Implementing optimal allocation in sequential binary response experiments

For sequential experiments with K treatments, we establish two formal optimization criteria to find optimal allocation strategies. Both criteria involve the sample sizes on each treatment and a concave noncentrality parameter from a multivariate test. We show that these two criteria are equivalent. We apply this result to specific questions: (1) How do we maximize power of a multivariate test of homogeneity with binary response?, and (2) for fixed power, how do we minimize expected treatment failures? Because the solutions depend on unknown parameters, we describe a response-adaptive randomization procedure that targets the optimal allocation and provides increases in power along the lines of 2-4% over complete randomization for equal allocation. The increase in power contradicts the conclusions of other authors who have explored other randomization procedures for K = 2 and have found that the variability induced by randomization negates any benefit of targeting an optimal allocation.