The Safe Logrank Test: Error Control under Continuous Monitoring with Unlimited Horizon

We introduce the safe logrank test, a version of the logrank test that provides type-I error guarantees under optional stopping and optional continuation. The test is sequential without the need to specify a maximum sample size or stopping rule and allows for cumulative meta-analysis with Type-I error control. The method can be extended to define anytime-valid confidence intervals. All these properties are a virtue of the recently developed martingale tests based on E-variables, of which the safe logrank test is an instance. We demonstrate the validity of the underlying nonnegative martingale in a semiparametric setting of proportional hazards and show how to extend it to ties, Cox' regression and confidence sequences. Using a Gaussian approximation on the logrank statistic, we show that the safe logrank test (which itself is always exact) has a similar rejection region to O'Brien-Fleming alpha-spending but with the potential to achieve 100% power by optional continuation. Although our approach to study design requires a larger sample size, the expected sample size is competitive by optional stopping.