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CROP BREEDING, GENETICS & CYTOLOGY

Additive Main Effects and Multiplicative Interaction Model

I. Theory on Variance Components for Predicting Cell Means

^a Centro de Investigaciones Agrarias de Mabegondo, Apartado 10, A Coruña, Spain
^b Biometrics and Statistics Unit, International Maize and Wheat Improvement Center (CIMMYT), Apdo. Postal 6-641, 06600 Mexico DF, Mexico
^c Dep. of Agronomy and Dep. of Statistics, Univ. of Kentucky, Lexington, KY 40546-0091

^* Corresponding author (j.crossa{at}cgiar.org)

Many studies have shown the practical advantages of applying the additive main effects and multiplicative interaction (AMMI) model to multienvironment trials; however, a theory about the contributions of error and genotype x environment interaction (GEI) variance components to interaction principal components (PCs) in AMMI models is needed. The objectives of this work were to (i) develop an eigenvalue partition (EVP) method for separating variation attributable to each AMMI interaction PC into interaction variance and error variance components: (ii) develop root mean square predictive difference (RMSPD) on the basis of the EVP theory (RMSP_EVP), for selecting the best truncated AMMI model; (iii) apply the RMSPD_EVP criterion to three multienvironment cultivar trials and to simulation data generated for selecting the best truncated AMMI model; and (iv) validate the EVP method by comparing results of the RMSPD_EVP criterion with those obtained with the criterion conventionally used to choose a truncated AMMI model by cross validation (RMSPD_CV). A data resampling method was devised to estimate the contribution of error variance to the eigenvalues. The coefficients of the structural GEI variance component were always larger than those of the error variance component for the earlier PC axes. As the error associated with the cell means decreased and the number of replications increased, the portion of the cumulative GEI explained by the earlier AMMI PC axes generally increased, whereas the portion of the error sum of squares (SS) explained by the earlier AMMI PC axes decreased. The RMSPD_EVP and RMSPD_CV methods selected similar truncated AMMI models. The RMSPD_EVP criterion is useful for selecting the best truncated AMMI models with the advantage that it can be applied to all trial replications.

Abbreviations: AMMI, additive main effect and multiplicative interaction • BLUP, best linear unbiased predictor • COMM, completely multiplicative model • EVP, eigenvalue partition • GEAR, genotypes, environment, attribute regression model • GREG, genotypes regression model • GEI, genotype x environment interaction • PC, principal component • SREG, sites regression model • SHMM, shifted multiplicative model • RMSPD, root mean square predictive difference

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