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

Bayesian adaptive designs enable flexible clinical trials by adapting features based on accumulating data. Among these, Bayesian Response-Adaptive Randomisation (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.