Quentin F. Stout
University of Michigan
Abstract: Suppose one wishes 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. We assume that the maximal population size for the experiment is fixed, bbut 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. Our approach can also be applied to minimizing the expected cost of the experiment when the options have different costs, and in clinical trials to optimize the ethical goals of minimizing the expected number of failures or the expected number of patients assigned to the inferior treatment.
We show that this experimental design problem reduces to a problem of optimal response adaptive allocation which can be solved exactly using dynamic programming. The problem, and solution approach, is similar to the well-known bandit problem.
We compare the optimal allocation procedure to the commonly-used approach of curtailed alternating allocation and show that the optimal response adaptive allocation procedure is noticeably superior. The evaluations of allocation procedures are all exact, calculated via backward induction. Since the optimal response 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 simple alternating allocation.
Keywords: statistical computing, response adaptive sampling designs, sequential allocation procedure, dynamic programming, curtailment, probability of correct selection, equal allocation, vector at a time, ethical clinical trial
Complete paper. This paper appears in Computing Science and Statistics 24, (1992), pp. 597-601.
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