Thompson, Ulam, or Gauss? Multi-criteria recommendations for posterior probability computation methods in Bayesian response-adaptive trials

Bayesian adaptive designs enable flexible clinical trials by adapting features based on accumulating data. Among these, Bayesian Response-Adaptive Randomization (BRAR) skews patient allocation towards more promising treatments based on interim data. Implementing BRAR requires the relatively quick evaluation of posterior probabilities. However, the limitations of existing closed-form solutions mean trials often rely on computationally intensive approximations which can impact accuracy and the scope of scenarios explored. While faster Gaussian approximations exist, their reliability is not guaranteed. Critically, the approximation method used is often poorly reported, and the literature lacks practical guidance for selecting and comparing these methods, particularly regarding the trade-offs between computational speed, inferential accuracy, and their implications for patient benefit. In this paper, we focus on BRAR trials with binary endpoints, developing a novel algorithm that efficiently and exactly computes these posterior probabilities, enabling a robust assessment of existing approximation methods in use. Leveraging these exact computations, we establish a comprehensive benchmark for evaluating approximation methods based on their computational speed, patient benefit, and inferential accuracy. Our comprehensive analysis, conducted through a range of simulations in the two-armed case and a re-analysis of the three-armed Established Status Epilepticus Treatment Trial, reveals that the exact calculation algorithm is often the fastest, even for up to 12 treatment arms. Furthermore, we demonstrate that commonly used approximation methods can lead to significant power loss and Type I error rate inflation. We conclude by providing practical guidance to aid practitioners in selecting the most appropriate computation method for various clinical trial settings.