Comparing Survival Distributions in the Presence of Dependent Censoring: Asymptotic Validity and Bias-corrections of the Logrank Test

Abstract

We study the asymptotic properties of the logrank and stratified logrank tests under different types of assumptions regarding the dependence of the censoring and the survival times. When the treatment group and the covariates are conditionally independent given that the subject is still at risk, the logrank statistic is asymptotically standard normally distributed under the null hypothesis of no treatment effect. Under this assumption, the stratified logrank statistic has asymptotic properties similar to logrank statistic. However, if the assumption of conditional independence of the treatment and covariates given the at risk indicator fails, then the logrank test statistic is generally biased and the bias generally increases in proportional to the square root of the sample size. We provide general formulas for the asymptotic bias and variance. We also establish a contiguous alternative theory regarding small violations of the assumption as well as of the usually considered small differences between treatment and control group survival hazards. We discuss and extend an available bias-correction method of DiRienzo and Lagakos (2001a), especially with respect to the practical use of this method with unknown and estimated distribution function for censoring given treatment group and covariates. We obtain the correct asymptotic distribution of the bias-corrected test statistic when stratumwise Kaplan-Meier estimators of the conditional censoring distribution are substituted into it. Within this framework, we prove the asymptotic unbiasedness of the corrected test and find a consistent variance estimator. Major theoretical results and motivations of future studies are confirmed by a series of simulation studies.