Duncan's k-ratio Bayes rule approach to multiple comparisons: an overview

Abstract

An alternative to frequentist approaches to multiple comparisons is Duncan's k-ratio Bayes rule approach. The purpose of this paper is to compile key results on k-ratio Bayes rules for a number of multiple comparison problems that heretofore, have only been available in separate papers or doctoral dissertations. Among other problems, multiple comparisons for means in one-way, two-way, and treatments-vs.-control structures will be reviewed. In the k-ratio approach, the optimal joint rule for a multiple comparisons problem is derived under the assumptions of additive losses and prior exchangeability for the component comparisons. In the component loss function for a comparison, a balance is achieved between the decision losses due to Type I and Type II errors by assuming that their ratio is k. The component loss is also linear in the magnitude of the error. Under the assumption of additive losses, the joint Bayes rule for the component comparisons applies to each comparison the Bayes test for that comparison considered alone. That is, a comparisonwise approach is optimal. However, under prior exchangeability of the comparisons, the component test critical regions adapt to omnibus patterns in the data. For example, for a balanced one-way array of normally distributed means, the Bayes critical t value for a difference between means is inversely related to the F ratio measuring heterogeneity among the means, resembling a continuous version of Fisher's F-protected least significant difference rule. For more complicated treatment structures, the Bayes critical t value for a difference depends intuitively on multiple F ratios and marginal difference(s) (if applicable), such that the critical t value warranted for the difference can range from being as conservative as that given by a familywise rule to actually being anti-conservative relative to that given by the unadjusted 5%-level Student's t test.