Interpreting Genotype × Environment Interaction in Wheat by Partial Least Squares Regression Mateo Vargas, Jos6 Crossa,* Ken Sayre, Matthew Reynolds, Martha E. Ramffez, and Mike Talbot ABSTRACT The partial least squares (PLS) regression method relates geno- type × environment interaction effects (GEl) as dependent variables (Y) to external environmental (or cultivar) variables as the explana- tory variables (X) in one single estimation procedure. We applied PLS regression to two wheat data sets with the objective of determining the most relevant cultivar and environmental variables that explained grain yield GEl. Onedata set had two field experiments,one including seven durum wheat (Triticum turgidum L. var. durum) cultivars and the other, seven bread wheat (Triticum aestivum L.) cultivars, both tested for 6 yr. In durum wheat cultivars, sun hours per day in Decem- ber, February, and Marchas well as maximum temperature in March were related to the factor that explained more than 39% of GEl, while in bread wheat cultivars, minimum temperature in December and January as well as sun hours per day in January and February were the environmental variables related to the factor that explained the largest portion (>41%)of GEl. The second data set had eight bread wheat cultivars evaluated in 21 low relative humidity (RH) environments and 12 high RHenvironments. For both low and high RH environments, results indicated that relative performance of culti- vars is influenced by differential sensitivity to minimum temperatures during the spike growth period. The PLS method was effective in detecting environmental and cultivar explanatoryvariables associated with factors that explained large portions of GEl. Wa nEN ASSESS~N6 grain yield of a set of cultivars in multi-environment trial, changes are commonly observed in the relative yield performance of cultivars with respect to each other across sites. This differential yield response of cultivars from one environment to another is called genotype × environment interaction (GEI) and can be studied, described, and interpreted by statistical models(Crossa, 1990). A commonly used procedure for modeling statistical interaction is a simple regression of the cultivar perfor- mance on the site mean (Yates and Cochran, 1938; Fin- lay and Wilkinson, 1963; Eberhart and Russell, 1966). This model can be depicted in a set of straight lines with different slopes, one for each cultivar, and the heterogeneity of slopes accounts for the interaction. Since heterogeneity of slopes generally explains only a small proportion of the complex interaction, a more M. Vargas, Programa de Estadfstica del Instituto de Socioeconomla, Estadfstica e Inform~itica, Colegio de Postgraduados, CP 56230, Mon- tecillo, Mexico, Universidad Aut6nomaChapingo, CP 56230, Chap- ingo, Mexico, and International Maize and Wheat Improvement Cen- ter (CIMMYT), Lisboa 27, Apdo. Postal 6-641, 06600 Mexico, D.F., Mexico; J. Crossa, Biometrics and Statistics Unit, CIMMYT, Lisboa 27, Apdo. Postal 6-641, 06600 Mexico, D.F., Mexico; K. Sayre and M. Reynolds, Wheat Program, CIMMYT, Lisboa 27, Apdo. Postal 6-641, 06600 Mexico, D.F., Mexico; M.E. Ramfrez, Programa de Estad- istica del Instituto de Socioeconom/a, Estadfstica e Inform~itica, Cole- gio de Postgraduados, CP 56230, Montecillo, Mexico; M. Talbot, Bio- mathematics and Statistics Scotland, Univ. of Edinburgh, JCMB, Kings Buildings, Edinburgh EH93JZ, UK. *Corresponding author (JCROSSA@ALPHAC.CIMMYT.MX). Published in Crop Sci. 38:679~689 (1998). elaborate model would be necessary to describe the GEI. A generalization of the regression on the site mean model is the multiplicative model also called Principal Component Analysis of the GEI or Additive Main Ef- fect and Multiplicative Interaction (AMMI)model (Gollob, 1968; Mandel, 1971; Kempton, 1984; Gauch, 1988). Crossa et al. (1990a) investigated AMMI other procedures for grouping environments and wheat cultivars into homogeneous subsets and determining yield stability. The AMMI model provides more opportunity for modeling and interpreting GEI than the simple regres- sion on the site meanmodel because it allows modeling the GEI in more than one dimension; however, it esti- mates the environmental and cultivar interaction pa- rameters by statistics derived from the observed pheno- typic data themselves. When information on external environmental variables is available (i.e., precipitation, temperature, etc.), it can be correlated to or regressed on the AMMI environmental scores so that some inter- pretation of the causes of grain yield GEI can be at- tempted. However, external environmental information cannot be used directly in the AMMI model. When additional information is available on environ- ment, cultivars, or both, GEI can be modeled directly by the factorial regression model(Denis, 1988; van Eeu- wijk et al., 1996). Since a large number of external cova- riables maybe modelingjust noise, the most explanatory covariables maybe synthesized in one covariate by the reduced rank factorial regression (van Eeuwijk et al., 1996). Also, whenenvironmental information is avail- able, interpretation of GEImaybe possible by the prin- cipal component regression procedure that relates in- dividual environmental variables to the principal component scores of the GEI (Aastveit and Martens, 1986). However, this approach has several problems, given that (i) it is sensitive to multicollinearity and noise and is nonparsimonious, (ii) it is not easy to relate many environmental variables to several principal component factors simultaneously, and (iii) retaining the optimal number of principal components for interpretation may be difficult (Aastveit and Martens, 1986). To overcome some of these problems, Aastveit and Martens (1986) proposed the partial least squares (PLS) regression method as a more direct and parsimonious linear model. This method consists of relating X and Y matrices in one single estimation procedure. The Y ma- trix contains site x cultivar grain yield data as dependent variables and the X matrix has the external environmen- tal variables (or external cultivar variables) as the ex- planatory variables. In contrast to the principal compo- nent regression approach where each component is a Abbreviations: PLS, partial least squares; GEI, genotype × environ- ment interaction; AMMI, additive main effect and multiplicative inter- action. 679 Published May, 1998