Hierarchical Testing of a Primary and a Secondary Endpoint in a Group Sequential Design With Different Information Times

We consider a confirmatory clinical trial where a primary and a secondary endpoints are tested hierarchically to control the family-wise error rate at level α. The trial uses a group sequential design with an interim and a final analysis. When the information times at the interim analysis are the same for the primary and the secondary hypotheses, it has been shown in the literature that the secondary hypothesis has to be tested using a group sequential boundary at a level no more than α. In many event-driven trials, however, the information times are usually different because of different event rates for the two endpoints. The information times may also be different for a noninferiority hypothesis and a superiority hypothesis due to different analysis sets. We consider this general setup and derive a sharp upper bound on the probability of rejecting the secondary hypothesis at an interim and a final analysis. This bound suggests that the secondary boundary can be refined so that the group sequential design can be tested at a significance level greater than α while still controlling the family-wise error rate at level α. We carry out a simulation study to illustrate the power gain by using the refined boundary for different choices of boundaries for the two hypotheses. The proposed approach is illustrated in two oncology clinical trials with more than two analyses.