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Multiplicative Interaction in Generalized Linear Models

Fred A. van Eeuwijk
Biometrics
Vol. 51, No. 3 (Sep., 1995), pp. 1017-1032
DOI: 10.2307/2533001
Stable URL: http://www.jstor.org/stable/2533001
Page Count: 16
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Multiplicative Interaction in Generalized Linear Models
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Abstract

Bilinear or biadditive multiplicative models for interaction in two-way tables provide the major means for studying genotype by environment interaction problems. In applications the typical accompanying assumptions are those of a normally distributed error and an identity link. These assumptions are unnecessarily restrictive. Introduction of multiplicative terms for interaction in generalized linear models removes these restrictions. Parameter estimates can be obtained by an iterative process of alternating generalized row and column regressions within a quasi-likelihood 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, Biometrics 44, 705-715) and Goodman's RC-association models (Goodman, 1981, Journal of the American Statistical Association 76, 320-334). The multiplicative interaction part of GAMMI models can be visualized through biplots. Two applications of GAMMI models are presented for data coming from plant breeding experiments. The first illustration deals with a log-bilinear model for count data with (extra) Poisson variation. The second illustration concerns a logit-bilinear model for disease incidence data with a special type of variance function, an extension of a model presented by Wedderburn (1974, Biometrika 61, 439-447).

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