Optimal Adaptive Equal Allocation Rules

Suppose we wish to decide which of two treatments is better, where the outcomes are Bernoulli random variables, the success probabilities of which, themselves, are modeled as independent beta random variables. Assume that the maximal population size for the experiment is xed, but that the length of the study and the number and order of patients assigned to each treatment may be random. Our goal is to maximize the likelihood of making the correct decision by utilizing a curtailed equal allocation rule, but we wish to do so with a minimal average study length. We show that this experimental design problem reduces to a problem of optimal adaptive allocation which can be solved exactly using dynamic programming. We compare the optimal allocation procedure to the commonly-used approach of curtailed alternating allocation and show that the optimal allocation procedure is noticeably superior. The evaluations of allocation procedures are all exact, calculated via backward induction. Since the optimal adaptive allocation procedure can be easily determined and evaluated on workstations, and stored on personal computers for ready access during experiments, it is a practical improvement over alternating allocation.