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.
Testing by betting can also help us with descriptive data analysis. To obtain a purely and honestly descriptive analysis using competing probability distributions, we have them bet against each other using the principle. The place of confidence intervals is then taken by sets of distributions that do relatively well in the competition. In the simplest implementation, these sets coincide with R. A. Fisher’s likelihood ranges.
Clinical trials usually involve sequential patient entry. When designing a clinical trial, it is often desirable to include a provision for interim analyses of accumulating data with the potential for stopping the trial early. We review Bayesian sequential clinical trial designs based on posterior probabilities, posterior predictive probabilities, and decision-theoretic frameworks. A pertinent question is whether Bayesian sequential designs need to be adjusted for the planning of interim analyses. We answer this question from three perspectives: a frequentist-oriented perspective, a calibrated Bayesian perspective, and a subjective Bayesian perspective. We also provide new insights into the likelihood principle, which is commonly tied to statistical inference and decision making in sequential clinical trials. Some theoretical results are derived, and numerical studies are conducted to illustrate and assess these designs.