A Study of Bootstrap Confidence Intervals in a Cox Model

Abstract

Abstract : We study bootstrap confidence intervals for three types of parameters in Cox's proportional hazards model: the regression parameter, the survival function at fixed time points, and the median survival time at fixed values of a covariate. Several types of bootstrap confidence intervals are studied, and the type of interval is determined by two factors. One factor is the method of drawing the bootstrap sample. We consider three such methods, which may be briefly described as follows: (1) Ordinary resampling from the empirical cumulative distribution function, (2) Resampling conditional on the covariates, and (3) Resampling conditional on the covariates and the censoring pattern. Another factor is the method of forming the confidence interval from a given sample; the methods considered are the percentile, hybrid, and bootstrap-t. We provide a theorem on the asymptotic validity of the third method of bootstrap resampling. All the methods of forming confidence intervals are compared to each other and to the standard asymptotic method via a Monte Carlo study. The data sets for this Monte Carlo study are simulated conditionally on the covariates and the censoring pattern, the situation appropriate for the third method of resampling. One conclusion drawn from the Monte Carlo study is that the asymptotic method is best for the regression parameter, but not for the survival function or the median survival time. Conclusions about the bootstrap methods include the surprising result that, overall, the second method of drawing the samples outperforms the third method.