With censored event time observations, the logrank test is the most popular tool for testing the equality of two underlying survival distributions. Although this test is asymptotically distribution-free, it may not be powerful when the proportional hazards assumption is violated. Various other novel testing procedures have been proposed, which generally are derived by assuming a class of specific alternative hypotheses with respect to the hazard functions. The test considered by Pepe and Fleming (1989) is based on a linear combination of weighted differences of two Kaplan-Meier curves over time and is a natural tool to assess the difference of two survival functions directly. In this article, we take a similar approach, but choose weights which are proportional to the observed standardized difference of the estimated survival curves at each time point. The new proposal automatically makes weighting adjustments empirically. The new test statistic is aimed at a one-sided general alternative hypothesis, and is distributed with a short right tail under the null hypothesis, but with a heavy tail under the alternative. The results from extensive numerical studies demonstrate that the new procedure performs well under various general alternatives. The survival data from a recent cancer comparative study are utilized for illustrating the implementation of the process.
Uno, Hajime; Tian, Lu; Claggett, Brian; and Wei, L. J., "A versatile test for equality of two survival functions based on weighted differences of Kaplan-Meier curves" (May 2013). Harvard University Biostatistics Working Paper Series. Working Paper 159.