CROP BREEDING, GENETICS & CYTOLOGY A Multivariate Method for Classifying Cultivars and Studying Group Environment Trait Interaction

each observation with p traits measured in each environment is normally distributed. For r environments, the Classification methods are used in genetic resource conservation vector of each observation has dimensions 1 rp, where and plant breeding. The two-stage Ward-Modified Location Model (Ward-MLM) clustering strategy defines initial groups by the Ward rp trait–environment combinations (variables). method and the MLM is then used to improve those groups. The Lawrence and Krzanowski (1996) used the Location three-way clustering strategy considers the same trait measured in Model (LM), originally proposed by Olkin and Tate different environments as different variables (environment–trait com(1961), for classifying n individuals when p continuous bination), so that the resulting clusters of cultivars have a small traits and q categorical traits are measured in one envigroup environment interaction (GEI) for all the traits included in ronment. The LM combines the levels of all q categorical the study. An important component of the GEI is the imperfect traits in one multinomial variable, W, with m levels (w genotypic correlation or crossover interaction (COI). This study used 1,2,...,m). Franco et al. (1998) modified the LM and the three-way Ward-MLM clustering strategy in three different data proposed the Modified Location Model (MLM), by assets with the objectives of (i) evaluating the ability of the Ward-MLM suming that the m levels of the W variable and the pmethodology for clustering cultivars into groups with low COI, (ii) obtaining a graphical representation of the variables in a low dimenmultinormal variables for each subpopulation, are indesional scatter plot produced by the multidimensional scaling method pendent. The authors proposed using the MLM in a (MDS) in which the GEI of continuous and categorical traits and two-stage clustering strategy where initial groups are environments can be visualized, and (iii) studying how the relationship defined by the Ward (or any other hierarchical clusterbetween a pair of traits changes across environments. The three-way ing strategy) method and the MLM is then used with Ward-MLM strategy produced groups of cultivars with low levels of those groups (Ward-MLM). COI. The increment of the correlation coefficient values between Franco et al. (1999) extended the Ward-MLM stratgroups with respect to the total correlation coefficients indicated that egy to the case of clustering three-way data of cultivar the groups formed by the three-way Ward-MLM strategy comprised environment trait. Thus, the vector of each observasubsets of individuals that had similar trait responses across envition for r environments is 1 (rp 1). The three-way ronments. Ward-MLM clustering strategy considers the same trait measured in different environments as different variables (environment-trait combinations). Since the I genetic resources conservation and plant breeding Ward-MLM strategy clusters individuals with consistent research, classification methods are used for studying response across all variables in all the environments, it is phenotypic and genetic diversity, formation of core subreasonable to expect that the three-way cluster strategy sets, and grouping cultivars into homogeneous clusters should form groups of cultivars with negligible group (Crossa et al., 1995; Franco et al., 1999). Continuous GEI for all the traits included in the study. and categorical traits are measured in each of the gene Group environment interaction can be expressed bank accessions or cultivars evaluated in one or several as imperfect genotypic (or environmental) correlation environments, and hierarchical and statistical classificaor COI, or as heterogeneity of variance across environtion methods are employed to recover, as much as possiments or non-COI (Moll et al., 1978; Muir et al., 1992). ble, the underlying subpopulations of individuals. The crossover GEI is the most important component The method for clustering n individuals on the basis of the GEI to be considered. Thus, the three-way Wardof p continuous and q categorical traits measured in r MLM strategy should tend to form groups of cultivars different environments is known as three-way (cultiwith high genotypic correlations across environments var environment trait) cluster analysis. Basford and McLachlan (1985) proposed a classification method Abbreviations: GM, Gaussian Mixture; LM, Location model; MLM, when all measured traits are continuous. The authors Modified location model; EM, Expectation maximization; p.d.f., probutilized the Gaussian Model (GM) that assumes that ability density function; UPGMA, unweighted pair grouping with the arithmetic means; COI, crossover interaction; GEI, group environment interaction. VT, total variance-covariance matrix; VB, betweenJ. Franco, Facultad de Agronomia, Universidad de la República Origroups variance-covariance matrix; VW, pooled within-group varianceental del Uruguay, Avd. Garzon 780 CP 12900. Montevideo, Uruguay; covariance matrix; RT, total correlation matrix; RB, between-groups J. Crossa, Biometrics and Statistics Unit, CIMMYT, Apdo. Postal correlation matrix; RW, pooled within-group correlation matrix; DA, 6-641, 06600, México DF, México; S. Taba, Maize Genetic Resources days to anthesis; P, plant height; DS, days to senescence; EL, ear Unit, CIMMYT, México; and H. Shands, National Seed Storage Labolength; A, anthesis-silking interval; E, number of rows per ear; F, days ratory, USDA, ARS, Fort Collins, CO 80523. Received 20 Sept. 2002. to silking; DM, days to maturity; AS, agronomic scale; Y, grain yield; *Corresponding author ([email protected]). SP, Septoria leaf blotch; LR, leaf rust; PM, powdery mildew; SE, proportion of selected plants; MDS, multidimensional scaling. Published in Crop Sci. 43:1249–1258 (2003).