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

We study the asymptotic properties of the logrankand stratified logrank tests under different types of assumptionsregarding the dependence of the censoring and the survival times.When the treatment group and the covariates are conditionallyindependent given that the subject is still at risk, the logrankstatistic is asymptotically standard normally distributed under thenull hypothesis of no treatment effect. Under this assumption, thestratified logrank statistic has asymptotic properties similar tologrank statistic.However, if the assumption of conditional independence of thetreatment and covariates given the at risk indicator fails, then thelogrank test statistic is generally biased and the bias generallyincreases in proportional to the square root of the sample size. Weprovide general formulas for the asymptotic bias and variance. Wealso establish a contiguous alternative theory regarding smallviolations of the assumption as well as of the usually consideredsmall differences between treatment and control group survivalhazards.We discuss and extend an available bias-correction method ofDiRienzo and Lagakos (2001a), especially with respect to thepractical use of this method with unknown and estimated distributionfunction for censoring given treatment group and covariates. Weobtain the correct asymptotic distribution of the bias-correctedtest statistic when stratumwise Kaplan-Meier estimators of theconditional censoring distribution are substituted into it. Withinthis framework, we prove the asymptotic unbiasedness of thecorrected test and find a consistent variance estimator.Major theoretical results and motivations of future studies areconfirmed by a series of simulation studies.