Note on a stepwise procedure for rejecting at least k out of m hypotheses: A simple Holm-type formulation and proof

Mielke et al. (2021) proposed a stepwise multiple testing procedure (MTP) based on marginal p-values for rejecting at least k out of m null hypotheses. Briefly, the MTP cannot reject less than k hypotheses, but this property improves the power to reject k or more hypotheses relative to the stepdown MTP of Holm (1979). Mielke et al. discussed why such an MTP is of interest in the context of biosimilarity developments. This article describes how a slight modification of Holm's simple direct arguments can be used as an alternative to the closed-testing arguments of Mielke et al. to show the strong control of the family-wise error rate. This modification is based on a Holm-type formulation of the rejection algorithm where each step involves a single ordered p-value. With k equal to one, the stepwise MTP reduces to Holm's stepdown MTP. The MTP is valid quite generally. A version of the MTP with weights for hypotheses is also described and discussed. As in the case without weights: (a) a modification of Holm's arguments can be used; (b) with k equal to one, the MTP reduces to Holm's MTP with weights; and (c) the MTP is valid quite generally.