Testing One Primary and Two Secondary Endpoints in a Two-Stage Group Sequential Trial With Extensions

We study the problem of testing multiple secondary endpoints conditional on a primary endpoint being significant in a two-stage group sequential procedure, focusing on two secondary endpoints. This extends our previous work with one secondary endpoint. The test for the secondary null hypotheses is a closed procedure. Application of the Bonferroni test for testing the intersection of the secondary hypotheses results in the Holm procedure while application of the Simes test results in the Hochberg procedure. The focus of the present paper is on developing normal theory analogs of the abovementioned $p$ -value based tests that take into account (i) the gatekeeping effect of the test on the primary endpoint and (ii) correlations between the endpoints. The normal theory boundaries are determined by finding the least favorable configuration of the correlations and so their knowledge is not needed to apply these procedures. The $p$ -value based procedures are easy to apply but they are less powerful than their normal theory analogs because they do not take into account the correlations between the endpoints and the gatekeeping effect referred to above. On the other hand, the normal theory procedures are restricted to two secondary endpoints and two stages mainly because of computational difficulties with more than two secondary endpoints and stages. Comparisons between the two types of procedures are given in terms of secondary powers. The sensitivity of the secondary type I error rate and power to unequal information times is studied. Numerical examples and a real case study illustrate the procedures.