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CROP BREEDING, GENETICS & CYTOLOGY

Additive Main Effects and Multiplicative Interaction Model

II. Theory on Shrinkage Factors 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).

A shrinkage estimation method, herein named the CCC method, for multiplicative models has been shown to improve predictions of cell means in multienvironment cultivar trials; however, better estimations of shrinkage factors are needed. Objectives of this study were (i) to develop shrinkage factors for AMMI (Additive Main effect and Multiplicative Interaction) multiplicative terms based on the eigenvalue partition (EVP) method and (ii) to compare AMMI models fitted by EVP and CCC shrinkage methods, unshrunken AMMI models chosen by cross validation, and best linear unbiased prediction (BLUP) based on a fixed main effects and random genotype x environment interaction (GEI) model. The prediction methods were applied to three multienvironment cultivar trials, and also to simulated data. The four alternative prediction methods were compared by cross validation with the root mean squared predictive difference (RMSPD). According to the RMSPD cross validation criterion, the EVP shrinkage estimation method produced the most accurate predictions in almost all cases. When error variance was of a small to moderately large magnitude, results for EVP and CCC methods were in close agreement, CCC being slightly worse; EVP became better as error variance became very large, probably because of the unreliability of Gollos degrees of freedom as a measure of error variance absorption by the principal components (PC) when data are extremely noisy. Shrunken EVP models were generally more predictively accurate than truncated least squares-fitted AMMI models and BLUPs, which was also true for CCC except when error variance was large. The EVP shrinkage method appears to be promising for obtaining improved predictions of cultivar performance in multienvironment cultivar trials.

Abbreviations: AMMI, additive main effect and multiplicative interaction • BLUP, best linear unbiased predictor • CCC, work of Cornelius, Crossa, and associates • COMM, completely multiplicative model • EVP, eigenvalue partition • GEAR, genotype, environment, attribute model • GEI, genotypes x environment interaction • GREG, genotype regression model • MSEPM, mean squared error of predicted means • PC, principal component • RCBD, randomized complete block design • RMSPD, root mean squared predictive difference • SEPM, standard error of predicted means • SREG, sites regression model

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