Parsimonious estimation of multiplicative interaction in analysis of variance using Kullback-Leibler Information

Many standard methods for modeling interaction in two-way ANOVA require mn interaction parameters, where m and n are the number of rows and columns in the table. By viewing the interaction parameters as a matrix and performing a singular value decomposition, one arrives at the additive main effects and multiplicative interaction (AMMI) model which is commonly used in agriculture. By using only those interaction components with the largest singular values, one can produce an estimate of interaction that requires far fewer than mn parameters while retaining most of the explanatory power of standard methods. The central inference problems of estimating the parameters and determining the number of interaction components has been difficult except in "ideal" situations (equal cell sizes, equal variance, etc.). The Bayesian methodology developed in this paper applies for unequal sample sizes and heteroscedastic data, and may be easily generalized to more complicated data structures. We illustrate the proposed methodology with two examples.