Refined critical boundary with enhanced statistical power for non-directional two-sided tests in group sequential designs with multiple endpoints

Group sequential designs with multiple endpoints are widely used in clinical trials and medical research. In the literature of group sequential designs with a single primary hypothesis, it was recommended to calculate the critical boundary for a symmetric non-directional two-sided test at the significance level $$\alpha $$ α by constructing a one-sided critical boundary at the significance level $$\alpha /2$$ α / 2 . As a natural but unvalidated extension, researchers may think the same conclusion holds for testing multiple endpoints in group sequential designs, that is, the critical boundary calculated to control the type I error at level $$\alpha /2$$ α / 2 for one-sided test also controls the type I error at level $$\alpha $$ α for non-directional two-sided tests. In this paper, we demonstrate that using one-sided $$\alpha /2$$ α / 2 -level critical boundaries for non-directional two-sided $$\alpha $$ α -level tests leads to notable and unnecessary conservativeness when multiple endpoints are hierarchically tested in group sequential designs. To the best of our knowledge, we are the first to reveal the relationship between the type I error rates of the two-sided and one-sided tests in hierarchical testing of multiple endpoints in group sequential designs. We consider three commonly used hierarchical structures for the primary and secondary endpoints, namely the stagewise hierarchical, overall hierarchical, and partially hierarchical structures. Under all three hierarchical structures, we refine the critical boundary for the secondary hypothesis. In addition, we prove that our boundary refinement methods yield enhanced statistical power. The proposed methodology is illustrated using a randomized clinical trial examining the effect of energy-reduced diets with different macronutrient compositions on body weight.