Stepwise multiple test procedures and control of directional errors

Abstract

One of the most difficult problems occurring with stepwise multiple test procedures for a set of two-sided hypotheses is the control of direc-tional errors if rejection of a hypothesis is accomplished with a directional decision. In this paper we generalize a result for so-called step-down procedures derived by Shaffer to a large class of stepwise or closed multiple test procedures. In a unifying way we obtain results for a large class of order statistics procedures including step-down as well as step-up procedures (Hochberg, Rom), but also a procedure of Hommel based on critical values derived by Simes. Our method of proof is also applicable in situations where directional decisions are mainly based on conditionally independent $t$-statistics. A closed $F$-test procedure applicable in regression models with orthogonal design, the modified $S$-method of Scheffé applicable in the Analysis of Variance and Fisher’s LSD-test for the comparison of three means will be considered in more detail.