On stepwise MTPs between Holm's step-down MTP and the max-p-value MTP for co-primary endpoints: A further extension for free.

This article extends a recently discussed p-value-based multiple testing procedure (MTP) that can be used if study-success requires that at least k out of m primary/important hypotheses become rejected. With k strictly between one and m, this previous MTP is between the two practically important MTPs in the title. It has a certain initial gatekeeping step/test, followed by m steps where primary hypotheses are rejected in a step-down manner. The extension discussed in this article consists in replacing the initial step/test with a Fixed-Sequence MTP for a family of increasing hypotheses about the number of false primary hypotheses. The last step of this gatekeeping family equals the initial step/test of the previous MTP. Thus, even if this last step does not open the gate to the step-down part, it is possible to make inferences about the number of false primary hypotheses through preceding fixed-sequence steps. This extension is for free in that the extended MTP inherits all nice properties of the previous MTP. Notably, it: (a) is as generally valid as Holm's MTP; (b) can reject k or more primary hypotheses even if Holm's MTP stops before its kth step; and (c) has optimal critical constants in that if at least one is increased, the general strong familywise error rate control is lost. A simple generally valid confidence lower bound for the number of false primary hypotheses is discussed. An R-function is provided that calculates all the adjusted p-values needed.