Bayesian optimal experimental designs for binary responses in an adaptive framework

Bayesian designs make formal use of the experimenter's prior information in planning scientific experiments. In their 1989 paper, Chaloner and Larntz suggested to choose the design that maximizes the prior expectation of a suitable utility function of the Fisher information matrix, which is particularly useful when Fisher's information depends on the unknown parameters of the model. In this paper, their method is applied to a randomized experiment for a binary response model with two treatments, in an adaptive way, that is, updating the prior information at each step on the basis of the accrued data. The utility is the A‐optimality criterion and the marginal priors for the parameters of interest are assumed to be beta distributions. This design is shown to converge almost surely to the Neyman allocation. But frequently, experiments are designed with more purposes in mind than just inferential ones. In clinical trials for treatment comparison, Bayesian statisticians share with non‐Bayesians the goal of randomizing patients to treatment arms so as to assign more patients to the treatment that does better in the trial. One possible approach is to optimize the prior expectation of a combination of the different utilities. This idea is applied in the second part of the paper to the same binary model, under a very general joint prior, combining either A‐ or D‐optimality with an ethical criterion. The resulting randomized experiment is skewed in favor of the more promising treatment and can be described as Bayes compound optimal. Copyright © 2016 John Wiley & Sons, Ltd.