The New England Journal of Statistics in Data Science

We introduce the anytime-valid (AV) 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. The logrank test is an instance of the martingale tests based on E-variables that have been recently developed. We demonstrate type-I error guarantees for the test in a semiparametric setting of proportional hazards, show explicitly how to extend it to ties and confidence sequences and indicate further extensions to the full Cox regression model. Using a Gaussian approximation on the logrank statistic, we show that the AV logrank test (which itself is always exact) has a similar rejection region to O’Brien-Fleming α-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.

When testing a statistical hypothesis, is it legitimate to deliberate on the basis of initial data about whether and how to collect further data? Game-theoretic probability’s fundamental principle for testing by betting says yes, provided that you are testing the hypothesis’s predictions by betting and do not risk more capital than initially committed. Standard statistical theory uses Cournot’s principle, which does not allow such optional continuation. Cournot’s principle can be extended to allow optional continuation when testing is carried out by multiplying likelihood ratios, but the extension lacks the simplicity and generality of testing by betting.

In addition to scientific questions, clinical trialists often explore or require other design features, such as increasing the power while controlling the type I error rate, minimizing unnecessary exposure to inferior treatments, and comparing multiple treatments in one clinical trial. We propose implementing adaptive seamless design (ASD) with response adaptive randomization (RAR) to satisfy various clinical trials’ design objectives. However, the combination of ASD and RAR poses a challenge in controlling the type I error rate. In this paper, we investigated how to utilize the advantages of the two adaptive methods and control the type I error rate. We offered the theoretical foundation for this procedure. Numerical studies demonstrated that our methods could achieve efficient and ethical objectives while controlling the type I error rate.