2000).Yan et al.(2000)referred to biplots based on sin- scaling).Mathem gular value ition (SVD)of nk 2 lea ank n natrix Z.This rep ntation anique except for hinlots”bee chan on all o nd/ which are the two sources of variation that are relevant to or all 6 andAn important property of the biplot is cultivar evaluation (Kang.1988.1993:Gauch and Zobel that the rank 2 approximation of any entry in the original 1996:Yan and Kang.2003) matrix Z can be computed by taking the inner product The commonly used GGE biplot is based on the Sites of the corresponding genotype and environment vectors, Regression(SREG)linear-bilinear(multiplicative)mode ie.6a,2)-,-2)=，di+di2 (Cornelius et al.,1996).which can be written as -,=∑入40+回 This is know -product property of the biplo logy (Yan e nean of genotype i in environm en 200 200 ran an ng,200 8 biplot interpreta and t is C) ined in the nd oth 1).The model is subied t to the found GGE biplots useful in mega ent analys 入，≥0 and to orthonormality Yan and rai s et al 2005-Sam scores that is et al.2005:Yan and Tinker.2005b:Dardanellia et al. =0 if with similar constraints on the 2006),genotype evaluation(Bhan et al.,2005;Malvar et [defined by replacing symbols (i.g,o)with (je,).The al.,2005:Voltas et al.,2005;Kang et al,2006),test-envi- e:are assumed NID(0.2/r).where r is the number of ronment evaluation (Yan and raican.2002:Blanche and replications within an environment. Myers,2006;Thomason and Phillips,2006 trait-as Least squares solution for is the empirical mean ciation and trait-profile analyses(Yan and Rajcan,2002 for the jth environme and th least squares solutions to M 004:Ober et a and heter para the ter analy and Hunt ble for rom tl th(for is the et 2006 mac del Rank(Z).In gen forGED alit super =1 also T of this nd inte multinlicative effects of the ith cultivar and ith enviro are:(i)to compare GGE biplotanalysis and AMMlanaly ment(for first usage of such terminology in a multiplic on three aspects of ged analysis namely mega-environ- tive model context,see Seyedsadr and Cornelius,1992). ment analysis,genotype evaluation,and test-environment Thus,Eq.[1]may be described as modeling the deviations evaluation:(ii)to discuss whether g and ge should be of the cell means from the environment means as a sum of combined or separated in GED analysis;and(iii)to discuss PCs,each of which is the product of a cultivar score the importance of model diagnosis in SVD-based analy. an environment score ()and a scale factor(the singular sis of GED.This disc ussion should enhance agrict ultura researchers'understanding of biplot analysis of GED GE biplot is onstr ed from PC the n cu THREE ASPECTS OF GED ANALYSIS USING GGE BIPLOTS and i-/ ent The of GED G e.,MET data for ingle e trai 0<f<1.i ale the s to enhance visual in of the biplot for nd fii particular purpose.Specifically,singular values are allo- type evaluation (Yan and Kang.2003).We use the vield cated entirely to cultivar scores if f=1 ithis is"cultivar. data of 18 winter wheat(Triticum aestivum L)genotypes focused"scaling (Yan,2002),or entirely to environment (Gl to G18)tested at nine Ontario locations(El to E9) scores if f=0("environment-focused"scaling):and f= (Table 1)as an example to illustrate the three aspects of 0.5 will allocate the square roots of the X values to cul biplot analysis.The same dataset was used extensively in tivar scores and also to environment scores ("symmetric Yan and Kang (2003)and Yan and Tinker (2006).When 644 WWW.CROPS.ORG CROP SCIENCE,VOL.47,MARCH-APRIL 2007 Reproduced from Crop Science. Published by Crop Science Society of America. All copyrights reserved. 644 WWW.CROPS.ORG CROP SCIENCE, VOL. 47, MARCH–APRIL 2007 2000). Yan et al. (2000) referred to biplots based on sin￾gular value decomposition (SVD) of environment-cen￾tered or within-environment standardized GED as “GGE biplots,” because these biplots display both G and GE, which are the two sources of variation that are relevant to cultivar evaluation (Kang, 1988, 1993; Gauch and Zobel, 1996; Yan and Kang, 2003). The commonly used GGE biplot is based on the Sites Regression (SREG) linear-bilinear (multiplicative) model (Cornelius et al., 1996), which can be written as 1 t ij j k ik jk ij k y = − μ = ∑λα γ + ε [1] where y – ij is the cell mean of genotype i in environment j; μj is the mean value in environment j; i = 1, ∙ ∙ ∙ g; j = 1, ∙ ∙ ∙ e, g and e being the numbers of cultivars and envi￾ronments, respectively; and t is the number of principal components (PC) used or retained in the model, with t ≤ min(e,g − 1). The model is subject to the constraint λ1 ≥ λ2 ≥ ∙ ∙ ∙ λt ≥ 0 and to orthonormality constraints on the αik scores, that is, 1 ' g ik ik i= ∑ α α = 1 if k = k' and 1 ' g ik ik i= ∑ α α = 0 if k ≠ k', with similar constraints on the γjk scores [defi ned by replacing symbols (i,g,α) with (j,e, γ)]. The eij are assumed 2 NID(0, / ) σ r , where r is the number of replications within an environment. Least squares solution for μj is the empirical mean (y – .j) for the jth environment, and the least squares solutions for parameters in the term λk αikγjk (for i = 1, ∙ ∙ ∙ ,g; j = 1,…,e) are obtained from the kth PC of the SVD of the matrix Z = [zij], where zij = y – ij – y – .j. The maximum number of PCs available for estimating the model parameters is p = Rank(Z). In general, p ≤ min(e, g − 1), with equality hold￾ing in most cases. For k = 1, 2, 3, ∙ ∙ ∙ , αik and γjk have also been characterized as primary, secondary, tertiary, etc., multiplicative eff ects of the ith cultivar and jth environ￾ment (for fi rst usage of such terminology in a multiplica￾tive model context, see Seyedsadr and Cornelius, 1992). Thus, Eq. [1] may be described as modeling the deviations of the cell means from the environment means as a sum of PCs, each of which is the product of a cultivar score (αik), an environment score (γjk), and a scale factor (the singular value, λk ). The GGE biplot is constructed from the fi rst two PCs from the SVD of Z with “markers,” one for each cultivar, plotted with 1 1 ˆ ˆ f λ αi as abscissa and 2 2 ˆ ˆ f λ αi as ordinate. Simi￾larly, markers for environments are plotted with 1 1 1 ˆ ˆf j − λ γ as abscissa and 1 2 2 ˆ ˆf j − λ γ as ordinate. The exponent f, with 0 ≤ f ≤ 1, is used to rescale the cultivar and environment scores to enhance visual interpretation of the biplot for a particular purpose. Specifi cally, singular values are allo￾cated entirely to cultivar scores if f = 1 [this is “cultivar￾focused” scaling (Yan, 2002)], or entirely to environment scores if f = 0 (“environment-focused” scaling); and f = 0.5 will allocate the square roots of the λˆ k values to cul￾tivar scores and also to environment scores (“symmetric” scaling). Mathematically, a GGE biplot is a graphical rep￾resentation of the rank 2 least squares approximation of the rank p matrix Z. This representation is unique except for possible simultaneous sign changes on all 1 ˆαi and 1 ˆ j γ and/ or all 2 ˆαi and 2 ˆ j γ . An important property of the biplot is that the rank 2 approximation of any entry in the original matrix Z can be computed by taking the inner product of the corresponding genotype and environment vectors, i.e., ( )( ) 1 1 1 1 2 2 1 1 2 2 111 22 2 ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆˆ , , ff f f i i j j ij ij − − ′ λα λα λ γ λ γ = λα γ + λα γ . This is known as the inner-product property of the biplot. The GGE biplot methodology (Yan et al., 2000; Yan, 2001, 2002; Yan and Kang, 2003; Yan and Tinker, 2006) consists of a set of biplot interpretation methods, whereby important questions regarding genotype evaluation and test-environment evaluation can be visually addressed. Increasingly, plant breeders and other agronomists have found GGE biplots useful in mega-environment analysis (Yan and Rajcan, 2002; Casanoves et al., 2005; Samonte et al., 2005; Yan and Tinker, 2005b; Dardanellia et al., 2006), genotype evaluation (Bhan et al., 2005; Malvar et al., 2005; Voltas et al., 2005; Kang et al., 2006), test-envi￾ronment evaluation (Yan and Rajcan, 2002; Blanche and Myers, 2006; Thomason and Phillips, 2006), trait-asso￾ciation and trait-profi le analyses (Yan and Rajcan, 2002; Morris et al., 2004; Ober et al., 2005), and heterotic pat￾tern analysis (Yan and Hunt, 2002; Narro et al., 2003; Andio et al., 2004; Bertoia et al., 2006). The legitimacy of GGE biplot analysis was, however, recently questioned by Gauch (2006), who concluded that, for GED analyses, AMMI analysis was either superior or equal to GGE bip￾lot analysis. The objectives of this review and interpretation paper are: (i) to compare GGE biplot analysis and AMMI analysis on three aspects of GED analysis, namely, mega-environ￾ment analysis, genotype evaluation, and test-environment evaluation; (ii) to discuss whether G and GE should be combined or separated in GED analysis; and (iii) to discuss the importance of model diagnosis in SVD-based analy￾sis of GED. This discussion should enhance agricultural researchers’ understanding of biplot analysis of GED. THREE ASPECTS OF GED ANALYSIS USING GGE BIPLOTS The analysis of GED (i.e., MET data for a single trait) should include three major aspects: (i) mega-environment analysis; (ii) test-environment evaluation, and (iii) geno￾type evaluation (Yan and Kang, 2003). We use the yield data of 18 winter wheat (Triticum aestivum L.) genotypes (G1 to G18) tested at nine Ontario locations (E1 to E9) (Table 1) as an example to illustrate the three aspects of biplot analysis. The same dataset was used extensively in Yan and Kang (2003) and Yan and Tinker (2006). When