Optimal allocation to maximize the power of two-sample tests for binary response

We study allocations that maximize the power of tests of equality of two treatments having binary outcomes. When a normal approximation applies, the asymptotic power is maximized by minimizing the variance, leading to a Neyman allocation that assigns observations in proportion to the standard deviations. This allocation, which in general requires knowledge of the parameters of the problem, is recommended in a large body of literature. Under contiguous alternatives the normal approximation indeed applies, and in this case the Neyman allocation reduces to a balanced design. However, when studying the power under a noncontiguous alternative, a large deviations approximation is needed, and the Neyman allocation is no longer asymptotically optimal. In the latter case, the optimal allocation depends on the parameters, but is rather close to a balanced design. Thus, a balanced design is a viable option for both contiguous and noncontiguous alternatives. Finite sample studies show that a balanced design is indeed generally quite close to being optimal for power maximization. This is good news as implementation of a balanced design does not require knowledge of the parameters.