  We consider the classical problem of selecting the best of two treatments in
clinical trials with binary response. The target is to find the design that
maximizes the power of the relevant test. Many papers use a normal
approximation to the power function and claim that Neyman allocation that
assigns subjects to treatment groups according to the ratio of the responses'
standard deviations, should be used. As the standard deviations are unknown, an
adaptive design is often recommended. The asymptotic justification of this
approach is arguable, since it uses the normal approximation in tails where the
error in the approximation is larger than the estimated quantity. We consider
two different approaches for optimality of designs that are related to Pitman
and Bahadur definitions of relative efficiency of tests. We prove that the
optimal allocation according to the Pitman criterion is the balanced allocation
and that the optimal allocation according to the Bahadur approach depends on
the unknown parameters. Exact calculations reveal that the optimal allocation
according to Bahadur is often close to the balanced design, and the powers of
both are comparable to the Neyman allocation for small sample sizes and are
generally better for large experiments. Our findings have important
implications to the design of experiments, as the balanced design is proved to
be optimal or close to optimal and the need for the complications involved in
following an adaptive design for the purpose of increasing the power of tests
is therefore questionable.
