The New England Journal of Statistics in Data Science

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.

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.