Bayesian Adaptive Randomization with Compound Utility Functions

Bayesian adaptive designs formalize the use of previous knowl-edge 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 adap-tive 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 sci-entific 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 ex-tension 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 (ER-ADE). Numerical simulation studies compare BRAC with BRACE when D -optimality is the information criterion chosen. In analogy with the frequen-tist theory, the BRACE-D design appears more efficient than the BRAC-D design.

Automated summary

Bayesian adaptive designs allow recursive updating of prior information and the use of utility functions, combining scientific knowledge acquisition and ethical/utilitarian gain. They can also incorporate frequentist adaptive design methods. In this study, we focused on binary response models with independent Beta prior distributions and showed that the treatment allocation converges to the maximum utility. Numerical simulations compared different designs and found that the BRACE-D design is more efficient than the BRAC-D design when D-optimality is the chosen information criterion.