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In this paper we describe Bonferroni-based multiple testing procedures (MTPs) as strategies to split and recycle test mass. Here, 'test mass' refers to (parts of) the nominal level alpha at which the family-wise error rate is controlled. Briefly, test mass is split between different null hypotheses, and whenever a null hypothesis is rejected, the part of alpha allocated to it may be recycled to the testing of other hypotheses. These recycling MTPs are closed testing procedures based on raw p-values associated with testing the individual null hypotheses, and the class of such MTPs includes, for example, serial and parallel gatekeeping, fallback and Holm procedures. Graphical displays and a concise algebraic notation are provided for such MTPs. This recycling approach has pedagogical advantages and may facilitate the tailoring of MTPs for different purposes.
Burman et al. (Tue,) studied this question.