Abstract
Test environment evaluation has become an increasingly important issue in plant breeding. In the context of indirect selection, a test environment can be characterized by two parameters: the heritability in the test environment and its genetic correlation with the target environment. In the context of GGE biplot analysis, a test environment is similarly characterized by two parameters: its discrimination power and its similarity with other environments. This paper investigates the relationships between GGE biplots based on different data scaling methods and the theory of indirect selection, and introduces a heritabilityadjusted (HA) GGE biplot. We demonstrate that the vector length of an environment in the HAGGE biplot approximates the square root heritability (\( \sqrt H \)) within the environment and that the cosine of the angle between the vectors of two environments approximates the genetic correlation (r) between them. Moreover, projections of vectors of test environments onto that of a target environment approximate values of \( r\sqrt H \), which are proportional to the predicted genetic gain expected in the target environment from indirect selection in the test environments at a constant selection intensity. Thus, the HAGGE biplot graphically displays the relative utility of environments in terms of selection response. Therefore, the HAGGE biplot is the preferred GGE biplot for test environment evaluation. It is also the appropriate GGE biplot for genotype evaluation because it weights information from the different environments proportional to their withinenvironment square root heritability. Approximation of the HAGGE biplot by other types of GGE biplots was discussed.
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Test environment evaluation is an important research area in plant breeding, because appropriate choice of test environments can reduce test cost and improve breeding efficiency. Test environment evaluation has been discussed mainly in the context of indirect selection (Cooper et al. 1996; GuillenPortal et al. 2004). According to Falconer and Mackay (1996), the response (G) observed in target environment j′ due to indirect selection in test environment j is:
where i _{ j } is the selection intensity imposed in the test environment, h _{ j } and h _{j′} are the square roots of the narrowsense heritability (h ^{2}) in the test environment and the target environment, respectively, r _{g(jj′)} is the additive genetic correlation between the test environment and the target environment, σ _{p(j′)} is the square root of the phenotypic variance in the target environment, and σ _{g(j′)} is the square root of the additive genetic variance in the target environment. From Eq. 1, it is clear that the usefulness of the test environment in indirect selection for the target environment has to be evaluated with regard to two aspects: (1) the heritability for the trait of interest in the environment (h ^{2}_{ j } ), and (2) its genetic correlation with the target environment (r _{g(jj′)}). In the terminology of Allen et al. (1978), the proper measure of the value of a test environment is \( r\sqrt H \) where r is the correlation between genotypic performance in the test environment and the target environments and H = h ^{2} is the heritability in the test environment.
In addition to identifying superior genotypes, genotypebyenvironment data from multienvironment trials are also valuable for evaluating the test environments (Cooper et al. 1996; Yan et al. 2007). Yan (2001) proposed that each test environment can be graphically evaluated in a GGE biplot for (1) its power to discriminate the genotypes, measured by its vector length in the biplot, and (2) its representativeness of other test environments, measured by its angle with the “average” environment. Intuitively, these two aspects should correspond to the heritability and the genetic correlation in the indirect selection theory but a clear connection between the two concepts has not been established.
A GGE biplot is a biplot (Gabriel 1971) based on environmentcentered data, which removes the environment main effect and integrates the genotypic main effect with the genotypebyenvironment interaction effect of a genotypebyenvironment dataset (Yan et al. 2000). For a given dataset, different GGE biplots can be generated, depending on how the data are scaled prior to singular value decomposition. The most commonly used data scaling methods include (1) no scaling, (2) scaling by the standard deviation of genotype means within environments (SDscaled), and (3) scaling by the standard error within environments (SEscaled) (Yan et al. 2000). The SEscaled GGE biplot has been identified as most appropriate for genotype and test environment evaluation because it accounts for heterogeneity among environments in experimental errors (Yan et al. 2000; Yan and Kang 2003; Blanche and Myers 2006). However, often a GGE biplot is used in genotype and test environment evaluation without mentioning the scaling method, which may lead to misinterpretation of the data. This occurs mainly due to the lack of understanding of the links between biplot interpretation and the indirect selection theory.
The objectives of this paper were to (1) establish mathematical connections between key parameters in the indirect selection theory and GGE biplot analysis; (2) introduce a heritabilityadjusted (HA) GGE biplot, in which the expected vector length of the test environments is the square root heritability in the environments, and (3) demonstrate the use of this HAGGE biplot in test environment evaluation and genotype evaluation using an oat (Avena sativa L.) multienvironment trial dataset.
Theory development
Common statistics characterizing the test environment
In a multienvironment trial framework, each test environment is typically characterized by the following statistical parameters relative to the trait of interest:

The mean, i.e., average value across genotypes in the environment. It is the environment main effect, which is not pertinent to genotype evaluation. It is not pertinent to test environment evaluation either because it does not affect the relative differences among genotypes.

The standard error (SE), which is the square root of the error variance in the environment (SE = σ _{ e }), which can be reduced by various methods, including improved experimental design, improved experimental execution, and improved data analyses (Casanoves et al. 2005; Gilmour et al. 1997; Smith et al. 2002).

The standard deviation of genotype means (SD), which equals the square root of the phenotypic variance within the environment (σ ^{2}_{ p } ), with
where σ ^{2}_{ g } is the genotypic variance and n is the number of replicates within the environment.

The coefficient of variation (CV), which is the ratio of standard error over the environmental mean: CV% = (SE/Mean) × 100%. CV is sometimes used as a measure of validity of the trial and a criterion to exclude trials with low precision, although such use has been criticized (Bowman and Watson 1997; Taylor et al. 1999).

The heritability in the broad sense (H), which is calculated as
H is an indication of the validity or usefulness of the trial in genotype evaluation. H = 1 means that the observed differences in genotypic means in the trial are entirely due to genetic effects; H = 0 indicates that the observed differences are completely due to random error.
Basic types of GGE biplots
The general model for GGE biplots is:
where p _{ ij } represents the genotypebyenvironment twoway table of GGE effects with i = 1,…,m genotypes and j = 1,…,e environments, which is decomposed into k = 1 to t principal components (PC), with t ≤ min(e, m − 1). \( \bar{y}_{ij} \) is the cell mean of genotype i in environment j; μ _{ j } is the mean value in environment j. The operation \( (\bar{y}_{ij}  \mu_{j} ) \) is referred to as “environmentcentering”, which results in removal of environment main effects (E) from the original data. The model is subject to the constraint \( \lambda_{1} \ge \lambda_{2} \ge \cdots \ge \lambda_{t} \ge 0 \) and to orthonormality on the α _{ ik } scores, i.e., \( \sum\nolimits_{i = 1}^{m} {\alpha_{ik} \alpha_{ik'} } = 1 \) if k = k′ and \( \sum\nolimits_{i = 1}^{m} {\alpha_{ik} \alpha_{ik'} } = \, 0 \) if \( k \ne k' \), with similar constraints on the γ _{ jk } scores [defined by replacing symbols (i, m, α) with (j, e, γ)]. The \( \bar{\varepsilon }_{ij} \) within an environment are assumed \( NID(0,\;\sigma^{2} /n) \), where σ ^{2} is the experimental error variance and n is the number of replicates within an environment. A twodimensional GGE biplot is constructed using the first two principal components (PC1 and PC2), thus t = 2 (see Yan et al. 2007 for more detailed descriptions).
Of particular relevance to this study is the parameter s _{ j }, which is referred to as the scaling factor, and the operation of division of the data matrix by s _{ j } is referred to as “data scaling”. For a given genotypebyenvironment twoway table, different GGE biplots can be generated, depending on how s _{ j } is defined (Yan and Kang 2003; Yan and Tinker 2006).
Unscaled GGE biplot
It is called unscaled GGE biplot if
SEscaled GGE biplot
It is called SEscaled GGE biplot if
where SE _{ j } is the standard error within environment j. As a variant of the SEscaled GGE biplot, it is called Pairwise SEscaled GGE biplot if
which is the standard error for pairwise genotype comparisons in an environment. It is called SEMscaled GGE biplot if
which is the standard error of means (SEM) in an environment. A further variant of the SEscaled GGE biplot is to scale the data with the least significant difference (LSD) within each environment:
SDscaled GGE biplot
The biplot is referred to as SDscaled GGE biplot if
where SD _{ j } is the standard deviation of the distribution of genotype means within environment j.
Interpretation of the environmental vector length in different GGE biplots
For an unscaled GGE biplot, the vector length of environment j is
When the data are environmentcentered such that the environment means \( \bar{y}_{j} \) become 0, the SD in an environment is
From Eqs. 8 and 9, it can be seen that when the data are environmentcentered but not scaled (Eq. 5), the vector length of an environment is \( \sqrt {m  1} \) times the standard deviation of genotype means within the environment (Kroonenberg 1995; Yan and Tinker 2006):
Considering the relationship between SD and H (Eq. 3), Eq. 10 can be written as
Thus, the vector length of an environment (L _{ j }) in the unscaled GGE biplot (Eq. 5) is proportional to the phenotypic variation among genotypes (SD _{ j }, Eq. 2), which is positively associated with both the experimental error (SE _{ j }) and the heritability (H _{ j }) in the environment. The number of replicates n is also a factor to affect the vector length of the environment.
In the SEscaled GGE biplot (Eq. 6), the vector length of environment j is determined solely by the heritability in the environment, though in a curvilinear fashion, given the same number of genotypes m and the same number of replications n:
The factor n can be removed from this relationship if the data are scaled by the pairwise standard error (Eq. 6a) because:
Similarly, if the data are scaled by the SEM (Eq. 6b), then:
Equations 12a and 12b provide a clear connection between the interpretation of the vector length of test environments in the GGE biplot (Eq. 8) and the heritability (H) in the theory of indirect selection (Eq. 2). That is, when all environments have the same number of genotypes (m), the vector length of the environments in the pairwise SE or SEMscaled GGE biplot is solely determined by, and therefore represents, the heritability or repeatability of genotype means in the environments, assuming a perfect fit of the GGE biplot. However, the relationship between the vector length and H is curvilinear rather than linear. When H is small, say less than about 0.75, the relationship appears approximately linear, but when H is large, say above about 0.95, the discriminatory power of that environment will be overstated in the GGE biplot (data not shown).
In the SDscaled GGE biplot (Eq. 7), the vector length of all environments is a constant if the same set of genotypes are tested in all environments, since
Therefore, unlike the unscaled or SEscaled GGE biplots, the vector length of the environments in the SDscaled GGE biplot is not a measure of the discriminating power of the environments. Rather, all environments are expected to have the same or similar vector length if the GGE biplot adequately approximates the SDscaled genotypebyenvironment data. For the same reason, if some environments have considerately shorter vectors than others, it indicates that the SDscaled GGE biplot does not adequately display the patterns regarding these environments. One consequence of this is that the correlations between environments with shorter vectors and other environments may not be correctly displayed by the angles between them (for an example see Yan and FrégeauReid 2008).
The heritabilityadjusted GGE biplot
Based on Eq. 3, we have
Therefore, scaling by SEM is equivalent to scaling by \( SD\sqrt {1  H} \), which leads to an expected vector length of the environment proportional to \( \sqrt {1  H} \) (Eq. 12b). If the data are scaled by the factor \( SD/\sqrt H \), which is equivalent to multiplying \( \sqrt H \) in each environment to the SDscaled data (standardized data), the expected vector length of the environments in the resulting GGE biplot would be:
Verbally, the vector length of a test environments will be proportional to the square root of the heritability in the environment, \( \sqrt H \). Such a biplot will be referred to as heritabilityadjusted (HA) GGE biplot. Equation 14 establishes a strictly linear relationship between \( \sqrt H\) and the expected vector length in the HAGGE biplot.
Interpretation of the angle between two environment vectors in different GGE biplot
When the GGE biplot has a perfect goodness of fit, the cosine of the angle (α _{jj′}) between two environments is the genetic correlation between them (Gabriel 1971; Kroonenberg 1995; Yan and Tinker 2006):
This relationship is not altered by data scaling, as scaling with a positive value does not alter the relative differences among genotypes within an environment. Note that in the GGE biplot the target environment j′ can be an individual test environment or the “average” test environment (Yan 2001).
Test environment evaluation based on heritabilityadjusted GGE biplot
In a HAGGE biplot, since the cosine of the angle between the vectors of two test environments (a test environment and a target environment) approximates the genetic correlation between them (Eq. 15), and since the vector length of the test environments are proportional to their \( \sqrt H \) (Eq. 14), the projection of test environment onto the target environment should approximate \( r\sqrt H \), which is an overall measure of the usefulness of a test environment (Allen et al. 1978). Thus, a HAGGE biplot graphically displays not only r and \( \sqrt H \) but also \( r\sqrt H \). This relationship is graphically illustrated in Fig. 1.
Since the genetic gain observed in target environment j′ from indirect selection practiced in environment j is given in Eq. 1 as:
the values of h _{ j } r _{g(jj′)} for different test environments (j) are directly proportional to the predicted indirect genetic gains in the target environment from selection in the test environments at a constant selection intensity, because σ _{g(j′)} is a constant for a given target environment. \( r\sqrt H \) is a valid approximation of h _{ j } r _{g(jj′)} if most genetic variation and covariation is additive, as expected for inbred lines. This may not be true for clonally propagated or fullsib families in species where dominance variation is an important component of genetic variance. In cases of predominantly additive genetic variation, however, projections of test environment vectors onto target environment vectors in a HAGGE biplot provide a means to compare the relative utility of test environments for indirect selection for a common target environment.
Case study
A case study is presented here to demonstrate the usefulness of the HAGGE biplot in genotypebyenvironment data analysis, in comparison with the commonly used GGE biplot based on unscaled data or standardized data. The focus here is on test environment evaluation; however, as an integral part of genotypebyenvironment data analysis, genotype evaluation based on the HAGGE biplot will also be demonstrated.
Materials and methods
The data used in this case study are grain yield data of 27 covered spring oat lines and nine check cultivars tested at nine locations across eastern Canada in 2006. The locations were Bornholm (ON1), Nairn (ON2), Ottawa (ON3) and New Liskeard (ON4) in Ontario; Hebertville (QC1), Normandin (QC2), Princeville (QC3), and Amqui (QC4) in Quebec; Charlottetown in Prince Edward Island (PEI); and Hartland in New Brunswick (NB). At each location, the experimental design was a randomized complete block design. The number of replicates was four at the Ontario locations and three at other sites except QC4, where the test was not replicated (Table 1). All 27 experimental lines were tested at all sites, but the check varieties used in Ontario, Quebec, and the Maritimes (which includes New Brunswick and Prince Edward Island) were different, and no check varieties were included at ON4 and NB. The ANOVA table and basic statistics for each test environment are presented in Table 1. ANOVA was conducted based on the general linear model for randomized complete blocks and the significance of various effects was based on F test against the experimental errors within environments.
Twodimensional unscaled GGE biplot (Fig. 2), HAGGE biplot (Fig. 3), and SDscaled GGE biplot (Fig. 4) were generated using the GGEbiplot software (Yan 2001). Only the 27 breeding lines, which were tested in all environments (i.e., sites), were included in the biplots because biplots require a balanced genotypebyenvironment table. All biplots in Figs. 2, 3, 4 and 5 were based on environmentfocused singular value partitioning (“SVP = 2”, Yan 2002) so that they are appropriate for test environment evaluation. An environmentbyparameter biplot (Fig. 6) was also constructed to help visualize the interrelationships between basic statistics characterizing the test environments and the vector length of the environments from the various GGE biplots. Test environment evaluation and genotype evaluation are meaningful only within a megaenvironment (Yan et al. 2007). Two HAGGE biplots were constructed for the Quebec and Maritime sites to illustrate their application in test environment evaluation (Fig. 7) and genotype evaluation (Fig. 8) within a megaenvironment.
Interpretation of the environmental vector length in different GGE biplots
The vector lengths of the test environments in the unscaled GGE biplot (Fig. 2) approximate the phenotypic variances within the environments (Eq. 11), as demonstrated by the close positioning of “SD” and “LUS” in the environmentbyparameter biplot (Fig. 6). QC3 exhibited the greatest phenotypic variance, but it also had the highest experimental error and CV (Table 1; Figs. 2, 6).
The vector length of the environments in the HAGGE biplot (Fig. 3) approximates their square root heritabilities (Equation [12b]), as is confirmed graphically in Fig. 6, where the vector length in the HAGGE biplot (“LHA”) and \( \sqrt H \) had a very small angle. Figure 3 shows that ON1 and ON2 were most discriminating (with the highest \( \sqrt H \)), followed by QC1 and QC2. ON3 and ON4 were considerably less discriminating. The difference in vector length among environments was not manifest in the SEscaled GGE biplot (Fig. 5), with clearly long vectors for ON1, ON2, QC1, and QC4 and shorter vectors for others. The correlation between environmental vector length and \( \sqrt H \) for the SEscaled GGE biplot was even slightly higher than that for the HAGGE biplot (Fig. 6). According to Fig. 6, the vector length of the environments in the SDscaled GGE biplot were also, quite unexpectedly, highly correlated with \( \sqrt H \). Examination of other datasets (not reported here), however, indicates that this was merely a coincident for this particular dataset.
The unscaled GGE biplot is an adequate approximation of the HAGGE biplot only when the SD and H are similar among environments. For this particular dataset, the phenotypic variance (SD) and the square root heritability \( \sqrt H \) were only loosely correlated as indicated by the large angle between them (Fig. 6), suggesting that the unscaled GGE biplot was not a good substitution for test environment evaluation and genotype evaluation.
The vector length of the environments in the SDscaled GGE biplot (Fig. 4) was similar for all environments except ON3 and ON4. This is not an indication that ON3 and ON4 were less discriminating; rather, it indicates that patterns related to these two environments were not fully displayed in the biplot, which indicates that these two environments were less associated with other environments than the angles in the biplot would suggest (Yan and FrégeauReid 2008).
A close positive association between CV and “SE” and a loose negative association between CV and \( \sqrt H \) were observed (Fig. 6). This indicates that CV was a poor representation of H. Therefore, CV is not a good measure of the validity of the trials and should not be used in judging the validity of trials in multienvironment data analyses (Bowman and Watson 1997; Taylor et al. 1999).
Interpretation of the angles between environments in different GGE biplots
According to Eq. 15, the cosine of the angle between two environments in the GGE biplot approximates the genotypic correlation between them, regardless of the data scaling methods. Although there are no strict relations, the goodness of approximation for the correlation coefficients by the angles is related to the goodness of fit of the biplot. The unscaled GGE biplot (Fig. 2) reveals two groups of environments: the four Ontario environments as one group and the Quebec and Maritime environments as the other. Between groups, the environments were unassociated (right angles) or negatively associated (obtuse angles). Within each group, the environments were more or less positively correlated (acute angles). Within the Quebec and Maritime group, QC3 were apparently less associated with other sites. The same statements hold for the HA (Fig. 3), the SDscaled (Fig. 4), and the SEscaled (Fig. 5) GGE biplots. However, the environment grouping is most clear in the SDscaled biplot due to similar vector length among environments. This property makes the SDscaled GGE biplot the preferred biplot if the main purpose is to investigate the associations among test environments. For simultaneous evaluation of both the discrimination power and the representativeness of the test environments, however, the HAGGE biplot is most appropriate. Note that the angles among the test environments are not identical in the four GGE biplots (Figs. 2, 3, 4, 5), this is probably due to the fact that the biplots differ in goodness of fit. For the current dataset, the goodness of fit was 66.6% for the unscaled GGE biplot, 57.8% for the HAscaled GGE biplot, 55.6% for the SDscaled GGE biplot, and 72.2% for the SEscaled GGE biplot. The interpretation of the angle between the vectors of two environments is also true for the biplot generated from Factor Analytic models (Smith et al. 2002), which may be regarded as the random effect version of the GGE biplot.
Interpretation of the angle between a test environment and the average environment
From the indirect selection theory (Eq. 1) or the formula of Allen et al. (1978), it is clear that genetic correlation (r) and the square root heritability (\( \sqrt H \)) must be considered simultaneously in assessing the usefulness of a test environment. It is a desirable feature of the HAGGE biplot to graphically and simultaneously display both factors, both separately and jointly (Fig. 1). A test environment is not useful if its \( \sqrt H \) is very low or its genetic correlation with the target environment is small or negative. Test environment evaluation in the context of multienvironment variety trials is different in that the target environment is not a single, well defined environment. Rather, it is a population of environments that is represented (presumably) by all the test environments. To represent the target environment, Yan (2001) defined a virtual “average environment”, which is the point on the biplot that has the average coordinates of all test environments (the small circle, Fig. 7). Smith et al. (2002) similarly defined an “average environment” as a reference for selecting representative test sites based on a Factor Analytic plot of test environments.
Yan et al. (2007) proposed that the angle between a test environment and the average environment in the GGE biplot (the line that passes through the biplot origin and the average environment) is a measure of its representativeness of the target environment. Based on the discrimination power (vector length) and the correlation with the average environment, test environments can be classified into four types: (1) nondiscriminating and therefore useless; (2) discriminating and representative, ideal for selecting superior genotypes; (3) discriminating but nonrepresentative, useful for culling unstable genotypes; and (4) discriminating, but negatively correlated with the average environment, which are misleading if used in genotype selection. This classification of test environments is meaningful only when all test environments belong to a common target megaenvironment (Yan et al. 2007). When strongly negative correlations exist among test environments, one should investigate whether the target environments can be divided into meaningful megaenvironments.
Test environment evaluation based on the HAGGE biplots
Figure 7 was constructed for the Quebec and Maritime subregion to show how test environments can be evaluated based on a HAGGEbiplot. The singlearrowed line passes through the biplot origin and the “average” environment. It represents the target environment and may be referred to as the Target Environment Axis (TEA). The projection of each test environments onto TEA approximate its \( r\sqrt H \), and are a measure of its usefulness in selecting superior genotypes for the megaenvironment (Fig. 1). Thus, the usefulness of the five environments can be ranked as: QC1 > QC2 > NB > PEI > QC3 (Fig. 7). The line with two arrows points away from representativeness, regardless of directions. Thus QC3 was the least representative environment; such an environment is not good for selecting superior genotypes but can be useful for culling unstable genotypes, if it is sufficiently discriminating.
Genotype evaluation based on the HAGGE biplots
Although the focus of this study is on test environment evaluation, the ultimate goal is to improve the efficiency of genotype evaluation. The HAGGE biplot is the most appropriate for genotype evaluation as well as for test environment evaluation. However, the form of GGE biplot in Fig. 7 is not the best for genotype evaluation because it is based on environmentfocused singular value partitioning (“SVP = 2”), which is appropriate only for evaluating test environments. Genotypefocused singular value partitioning (“SVP = 1”, Yan 2002) is required for the latter purpose (Fig. 8). For the Quebec and Maritime region, the best genotypes were “10638” and “11301” (Fig. 8). While 10638 was stable across environments, 11301 was highly variable. It is interesting to mention that 10638 was registered as “Dieter” for its excellent performance across years for Quebec. “11301” continued to perform well in 2007 and 2008 in Northern Quebec represented by QC1 ad QC2.
Discussion
Multiyear data are essential for test site evaluation
In the case study, a single year multisite test was used for the purpose of demonstration. However, multiyear data are critical for delineating megaenvironments and in selecting test locations. In practice, multiyear data are rarely balanced due to the change of genotypes and locations each year. DeLacy et al. (1996) proposed four strategies in dealing with this problem. The most convenient strategy appears to be one in which multisite data are analyzed yearly and summarized across years. Only patterns that are consistent cross years can be used in delineating megaenvironments, selecting superior test sites, and discarding noninformative and redundant test sites.
Approximation of the HAGGE biplot by other types of GGE biplots
Prior to the development of the HAGGE biplot, three types of GGE biplots were widely used: the unscaled GGE biplot, the SDscaled GGE biplot, and the SEscaled GGE biplot. Given the conclusion that HAGGE biplot is the most desired type of GGE biplot for test environment and genotype evaluation, it is relevant to discuss its relationships with the other types of GGE biplot. From the theory development section, it is clear that the SDscaled GGE biplot is identical to the HAGGE biplot if all environments have the same heritability H. The unscaled GGE biplot is identical to the HAGGE biplot if all environments have the same SD and H. Therefore, the SDscaled GGE biplot has more chances to be a better approximation of the HAGGE biplot than the unscaled GGE biplot (compare Figs. 2, 3, 4, 5). According to Eq. 3, scaling by SE is equivalent to scaling by \( SD\sqrt {n(1  H)} \) or its variants. Thus the SEscaled GGE biplot uses the same information as the HAGGE biplot but in a different way. The consequence is that while there is a linear relationship between the expected environment vector length and \( \sqrt H \) in the HAGGE biplot, this relationship for the SEscaled GGE biplot is curvilinear. If the H values of the environments are within a reasonable range, the difference between the two types of GGE biplot should be quite small. In fact, the empirical correlation between the environmental vector length and the \( \sqrt H \) was higher for the SEscaled GGE biplot than for the HAGGE biplot for the current dataset and two other datasets not reported here, an observation we don’t fully understand. Therefore, although the HAGGE biplot has a more direct link to the theory of indirect selection, conclusions derived from SEscaled GGE biplot, such as those reported by Blanche and Myers (2006), should remain largely relevant.
GGE biplot versus scatter plot
Gauch et al. (2008) criticized the use of GGE biplot in test environment evaluation and genotype evaluation as described in Yan et al. (2007). For test environment evaluation, they presented a scatter plot of discriminating power versus representativeness and argued that the scatter plot was superior to the GGE biplot approach because the values plotted were exact values rather than approximations and the process was simpler. Similarly, for genotype evaluation, Gauch et al. (2008) presented a scatter plot of mean versus stability and argued against the GGE biplot approach. We agree that scatter plot approach are valid alternatives of the GGE biplot approach for test environment evaluation and genotype evaluation, we also agree that GGE biplots are approximate displays of the data while the scatter plots display some “exact” summaries of the data. However, GGE biplots convey information that cannot be gleaned from these scatter plots due to their simultaneous display of both genotypes and environments and their innerproduct property (Gabriel 1971). GGE biplots not only display the quantities that are used in test environment and genotype evaluation but also preserves the original information from which the quantities are derived. For example, Fig. 6 demonstrates that QC3 was a poor test environment; this was partially due to the low yield of genotypes 11301 and 11683 at that site. Similarly, Fig. 8 shows that 11301 was a high yielding but unstable genotype resulting from its good adaptation to northern Quebec (QC1 and QC2) but poor adaptation to central Quebec (QC3). Such information is not conveyed in the scatter plots.
Single quantity versus two quantities for test environment evaluation
Gauch et al. (2008) also criticized the GGE biplot approach to test environment evaluation by citing Cooper et al. (2006): “The optimal site maximizes the phenotypic correlation between entry yields on the site and entry yields over the target environments (as represented by the test sites for a given megaenvironment) because the phenotypic correlation includes the genetic correlation and the broadsense heritability of the sites.” The HAGGE biplot proposed in this paper accomplishes precisely this (Figs. 1, 7) because the phenotypic correlation between environments j and j′ is \( r_{p(jj')} = h_{j} h_{j'} r_{g(jj')} \) (Cooper and DeLacy 1994). The HAGGE biplot displays h _{ j } and h _{j′} as the lengths of the two environment vectors and r _{g(jj′)} by the cosine of the angle between the two vectors. This is additional to the fact that the HAGGE biplot (Fig. 7) graphically displays r, \( \sqrt H \), and \( r\sqrt H \) for each test environment, which allows test environment evaluation based on r and\( \sqrt H \) both jointly and separately.
Conclusions
We demonstrated both theoretically and empirically that the HAGGE biplot graphically displays the square root heritability of each test environment, \( \sqrt H \), and its genetic correlation with other test environments, r, which are the two key elements for test environment evaluation in the framework of indirect selection theory. Moreover, the HAGGE biplot allows test environments to be ranked graphically based on their \( r\sqrt H \) values with respect to a common target environment. Therefore, the HAGGE biplot appears to be most appropriate of all GGE biplots for visual evaluation of the test environments. This biplot is also most appropriate for genotype evaluation, because it takes into consideration any heterogeneity among environments by giving weights to the test environments proportional to their \( \sqrt H \). The SEscaled GGE biplot is a good approximation of the HAGGE biplot when the H values of the environments are within a reasonably small range; the SDscaled GGE biplot is a good approximation when the environments have the same or similar H values; and the unscaled GGE biplot is a good approximation only when all environments are similar in both SD and H values.
Abbreviations
 H:

Heritability or repeatability of genotypic differences within an environment
 HA:

Heritabilityadjusted
 GGE:

Genotype main effect plus genotypebyenvironment interaction
 SD:

Standard deviation of genotype means within an environment
 SE:

Standard error within an environment
 SEM:

Standard error of means within an environment
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Acknowledgment
We thank Judith FregeauReid, Brad de Haan, Mark Etienne, John Rowsell, Richard Martin, Allan Cummiskey, Peter Scott, Denis Pageau, and Julie Durand for planning and conducting the 2006 oat yield trials at various research stations.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/bync/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Yan, W., Holland, J.B. A heritabilityadjusted GGE biplot for test environment evaluation. Euphytica 171, 355–369 (2010). https://doi.org/10.1007/s1068100900305
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DOI: https://doi.org/10.1007/s1068100900305