Bayesian Adaptive Randomization with Compound Utility Functions

Abstract

Bayesian adaptive designs formalize the use of previous knowledge at the planning stage of an experiment, permitting recursive updating of the prior information. They often make use of utility functions, while also allowing for randomization. We review frequentist and Bayesian adaptive design methods and show that some of the frequentist adaptive design methodology can also be employed in a Bayesian context. We use compound utility functions for the Bayesian designs, that are a trade-off between an optimal design information criterion, that represents the acquisition of scientific knowledge, and some ethical or utilitarian gain. We focus on binary response models on two groups with independent Beta prior distributions on the success probabilities. The treatment allocation is shown to converge to the allocation that produces the maximum utility. Special cases are the Bayesian Randomized (simply) Adaptive Compound (BRAC) design, an extension of the frequentist Sequential Maximum Likelihood (SML) design and the Bayesian Randomized (doubly) Adaptive Compound Efficient (BRACE) design, a generalization of the Efficient Randomized Adaptive DEsign (ERADE). Numerical simulation studies compare BRAC with BRACE when D-optimality is the information criterion chosen. In analogy with the frequentist theory, the BRACE-D design appears more efficient than the BRAC-D design.