Hypothesis Testing for Two-Arm Proportions with Two Binary Endpoints

Many studies require evidence that a new treatment improves efficacy and maintains or improves safety. Composite endpoints can obscure trade-offs and complicate interpretation. We propose a single-stage hypothesis test that directly evaluates two binary endpoints against a concurrent control, offering a transparent alternative to composite endpoints. The test rejects only if the observed improvements on both endpoints exceed pre-specified paired thresholds. The joint distribution of efficacy and safety is modeled with a four-category multinomial, yielding probabilities for all the outcome combinations. This enables exact computation of rejection probabilities and identification of least-favorable parameter configurations to control type I error at the nominal level while retaining adequate power. Design tables map the target significance level and power, together with predefined effect sizes for each endpoint, to the required sample size and decision thresholds. Simulations and one case study illustrate design selection and interpretation. The proposed test provides an exact and practical tool for early-phase trials with dual binary endpoints, particularly when efficacy and safety must be evaluated simultaneously.