Multiple Contrasts and Ordered Bonferroni Procedures

Abstract

This article presents a general yet simple system for avoiding the increases in Type I errors that typically occur when an increasing number of contrasts is to be computed. The procedures described are all based on the Bonferroni inequality, which leads to methods of correcting for the number of contrasts tested. The three types of Bonferroni tests presented differ in the degree to which we specify beforehand the relative importance we attach to each of the planned contrasts. Although a quite conservative procedure in its typical application, the Bonferroni system of procedures is recommended for its flexibility, simplicity, and generality. When the power of the basic Bonferroni method is focused by ordering the contrasts (or any other tests of significance) by their importance, the disadvantage of conservatism can be, to a great extent, overcome.