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To implement a Bayesian response-adaptive trial it is necessary to evaluate a sequence of posterior probabilities. This sequence is often approximated by simulation due to the unavailability of closed-form formulae to compute it exactly. Approximating these probabilities by simulation can be computationally expensive and impact the accuracy or the range of scenarios that may be explored. An alternative approximation method based on Gaussian distributions can be faster but its accuracy is not guaranteed. The literature lacks practical recommendations for selecting approximation methods and comparing their properties, particularly considering trade-offs between computational speed and accuracy. In this paper, we focus on the case where the trial has a binary endpoint with Beta priors. We first outline an efficient way to compute the posterior probabilities exactly for any number of treatment arms. Then, using exact probability computations, we show how to benchmark calculation methods based on considerations of computational speed, patient benefit, and inferential accuracy. This is done through a range of simulations in the two-armed case, as well as an analysis of the three-armed Established Status Epilepticus Treatment Trial. Finally, we provide practical guidance for which calculation method is most appropriate in different settings, and how to choose the number of simulations if the simulation-based approximation method is used., Comment: 16 pages, 3 figures, 3 tables

Dose selection is critical in pharmaceutical drug development, as it directly impacts therapeutic efficacy and patient safety of a drug. The Generalized Multiple Comparison Procedures and Modeling (MCP-Mod) approach is commonly used in Phase II trials for testing and estimation of dose-response relationships. However, its effectiveness in small sample sizes, particularly with binary endpoints, is hindered by issues like complete separation in logistic regression, leading to non-existence of estimates. Motivated by an actual clinical trial using the MCP-Mod approach, this paper introduces penalized maximum likelihood estimation (MLE) and randomization-based inference techniques to address these challenges. Randomization-based inference allows for exact finite sample inference, while population-based inference for MCP-Mod typically relies on asymptotic approximations. Simulation studies demonstrate that randomization-based tests can enhance statistical power in small to medium-sized samples while maintaining control over type-I error rates, even in the presence of time trends. Our results show that residual-based randomization tests using penalized MLEs not only improve computational efficiency but also outperform standard randomization-based methods, making them an adequate choice for dose-finding analyses within the MCP-Mod framework. Additionally, we apply these methods to pharmacometric settings, demonstrating their effectiveness in such scenarios. The results in this paper underscore the potential of randomization-based inference for the analysis of dose-finding trials, particularly in small sample contexts.