Comparing Kaplan–Meier curves with delayed treatment effects: applications in immunotherapy trials

We consider a comparison of Kaplan–Meier curves from clinical trials in which there may be a delayed treatment effect. Any such delay takes us outside the umbrella of a proportional hazards structure and therefore outside the setting in which the log‐rank test would be optimal. The approach of Chauvel and O’Quigley based on Brownian motion approximations enables the construction of powerful tests in situations of non‐proportionality and, in particular, a powerful test in the situation of delayed effect. The power of this test is seen to be very close to that of the most powerful test, which, however, is unavailable in practice. We show that the test is unbiased and consistent under general conditions. Under the null, we obtain identical large sample behaviour to the log‐rank test so the type 1 error is correctly controlled. Under proportional hazards departures from the null we obtain results that indicate a manageable loss in power compared with the log‐rank test. The usual sample size calculations can still provide a useful guide. Support for the theoretical findings are provided by simulations as well as illustrations from three immunotherapy clinical trials.