Testing components of two-way interaction in multi-environment trials

Experiments with two factors are commonly analyzed using two-way analysis of variance, where testing significance of interaction is straightforward. However, using bilinear models, interaction can be analyzed further. The additive main effects and multiplicative interaction (AMMI) model uses singular value decomposition for partitioning interaction into multiplicative terms, such that the first terms typically account for a large portion of the sum of squares, whereas the last terms are of minor importance. This model is used extensively for analysis of genotype-by-environment interaction in multi-environment trials. A recurring question is how to determine the number of terms to retain in the model. If data is replicated, which is usually the case, the FR test can be used for this purpose. The simple parametric bootstrap method is another option, although this test was developed for unreplicated data. Since both of these tests of significance may be applied in cases with replication, researchers need advice on which of the methods to use. We discuss several statistical models and show that the two methods address different questions.