农业院校《试验设计与分析》课程参考资料（多变量分析方法）Two Types of GGE Biplots for Analyzing Multi-Environment Trial Data.pdf

Two Types of GGE Biplots for Analyzing Multi-Environment Trial Data Weikai Yan,* Paul L. Cornelius, Jose Crossa, and L. A. Hunt ABSTRACT on scaled or non-scaled data. When replicated data are available, SREG on scaled data (Crossa and Cornelius, SA genotype main effect plus genotype 3 environment interaction 1997) is more desirable because it deals with any hetero- (GGE) biplot graphically displays the genotypic main effect (G) and geneity of within-site error variance. the genotype 3 environment interaction (GE) of the multienvironment trial (MET) data and facilitates visual evaluation of both the One unique merit of a GGE biplot is that it can graphgenotypes and the environments. This paper compares the merits of ically show the which-won-where patterns of the data, two types of GGE biplots in MET data analysis. The first type is as first described in Yan et al. (2000). Briefly, markers constructed by the least squares solution of the sites regression model of the cultivars furthest from the plot origin (0,0) are (SREG2 ), with the first two principal components as the primary and connected with straight lines to form a polygon such secondary effects, respectively. The second type is constructed by Man- that markers of all other cultivars are contained in the del’s solution for sites regression as the primary effect and the first polygon. To each side of the polygon, a perpendicular principal component extracted from the regression residual as the line, starting from the origin of the biplot, is drawn secondary effect (SREGM11 ). Results indicate that both the SREG2 and extended beyond the polygon so that the biplot is biplot and the SREGM11 biplot are equally effective in displaying the divided into several sectors and the markers of the test “which-won-where” pattern of the MET data, although the SREG2 sites are separated into different sectors. The cultivar biplot explains slightly more GGE variation. The SREGM11 biplot is more desirable, however, in that it always explicitly indicates the at the vertex for each sector is the best performer at average yield and stability of the genotypes and the discriminating sites included in that sector, provided that the GGE is ability and representativeness of the test environments. sufficiently approximated by PC1 and PC2. Thus, groups of sites that share the same best performers are graphically identified. Multienvironment trials are conducted for all ma- If the which-won-where patterns identified by a biplot jor crops throughout the world. The main pur- are repeatable over years, different mega-environments pose of MET is to identify superior cultivars for recom- (subregions) can be defined. By selecting superior cultimendation to farmers and to identify sites that best vars for each mega-environment, both G and GE can represent the target environment. Usually, a large num- be effectively exploited. The GGE biplot is still useful ber of genotypes are tested over a number of sites and even in cases where the which-won-where patterns are years, and it is often difficult to determine the pattern not repeatable over years, which suggests that the tested of genotypic responses across environments without the environments belong to a single mega-environment. It help of graphical display of the data. can be used to identify superior cultivars and test enviYan et al. (2000) developed a “GGE biplot” method- ronments that facilitate identification of such cultivars, ology for graphical analysis of MET data. “GGE” refers provided that the target mega-environment is suffito the genotype main effect (G) plus the genotype 3 ciently sampled and that the genotype PC1 scores have environment interaction (GE), which are the two near-perfect correlation (say, r . 0.95) with the genosources of variation that are relevant to cultivar evalua- type main effects. Ideal cultivars should have large PC1 tion. A biplot (Gabriel, 1971) is a plot that simultane- scores (higher average yield) and near zero PC2 scores ously displays both the genotypes and the environments (more stable). Similarly, ideal test environments should (or in more general terms, both the row and the column have large PC1 scores (more discriminating of the cultifactors). The GGE biplot is a biplot that displays the vars) and near zero PC2 scores (more representative of GGE of MET data. It is constructed by plotting the an average environment). (Note that a “test environment” refers to a year-site combination; it does not first two principal components (PC1 and PC2, also referred to as primary and secondary effects, respectively) necessarily correspond to a “test site”.) Thus, the GGE derived from singular value decomposition (SVD) of biplot allows many important questions to be addressed effectively and graphically. the environment-centered data. Models that decompose However, the requirement for a near-perfect correla- the environment-centered data are commonly referred tion between genotype PC1 scores and genotype main to as sites regression models or SREG, and SREG with effects is not always met, which restricts to the utility two PCs is referred to as SREG2. SREG can be used of the SREG2 based GGE biplot. Analysis of the yearly MET data of the Ontario winter wheat performance W. Yan and L.A. Hunt, Dep. of Plant Agriculture, Univ. of Guelph, trials during 1989-1999, and of winter wheat perforGuelph, Ontario, Canada N1G 2W1; P.L. Cornelius, Dep. of Agronomy and Dep. of Statistics, Univ. of Kentucky, Lexington, KY 40546- 0091; Jose Crossa, Biometrics and Statistics Unit, International Maize Abbreviations: G, genotypic main effect; GE, genotype 3 environand Wheat Improvement Center (CIMMYT), Lisboa 27, Apdo. Postal ment interaction; GGE, Genotype main effects plus genotype 3 environment interaction; E, environment main effect; SREGM11 6-641, 06600 Mexico D.F., Mexico. Received 14 Feb. 2000. *Corre- , Mandel’s sponding author (wyan@uoguelph.ca). sites regression model with one additional multiplicative term; PC, principle component; SREG2, Sites regression model with two multiPublished in Crop Sci. 41:656–663 (2001). plicative terms; SVD, singular value decomposition. 656

YAN ET AL.: BIPLOT ANALYSIS OF MULTI-ENVIRONMENT TRIAL DATA 657 mance trials from several states of the USA (Yan, un- max(j* in) 2 min(j* in) 5 max(h* jn) 2 min(h* jn), published) indicates that the genotype PC1 scores are i.e., usually highly correlated with the genotype main effect. lAn n (max(jin) 2 min(jin) 5 l12An Poor correlations between genotype PC1 scores and ge- n (max(hjn) 2 (hjn)). notype main effects, however, do occur for some years. Thus, Moreover, when multiple years of data are analyzed together, this becomes a norm rather than an exception because of large and complex GE interaction (discussed An 5 0.551 1 1n1 max(hjn) 2 min(hjn) max(jin) 2 min(jin)2 1nln 6. [2] later). In such cases, the genotype PC1 scores cannot be interpreted as representing the same information as the genotype main effects. Consequently, the yielding ability and stability of the genotypes, and the discrimi- The SREGM11 Biplot nating ability and the representativeness of the test envi- Mandel (1961) presented the following model for analysis ronments cannot be readily visualized. of non-additivity of two-way data: To avoid these possible exceptions, in this paper we report an alternative GGE biplot, which is constructed Yij 5 bj 1 bjai 1 εij [3] by Mandel’s sites regression on genotype main effects where Yij and bj are the same as in Eq. [1], ai is the main as the primary effect and the first principal component effect of Genotype i, and bj is the regression coefficient of the derived from subjecting that residual to SVD as the environment centered yields (i.e., Yij 2 bj) within Environ- secondary effect. Such a GGE biplot is referred to as ment j on the genotype main effects (ai). Equation [3] is similar a SREGM1 to the well-known model of Finlay and Wilkinson (1963), but 1 biplot, with the subscript "M" referring to Mandel’s solution. In a SREGM1 the roles of cultivars and sites are exchanged. 1 biplot, the primary effects are the genotype main effects per se; it is, there- If the first principal component (l1ji1hj1) from SVD of the residual from Eq. [3], i.e., (Yij 2 bj 2 bjai), is added, then fore, free from the problem discussed above for the SREG2 biplot. However, it is not clear if a SREGM11 Yij 5 bj 1 bjai 1 l1ji1hj1 1 εij or biplot is as effective as the SREG2 biplot in explaining Yij 2 bj 5 bjai 1 l1ji1hj1 1 εij [4] the GGE and in displaying the which-won-where patterns of the data. This study was initiated to answer where all terms are the same as defined in Eq. [1] or [3]. To these questions by comparing the SREG construct a SREGM11 biplot, Eq. [4] is written as 2 biplot and the SREGM11 biplot applied to several datasets that showed Yij 2 bj 5 b* ja* i 1 j* i1h* j1 1 εij [5] different relations between genotype PC1 scores of with j* i1 5 lA1 1 jil ,h* j1 5 l12A1 1 hj1,b* j 5 Bbj , and a* i 5 B21 ai , where SREG2 and the genotype main effects. A1 is defined by Eq. [2], and MATERIALS AND METHODS B 5 ! max(ai) 2 (min(ai) max(bj) 2 min(bj) . [6] The SREG2 Biplot The SREG2 based GGE biplot is derived from Eq. [1] A1 and B are chosen such that the plot space used by genotypes are the same as that by environments. Analogous to PC1 and Yij 2 bj 5 o 2 n51 lnjinhjn 1 εij 5 o 2 n51 j* inh* jn 1 εij [1] PC2 in the SREG2 model, b* ja* i and j* j1h* i1 are referred to as the primary and secondary effects, respectively. All analyses were where Yij is the average yield of Genotype i in Environment conducted using SAS (SAS Institute, 1996). j, bj is the average yield of all genotypes in Environment j, ln is the singular value for principal component PCn, jin and h The Data jn are scores for Genotype i and Environment j on PCn, respectively, and εij is the residual associated with Genotype The data used in this study were from the 1989 to 1999 i in Environment j. The values of ln, jin, and hjn are simultane- Ontario winter wheat performance trials (Yan, 1999). Each ously obtained by subjecting the environment-centered yield year, 10 to 33 winter wheat (Triticum aestivum L.) cultivars (i.e., Yij2bj) to SVD. This can be achieved by principal compo- are tested with four to six replicates in seven to 14 sites repre- nent analysis of the environment-centered yield using the SAS senting the Ontario winter wheat growing areas. Previous anal- procedure PRINCOMP. The PRINCOMP generates jin as ysis indicated that the yearly variance components due to the genotype scores and (ln jin ) as the environment scores. environment (E) dominated the total yield variation, ranging Alternatively, ln, jin and hjn can be obtained by the SVD from 55 to 91% and averaging 80% of the total variance. function within the SAS procedure IML, which is a basic The variance component due to G ranged from 1.8 to 28.5%, function in many SAS procedures related to principal compo- whereas that due to GE ranged from 7.3 to 15.1% (Yan, 1999). nent analysis. A SAS program for principal component analy- G ranged from 13 to 65% of the total GGE. Analysis with sis of MET data is available from the senior author of this the SREG2 biplot revealed that in all years except 1995 the paper. environmental PC1 scores were of the same sign; and in all To display results of fitting Eq. [1] in a biplot, the singular years except 1995 and 1996 the genotype PC1 scores showed value ln has to be absorbed by the singular vector for cul- high correlation with the mean yield of the genotypes (r . tivars hjn and that for environments jin. That is, j* in 5 lAn n jin and 0.93). Thus, in this study the 1995, 1996, and 1998 datasets, h* jn 5 l12An n hjn . An is chosen such that the range of the environ- representing different types of relations between genotype ment markers is equal to the range of the cultivar markers: PC1 versus genotype main effects, were chosen to compare

658 CROP SCIENCE, VOL. 41, MAY–JUNE 2001 Table 1. Proportions of GGE SS explained by SREG2 and SREGM11 for 12 datasets from the 1989–1999 Ontario winter wheat performance trials. % of GGE explained SREG2 SREGM11 No. of No. of Degrees of Year cultivars sites freedom PC1 PC2 Total Primary Secondary Total 1989 10 9 32 42.5 21.3 63.8 40.7 21.9 62.6 1990 10 7 28 59.7 21.2 80.9 53.5 25.1 78.6 1991 10 9 32 53.3 20.7 74.0 49.1 22.1 71.2 1992 10 10 34 57.0 19.9 76.9 56.4 20.1 76.5 1993 18 9 48 56.8 20.0 76.8 55.4 21.2 76.6 1994 14 11 44 45.6 16.2 61.8 41.6 16.8 58.4 1995 14 14 50 54.2 13.4 67.6 40.8 25.2 66.0 1996 23 9 56 29.6 24.5 54.1 26.7 25.3 52.0 1997 28 8 66 55.0 15.9 70.9 54.0 15.9 69.9 1998 33 8 76 71.5 14.7 86.2 71.0 15.2 86.2 1999 31 9 74 51.5 17.4 68.9 50.7 17.7 68.4 1996–99 11 34 84 24.5 22.7 47.2 23.0 23.9 46.9 Average – – – 50.1 19.0 69.1 46.9 20.9 67.8 the GGE biplot based on SREGM11 with one based on SREG2. dicular to the sides of the polygon are drawn to, and In addition, a complete subset of 11 cultivars by 34 environ- extended beyond, each side of the polygon dividing the ments (year-site combinations) extracted from the 1996 to plot into several sectors; each site will fall into one of 1999 trials was also used in the comparison. the sectors (note that only perpendiculars relevant to discussion were drawn). Assuming that the biplot suffi- RESULTS ciently approximates the variation of GGE, it can be For all datasets, both SREG2 and SREGM11 use the mathematically proven that all sites in the same sector same number of degrees of freedom [(g1e22)1 (g1e2 share the same winning genotype, which is the vertex 4) or 2(g1e)26, where g is the number of genotypes genotype for that sector (Yan et al., 2000). and e the number of the environments] (Table 1). With In Fig. 1A, the sites fell into three sectors: the winning the same number of degrees of freedom, SREG2 is theo- genotype for sites RN, WE, ID, and NN was Genotype retically the most effective model for explaining the 6; the winning genotype for sites WK, HN, and EA was variation due to GGE, because the first two principal Genotype 9; and the winning genotype for site OA was components are computed to explain the maximum Genotype 29. Note that Genotype 9 was the best perforamount of variation. Nevertheless, SREGM11 explained mer for WK, HN, and EA because markers of these only slightly smaller amounts of GGE. When averaged sites were on Genotype 9’s side of the perpendicular to over 12 datasets, SREG2 explained 69.1%, whereas the line that connects Genotypes 9’s marker and that SREGM11 explained 67.8% of the total GGE (Table of genotype 6. Vertex genotypes without any site in 1). Thus, SREGM11 is nearly as effective as SREG2 in their sectors were not the highest yielding genotypes at explaining the variation of GGE. So the discussion will any site; moreover, they were the poorest genotypes at be focused on whether the SREGM11 biplot displays all or some sites. Genotypes within the polygon, particusimilar which-won-where pattern as the SREG2 biplot. larly those located near the plot origin, were less responsive than the vertex genotypes. It can be appreciated 1998 Data that the supplementary lines on the biplot are critical for visual analysis of the MET data. The PC1 scores of the SREG2 model had near-perfect In addition, a near-perfect correlation between geno- correlation (r 5 0.99) with the genotypic main effects type primary effect scores and the genotype main effects for this dataset. Consequently, the SREG2 biplot and allows both biplots, Fig. 1A, as well as Fig. 1B, to be the SREGM11 biplot look almost exactly alike. They were, therefore, equally effective in displaying the GGE used to evaluate cultivars for their yielding ability and stability and to evaluate environments for their discrimi- information (Fig. 1A and 1B). The GGE biplot is constructed by plotting the pri- nating ability and representiveness. Genotypes 6 and 9 mary effect scores of each genotype (as x gave the highest average yields (largest primary scores) -axis) and each site against their respective secondary effect scores (as and were relatively stable over the sites (small absolute y secondary scores). In contrast, three non-adapted Geno- -axis) such that each genotype and each test site is represented by a “marker.” For visualizing the which- types 27, 28, and 31 yielded poorly at all sites, as indiwon-where pattern, the genotype markers located away cated by their small primary scores (low yielding) and from the plot origin were first visually identified and relatively small secondary scores (relatively stable). The connected with straight lines to form a polygon, within average yield of Cultivars 1 and 20 were below average which the markers of all other genotypes are contained. (primary scores ,0) and highly unstable (large absolute These away-from-origin genotypes, namely 6, 9, 29, 33, secondary values). The biplots show not only the aver- 27, 28, 20, and 2 in Fig. 1A, are called “corner” or age yield of a genotype (the primary effect), but also “vertex” genotypes because they are at the corners of how it was achieved. That is, the biplots also show the the polygon. Next, starting from the origin, lines perpen- yield of a genotype at individual sites. For example

YAN ET AL.: BIPLOT ANALYSIS OF MULTI-ENVIRONMENT TRIAL DATA 659 (i.e., genotype main effects since they were perfectly correlated in this dataset) and the differences among genotypes are in proportion to the primary effect scores of the sites. Thus, a genotype that yielded well at such a site has a large average yield. On the contrary, site OA was neither discriminating (small primary effect score) nor representative (large secondary effect score); and therefore, cultivars had high yield at OA did not necessarily give high average yield over sites. Analysis of multiple year data indicated that OA represented a different mega-environment (eastern Ontario) from the major winter wheat growing regions in Ontario (Yan et al., 2000; Yan, 1999). 1996 Data As with most datasets, the SREG2 biplot (Fig. 2A) for 1996 indicates that all PC1 scores of the sites were of the same sign, which was arbitrarily assigned positive so that the genotype PC1 scores correlated positively with the genotype main effect. However, as mentioned earlier, the correlation between the genotype PC1 scores and the genotype main effects for this dataset was only 0.85. The relatively poor correlation is associated with the fact that the GGE explained by PC1 is only slightly greater than that by PC2 (29.6 vs. 24.5%). The poor correlation prevents the genotype PC1 scores of the SREG2 solution being interpreted as representing the genotype main effect; in fact, it alone is not interpretable in known biological and agricultural terms. In such cases, the utility of a SREG2 biplot is limited to investigation of the which-won-where patterns. Based on Fig. Fig. 1. SREG2 and SREGM11 biplots for the 1998 Ontario winter wheat 2A, Cultivar 1 was the best performer at sites RN, LN, performance trial data. The numbers are different cultivars; the ID, and WE; and Cultivar 2 was the best performer at sites are EA 5 Elora, HN 5 Harriston, ID 5 Inwood, NN 5 Nairn, OA 5 Ottawa, RN 5 Ridgetown, WE 5 Woodslee, WK 5 sites EA, WK, CA, and OA, and nearly the best at HW. Woodstock. The SREGM11 biplot (Fig. 2B) explained slightly less GGE, but revealed the same which-won-where patterns as the SREG2 biplot. It indicates that Cultivar 1 won at Cultivar 6 had the highest average yield because it sites RN, LN, WE, and ID, and Cultivar 2 won at sites yielded the highest at sites RN, WE, ID, and NN, and yielded above average at all other sites. On the other EA, WK, CA, HW, and OA. In addition, the SREGM11 biplot is more interpretable. By definition, the primary hand, the average yield of Cultivar 20 was below aver- effects of the SREGM11 biplot are the cultivar main age, because it yielded below average at sites OA, EA, effects, and its secondary effects are deviations from the HN, WK, and NN, even though it was quite good at main effects of the cultivars. Thus, the SREGM11 biplot RN. A below-average yield is indicated if the virtual explicitly showed that Cultivars 1 and 2 were the highest line from the origin to the marker of a genotype has an yielding cultivars on average, but neither was very sta- obtuse angle with the virtual line from the origin to the ble, as evidenced by their relatively large secondary marker of a test site. Likewise, an above-average yield effects. With respect to the sites, the SREGM11 biplot is indicated by an acute angle. Supplementary lines, indicated that site EA was highly discriminating, but not presented in the biplots, are required to explicitly not representative of the average environment, whereas determine these relationships. WK and RN were both discriminating and represen- With respect to the test sites, RN was most discrimi- tative. nating as indicated by the longest distance between its marker and the origin. However, due to its large second- 1995 Data ary score, cultivar differences observed at RN may not exactly reflect the cultivar differences in average yield The 1995 dataset was the only dataset found during over all sites. Site NN was not the most discriminating, the 1989 to 1999 Ontario winter wheat performance but cultivar differences at NN should be highly consis- trials in which the site PC1 scores of the SREG2 differ tent with those averaged over sites because it had a in sign (Fig. 3A). Among the 14 test sites, four (Sites near-zero secondary effect score. At a site with a near- 4, 6, 7, and 10) had negative PC1 scores, though their zero secondary effect score, the genotypes are essen- absolute values were small. This led to poor a correlatially ranked according to their primary effect scores tion between the cultivar PC1 scores and the cultivar