Fixed sample size subset selection among treatments with two binary endpoints in comparison with a control

In this article, we propose a procedure for selecting a random-sized subset that includes all experimental treatments that are superior to the control treatment. The comparison is based on two binary endpoints. An experimental treatment is deemed superior if it demonstrates higher success probabilities than the control on both endpoints. While responses across different treatments are assumed to be independent, responses within the same treatment may exhibit dependence. To model this within-treatment association, we use the odds ratio and employ the bivariate binomial distribution to compute the probability of a correct selection. We derive design parameters for three scenarios: (1) independent endpoints, (2) dependent endpoints with known association, and (3) dependent endpoints with unknown association. For each case, we provide tables indicating the minimum sample size required to satisfy the specified probability criteria.