A sharp upper bound for the expected number of false rejections

We consider the class of monotone multiple testing procedures (monotone MTPs). It includes, among others, traditional step-down (Holm type) and step-up (Benjamini–Hochberg type) MTPs, as well as their generalization–step-up-down procedures (Tamhane et al., 1998). Our main result–the All-or-Nothing Theorem–allows us to explicitly calculate, for each MTP in those classes, its per-family error rate–the exact level at which the procedure controls the expected number of false rejections under general and unknown dependence structure of the individual tests. As an illustration, we show that, for any monotone step-down procedure (where the term “step-down” is understood in the most general sense), the ratio of its per-family error rate and its familywise error rate (the exact level at which the procedure controls the probability of one or more false rejections) does not exceed 4 if the denominator is less than 1.