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Abstract:
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Holm's and Hochberg's methods are popular multiple tests. They are viewed as step-down and step-up versions of the Bonferroni test. We will give the insight that both are special cases of partition testing. The difference is that, while Holm's method tests each partition hypothesis using the largest order statistic, setting a critical value based on the Bonferroni inequality, Hochberg's method tests each partition hypothesis using all the order statistics, setting a series of critical values based on Simes' inequality. Geometrically, Hochberg's step-up method 'cuts corners' off the acceptance regions of Holm's step-down method by making assumption on the joint distribution of the test statistics. As can be expected, partition testing making use of the joint distribution of the test statistics is more powerful than partition testing using probabilistic inequalities.
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