Multiplicative Interaction in Generalized Linear Models

Bilinear oI biadditive multiplicative models for interaction in two-way tables provide the major means for Studyiing g ,eniotype by cnvlirOnllllcmlt initer'actioni pioblems. In applicatioIns the typical accompanying assumptions are those of a noirnally clistributecl error and an identity link. These assumptioins are uninecessarily restrictive. Introduction of multiplictive terms foi i'nteraction in generalized linear models removes these restrictions. Parameter estimates can be obtained by ain iterative pr ocess of altcrInatinlg generalized Irow anid columin Iregr-essions withlini a quasi-likclihood set-up. The best known examples of this class of generalized additive main effects and multiplicative interaction effects (GAMMI) models are the AMMI models (Gauch, 1988, Biomiietrics 44, 705-715) and Goodmain's RC-association imiodels (Goodman, 1981, Joil{Z-lr7 of thle Amnericain Stcatistical Associaition 76, 320-334). The multiplicative inteiactioin part of GAMMI models canl be visualized througlh biplots. Two applications of GAMMI models are presented for data coming from planlt breeding experimenits. The first illustration deals with a log-bilinear model for couInt data with (extra) Poisson variation. The second ilILustiatioIn concerniis a logit-bilinear model for disease incidence data with a special type of variance function, anl extensioin of a model presented by Wedderbunii (1974 Biomiietrikal 61, 439-447).