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.

翻译：实施贝叶斯响应自适应试验需要计算一系列后验概率。由于缺乏精确计算的闭式公式，该序列常通过模拟进行近似。通过模拟近似这些概率可能计算成本高昂，并影响可探索场景的准确性或范围。基于高斯分布的替代近似方法速度更快，但其准确性无法保证。现有文献缺乏选择近似方法及比较其性质（特别是权衡计算速度与准确性时）的实用建议。本文聚焦于试验具有二值终点且采用Beta先验的情形。我们首先概述了针对任意治疗组数精确计算后验概率的有效方法。随后利用精确概率计算，展示了如何基于计算速度、患者获益和推断准确性等多重考量对计算方法进行基准测试。这项工作通过双臂情形的系列模拟实验，以及对三臂既定癫痫持续状态治疗试验的分析完成。最后，我们为不同场景下选择最合适的计算方法提供了实用指导，并说明了若采用基于模拟的近似方法应如何确定模拟次数。