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CROP BREEDING, GENETICS & CYTOLOGY

Model Selection and Cross Validation in Additive Main Effect and Multiplicative Interaction Models

^a Dep. of Ciências Exatas, Univ. of São Paulo/ESALQ, Av. Padua Dias 11, Cx.P.09, 13418-900, Piracicaba-SP, Brazil
^b School of Mathematical Sciences, Laver Building, North Park Road, Exeter, EX4 4QE, UK

^* Corresponding author (ctsdias{at}carpa.ciagri.usp.br)

The additive main effects and multiplicative interaction (AMMI) model has been proposed for the analysis of genotype–environmental data. For plant breeding, the recovery of pattern might be considered to be the principal objective of analysis. However, some problems still remain with the analysis, notably in selecting the number of multiplicative components in the model. Methods based on distributional assumptions do not have a sound methodological basis, while existing data-based approaches do not optimize the cross-validation process. This paper first summarizes the AMMI model and outlines the available methodology for selecting the number of multiplicative components to include in it. Then two new methods are described that are based on a full "leave-one-out" procedure optimizing the cross-validation process. Both methods are illustrated and compared on some unstructured multivariate data. Finally, their applications to analysis of genotype x environment interaction (GEI) are demonstrated on experimental grain yield data. Conclusions of the study are that the "leave-one-out" procedure is preferable in practice to either distributional F-test or cross-validation randomization methods, and of the two "leave-one-out" procedures the Eastment-Krzanowski method exhibits the greater parsimony and stability.

Abbreviations: AMMI, additive main effects and multiplicative interaction model • COMM, completely multiplicative model • DF, degrees of freedom • GEI, genotype x environment interaction • GREG, genotype regression model • IPCA, interaction principal component analysis • MET, multi-environment trials • NID, normally and independently distributed • PCA, principal components analysis • PRESS, predictive sum of squares • PRECORR, predictive correlation • RMSPD, root mean square predictive difference • SHMM, shifted multiplicative model • SREG, sites regression model • SS, sum of squares • SVD, singular value decomposition

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