Tests for comparing estimated survival functions

We describe a class of statistical tests for the comparison of two or more survival curves, typically estimated using the Kaplan–Meier method. The class is based on the construction of <cross-ref type="bib" refid="ASU015C14">O’Quigley (2003)</cross-ref>, and some special cases are of particular interest. Underlying the inferential development are the arguments of <cross-ref type="bib" refid="ASU015C6">Efron & Hinkley (1978)</cross-ref>, leading to a theoretical sampling model that is in some sense closer to the observed data. The log-rank and weighted log-rank tests arise as special members of the class. In practice the log-rank test will often be a suboptimal, even poor, test due to the presence of non-proportional hazards. The proposed test maintains good power and, in all the cases considered, has greater power than the log-rank test under non-proportional hazards. The power will depend on the alternatives being considered, and under reasonable assumptions on the alternatives, we conclude that the proposed test is more powerful than the log-rank test. Simulations support these conclusions. An example is given as an illustration.