A Graphical Framework for Testing Hierarchically Structured Hypothesis Families

In clinical trials, hypotheses are frequently organized into hierarchically ordered families, requiring specialized testing strategies that account for these structured relationships. Existing gatekeeping methods-including serial, parallel, and tree-structured approaches-provide important solutions but are often either too rigid or insufficiently intuitive to accommodate increasingly complex logical dependencies among hypothesis families. To address these limitations, we propose a novel family-based graphical approach that unifies the derivation and visualization of diverse gatekeeping strategies. In this framework, procedures are represented as directed, weighted graphs, where nodes correspond to hypothesis families. Two simple updating rules govern the allocation of significance levels within families and the propagation of significance levels between them. We establish that the proposed method strongly controls the familywise error rate (FWER) at a pre-specified level. Simulation studies under representative configurations indicate that the proposed procedure achieves performance comparable to hypothesis-level graphical approaches and competitive with the superchain procedure, while providing a simpler and more interpretable family-level representation. Case studies and a real clinical trial application further illustrate its flexibility and practical advantages, making it a powerful tool for managing hierarchically structured multiple testing in clinical research.