Weighted Kaplan-Meier Statistics: Large Sample and Optimality Considerations

SUMMARY We propose a cumulative weighted difference in the Kaplan-Meier estimates as a test statistic for equality of distributions in the two-sample censored data survival analysis problem. For stability of such a statistic, the absolute value of the possibly random weight function must be bounded above by a multiple of (C −)1/2 + δ where 1 – C − is the left continuous censoring distribution function and δ > 0. For these weighted Kaplan–Meier (WKM) statistics, asymptotic distribution theory is presented along with expressions for the efficacy under a sequence of local alternatives. A simple censored data generalization of the two-sample difference in means test (z-test) is a member of this class and in large samples is seen to be quite efficient relative to the popular log-rank test under a range of alternatives including the proportional hazards alternative. Optimal weight functions are also calculated. The optimal WKM statistic is as efficient as the optimal weighted log-rank statistic for any particular sequence of local alternatives. Stratified statistics and trend statistics are also presented.