Multiplicative Models for Interaction in Three-Way ANOVA, with Applications to Plant Breeding

Abstract

SUMMARY In plant breeding, multiplicative models for two-way analysis of variance interaction have become a general means of describing genotype-by-environment interaction. These models offer parsimonious descriptions and facilitate interpretations in biological terms. A disadvantage of the prevailing dominance of two-way multiplicative models is that data with more complicated environmental structure are often forced to fit the two-way framework. As a partial solution to this problem, three-way multiplicative models are presented that can be used in addition to the more familiar two-way multiplicative models. Most importantly, a three-way generalization is given of the twoway singular-value decomposition that can be applied for closer inspection of three-way analysis of variance interaction, much in the same way as its two-way counterpart is used for two-way interaction. Two real data sets are analyzed to illustrate how two- and three-way multiplicative models for interaction jointly provide parsimonious descriptions of various types of genotype-byenvironment interaction. These parsimonious descriptions are often open to meaningful biological interpretations as well.