Closed procedures are better and often admit a shortcut

Abstract

Abstract It is ‘common knowledge’ that closed multiple test procedures are typically better than others in terms of power. In the first part of this paper we prove that closed procedures constitute a complete class in the class of all coherent procedures, and, also, that α-exhaustive procedures constitute a minimal complete class there. However, closed procedures are not always ‘monotone in p-values’ and are often computationally inefficient. Thus, in the second part of this paper we first investigate conditions for a closed procedure to be ‘monotone in p-values’, and second, we shed further light on the possibility of reducing a given closed procedure to a step-down sequentially rejective procedure. We give a sufficient condition for a closure based on some generalized Simes (1986) , (Biometrika 73, 751–754) type tests to be ‘monotone in p-values’ and clarify Hochberg and Tamhane (1987) , (Multiple Comparison Procedures, Wiley, New York) general result on the existence of a shortcut in families satisfying the free combination condition. We comment on the interrelationship between ‘stepwise’ (step-down and step-up), ‘monotone in p-values’, and ‘sequentially rejective’ procedures, in families satisfying the ‘free combinations condition’ as well as in families involving ‘logically related’ hypotheses.