Exact power and sample size in clinical trials with two co-primary binary endpoints

Binary endpoints are used widely to evaluate treatment effects during clinical trials. Although clinical trials in many therapeutic areas evaluate a single binary endpoint as the primary endpoint, clinical trials in certain therapeutic areas require two co-primary binary endpoints to evaluate treatment benefit multi-dimensionally. We consider the situation in which evidence of effects on both co-primary endpoints is necessary to conclude that the intervention is effective, which differs from approaches by which significance on at least one endpoint is sufficient for trial success. When designing clinical trials with two co-primary binary endpoints, consideration of correlation between the endpoints can increase trial power and consequently reduce the required sample size, leading to improved efficiency. For clinical trials with two co-primary binary endpoints, methods for calculating power and sample size have been proposed, but they are based on approximations or require Monte Carlo integration. Alternatively, we propose methods for calculating the exact power and sample size in clinical trials with two co-primary binary endpoints. The proposed methods are useful for any statistical test for binary endpoints. Numerical investigation under various scenarios demonstrated that our proposed methods can incorporate consideration of the correlation between two co-primary binary endpoints in sample size calculation, thereby allowing the required sample size to be reduced. We demonstrate that the exact power for the required sample size calculated using our proposed method is approximately equal to target power. Finally, we present application of our proposed methods to a clinical trial of relapsing or refractory eosinophilic granulomatosis with polyangiitis.