Randomization-based Test for Censored Outcomes: A New Look at the Logrank Test

Two-sample tests have been one of the most classical topics in statistics with wide application even in cutting edge applications. There are at least two modes of inference used to justify the two-sample tests. One is usual superpopulation inference assuming the units are independent and identically distributed (i.i.d.) samples from some superpopulation; the other is finite population inference that relies on the random assignments of units into different groups. When randomization is actually implemented, the latter has the advantage of avoiding distributional assumptions on the outcomes. In this paper, we will focus on finite population inference for censored outcomes, which has been less explored in the literature. Moreover, we allow the censoring time to depend on treatment assignment, under which exact permutation inference is unachievable. We find that, surprisingly, the usual logrank test can also be justified by randomization. Specifically, under a Bernoulli randomized experiment with non-informative i.i.d. censoring within each treatment arm, the logrank test is asymptotically valid for testing Fisher's null hypothesis of no treatment effect on any unit. Moreover, the asymptotic validity of the logrank test does not require any distributional assumption on the potential event times. We further extend the theory to the stratified logrank test, which is useful for randomized blocked designs and when censoring mechanisms vary across strata. In sum, the developed theory for the logrank test from finite population inference supplements its classical theory from usual superpopulation inference, and helps provide a broader justification for the logrank test.