The k-Ratio Multiple Comparisons Bayes Rule for the Balanced Two-Way Design

A comparisonwise alternative to the t testing of multiple comparisons is the k-ratio Bayes rule method. This method is comparisonwise in that the test of a mean difference does not depend on the prior choice of the number of other differences tested, but yet is F protective in that for the one-way design, the critical t value of the test rises as the between-means F ratio falls. In this article the k-ratio method is extended to the balanced two-way design. Attention is limited to differences between column entries within a row of the two-way mean table (or vice versa). For such a difference, the k-ratio rule F protects against homogeneity of the column effects and depends on the corresponding marginal difference over all rows. Dependence on the marginal difference increases as the interaction F ratio decreases. When positive, the marginal difference causes the critical t value for declaring significance in the positive direction to be less stringent than the one for declaring significance in the negative direction. The critical t value F protects against column-level homogeneity by becoming more stringent in both directions when the column F ratio is lowered and the interaction F ratio is accordingly lowered to maintain the same degree of marginal dependence. In the 2 2 design, the critical t values are much less dependent on the marginal difference due to the relationship between the marginal difference and the interaction F ratio.