Response adaptive randomization design for a two-stage study with binary response

Response adaptive randomization has the potential to treat more participants in better treatments in a trial to benefit participants. We propose optimal response adaptive randomization designs for a two-stage study with binary response, having the smallest expected sample size or the fewest expected number of failures. Equal randomization is used in the first stage, and data from the first stage is used to determine the adaptive sample size ratio in the second stage. In the proposed optimal designs, the type I error rate and the statistical power are calculated from the asymptotic normal distributions. The new designs that minimize the expected number of failures have the advantage over the existing optimal randomized designs to substantially reduce the number of failures.