Some Bayesian Perspectives on Clinical Trials

TL;DR

This paper applies a unified Bayesian framework to analyze landmark clinical trials, introduces exact backward induction for binary trials, and explores the trade-offs between sample size and power in Bayesian designs.

Contribution

It presents a novel exact backward induction method for two-arm binary trials and discusses Bayesian design trade-offs, with practical implications for clinical trial methodology.

Findings

Posterior probability of treatment superiority is robust across priors.

Predictive probability monitoring can reduce sample size significantly.

Bayesian designs can lower sample sizes but may reduce statistical power.

Abstract

We examine three landmark clinical trials -- ECMO, CALGB~49907, and I-SPY~2 -- through a unified Bayesian framework connecting prior specification, sequential adaptation, and decision-theoretic optimisation. For ECMO, the posterior probability of treatment superiority is robust across the range of priors examined. For CALGB, predictive probability monitoring stopped enrolment at 633 instead of 1800 patients. For I-SPY~2, adaptive enrichment graduated nine of 23 arms to Phase~III. These case studies motivate a methodological contribution: exact backward induction for two-arm binary trials, where Beta-Binomial conjugacy yields closed-form transitions on the integer lattice of success counts with no quadrature. A P\'olya-Gamma augmentation bridges this to covariate-adjusted logistic regression. Simulation reveals a fundamental tension: the optimal Bayesian design reduces expected sample sizes to 14--26 per arm (versus 42--100 for alternatives) but with substantially lower power. A calibrated variant embedding the declaration threshold in the terminal utility improves power while maintaining sample-size savings; varying the per-stage cost traces a power frontier for selecting the preferred operating point, with suitability highest in patient-sparing contexts such as rare diseases and paediatrics. The P\'olya-Gamma Laplace approximation is validated against exact calculations (mean absolute error below 0.01). We discuss implications for the 2026 FDA draft guidance on Bayesian methodology.