A recycling framework for the construction of Bonferroni-based multiple tests

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.