On the Development of Statistical Modeling in Plant Breeding: An Approach of Row-Column Interaction Models (RCIM) For Generalized AMMI Models with Deviance Analysis

Generalized AMMI (GAMMI) model has been widely used to model the Genotype × Environment Interaction (GEI) with categorical (or in general, non-normal) response variables. It was developed by introduce the concept of Generalized Linear Model (GLM) into Additive Main Effect & Multiplicative Interaction (AMMI) model. GAMMI model will provide two major results (i) the stability analysis of some genotypes across environments and (ii) determine some others that have site specific for particular environment through Biplot of Singular Value Decomposition (SVD) of the interaction terms. This research will focus on major studies on counting data that is to round up the previous work of first author's on the Row Column Interaction Models (RCIMs) for the GEI by VGAM package of an R implementation with an addition on the deviance analysis. A simple illustrative comparison of both approaches (RCIM vs. GAMMI) was conducted on Poisson counting data of 4 rows × 5 columns. The defiance analysis was provided by log-likelihood of the model and ones of the residual. Deviance analysis will provide a way to determine the complexity of interaction component in the model, named by “rank” of model. The Biplot of both approaches seem not quite different. Finally, we did show that RCIMs be relied upon to fit well the GAMMI model and then applied it in an illustrative example to a real dataset. In addition, a simple scheme of simulation, adding some outlier on Poisson count data, will show an easy way handling the over dispersion problems, but firstly, we will talk about some statistical framework of Reduce Rank Regression (RR-VGLMs), the RCIMs, and then the approach of RCIMs for GAMMI models.