Comparison of methods to testing for differential treatment effect under non-proportional hazards data

Many tests for comparing survival curves have been proposed over the last decades. There are two branches, one based on weighted log-rank statistics and other based on weighted Kaplan-Meier statistics. If we carefully choose the weight function, a substantial increase in power of tests against non-proportional alternatives can be obtained. However, it is difficult to specify in advance the types of survival differences that may actually exist between two groups. Therefore, a combination test can simultaneously detect equally weighted, early, late or middle departures from the null hypothesis and can robustly handle several non-proportional hazard types with no a priori knowledge of the hazard functions. In this paper, we focus on the most used and the most powerful test statistics related to these two branches which have been studied separately but not compared between them. Through a simulation study, we compare the size and power of thirteen test statistics under proportional hazards and different types of non-proportional hazards patterns. We illustrate the procedures using data from a clinical trial of bone marrow transplant patients with leukemia.