Optimal designs for clinical trials with dichotomous responses

We consider two-stage designs for clinical trials that involve two treatments with dichotomous responses. The first is the information-gathering stage; the treatment chosen as the better from first stage and prior data is used exclusively in the second stage. Determination of treatment allocation in the first stage results from weighing the anticipated gain in information with effective treatment; the objective is to maximize the expected number of successes in the entire trial. This is in contrast to randomized controlled trials with the restricted objective of obtaining information concerning treatment differences. We allow the length of the first stage to be arbitrary and fixed in advance, or optimized as a function of prior information and the 'patient horizon'. We can regard this patient horizon as either the number of patients in the trial or the number who have the condition under treatment. We consider two forms of prior information: both success probabilities known but the better of the two treatments is unknown, and one success probability known whereas the other has an arbitrary distribution. In many instances of the latter case the optimal first stage size is of the order of the square root of the patient horizon.