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.