Exact Randomized Two-Stage Phase 2 Clinical Trial Designs for Two Binary Co-Primary Endpoints

Phase 2 clinical trials typically rely on a single primary endpoint, yet in many settings, treatment efficacy must be demonstrated across multiple co-primary endpoints. Such settings require intersection-union hypothesis testing, in which the global null hypothesis is rejected only when all individual component hypotheses are rejected. Existing phase 2 designs for binary endpoints are largely restricted to single-arm or single-endpoint settings, or rely on large-sample normal approximations for two-arm trials, which may be inadequate when sample sizes are small. We developed exact and simulation-based methods for designing two-arm, two-stage phase 2 trials with two co-primary binary endpoints using the bivariate binomial distribution, allowing different correlations between endpoints across treatment arms. The proposed framework identifies optimal designs that minimize the expected sample size and minimax designs that minimize the maximum sample size, while allowing early termination for futility at the first stage. For small sample sizes (approximately $n\le 25$ per arm), the exact method is computationally feasible and yields analytically exact operating characteristics under the assumed bivariate binomial distribution. For larger sample sizes, however, exact computations become intensive. To address this, we evaluated a simulation-based approach and a normal approximation by comparing their performance against the exact method. The simulation approach selected designs that closely matched those from the exact method, whereas the normal approximation often deviated. Therefore, we recommend the simulation-based method as a computationally efficient and accurate alternative for moderate-to-large sample size trials.