what is a censoring hazard ratio? I saw the following expression in paper [1] (equation 1)
\begin{equation}
\Pi_{i=1}^{n} \left[\frac{HR(t_i,x_{\sigma(i)})}{\sum_{j=1}^{n}Y_j(t_i)HR(t_i,x_{j})}\right]^{\delta_i}\left[\frac{HR^C(t_i,x_{\sigma(i)})}{\sum_{j=1}^{n}Y_j(t_i)HR^C(t_i,x_{j})}\right]^{1-\delta_i}
\end{equation}
In the text the function $HR^C$ is called the censoring hazard, which is
the ratio of the hazard of censoring at time t for an individual
with vector profile x versus an individual with a baseline vector
profile.
However I don't really get what this means?
[1]:Mackenzie, Todd, and Michal Abrahamowicz. "Marginal and hazard ratio specific random data generation: applications to semi-parametric bootstrapping." Statistics and Computing 12.3 (2002): 245-252.
 A: If we treat censoring as an event, then we can consider the time-to-censoring as a survival outcome and write the Cox partial likelihood for that censoring-event. The paper you link is concerned with simulating survival data. If you do this, you must simulate survival times as well as censoring times and observe the minimum of these two with a failure indicator taking one if a person dies before they are censored. The Cox Model is a natural way to estimate the censoring process if you treat the censoring as an event. The hazard ratio describes the relationship between covariates and the censoring process, where if the coefficient for certain values of X is greater than 1, that person is continually at a higher risk of being censored.
As an example, suppose we generate data from an exponential survival model where a treatment is effective but simultaneously increases the risk of censoring.
set.seed(123)
numpats <- 10000
trtind <- rep(0:1, each=numpats/2)
beta1 <- log(0.6) ## log rel risk for death trt:ctl
gamma1 <- log(0.8) ## log rel risk for censoring trt:ctl 
failtime <- rexp(numpats, exp(exp(beta1*trtind)) )
censtime <- rexp(numpats, exp(exp(gamma1*trtind)) )
obstime <- pmin(failtime, censtime)
dead <- obstime == failtime

## model failure process
failproc <- coxph(Surv(obstime, dead) ~ trtind)

## model censoring process: NB only works b/c noninformative censoring
censproc <- coxph(Surv(obstime, 1-dead) ~ trtind)

Gives
> exp(coef(censproc)) ## compare to 0.80
   trtind 
0.8416571 

