Bayesian Estimation of the Binomial Parameter in Adaptive Designs With Treatment Selection

It is well-known that results from adaptive multi-arm experiments with treatment selection are subject to the bias of the selection process. In this article, we consider designs with a binary outcome that begin with randomized parallel arms. After each of the preplanned interim looks at data, researchers proceed to the next stage with treatments selected according to predetermined rules. The experiment ends with the best arm, and the binomial parameter is estimated from all the observations accrued throughout the experiment in this arm. We present a comprehensive and unified Bayesian approach to the point and interval estimations, which uses a class of design-dependent priors obtained from the reference prior theory (Bernardo 1979). The approach is applicable regardless of the treatment selection rule, and is not affected by the inclusion of a control arm if the treatment selection is based on direct comparisons of the response rates relative to the control arm. The frequentist characteristics of the posterior estimators are studied and compared with alternative methods, where available. To this end, we introduce specific tools that facilitate the evaluation of estimation methods in binary data analysis. We also study the influence of the treatment selection rule and the effect of the prior correction on estimates through various examples. The approach is easy to implement in the experimental practice, and can be used in the interim analyses to help researchers in the treatment selection process.