order.y), Inf, order.y[index.1])
index.y2 <- ifelse(index.2>length(order.y), Inf, order.y[index.2])
index.y1.cdf <- ifelse(index.1==0, 0, m.cdf[cbind(1:dim(new.data)[1], index.1)])
index.y2.cdf <- ifelse(index.2>length(order.y), 1, m.cdf[cbind(1:dim(new.data)[1], index.2)])
quantile[,i] <- ifelse(index.1==index.2, index.y1,
(index.y2-index.y1)/(index.y2.cdf - index.y1.cdf)*(probs[i]-index.y1.cdf) + index.y1)
quantile[, i] <- ifelse(is.infinite(quantile[,i]), max(order.y), quantile[, i])
})
}
result <- quantile
if(se){
if(mod$family=="logistic") mod$trans$deriv <- function(x) exp(-x)/(1+exp(-x))^2
quantile.lb <- quantile.ub <- matrix(NA, nrow=dim(new.data)[1], ncol=length(probs))
lb.se <- matrix(NA, ncol=dim(lb)[2], nrow=dim(new.data)[1])
var <- as.matrix(solve(mod$info.matrix))
for(i in 1:dim(lb)[2]){
var.i <- var[c(i, which(names(coef(mod)) %in% colnames(new.data))),
c(i, which(names(coef(mod)) %in% colnames(new.data)))]
dcdf.dtheta <- cbind(-mod$trans$deriv(lb[,i]),
-mod$trans$deriv(lb[,i])*as.matrix(new.data) )
dlb.dtheta <- as.matrix(cbind(1, new.data))
lb.se[,i] <- sqrt(diag(dlb.dtheta%*%var.i%*% t(dlb.dtheta)))
}
ci.lb <- sapply(1:dim(lb)[2], FUN=function(i) { 1- mod$trans$cumprob(lb[, i] +qnorm(0.975)*lb.se[, i])})
ci.ub <- sapply(1:dim(lb)[2], FUN=function(i) { 1- mod$trans$cumprob(lb[, i] -qnorm(0.975)*lb.se[, i])})
ci.lb <- matrix(ci.lb, nrow=dim(new.data)[1])
ci.ub <- matrix(ci.ub, nrow=dim(new.data)[1])
ci.lb <- cbind(0, ci.lb, 1)
ci.ub <- cbind(0, ci.ub, 1)
for(i in 1: length(probs)){
try({
index.1 <- apply(ci.lb, 1, FUN=function(x){ max(which(x<=probs[i]))[1]} )
index.2 <- apply(ci.lb, 1, FUN=function(x){ min(which(x>=probs[i]))[1]} )
index.y1 <- ifelse(index.1>length(order.y), Inf, order.y[index.1])
index.y2 <- ifelse(index.2>length(order.y),Inf,order.y[index.2])
index.y1.cdf <- ifelse(index.1==0, 0, ci.lb[cbind(1:dim(new.data)[1], index.1)])
index.y2.cdf <- ifelse(index.2>length(order.y), 1, ci.lb[cbind(1:dim(new.data)[1], index.2)])
quantile.lb[,i] <- ifelse(index.1==index.2, index.y1,
(index.y2-index.y1)/(index.y2.cdf - index.y1.cdf)*(probs[i]-index.y1.cdf) + index.y1)
quantile.lb[, i] <- ifelse(is.infinite(quantile.lb[,i]), max(order.y), quantile.lb[, i])
index.1 <- apply(ci.ub, 1, FUN=function(x){ max(which(x<=probs[i]))[1]} )
index.2 <- apply(ci.ub, 1, FUN=function(x){ min(which(x>=probs[i]))[1]} )
index.y1 <- ifelse(index.1>length(order.y), Inf, order.y[index.1])
index.y2 <- ifelse(index.2>length(order.y),Inf,order.y[index.2])
index.y1.cdf <- ifelse(index.1==0, 0, ci.ub[cbind(1:dim(new.data)[1], index.1)])
index.y2.cdf <- ifelse(index.2>length(order.y), 1, ci.ub[cbind(1:dim(new.data)[1], index.2)])
quantile.ub[,i] <- ifelse(index.1==index.2, index.y1,
(index.y2-index.y1)/(index.y2.cdf - index.y1.cdf)*(probs[i]-index.y1.cdf) + index.y1)
quantile.ub[, i] <- ifelse(is.infinite(quantile.ub[,i]), max(order.y), quantile.ub[, i])
})
}
result <- list(quantile=quantile,
lb=quantile.ub,
ub=quantile.lb)
}
return(result)
}
#### function to estimate conditional CDF and its standard error for orm models
cdf.orm <- function(mod, new.data, at.y=0,se=TRUE){
if(is.null(mod$yunique)) {
stop("Need to set x=TRUE and y=TRUE for orm")
} else{
order.y <- mod$yunique
xb <- as.matrix(new.data)%*%matrix(coef(mod)[colnames(new.data)])
index <- sapply(at.y, FUN=function(x) {if(x<min(order.y)[1]) result <- Inf
else if (x==min(order.y)[1]) result <- 1
else if(x >= max(order.y)[1]) result <- -Inf
else which(order.y>=x)[1]-1})
m.alpha <- mod$coef[index]
m.alpha <- ifelse(is.infinite(index), index, m.alpha)
if(length(at.y)==1){
lb <- as.matrix(outer(m.alpha, xb, "+")[,,1])
} else lb <- t(outer(m.alpha, xb, "+")[,,1])
m.cdf <- 1- mod$trans$cumprob(lb)
if(se){
if(mod$family=="logistic") mod$trans$deriv <- function(x) exp(-x)/(1+exp(-x))^2
cdf.se <- matrix(NA, ncol=length(at.y), nrow=dim(new.data)[1])
lb.se <- matrix(NA, ncol=length(at.y), nrow=dim(new.data)[1])
var <- as.matrix(solve(mod$info.matrix))
for(i in 1:length(at.y)) {
var.i <- var[c(index[i], which(names(coef(mod)) %in% colnames(new.data))),
c(index[i], which(names(coef(mod)) %in% colnames(new.data)))]
dcdf.dtheta <- cbind(-mod$trans$deriv(lb[,i]),
-mod$trans$deriv(lb[,i])*as.matrix(new.data) )
dlb.dtheta <- as.matrix(cbind(1, new.data))
cdf.se[,i] <- sqrt(diag(dcdf.dtheta %*% var.i%*% t(dcdf.dtheta)))
lb.se[, i] <- sqrt(diag(dlb.dtheta%*%var.i%*% t(dlb.dtheta)))
}
ci.lb <- sapply(1:length(at.y), FUN=function(i) { 1- mod$trans$cumprob(lb[, i] +qnorm(0.975)*lb.se[, i])})
ci.ub <- sapply(1:length(at.y), FUN=function(i) { 1- mod$trans$cumprob(lb[, i] -qnorm(0.975)*lb.se[, i])})
result <- list(est=m.cdf,
se=cdf.se,
lb=ci.lb,
ub=ci.ub)
} else{
result <- list(est=m.cdf)
}
return(result)
}
}
############################ SIMULATIONS ##############
###########Simulation 1: orm with properly specified link functions under ####################
################### i) normal error ###################
generate.data.1 <- function(seed=1, n=200, p=0.5, alpha=0, beta=c(1, -0.5), sigma=1){
set.seed(seed)
z1 <- sample(c(0,1), size=n, replace=TRUE, prob=c(1-p, p))
z2 <- rnorm(n, 0, 1)
#y.star <- rnorm(n, alpha+beta*z, sigma)
#y <- exp(y.star)
log.y <- rnorm(n, alpha+beta[1]*z1 + beta[2]*z2, sigma)
data <- data.frame(y=log.y, z1=z1, z2=z2)
return(data)
}
######### Sim 1 (i) regression coefficients ################
sim1_coeff.fun <- function(sim=100,seed=1, n=50,
p=0.5, alpha=0, beta=c(1,-0.5), sigma=1,
log.y=c(-1, -0.33, 0.5, 1.33, 2), se=TRUE){
set.seed(seed)
seeds <- unique(round(runif(sim*10,0,10)*sim,0))[seq(sim)]
alpha.y <- matrix(NA, ncol=length(log.y), nrow=sim)
alpha.y.se <- matrix(NA, ncol=length(log.y), nrow=sim)
beta.est <- matrix(NA, ncol=length(beta), nrow=sim)
beta.se <- matrix(NA, ncol=length(beta), nrow=sim)
for(i in 1:sim){
try({
data <- generate.data.1(seed=seeds[i], n=n,p=p, alpha=alpha, beta=beta, sigma=sigma)
order.y <- data$y[order(data$y)]
m.orm <- orm(y~z1+z2, data=data, family="probit")
beta.est[i,] <- m.orm$coef[c(n, n+1)]
beta.se[i, 1] <- sqrt(m.orm$var["z1", "z1"])
beta.se[i, 2] <- sqrt(m.orm$var["z2", "z2"])
alpha.y[i,] <- -sapply(log.y, FUN=function(x) {if(x<=min(data$y)) result <- Inf
else if(x > max(data$y)) result <- -Inf
else m.orm$coef[which(order.y>=x)[1]-1]})
if(se){
var <- as.matrix(solve(m.orm$info.matrix))
alpha.y.se[i,] <- sqrt(sapply(log.y, FUN=function(x) {if(x<=min(data$y)) result <- Inf
else if(x > max(data$y)) result <- Inf
else var[which(order.y>=x)[1]-1, which(order.y>=x)[1]-1]}) )
}
})
}
return(list(beta.est=beta.est,
beta.se=beta.se,
alpha.y=alpha.y,
alpha.y.se=alpha.y.se))
}
seed=1
p <- 0.5
alpha <- 0
beta <- c(1, -0.5)
sigma <- 1
log.y <- c(-1, -0.33, 0.5, 1.33, 2)
sim=10^4
#sim=100
sim1_coeff.n25 <- sim1_coeff.fun(n=25, sim=sim)
sim1_coeff.n50 <- sim1_coeff.fun(n=50, sim=sim)
sim1_coeff.n100 <- sim1_coeff.fun(n=100, sim=sim)
sim1_coeff.n200 <- sim1_coeff.fun(n=200, sim=sim)
sim1_coeff.n500 <- sim1_coeff.fun(n=500, sim=sim)
sim1_coeff.n1000 <- sim1_coeff.fun(n=1000, sim=sim)
sim1_coeff.summary <- function(result, beta=c(1, -0.5),alpha.y=c(-1, -0.33, 0.5, 1.33, 2)){
mean.est <- c(apply(result$beta.est, 2, FUN=function(x) mean(x, na.rm=TRUE)),
apply(ifelse(is.infinite(result$alpha.y), NA, result$alpha.y), 2, FUN=function(x) mean(x, na.rm=TRUE)))
mean.se <- c(apply(result$beta.se, 2, FUN=function(x) mean(x, na.rm=TRUE)),
apply(ifelse(is.infinite(result$alpha.y.se), NA, result$alpha.y.se), 2, FUN=function(x) mean(x, na.rm=TRUE)))
emp.se <- c(apply(result$beta.est, 2, FUN=function(x) sd(x, na.rm=TRUE)),
apply(ifelse(is.infinite(result$alpha.y), NA, result$alpha.y), 2, FUN=function(x) sd(x, na.rm=TRUE)))
mse <- c(apply((result$beta.est-matrix(beta, nrow=dim(result$beta.est)[1], ncol=length(beta),byrow=TRUE))^2, 2, FUN=function(x) mean(x, na.rm=TRUE)),
apply((ifelse(is.infinite(result$alpha.y), NA, result$alpha.y)-matrix(alpha.y, nrow=dim(result$alpha.y)[1], ncol=length(alpha.y), byrow=TRUE))^2, 2 ,FUN=function(x) mean(x, na.rm=TRUE)))
beta.lb <-result$beta.est+qnorm(0.025, 0, 1)*result$beta.se
beta.ub <- result$beta.est-qnorm(0.025, 0, 1)*result$beta.se
alpha.lb <- cbind(result$alpha.y+qnorm(0.025, 0, 1)*result$alpha.y.se)
alpha.ub <- cbind(result$alpha.y-qnorm(0.025, 0, 1)*result$alpha.y.se)
alpha.lb <- ifelse(is.infinite(alpha.lb), NA, alpha.lb)
alpha.ub <- ifelse(is.infinite(alpha.ub), NA, alpha.ub)
alpha.coverage <- rep(NA, length(alpha.y))
for(i in 1:length(alpha.y)){
alpha.coverage[i] <-mean(apply(cbind(alpha.lb[,i], alpha.ub[,i]), 1, FUN=function(x) alpha.y[i]>=x[1] & alpha.y[i]<=x[2]), na.rm=TRUE)
}
beta.coverage <- rep(NA, length(beta))
for(i in 1:length(beta)){
beta.coverage[i] <-mean(apply(cbind(beta.lb[,i], beta.ub[,i]), 1, FUN=function(x) beta[i]>=x[1] & beta[i]<=x[2]), na.rm=TRUE)
}
coverage <- c(beta.coverage, alpha.coverage)
result <- cbind(format(c(beta, alpha.y), nsmall=1),
format(round(mean.est, 2), nsmall=2),
format(round(mean.se, 3) , nsmall=3),
format(round(emp.se, 3) , nsmall=3),
format(round(mse, 4) , nsmall=3),
format(round(coverage, 3) , nsmall=3))
return(result)
}
######## Table S.2; Figure 6 (i) error ~ Normal ################
sim1_coeff.probit.n25 <- sim1_coeff.summary(sim1_coeff.n25)
sim1_coeff.probit.n50 <- sim1_coeff.summary(sim1_coeff.n50)
sim1_coeff.probit.n100 <- sim1_coeff.summary(sim1_coeff.n100)
sim1_coeff.probit.n200 <- sim1_coeff.summary(sim1_coeff.n200)
sim1_coeff.probit.n500 <- sim1_coeff.summary(sim1_coeff.n500)
sim1_coeff.probit.n1000 <- sim1_coeff.summary(sim1_coeff.n1000)
######### Sim 1 (i) regression coefficients compared with Box-Cox transformation ######### ################
box.cox.trans <- function(x,lambda){
if(lambda==0){
x <- log(x)
}else{
x <- (x^lambda-1)/lambda
}
return(x)
}
generate.data.1 <- function(seed=1, n=200, p=0.5, alpha=0, beta=c(1, -0.5), sigma=1){
set.seed(seed)
z1 <- sample(c(0,1), size=n, replace=TRUE, prob=c(1-p, p))
z2 <- rnorm(n, 0, 1)
#y.star <- rnorm(n, alpha+beta*z, sigma)
#y <- exp(y.star)
log.y <- rnorm(n, alpha+beta[1]*z1 + beta[2]*z2, sigma)
data <- data.frame(y=exp(log.y), z1=z1, z2=z2)
return(data)
}
sim1_boxcox.fun <- function(sim=100,seed=1, n=50,
p=0.5, alpha=0, beta=c(1, -0.5), sigma=1,
log.y=c(-1, -0.33, 0.5, 1.33, 2), se=TRUE){
set.seed(seed)
seeds <- unique(round(runif(sim*10,0,10)*sim,0))[seq(sim)]
t.y <- matrix(NA, ncol=length(log.y), nrow=sim)
beta.est <- matrix(NA, ncol=length(beta), nrow=sim)
beta.se <- matrix(NA, ncol=length(beta), nrow=sim)
beta.confint <- matrix(NA, ncol=4, nrow=sim)
for(i in 1:sim){
try({
data <- generate.data.1(seed=seeds[i], n=n,p=p, alpha=alpha, beta=beta, sigma=sigma)
b.m <- boxcox(y~z1+z2, data=data, plotit=F, interp = TRUE)
lambda <- b.m$x[which(b.m$y==max(b.m$y))][1]
data$y.trans <- box.cox.trans(data$y, lambda)
mod <- ols(y.trans~z1+z2, data=data)
beta.est[i, ] <- mod$coef[c("z1", "z2")]
beta.se[i, 1] <- sqrt(mod$var["z1", "z1"])
beta.se[i, 2] <- sqrt(mod$var["z2", "z2"])
beta.confint[i,c(1,2)] <- confint(mod)["z1",]
beta.confint[i,c(3,4)] <- confint(mod)["z2",]
t.y[i,] <- box.cox.trans(exp(log.y), lambda)
})
}
return(list(beta.est=beta.est,
beta.se=beta.se,
beta.confint=beta.confint,
t.y=t.y))
}
seed=1
p <- 0.5
alpha <- 0
beta <- c(1,-0.5)
sigma <- 1
log.y=c(-1, -0.33, 0.5, 1.33, 2)
sim=10^4
sim1_boxcox.n25 <- sim1_boxcox.fun(n=25, sim=sim)
sim1_boxcox.n50 <- sim1_boxcox.fun(n=50, sim=sim)
sim1_boxcox.n100 <- sim1_boxcox.fun(n=100, sim=sim)
sim1_boxcox.n200 <- sim1_boxcox.fun(n=200, sim=sim)
sim1_boxcox.n500 <- sim1_boxcox.fun(n=500, sim=sim)
sim1_boxcox.n1000 <- sim1_boxcox.fun(n=1000, sim=sim)
###### relative efficiency: orm vs. box-cox
RE.summary <- function(result1, result2,beta=c(1, -0.5)){
MSE1 <- apply(result1$beta.est -matrix(beta, nrow=dim(result1$beta.est)[1], ncol=length(beta),byrow=TRUE), 2, FUN=function(x) mean(x^2, na.rm=TRUE))
MSE2 <- apply(result2$beta.est -matrix(beta, nrow=dim(result2$beta.est)[1], ncol=length(beta),byrow=TRUE), 2, FUN=function(x) mean(x^2, na.rm=TRUE))
re <- MSE2/MSE1
return(re)
}
sim1_beta.re.n25 <- RE.summary(sim1_coeff.n25, sim1_boxcox.n25)
sim1_beta.re.n50 <- RE.summary(sim1_coeff.n50, sim1_boxcox.n50)
sim1_beta.re.n100 <- RE.summary(sim1_coeff.n100, sim1_boxcox.n100)
sim1_beta.re.n200 <- RE.summary(sim1_coeff.n200, sim1_boxcox.n200)
sim1_beta.re.n500 <- RE.summary(sim1_coeff.n500, sim1_boxcox.n500)
sim1_beta.re.n1000 <- RE.summary(sim1_coeff.n1000, sim1_boxcox.n1000)
##### Figure 7: relative efficiency (a)
sim1_re.probit.beta.1 <- c(sim1_beta.re.n25[1],
sim1_beta.re.n50[1],
sim1_beta.re.n100[1],
sim1_beta.re.n200[1],
sim1_beta.re.n500[1],
sim1_beta.re.n1000[1])
sim1_re.probit.beta.2 <- c(sim1_beta.re.n25[2],
sim1_beta.re.n50[2],
sim1_beta.re.n100[2],
sim1_beta.re.n200[2],
sim1_beta.re.n500[2],
sim1_beta.re.n1000[2])
####### Sim 1 (i) cdf and mean
generate.data.1 <- function(seed=1, n=200, p=0.5, alpha=0, beta=c(1, -0.5), sigma=1){
set.seed(seed)
z1 <- sample(c(0,1), size=n, replace=TRUE, prob=c(1-p, p))
z2 <- rnorm(n, 0, 1)
#y.star <- rnorm(n, alpha+beta*z, sigma)
#y <- exp(y.star)
log.y <- rnorm(n, alpha+beta[1]*z1 + beta[2]*z2, sigma)
data <- data.frame(y=exp(log.y), z1=z1, z2=z2)
return(data)
}
sim1_cdfmean.fun <- function(sim=100,seed=1, n=50,
p=0.5, alpha=0, beta=c(1, -0.5), sigma=1,
log.y=c(-1, -0.33, 0.5, 1.33, 2), se=TRUE){
set.seed(seed)
seeds <- unique(round(runif(sim*10,0,10)*sim,0))[seq(sim)]
control.est <- matrix(NA, ncol=length(log.y)+1, nrow=sim)
trt.est <- matrix(NA, ncol=length(log.y)+1, nrow=sim)
control.est.1 <- matrix(NA, ncol=length(log.y)+1, nrow=sim)
trt.est.1 <- matrix(NA, ncol=length(log.y)+1, nrow=sim)
control.se <- matrix(NA, ncol=length(log.y)+1, nrow=sim)
trt.se <- matrix(NA, ncol=length(log.y)+1, nrow=sim)
control.se.1 <- matrix(NA, ncol=length(log.y)+1, nrow=sim)
trt.se.1 <- matrix(NA, ncol=length(log.y)+1, nrow=sim)
control.lb <- matrix(NA, ncol=length(log.y)+1, nrow=sim)
control.ub <- matrix(NA, ncol=length(log.y)+1, nrow=sim)
trt.lb <- matrix(NA, ncol=length(log.y)+1, nrow=sim)
trt.ub <- matrix(NA, ncol=length(log.y)+1, nrow=sim)
control.lb.1 <- matrix(NA, ncol=length(log.y)+1, nrow=sim)
control.ub.1 <- matrix(NA, ncol=length(log.y)+1, nrow=sim)
trt.lb.1 <- matrix(NA, ncol=length(log.y)+1, nrow=sim)
trt.ub.1 <- matrix(NA, ncol=length(log.y)+1, nrow=sim)
for(i in 1:sim){
try({
data <- generate.data.1(seed=seeds[i], n=n,p=p, alpha=alpha, beta=beta, sigma=sigma)
#order.y <- data$y[order(data$y)]
m.orm <- orm(y~z1+z2, data=data, family="probit", x=TRUE, y=TRUE)
cdf.result <- cdf.orm(m.orm, new.data=data.frame(z1=c(0,1, 0, 1), z2=c(0, 0, 1, 1)), at.y=exp(log.y),se=se)
mean.result <- data.frame(mean.orm(m.orm, new.data=data.frame(z1=c(0,1, 0, 1), z2=c(0, 0, 1, 1)), se=se))
control.est[i, ] <- c(mean.result$est[1], cdf.result$est[1,])
trt.est[i, ] <- c(mean.result$est[2], cdf.result$est[2,])
control.est.1[i, ] <- c(mean.result$est[3], cdf.result$est[3,])
trt.est.1[i, ] <- c(mean.result$est[4], cdf.result$est[4,])
if(se){
control.se[i, ] <- c(mean.result$se[1], cdf.result$se[1,])
trt.se[i, ] <- c(mean.result$se[2], cdf.result$se[2,])
control.se.1[i, ] <- c(mean.result$se[3], cdf.result$se[3,])
trt.se.1[i, ] <- c(mean.result$se[4], cdf.result$se[4,])
control.lb[i, ] <- c(mean.result$lb[1], cdf.result$lb[1,])
control.ub[i, ] <- c(mean.result$ub[1], cdf.result$ub[1,])
control.lb.1[i, ] <- c(mean.result$lb[3], cdf.result$lb[3,])
control.ub.1[i, ] <- c(mean.result$ub[3], cdf.result$ub[3,])
trt.lb[i, ] <- c(mean.result$lb[2], cdf.result$lb[2,])
trt.ub[i, ] <- c(mean.result$ub[2], cdf.result$ub[2,])
trt.lb.1[i, ] <- c(mean.result$lb[4], cdf.result$lb[4,])
trt.ub.1[i, ] <- c(mean.result$ub[4], cdf.result$ub[4,])
}
})
}
return(list(control.est=control.est,
trt.est=trt.est,
control.se=control.se,
trt.se=trt.se,
control.lb=control.lb,
trt.lb=trt.lb,
control.ub=control.ub,
trt.ub=trt.ub,
control.est.1=control.est.1,
trt.est.1=trt.est.1,
control.se.1=control.se.1,
trt.se.1=trt.se.1,
control.lb.1=control.lb.1,
trt.lb.1=trt.lb.1,
control.ub.1=control.ub.1,
trt.ub.1=trt.ub.1))
}
seed=1
p <- 0.5
alpha <- 0
beta <- c(1, -0.5)
sigma <- 1
log.y=c(-1, -0.33, 0.5, 1.33, 2)
sim=10^4
sim1_cdfmean.n25 <- sim1_cdfmean.fun(n=25, sim=sim)
sim1_cdfmean.n50 <- sim1_cdfmean.fun(n=50, sim=sim)
sim1_cdfmean.n100 <- sim1_cdfmean.fun(n=100, sim=sim)
sim1_cdfmean.n200 <- sim1_cdfmean.fun(n=200, sim=sim)
sim1_cdfmean.n500 <- sim1_cdfmean.fun(n=500, sim=sim)
sim1_cdfmean.n1000 <- sim1_cdfmean.fun(n=1000, sim=sim)
sim1_cdfmean.summary <- function(result, beta=c(1, -0.5),alpha=0,sigma=1,alpha.y=c(-1, -0.33, 0.5, 1.33, 2), family=c("probit", "loglog")){
if(family[1]=="probit"){
true.cdf.01 <- pnorm(log.y, alpha+c(0,1)%*%beta, sigma)
true.cdf.11 <- pnorm(log.y, alpha+c(1,1)%*%beta)
true.mean.01 <- exp(alpha+c(0,1)%*%beta+sigma^2/2)
true.mean.11 <- exp(alpha+c(1,1)%*%beta+sigma^2/2)
true.mean.00 <- exp(alpha+c(0,0)%*%beta+sigma^2/2)
true.mean.10 <- exp(alpha+c(1,0)%*%beta+sigma^2/2)
true.mean <- c(true.mean.00, true.mean.10, true.mean.01, true.mean.11)
} else if (family[1]=="loglog"){
true.cdf.01 <- pGumbelMin(log.y, alpha+c(0,1)%*%beta, sigma)
true.cdf.11 <- pGumbelMin(log.y, alpha+c(1,1)%*%beta, sigma)
true.mean.01 <- mean(exp(rGumbelMin(10^6, alpha+c(0,1)%*%beta, sigma)))
true.mean.11 <- mean(exp(rGumbelMin(10^6, alpha+c(1,1)%*%beta, sigma)))
true.mean.00 <- mean(exp(rGumbelMin(10^6, alpha+c(0,0)%*%beta, sigma)))
true.mean.10 <- mean(exp(rGumbelMin(10^6, alpha+c(1,0)%*%beta, sigma)))
true.mean <- c(true.mean.00, true.mean.10, true.mean.01, true.mean.11)
}
mean.est <- c(mean(result$control.est[,1], na.rm=TRUE),
mean(result$trt.est[,1], na.rm=TRUE),
mean(result$control.est.1[,1], na.rm=TRUE),
mean(result$trt.est.1[,1], na.rm=TRUE))
mean.emp.se <-c(sd(result$control.est[,1], na.rm=TRUE),
sd(result$trt.est[,1], na.rm=TRUE),
sd(result$control.est.1[,1], na.rm=TRUE),
sd(result$trt.est.1[,1], na.rm=TRUE))
mean.est.se <- c(mean(result$control.se[,1], na.rm=TRUE),
mean(result$trt.se[,1], na.rm=TRUE),
mean(result$control.se.1[,1], na.rm=TRUE),
mean(result$trt.se.1[,1], na.rm=TRUE))
mean.coverage <- c(mean(true.mean[1] >=result$control.lb[, 1] & result$control.ub[,1]>=true.mean[1], na.rm=TRUE),
mean(true.mean[2] >=result$trt.lb[, 1] & result$trt.ub[,1]>=true.mean[2], na.rm=TRUE),
mean(true.mean[3] >=result$control.lb.1[, 1] & result$control.ub.1[,1]>=true.mean[3], na.rm=TRUE),
mean(true.mean[4] >=result$trt.lb.1[, 1] & result$trt.ub.1[,1]>=true.mean[4], na.rm=TRUE) )
cdf.01.est <- apply(result$control.est.1[,-1], 2, FUN=function(x) mean(x, na.rm=TRUE))
cdf.11.est <- apply(result$trt.est.1[,-1], 2, FUN=function(x) mean(x, na.rm=TRUE))
cdf.01.est.se <- apply(result$control.est.1[,-1], 2, FUN=function(x) sd(x, na.rm=TRUE))
cdf.11.est.se <- apply(result$trt.est.1[,-1], 2, FUN=function(x) sd(x, na.rm=TRUE))
cdf.01.emp.se <- apply(result$control.se.1[,-1], 2, FUN=function(x) mean(x, na.rm=TRUE))
cdf.11.emp.se <- apply(result$trt.se.1[,-1], 2, FUN=function(x) mean(x, na.rm=TRUE))
cdf.01.coverage <- sapply(1:(length(log.y)), FUN=function(i) mean( true.cdf.01[i]>=result$control.lb.1[,i+1]&true.cdf.01[i]<=result$control.ub.1[, i+1], na.rm=TRUE))
cdf.11.coverage <- sapply(1:(length(log.y)), FUN=function(i) mean( true.cdf.11[i]>=result$trt.lb.1[,i+1]&true.cdf.11[i]<=result$trt.ub.1[, i+1], na.rm=TRUE))
mean.out <- cbind(format(round(true.mean, 2), nsmall=2),
format(round(mean.est, 3), nsmall=3),
format(round(mean.est.se, 3), nsmall=3),
format(round(mean.emp.se, 3), nsmall=3),
format(round(mean.coverage, 3), nsmall=3))
rownames(mean.out) <- c("mean|00", "mean|10", "mean|01", "mean|11")
cdf.01.out <- cbind(format(round(true.cdf.01, 4),4),
format(round(cdf.01.est, 4),4),
format(round(cdf.01.est.se, 4),4),
format(round(cdf.01.emp.se, 4),4),
format(round(cdf.01.coverage, 3),3)
)
cdf.11.out <- cbind(format(round(true.cdf.11, 4),4),
format(round(cdf.11.est, 4),4),
format(round(cdf.11.est.se, 4),4),
format(round(cdf.11.emp.se, 4),4),
format(round(cdf.11.coverage, 3),3)
)
result <- list(mean.out=mean.out,
cdf.01.out=cdf.01.out,
cdf.11.out=cdf.11.out)
return(result)
}
sim1_cdfmean.probit.n25 <- sim1_cdfmean.summary(sim1_cdfmean.n25)
sim1_cdfmean.probit.n50 <- sim1_cdfmean.summary(sim1_cdfmean.n50)
sim1_cdfmean.probit.n100 <- sim1_cdfmean.summary(sim1_cdfmean.n100)
sim1_cdfmean.probit.n200 <- sim1_cdfmean.summary(sim1_cdfmean.n200)
sim1_cdfmean.probit.n500 <- sim1_cdfmean.summary(sim1_cdfmean.n500)
sim1_cdfmean.probit.n1000 <- sim1_cdfmean.summary(sim1_cdfmean.n1000)
###### Figure 8 (i) error ~ Normal, Table S.3 (i) error ~ Normal
sim1_cdf.probit.out <- rbind(rep("", 5),
rbind(sim1_cdfmean.probit.n25$cdf.11.out),
rep("", 5),
rbind(sim1_cdfmean.probit.n50$cdf.11.out),
rep("", 5),
rbind(sim1_cdfmean.probit.n100$cdf.11.out),
rep("", 5),
rbind(sim1_cdfmean.probit.n200$cdf.11.out),
rep("", 5),
rbind(sim1_cdfmean.probit.n500$cdf.11.out),
rep("", 5),
rbind(sim1_cdfmean.probit.n1000$cdf.11.out))
###### Figure S.1 (i) error ~ Normal, Table S.4 (i) error ~ Normal
sim1_mean.probit.out <- rbind(rep("", 5),
sim1_cdfmean.probit.n25$mean.out,
rep("", 5),
sim1_cdfmean.probit.n50$mean.out,
rep("", 5),
sim1_cdfmean.probit.n100$mean.out,
rep("", 5),
sim1_cdfmean.probit.n200$mean.out,
rep("", 5),
sim1_cdfmean.probit.n500$mean.out,
rep("", 5),
sim1_cdfmean.probit.n1000$mean.out)
########### Sim 1 (i) quantiles
generate.data.1 <- function(seed=1, n=200, p=0.5, alpha=0, beta=c(1, -0.5), sigma=1){
set.seed(seed)
z1 <- sample(c(0,1), size=n, replace=TRUE, prob=c(1-p, p))
z2 <- rnorm(n, 0, 1)
#y.star <- rnorm(n, alpha+beta*z, sigma)
#y <- exp(y.star)
log.y <- rnorm(n, alpha+beta[1]*z1 + beta[2]*z2, sigma)
data <- data.frame(y=exp(log.y), z1=z1, z2=z2)
return(data)
}
sim1_quantile.fun <- function(sim=100,seed=1, n=50,
p=0.5, alpha=0, beta=c(1, -0.5), sigma=1,
probs=c(0.1, 0.25, 0.5, 0.75, 0.9), se=TRUE){
set.seed(seed)
seeds <- unique(round(runif(sim*10,0,10)*sim,0))[seq(sim)]
est.00 <- est.10 <- est.01<- est.11 <- matrix(NA, ncol=length(probs), nrow=sim)
lb.00 <- lb.10 <- lb.01<- lb.11 <- matrix(NA, ncol=length(probs), nrow=sim)
ub.00 <- ub.10 <- ub.01<- ub.11 <- matrix(NA, ncol=length(probs), nrow=sim)
for(i in 1:sim){
try({
data <- generate.data.1(seed=seeds[i], n=n,p=p, alpha=alpha, beta=beta, sigma=sigma)
m.orm <- orm(y~z1+z2, data=data, family="probit", x=TRUE, y=TRUE)
result <- quantile.orm(mod=m.orm, new.data=data.frame(z1=c(0,1, 0, 1), z2=c(0, 0, 1, 1)), probs=probs, se=TRUE)
est.00[i, ] <- result$quantile[1,]
est.10[i, ] <- result$quantile[2,]
est.01[i, ] <- result$quantile[3,]
est.11[i, ] <- result$quantile[4,]
lb.00[i, ] <- result$lb[1,]
lb.10[i, ] <- result$lb[2,]
lb.01[i, ] <- result$lb[3,]
lb.11[i, ] <- result$lb[4,]
ub.00[i, ] <- result$ub[1,]
ub.10[i, ] <- result$ub[2,]
ub.01[i, ] <- result$ub[3,]
ub.11[i, ] <- result$ub[4,]
})
}
result <- list(result.00=list(est=est.00,
lb=lb.00,
ub=ub.00),
result.10=list(est=est.10,
lb=lb.10,
ub=ub.10),
result.01=list(est=est.01,
lb=lb.01,
ub=ub.01),
result.11=list(est=est.11,
lb=lb.11,
ub=ub.11))
return(result)
}
seed=1
p <- 0.5
alpha <- 0
beta <- c(1, -0.5)
sigma <- 1
probs=c(0.1, 0.25, 0.5, 0.75, 0.9)
se=TRUE
sim=10^4
sim1_quantile.n25 <- sim1_quantile.fun(n=25, sim=sim)
sim1_quantile.n50 <- sim1_quantile.fun(n=50, sim=sim)
sim1_quantile.n100 <- sim1_quantile.fun(n=100, sim=sim)
sim1_quantile.n200 <- sim1_quantile.fun(n=200, sim=sim)
sim1_quantile.n500 <- sim1_quantile.fun(n=500, sim=sim)
sim1_quantile.n1000 <- sim1_quantile.fun(n=1000, sim=sim)
sim1_quantile.summary <- function(result, true.quantile){
true <- true.quantile
est <- apply(result$est, 2, FUN=function(x) mean(x, na.rm=TRUE))
emp.se <- apply(result$est, 2, FUN=function(x) sd(x, na.rm=TRUE))
mse <- apply(result$est -matrix(true.quantile, ncol=length(true.quantile), nrow=dim(result$est)[1], byrow=T ),
2, FUN=function(x) mean(x^2, na.rm=TRUE))
cover <- sapply(1:length(true.quantile), FUN=function(i) true.quantile[i]>=result$lb[,i] & true.quantile[i]<=result$ub[,i])
coverage <- apply(cover, 2, FUN=function(x) mean(x, na.rm=TRUE))
result <- cbind(format(round(true, 3), nsmall=3),
format(round(est, 3), nsmall=3),
format(round(emp.se, 4), nsmall=4),
format(round(coverage, 3), nsmall=3)
)
}
new.data <- rbind(c(0,1), c(1, 1))
true.mean <- beta %*% new.data
sim1_quantile.probit.n25 <- sim1_quantile.summary(result=sim1_quantile.n25$result.11,
true.quantile=exp(qnorm(c(0.1, 0.25, 0.5, 0.75, 0.9), mean=true.mean[2], sd=1)))
sim1_quantile.probit.n50 <- sim1_quantile.summary(result=sim1_quantile.n50$result.11,
true.quantile=exp(qnorm(c(0.1, 0.25, 0.5, 0.75, 0.9), mean=true.mean[2], sd=1)))
sim1_quantile.probit.n100 <- sim1_quantile.summary(result=sim1_quantile.n100$result.11,
true.quantile=exp(qnorm(c(0.1, 0.25, 0.5, 0.75, 0.9), mean=true.mean[2], sd=1)))
sim1_quantile.probit.n200 <- sim1_quantile.summary(result=sim1_quantile.n200$result.11,
true.quantile=exp(qnorm(c(0.1, 0.25, 0.5, 0.75, 0.9), mean=true.mean[2], sd=1)))
sim1_quantile.probit.n500 <- sim1_quantile.summary(result=sim1_quantile.n500$result.11,
true.quantile=exp(qnorm(c(0.1, 0.25, 0.5, 0.75, 0.9), mean=true.mean[2], sd=1)))
sim1_quantile.probit.n1000 <- sim1_quantile.summary(result=sim1_quantile.n1000$result.11,
true.quantile=exp(qnorm(c(0.1, 0.25, 0.5, 0.75, 0.9), mean=true.mean[2], sd=1)))
###### Figure S.2 (i) error ~ Normal, Table S.5 (i) error ~ Normal
sim1_quantile.probit.out <- rbind(rep("", 4),
rbind(sim1_quantile.probit.n25),
rep("", 4),
rbind(sim1_quantile.probit.n50),
rep("", 4),
rbind(sim1_quantile.probit.n100),
rep("", 4),
rbind(sim1_quantile.probit.n200),
rep("", 4),
rbind(sim1_quantile.probit.n500),
rep("", 4),
rbind(sim1_quantile.probit.n1000)
)
##ii) extreme value error distribution
##### Sim 1 (ii) regression coefficients ###########
rGumbelMin<- function(n, mu=0, sigma=1){
u <- runif(n, min=0, max=1)
x <- mu + sigma*log(-log(1-u))
return(x)
}
dGumbelMin <- function(x, mu=0, sigma=1){
df <- 1/sigma *exp((x-mu)/sigma)*exp(-exp((x-mu)/sigma))
return(df)
}
pGumbelMin <- function(x, mu=0, sigma=1){
pf <- 1-exp(-exp((x-mu)/sigma))
return(pf)
}
qGumbelMin <- function(p, mu=0, sigma=1){
x <- mu + sigma*(-log(-log(1-p)))
}
generate.data.2 <- function(seed, n, p=0.5,alpha=0, beta=c(1, -0.5), sigma=1){
set.seed(seed)
z1 <- sample(c(0,1), size=n, replace=TRUE, prob=c(1-p, p))
z2 <- rnorm(n, 0, 1)
y <- rGumbelMin(n, mu=alpha+beta[1]*z1+beta[2]*z2, sigma=sigma)
data <- data.frame(y, z1=z1, z2=z2)
return(data)
}
sim2_coeff.fun <- function(sim=100,seed=1, n=50,
p=0.5, alpha=0, beta=c(1,-0.5), sigma=1,
log.y=c(-1, -0.33, 0.5, 1.33, 2), se=TRUE){
set.seed(seed)
seeds <- unique(round(runif(sim*10,0,10)*sim,0))[seq(sim)]
alpha.y <- matrix(NA, ncol=length(log.y), nrow=sim)
alpha.y.se <- matrix(NA, ncol=length(log.y), nrow=sim)
beta.est <- matrix(NA, ncol=length(beta), nrow=sim)
beta.se <- matrix(NA, ncol=length(beta), nrow=sim)
for(i in 1:sim){
try({
data <- generate.data.2(seed=seeds[i], n=n,p=p, alpha=alpha, beta=beta, sigma=sigma)
order.y <- data$y[order(data$y)]
m.orm <- orm(y~z1+z2, data=data, family=loglog)
beta.est[i,] <- m.orm$coef[c(n, n+1)]
beta.se[i, 1] <- sqrt(m.orm$var["z1", "z1"])
beta.se[i, 2] <- sqrt(m.orm$var["z2", "z2"])
alpha.y[i,] <- -sapply(log.y, FUN=function(x) {if(x<=min(data$y)) result <- Inf
else if(x > max(data$y)) result <- -Inf
else m.orm$coef[which(order.y>=x)[1]-1]})
if(se){
var <- as.matrix(solve(m.orm$info.matrix))
alpha.y.se[i,] <- sqrt(sapply(log.y, FUN=function(x) {if(x<=min(data$y)) result <- Inf
else if(x > max(data$y)) result <- Inf
else var[which(order.y>=x)[1]-1, which(order.y>=x)[1]-1]}) )
}
})
}
return(list(beta.est=beta.est,
beta.se=beta.se,
alpha.y=alpha.y,
alpha.y.se=alpha.y.se))
}
seed=1
p <- 0.5
alpha <- 0
beta <- c(1, -0.5)
sigma <- 1
log.y=c(-1, -0.33, 0.5, 1.33, 2)
sim=10^4
#sim=100
sim2_coeff.n25 <- sim2_coeff.fun(n=25, sim=sim)
sim2_coeff.n50 <- sim2_coeff.fun(n=50, sim=sim)
sim2_coeff.n100 <- sim2_coeff.fun(n=100, sim=sim)
sim2_coeff.n200 <- sim2_coeff.fun(n=200, sim=sim)
sim2_coeff.n500 <- sim2_coeff.fun(n=500, sim=sim)
sim2_coeff.n1000 <- sim2_coeff.fun(n=1000, sim=sim)
sim2_coeff.summary <- function(result, beta=c(1, -0.5),alpha.y=c(-1, -0.33, 0.5, 1.33, 2)){
mean.est <- c(apply(result$beta.est, 2, FUN=function(x) mean(x, na.rm=TRUE)),
apply(ifelse(is.infinite(result$alpha.y), NA, result$alpha.y), 2, FUN=function(x) mean(x, na.rm=TRUE)))
mean.se <- c(apply(result$beta.se, 2, FUN=function(x) mean(x, na.rm=TRUE)),
apply(ifelse(is.infinite(result$alpha.y.se), NA, result$alpha.y.se), 2, FUN=function(x) mean(x, na.rm=TRUE)))
emp.se <- c(apply(result$beta.est, 2, FUN=function(x) sd(x, na.rm=TRUE)),
apply(ifelse(is.infinite(result$alpha.y), NA, result$alpha.y), 2, FUN=function(x) sd(x, na.rm=TRUE)))
mse <- c(apply((result$beta.est-matrix(beta, nrow=dim(result$beta.est)[1], ncol=length(beta),byrow=TRUE))^2, 2, FUN=function(x) mean(x, na.rm=TRUE)),
apply((ifelse(is.infinite(result$alpha.y), NA, result$alpha.y)-matrix(alpha.y, nrow=dim(result$alpha.y)[1], ncol=length(alpha.y), byrow=TRUE))^2, 2 ,FUN=function(x) mean(x, na.rm=TRUE)))
beta.lb <-result$beta.est+qnorm(0.025, 0, 1)*result$beta.se
beta.ub <- result$beta.est-qnorm(0.025, 0, 1)*result$beta.se
alpha.lb <- cbind(result$alpha.y+qnorm(0.025, 0, 1)*result$alpha.y.se)
alpha.ub <- cbind(result$alpha.y-qnorm(0.025, 0, 1)*result$alpha.y.se)
alpha.lb <- ifelse(is.infinite(alpha.lb), NA, alpha.lb)
alpha.ub <- ifelse(is.infinite(alpha.ub), NA, alpha.ub)
alpha.coverage <- rep(NA, length(alpha.y))
for(i in 1:length(alpha.y)){
alpha.coverage[i] <-mean(apply(cbind(alpha.lb[,i], alpha.ub[,i]), 1, FUN=function(x) alpha.y[i]>=x[1] & alpha.y[i]<=x[2]), na.rm=TRUE)
}
beta.coverage <- rep(NA, length(beta))
for(i in 1:length(beta)){
beta.coverage[i] <-mean(apply(cbind(beta.lb[,i], beta.ub[,i]), 1, FUN=function(x) beta[i]>=x[1] & beta[i]<=x[2]), na.rm=TRUE)
}
coverage <- c(beta.coverage, alpha.coverage)
result <- cbind(format(c(beta, alpha.y), nsmall=1),
format(round(mean.est, 2), nsmall=2),
format(round(mean.se, 3) , nsmall=3),
format(round(emp.se, 3) , nsmall=3),
format(round(mse, 4) , nsmall=3),
format(round(coverage, 3) , nsmall=3))
return(result)
}
######## Table S.2; Figure 6 (ii) error ~ extreme Type I ################
sim2_coeff.loglog.n25 <- sim2_coeff.summary(sim2_coeff.n25)
sim2_coeff.loglog.n50 <- sim2_coeff.summary(sim2_coeff.n50)
sim2_coeff.loglog.n100 <- sim2_coeff.summary(sim2_coeff.n100)
sim2_coeff.loglog.n200 <- sim2_coeff.summary(sim2_coeff.n200)
sim2_coeff.loglog.n500 <- sim2_coeff.summary(sim2_coeff.n500)
sim2_coeff.loglog.n1000 <- sim2_coeff.summary(sim2_coeff.n1000)
######## sim 1 (ii) regression coefficients compared with Cox proportional hazards model
sim2_cox.fun <- function(sim=100,seed=1, n=50,
p=0.5, alpha=0, beta=c(1,-0.5), sigma=1,
log.y=c(-1, -0.33, 0.5, 1.33, 2)){
set.seed(seed)
seeds <- unique(round(runif(sim*10,0,10)*sim,0))[seq(sim)]
alpha.y <- matrix(NA, ncol=length(log.y), nrow=sim)
alpha.y.se <- matrix(NA, ncol=length(log.y), nrow=sim)
beta.est <- matrix(NA, ncol=length(beta), nrow=sim)
beta.se <- matrix(NA, ncol=length(beta), nrow=sim)