Development of robustness on additive Main Effect – Multiplicative Interaction (AMMI) Models

Abstract

AMMI model for interactions in two-way table provide the major mean for studying stability and adaptability through GEI, which modeled by full interaction model. Eligibility of AMMI model depends on the assumption of normally independent distributed error with a constant variance. Nowadays, AMMI models have been developed for any condition of MET data i.e the violence of normality and homegeneity assumptions. We can mention in this class of modelling as MAMMI for Mixed AMMI model and GAMMI for Generalized AMMI model. GAMMI model handles non-normality i.e categorical response variables using an algorithm of alternating regression. While handling the non-homogeneity in mixed-models sense, one may use a model called factor analytic multiplicative for a MAMMI models. Outlier might be found in the data coincides with non-homogeneity variance. A method of handling outlier in additive and multiplicative modeling by applying Robust Alternating Regression (RAR) in FANOVA model. RAR FANOVA model was downweighting outlying scores and loadings in the k-dimensional spaces of scores and loadings, and robust estimator will be minimized under the constraints that are consistent with robust approach of the median of parameters. Application of GAMMI was found in several distribution of exponential family. The most interesting here is Poisson distribution which it has a unique property of equal mean and variance. Many zero observations make some dificulties and fatal consequence in Poisson modeling. We consider to facilitates an analysis of two-way tables of count with many zero observations in AMMI model. We develop GAMMI model for Poisson with zeroes problem, by a statistical framework of RCAM. Some link function apply to the mean of a cell equalling a row effect plus a column effect plus interaction terms are modelled as a reduced-rank regression with rank of 2, then it will be visualized by Biplot through SVD reparameterization. The ZIP-GAMMI model handle the excess-zero and also the overdispersion at once. The interaction structures are extracted from the non-zero cell. ZIP model provide us the probability become zero and the fitted value of the Poisson. The modelling scheme involves two important things: (1) the distribution (and link-function) and (2) the rank of model. Both are confounded, especially if there is overdispersion or excess-zero. Best-model fit can be provided by proper link function with respect to data’s distribution, at the same time it’s also possible by the rank of the model in decomposing the interaction terms. In this case the likelihood ratio test provides us the hypothesis testing.