A general multistage procedure for k-out-of-n gatekeeping

We generalize a multistage procedure for parallel gatekeeping to what we refer to as k-out-of-n gatekeeping in which at least k out of n hypotheses ( 1 ≤ k ≤ n) in a gatekeeper family must be rejected in order to test the hypotheses in the following family. This gatekeeping restriction arises in certain types of clinical trials; for example, in rheumatoid arthritis trials, it is required that efficacy be shown on at least three of the four primary endpoints. We provide a unified theory of multistage procedures for arbitrary k, with k=1 corresponding to parallel gatekeeping and k=n to serial gatekeeping. The theory provides an insight into the construction of truncated separable multistage procedures using the closure method. Explicit formulae for calculating the adjusted p-values are given. The proposed procedure is simpler to apply for this particular problem using a stepwise algorithm than the mixture procedure and the graphical procedure with memory using entangled graphs.