Finite Intersection Tests: A Paradigm for Optimizing Simultaneous and Sequential Inference

When testing a family of comparisons or contrasts across k treatment groups, researchers are often encouraged to maintain control over the familywise Type I error rate. For common families such as comparisons against a reference group, sets of orthogonal and/or nonorthogonal contrasts, and all possible pairwise comparisons, numerous simultaneous (and more recently sequential) testing methods have been proposed. Many of the simultaneous methods can be shown to be a form of Krishnaiah’s (e.g., 1979) finite intersection test (FIT) for simultaneous multiple comparisons, which controls the familywise error rate to precisely a under conditions assumed in standard ANOVA scenarios. Other methods, however, merely represent conservative approximations to a FIT procedure, yielding suboptimal power for conducting simultaneous testing. The purpose of the current article is threefold. First, we discuss how FIT methodology represents a paradigm that unifies many existing methods for simultaneous inference, as well as how it suggests an improved method for testing nonorthogonal contrasts. Second, we illustrate more powerful multiple comparison procedures that combine FIT methodology with sequential hypothesis testing strategies. Third, we present a simple simulation strategy for generating critical values necessary to conduct these more powerful FIT-based methods. Examples of these methods are given.