Critical Boundary Refinement in a Group Sequential Trial When the Primary Endpoint Data Accumulate Faster Than the Secondary Endpoint

  • Jiangtao GouEmail author
  • Oliver Y. Chén
Part of the ICSA Book Series in Statistics book series (ICSABSS)


We propose a generalized framework for critical boundary refinement when conducting hierarchical hypothesis test in a clinical trial involving multiple interim stages. When the hypothesis test follows the stagewise hierarchical rule or the partially hierarchical rule, we provide an improvement on the secondary boundary. This refinement boosts the power to reject the secondary hypothesis significantly. For a trial using a stage-wise hierarchical rule, we deliver a feasible region of information fractions under which an α-level boundary can be directly used in testing the secondary hypothesis at each interim stage. For a trial using a partially hierarchical rule, we recommend using the refined O’Brien-Fleming boundary for both the primary and the secondary endpoint. To evaluate the efficacy of the framework, we present the theoretical underpinning for the boundary refinement, and prove the uniform monotonicity of as well as the upper bound for the type I error rate. The framework has particular advantage when the primary endpoint data can be assessed earlier than the secondary endpoint data. Finally, we extend the framework to include an adaptive update on the refined boundary when the attained sample sizes are different from what they are originally planned.



We thank Ajit C. Tamhane and Dong Xi for comments that greatly improved the manuscript. This work was partially supported by the Professional Staff Congress-City University of New York (PSC-CUNY) research grant, Cycle 48 (2017–2018). This research article extended the framework that was present at the 2017 ICSA Applied Statistics Symposium, Session 148, Recent Developments in Theory and Application of Multiple Comparison Methods, A gatekeeping test on a primary and a secondary endpoint in a group sequential design, by Dr. Ajit C. Tamhane. It was also present at the 2017 ICSA Applied Statistics Symposium, Session 121, Multiplicity in Clinical Trials, A gatekeeping test in a group sequential design with multiple interim looks, by Dr. Jiangtao Gou. The authors thank editor Dr. Lanju Zhang and an anonymous referee for suggestions that improved this paper.

Conflict of Interest

The authors have declared no conflict of interest.


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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsHunter College of CUNYNew YorkUSA
  2. 2.Department of Mathematics and StatisticsVillanova UniversityVillanovaUSA
  3. 3.Institute for Biomedical EngineeringUniversity of OxfordOxfordUK

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