A new optimality property of the Holm step-down procedure

The celebrated Holm step-down multiple testing procedure was introduced as an extension of the Bonferroni procedure that still controls the familywise error rate (FWER)–the probability of one or more false rejections–at the same level as the Bonferroni procedure. It follows from a recent result (Gordon and Salzman, 2008) that the Holm procedure cannot be further extended in the class of monotone step-down procedures without weakening the control of the FWER. Here the term “step-down” is understood in the most general sense, and monotonicity means that reduction in some (or all) observed pp-values of the individual tests can only increase the number of rejections. What if we look for an extension of the Holm procedure beyond the class of step-down procedures? The result of the present work states that even in the entire class of monotone procedures the Holm procedure still cannot be extended without weakening the control of the FWER.