Allocating recycled significance levels in group sequential procedures for multiple endpoints

Graphical approaches have been proposed in the literature for testing hypotheses on multiple endpoints by recycling significance levels from rejected hypotheses to unrejected ones. Recently, they have been extended to group sequential procedures (GSPs). Our focus in this paper is on the allocation of recycled significance levels from rejected hypotheses to the stages of the GSPs for unrejected hypotheses. We propose a delayed recycling method that allocates the recycled significance level from Stage r onward, where r is prespecified. We show that r cannot be chosen adaptively to coincide with the random stage at which the hypothesis from which the significance level is recycled is rejected. Such an adaptive GSP does not always control the FWER. One can choose r to minimize the expected sample size for a given power requirement. We illustrate how a simulation approach can be used for this purpose. Several examples, including a clinical trial example, are given to illustrate the proposed procedure.