I am trying to understand the different ways of simulating survival times in a cox proportional hazards model. A simple example consists in simulating the event times following a Weibull distribution. I refer to this answer (https://stats.stackexchange.com/q/135129) for more deatils on the simulation procedure:
I paste part of the answer that details the simulation of event times in the Weibull case below :
Example [Weibull baseline hazard]
Let $h_0(t) = \lambda \rho t^{\rho - 1}$ with shape $\rho > 0$ and scale $\lambda > 0$. Then $H_0(t) = \lambda t^\rho$ and $H^{-1}_0(t) = (\frac{t}{\lambda})^{\frac{1}{\rho}}$. Following the inverse probability method, a realisation of $T \sim S(\cdot \,|\, \mathbf{x})$ is obtained by computing $$ t = \left( - \frac{\log(v)}{\lambda \exp(\mathbf{x}^\prime \mathbf{\beta})} \right)^{\frac{1}{\rho}} $$ with $v$ a uniform variate on $(0, 1)$. Using results on transformations of random variables, one may notice that $T$ has a conditional Weibull distribution (given $\mathbf{x}$) with shape $\rho$ and scale $\lambda \exp(\mathbf{x}^\prime \mathbf{\beta})$.
Right censoring of the event times implies that we observe min($t$,$C$) where $C$ is the censoring variable. I try out two ways of generating the censoring times below (20 percent censoring).
I also start from the same code in the answer with a different way of dealing with the censoring mechanism. I only changed the line dealing with generation of the censoring variable C in the code. Instead of simulating following an exponential distribution (as in the answer cited), I randomly generate 0's and 1's to censor the event time variable. :
simulWeib <- function(N, lambda, rho, beta, rateC, method) {
# covariate --> N Bernoulli trials
x <- sample(x=c(0, 1), size=N, replace=TRUE, prob=c(0.5, 0.5))
# Weibull latent event times
v <- runif(n=N)
Tlat <- (- log(v) / (lambda * exp(x * beta)))^(1 / rho)
# censoring times
if (method == "method 1") {
C = c(rep(0,N*rateC),rep(1,N*(1-rateC)))
C <- sample(C)
# follow-up times and event indicators
time <- Tlat
status <- C
} else if (method == "method 2") {
# censoring times
C <- rexp(n=N, rate=rateC)
# follow-up times and event indicators
time <- pmin(Tlat, C)
status <- as.numeric(Tlat <= C)
}
# data set
data.frame(id=1:N,
time=time,
status=status,
x=x)
}
set.seed(1234)
betaHat_1 <- rep(NA, 1e3)
betaHat_2 <- rep(NA, 1e3)
for(k in 1:1e3)
{
dat1 <- simulWeib(N=100, lambda=0.01, rho=1, beta=-0.6, rateC=0.2, method = "method 1")
fit1 <- coxph(Surv(time, status) ~ x , data=dat1)
betaHat_1[k] <- fit1$coef
dat2 <- simulWeib(N=100, lambda=0.01, rho=1, beta=-0.6, rateC=0.001, method = "method 2")
fit2 <- coxph(Surv(time, status) ~ x , data=dat2)
betaHat_2[k] <- fit2$coef
}
mean(betaHat_1)
> -0.6169394
mean(betaHat_2)
> -0.613186
With method 1, rateC gives directly the censoring rate but I am not sure this could still be considered right censoring. The common way of generating censored times is by method 2. In this example, the two estimates are similar though. Method 2 can be considered right censoring but what about method 1 ?
If the question or part of the code is not clear, feel free to ask for edits.
Thank you !
EDIT : I am mainly interested in random right censoring. If we look at "method 2" in the code, this is the classic way of simulating right censoring times. That is, you simulate event times using the inverse probability method as shown in the answer. Then the censoring times are simulated following an exponential distribtuion with a rate parameter that is adjusted so that we get the desired censoring level when we compute the right censored time : right-censored time = min(event_time,censoring_time). This part corresponds to "method 2" in the code and is clear.
When I was less well versed in the subject of survival analysis (3 years ago), I naively censored the event times by generating a censor variable with 0's (censored) and 1's (not censored) and randomly allocated the 0's and 1's to the event times. This corresponds to "method 1" in the code. What type of censoring does this correspond to ? Is this even considered censoring ? These are my questions.
