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Published online 31 May 2007
Published in Crop Sci 47:1051-1062 (2007)
© 2007 Crop Science Society of America
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CROP BREEDING & GENETICS

Mixed-Model Analysis of Crossover Genotype–Environment Interactions

Rong-Cai Yang*

Alberta Agriculture, Food and Rural Development, no. 300, 7000-113 St., Edmonton, AB, Canada T6H 5T6, and Dep. of Agricultural, Food and Nutritional Science, Univ. of Alberta, Edmonton, AB, Canada T6G 2P5

* Corresponding author (rong-cai.yang{at}ualberta.ca).


   ABSTRACT
TOP
 NOTES
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Genotype–environment interactions (GEI) are important in crop improvement if genotype ranks change across environments. Current tests for crossover (rank changing) interactions (COI) assume that effects are all fixed or all random. The objective of this study was to develop a new test for COI under the model with a mixture of fixed and random genotypic, environmental, and GEI effects. The key part of this new test is that the difference between a pair of genotypes at a random environment or the difference between a pair of environments for a random genotype involves the linear combinations (predictable functions) of both best linear unbiased estimates (BLUEs) of fixed effects and best linear unbiased predictors (BLUPs) of random effects. The predictable functions are used in the same way as the usual estimable functions for the fixed effects in hypothesis testing except that the BLUPs of random effects are adjusted by accounting for the uncertainty arising from the distributions of these effects. Strategies are proposed to implement the procedure using the SAS system. The procedure was used to analyze barley (Hordeum vulgare L.) and field pea (Pisum sativum L.) cultivar trials. The analyses show that treating random effects as fixed, as may happen with previous analysis procedures, results in detection of more COI than mixed- or random-effect models. Therefore, significant COI may be overemphasized when random GEI effects are treated as fixed.

Abbreviations: BLUE, best linear unbiased estimator • BLUP, best linear unbiased predictor • COI, crossover interactions • EBLUE, empirical best linear unbiased estimator • EBLUP, empirical best linear unbiased predictor • GEI, genotype–environment interactions • LR, likelihood ratio • MET, multiple-environment trial • ML, maximum likelihood • RCBD, randomized complete block design • REML, restricted maximum likelihood.


   INTRODUCTION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
THE PRESENCE OF GENOTYPE–ENVIRONMENT interactions (GEI) remains one of the major impediments for crop improvement and production. It has been long recognized (e.g., Haldane, 1947; Gregorius and Namkoong, 1986; Baker, 1988, 1996) that GEI are of consequence in selection programs when genotype ranks change and there is no clear superiority of a single genotype across environments in a multiple-environment trial (MET). Thus, interactions involving rank changes (crossover GEI) are much more important than those that only reflect differences in scale. There are currently two approaches to detecting crossover interactions (COI), depending on whether a fixed-effect or a random-effect model is used. Treating both genotypes and environments as fixed effects, Baker (1988) and Cornelius et al. (1992) have used the test of Azzalini and Cox (1984) to evaluate all possible 2 x 2 subtables (quadruples) for the presence of COI from a two-way genotype x environment table. A different strategy, also based on the fixed-effect model, is to minimize the presence of COI by clustering genotypes or environments into homogeneous subsets using the criteria derived from the multiplicative model or performance-based analyses (Crossa et al., 2004; Yang et al., 2005; Navabi et al., 2006). The discussion from recent works suggests, however, that either genotypic or environmental effects (and thus GEI effects) should be random (Baker, 1996; Piepho, 1998; Balzarini, 2002; Smith et al., 2005). Whether these effects are fixed or random determines (i) if the focus of the crop improvement work should be on which environments or on how many environments to test in the future (Baker, 1996) or (ii) if the purpose of the breeding and cultivar testing programs is to identify the best cultivars or to detect the difference between a pair of cultivars (Smith et al., 2005).

The second approach to characterizing GEI and detecting COI assumes that both genotypic and environmental effects (and thus GEI effects) are random. In this approach, the nature of GEI can be assessed by partitioning the total GEI variability into two components: (i) change in scale of a trait measured in different environments, that is, heterogeneity of variances, and (ii) imperfect genetic correlation of the same trait across environments (or genotypes) (e.g., Yang and Baker, 1991). When most of the GEI variability is explained by the heterogeneous variances, the GEI is considered to be unimportant since it suggests non-COI; however, when the variability due to imperfect genetic correlations between environments describes a large proportion of the total GEI it may indicate the presence of COI. Since these two components of the GEI sum of squares are not chi-square distributed, the usual ANOVA provides no direct statistical tests of significance. Yang (2002) and Crossa et al. (2004) have subsequently developed a likelihood ratio (LR) test based on the mixed-model theory for the significance of the two components of the GEI variability, thereby assessing the presence of COI. These LR tests, however, require estimating variances and covariances (second-order statistics). Hypothesis testing involving second-order statistics is more sensitive to departures from model assumptions, thereby rendering the LR tests less robust than the statistical test involving means or estimated model effects (first-order statistics). In addition, a significant imperfect genetic correlation between environments does not imply the presence of COI (Crossa et al., 2004).

I have developed a third approach to detecting COI based on the model with a mixture of fixed and random genotypic, environmental, and GEI effects. Currently the Azzalini–Cox test has been applied or developed strictly under the fixed-effect model even though, in many METs including regional cultivar or breeding trials, either genotypes or environments should be considered random (Baker, 1996; Piepho, 1998; Balzarini, 2002; Smith et al., 2005). The key part of this new approach is the recognition that the differences between genotypic effects at a random environment or the differences between environmental effects for a random genotype are the linear combinations of both best linear unbiased estimators (BLUEs) of fixed effects and best linear unbiased predictors (BLUPs) of random effects. These linear combinations are called predictable functions (Henderson, 1984; Stroup, 1989). The use of these predictable functions enables the Azzalini–Cox test to be applied directly to the mixed- and random-effect models. The new mixed-model test for COI is illustrated through the analysis of a barley (Hordeum vulgare L.) MET and a field pea (Pisum sativum L.) MET.


   MATERIALS AND METHODS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Mixed Models for Multiple-Environment Trials
I consider a MET data set where g genotypes are tested in each of e environments. These environments can be a set of sites tested in a given year or a combination of multiple sites tested in multiple years. In each environment, the genotypes can be arranged in a variety of complete and incomplete block designs. For clarity, I consider a randomized complete block design (RCBD) in each environment, but each environment may have a different number of replications or blocks (i.e., rj replications in the jth environment) to accommodate the commonly encountered situation in many breeding and cultivar testing programs (in the special case of the equal number of replications across all environments, rj = r for all j). For this MET, the conventional ANOVA model is given by

Formula 1[1]
where yijk is measured response (i.e., yield) of the kth replication of the ith genotype in the jth environment (i = 1, 2, ..., g; j = 1, 2, ..., e; k = 1, 2, ..., rj), µ is the overall mean, {tau}i is the effect of the ith genotype, {delta}j is the effect of the jth environment, ({tau}{delta})ij is the interaction effect of the ith genotype with the jth environment, {gamma}jk is the effect of kth replication in the jth environment, and {varepsilon}ijk is the random error. Averaging across replications within an environment, Eq. [1] reduces to the model that is commonly used to describe the genotype x environment cell means (e.g., Cornelius and Crossa 1999): Formula 1ij.=µ+{tau}i+{delta}j+({tau}{delta})ij+eij., where eij.={Sigma}k=1rj({gamma}jk+{varepsilon}ijk)/rj .

Three versions of the model in Eq. [1] are considered: (i) fixed-effect model with all effects except {gamma}jk and {varepsilon}ijk being fixed; (ii) random-effect model with all effects except µ being random; (iii) mixed-effect model arising from the situation where either of genotypic and environmental effects is fixed whereas the other is random. If the genotypic effect is fixed and the environmental effect is random, then, from Eq. [1], µ and {tau}i are fixed effects and {delta}j, ({tau}{delta})ij, {gamma}jk, and {varepsilon}ijk are independent and normally distributed random effects that have expectations of zero and variances {sigma}{delta}2, {sigma}{tau}{delta}2, {sigma}{gamma}2, and {sigma}{varepsilon}2, respectively; on the other hand, if the genotypic effect is random and the environmental effect is fixed, then µ and {delta}j are fixed effects and {tau}i, ({tau}{delta})ij, {gamma}jk, and {varepsilon}ijk are independent and normally distributed random effects that have expectations of zero and variances {sigma}{tau}2, {sigma}{tau}{delta}2, {sigma}{gamma}2, and {sigma}{varepsilon}2, respectively. Obviously, the two variants of the mixed-effect model are reciprocal with the nature (fixed or random) of effects {tau}i or {delta}j being swapped. While the mixed-effect model is the focus of the subsequent analysis, the fixed- and random-effect models are included for comparison.

Since each of the three models has one or more fixed and random effects, Eq. [1] can be conveniently written in the standard linear mixed model (Henderson, 1984; Littell et al., 2006),

Formula 2[2]
where, under the mixed-effect model with fixed genotypic and random environmental effects, y is an m [= {sigma}j=1e(grj)] x 1 vector of observations y = [y111, y112, ...,ygere ]'; ß is a (g + 1) x 1 vector of unknown fixed effects, ß = [µ, {tau}1, {tau}2, ..., {tau}g]'; u is an n ( = e + ge + {sigma}j=1erj) x 1 vector of random effects, u = [{delta}1, {delta}2, ..., {delta}e, ({tau}{delta})11, ({tau}{delta})12, ..., ({tau}{delta})ge, {gamma}11, {gamma}12, ..., gere]', X is an m x (g + 1) design matrix of 1s and 0s relating y to ß, Z is an m x n design matrix of 1s and 0s relating y to u, {varepsilon} is an m x1 vector of random errors {varepsilon} = [{varepsilon}111, {varepsilon}112, ..., {varepsilon}gere]', and the prime (') represents vector or matrix transposition. Random vectors u and {varepsilon} are assumed to be normally and independently distributed with zero mean vectors and variance–covariance matrices G and R, respectively, such that

Formula 2
where ~N means normally distributed. Thus, the expectation and variance of y are E(y) = Xß and var(y) = V = ZGZ' + R. Different aspects of GEI have been characterized by allowing G and R to take different forms of covariance structure (e.g., Piepho, 1998; Yang, 2002; Crossa et al., 2004), but here I take a simple form of G and R for a comparison with the conventional ANOVA model:

Formula 2
and R = {sigma}{varepsilon}2Im, where Ie, Ige, Ib, and Im are the identity matrices of orders e, g x e, b(= {sigma}j=1erj), and m, respectively. For the mixed-effect model with random genotypic and fixed environmental effects, Eq. [2] is modified with effects {tau}i or {delta}j being swapped, where ß = [µ, {delta}1, {delta}2, ..., {delta}e]', u is an n (=g + g x e + {sigma}j=1erj) x 1 vector of random effects, and u = [{tau}1, {tau}2, ..., {tau}g, ({tau}{delta})11, ({tau}{delta})12, ..., ({tau}{delta})ge, {gamma}11, {gamma}12, ..., {gamma}ere]'. The fixed- and random-effect models can be similarly considered with different numbers of elements in ß and u. In the fixed-effect model, ß = [µ, {tau}1, {tau}2, ..., {tau}g, {delta}1, {delta}2, ..., {delta}e, ({tau}{delta})11, ({tau}{delta})12, ..., ({tau}{delta})ge]', and u = [{gamma}11, {gamma}12, ..., {gamma}ere]' whereas in the random-effect model, ß = [µ] and u = [{tau}1, {tau}2, ..., {tau}g, {delta}1, {delta}2, ..., {delta}e, ({tau}{delta})11, ({tau}{delta})12, ..., ({tau}{delta})ge, {gamma}11, {gamma}12, ..., gere]'. Of course, the dimensions of design matrices X and Z are changed accordingly under these different models.

Predictable Functions and Test for Crossover Interactions
Statistical inference under the mixed-effect model involves both ß and u. The general problem is to predict and test a linear combination of fixed and random effects, K'ß + M'u, known as a predictable function, given that K'ß is an estimable function (Henderson, 1984; Stroup, 1989; Littell et al., 2006), where K and M are vectors of known coefficients that determine desired comparisons among a mixture of fixed and random effects. It has been shown (Henderson, 1984, p. 41–42; Schaeffer, 2006) that a predictor (a linear combination of observations), L'y, is a BLUP of Xß + Zu if L'y = K'ß + M'û, where

Formula 2
and

Formula 2
and Formula 2 and û are simply solutions to the well-known mixed model equations (Henderson, 1984):

Formula 3[3]
where superscript "–1" and superscript "–" represent matrix and generalized inverses, respectively. With known G and R, Formula 3 is the BLUE of ß as often obtained using the generalized least squares estimation procedure, and û is the BLUP of u, which shrinks the fixed-effect estimates of u toward the expected value of zero (e.g., McLean et al., 1991; Robinson, 1991). The values of G and R are usually unknown, however, and their estimates, Formula 3 and Formula 3, are substituted into Eq. [3] to obtain the empirical BLUE (EBLUE) of ß and the empirical BLUP (EBLUP) of u.

The sampling variability of the predictable function, K'Formula 3 + M'û, is measured by Var[K'(Formula 3 – ß) + M'(û – u)]. To evaluate this variance, one needs to recognize that (i) both estimated fixed effects (Formula 3) and predicted random effects (û) carry the sampling variability; (ii) while parametric values of fixed effects (ß) have zero variances and covariances with other effects, those of unobservable random effects (u) do not because these effects themselves have probability distributions. The sampling variance of K'Formula 3 + M'û (Stroup 1989) is

Formula 4[4]
where Var(Formula 4), Cov[Formula 4,(û – u)'], Cov[(û – u), Formula 4'], and Var(û – u) are the terms of the so-called C matrix, which is equal to the coefficient matrix in Eq. [3] (Henderson, 1984; McLean et al., 1991):

Formula 5[5]
with Cßu = Cuß' and

Formula 5
The standard error of the predictable function (SEP) is given by

Formula 6[6]
It should be pointed out that C would underestimate the true sampling variability of Formula 6 and û because the uncertainty arising from estimating G and R is not considered. Different inflation factors (e.g., Prasad and Rao, 1990; Harville and Jeske, 1992; Kenward and Roger, 1997) have been proposed to account for the underestimation but they are often very small unless a data set is poorly balanced.

The key step of testing for COI under the mixed-effect model is to identify a predictable function that allows a comparison between a pair of genotypes evaluated at a random environment or for a comparison between a pair of environments used to evaluate a random genotype. Like the Azzalini–Cox test in the fixed-effect model (Baker, 1988; Cornelius et al., 1992), a two-way (g x e) genotype–environment table is created to facilitate such comparisons. Unlike in the fixed-effect model, however, it is not the difference between the two cell means from the two-way table but rather the difference between their BLUPs that should be used. For example, to compare genotypes i and i' at environment j, one needs the difference between BLUPs of the ijth cell mean [µij = E(Formula 6ij.) = µ + {tau}i + {delta}j + ({tau}{delta})ij] and the i'jth cell mean [µi'j = E(Formula 6i'j.) = µ + {tau}i' + {delta}j + ({tau}{delta})i'j], BLUP(µij – BLUP(µi'j). For clarity, I will show the BLUPs of the cell means from a balanced data set (i.e., rj = r for all j) using the development of Cornelius and Crossa (1999). Thus, the BLUP of the ijth cell mean is

Formula 7[7]
where BLUE(µi) = BLUE(µ + {tau}i ) = the simple mean of the ith genotype (Formula 7i...),

Formula 7
and

Formula 7
with E(MS{delta}) and E(MS{tau}{delta}) being the expected mean squares for environmental and GEI factors from the ANOVA table, respectively, E(MS{delta}) = {sigma}e2 + g{sigma}{gamma}2 + r{sigma}{tau}{delta}2 + rg{sigma}{delta}2 and E(MS{tau}{delta}) = {sigma}e2 + r{sigma}{tau}{delta}2. Thus,

Formula 8[8]
where S{delta} = [rg{sigma}{delta}2 + r{sigma}{tau}{delta}2]/E(MS{delta}) and S{tau}{delta} = r{sigma}{tau}{delta}2/E(MS{tau}{delta}) are the shrinkage factors for environmental and GEI effects, respectively. If the variance components and thus expected mean squares are unknown and need to be estimated, then the shrinkage factors can be substituted by their estimates, S{tau}{delta}=(rgFormula 8{delta}2+rFormula 8{tau}{delta}2)/MS{delta} and S{tau}{delta}=rFormula 8{tau}{delta}2/MS{tau}{delta}, and BLUP of the ijth cell mean in Eq. [8] is replaced by its EBLUP, EBLUP(µij)=Formula 8i··+S{delta}(Formula 8·j·–Formula 8...)+S{tau}{delta}(Formula 8ij·–Formula 8i··–Formula 8·j·+Formula 8...).

Similarly, BLUP(µi'j) is obtainable by simply substituting the subscript i with i'. Furthermore, the BLUP values of these cell means under the random-effect model can also be obtained as done by Cornelius and Crossa (1999), whose derivation was based directly on the cell means rather than the individual observations used here. With the BLUPs of the cell means of fixed genotypes and random environments derived above, the desired comparison between genotypes i and i' at a random environment (j) is

Formula 9[9]
The first part of Eq. [9] leads directly to the construction of the desired predictable function for the comparison with the elements of vectors K = [{Kl}]' and M = [0e {Mlj} 0b]' being

Formula 10[10]
and 0e and 0b denote e x 1 and b x 1 vectors of 0s. It is also evident from the second part of Eq. [9] that the differences between the cell and marginal means are shrunken to the extent determined by the GEI and error variances only. Similarly, the BLUPs of the cell means and the differences between pairs of environments for a random genotype can be derived under the mixed-effect model involving random genotypic and fixed environmental effects, with the genotypic and environmental effects being swapped in Eq. [7–10]. Further, the same procedure can be used to obtain the BLUEs of the cell means and pairwise differences between BLUEs of the cell means under the fixed-effect model and to obtain the BLUPs of the cell means and pairwise differences between BLUPs of the cell means under the random-effect model. These BLUEs and BLUPs enable the desired predictable functions to be constructed, resulting in appropriate sets of vectors K and M.

The null hypothesis of no difference between the two genotypes (i and i') at random environment j or no difference between the two environments (j and j') for random genotype i (i.e., K'ß + M'u = 0) can be tested using a generalized t statistic as follows:

Formula 11[11]
where SEP is given in Eq. [6]. The statistic in Eq. [11] is approximately t distributed, with the approximate degrees of freedom being estimated according to McLean and Sanders (1988) and Kenward and Roger (1997). As pointed out above, the true sampling variability of Formula 11 and û and thus SEP may be underestimated because the uncertainty arising from estimating G and R is not accounted for. Thus the t statistic in Eq. [11] may be slightly biased upward if there is no correction for the possible underestimation of SEP (e.g., Kenward and Roger, 1997). Likewise, the t statistic for the comparison of the same pair of genotypes in another random environment (j') or for the comparison of the same pair of environments for another random genotype (i') can be similarly calculated. The presence of COI will then be evaluated by determining if the genotypic or environmental difference is significantly greater than zero in one environment or for one genotype and significantly less than zero in the other.

Following Cornelius et al. (1992), the critical values for assessing significant COI are calculated using three different t-tests, depending on if an experiment-wise (the original Azzalini–Cox test), comparison-wise, or interaction-wise error rate ({theta}) is used. For a given significance level ({alpha}), the respective {theta} values are given by

Formula 11
to control the overall error rate for all the comparisons, the individual or comparison-wise error rate for each comparison, and the error rate per quadruple for COI, respectively. The powers of the three t-tests for COI are different, with the test based on the experiment-wise error rate being the most conservative and its power decreasing with the increasing number of genotypes and environments. The test using the interaction-wise error rate is the most sensitive and the test based on the comparison-wise error rate has intermediate power and Type I error rate. Regardless of which error rate is protected, a test for significant COI is essentially the test for the null hypothesis that no COI exists in any one of all possible quadruples vs. the alternative hypothesis that some COI exist. Cornelius et al. (1992) argued against the use of the original Azzalini–Cox test because it gives experiment-wise error rate protection against rejecting a true null hypothesis (lower Type I error rate) at a cost of high Type II error rate (i.e., low power to detect the true COI). In other words, a Type I error may not be serious because follow-up cultivar trials will reveal spurious COI, but a Type II error is serious because a potentially important COI may go undetected.

As evident in Eq. [8], different variance components for random effects are needed to compute shrinkage factors for deriving BLUPs of the cell means. I used two ANOVA-based methods, SAS Type 1 (based on computation of sequential sum of squares for each random effect) and SAS Type 3 (based on computation of partial sum of squares for each random effect) and two likelihood-based methods, maximum likelihood (ML) and restricted maximum likelihood (REML) to assess the effects of different estimation methods on the BLUPs and thus on the test for COI. These estimation methods are all available in the SAS MIXED procedure (SAS Institute, 2004). The REML and ML methods are preferred methods for estimating variance components because they are able to accommodate data sets with unbalanced or complicated data structures and they possess the properties of consistency and asymptotic normality of the estimators desirable for hypothesis testing (Searle et al., 1992). The ANOVA estimators are included, however, because the ANOVA or least squares analysis of METs continues to dominate the current GEI literature (e.g., Cornelius and Crossa, 1999; Gauch, 2006; Yan and Tinker, 2006). In addition, REML and ML estimators may be biased when the constraint of non-negative estimates of variance components needs to be imposed. In contrast, ANOVA estimators are always unbiased. While the debate remains whether negative estimates of variance components should be reported as such or should be constrained to zero, the use of ANOVA estimators would avoid biased F tests and standard errors for different effects (Searle et al., 1992; Littell et al., 2006). The comparison between the two ANOVA estimators provides information on the effect of the lack of balance in the data, whereas the comparison between REML and ML estimators assesses the extent to which the ML estimator is biased due to the presence of fixed effects. With a balanced data set and non-negative estimates of variance components, REML estimators are identical to ANOVA estimators; otherwise, all four estimators may be different (Searle et al., 1992).

Computing Predictable Functions and Testing Crossover Interactions
The test for COI described above can be implemented in the SAS MIXED procedure (SAS Institute, 2004). The usual overall tests for fixed and random effects are achieved by specifying fixed effects in the MODEL statement and random effects in the RANDOM statement. For example, under the mixed-effect model involving fixed genotypic and random environmental effects, the genotypic effects are given in the MODEL statement and environmental, GEI, and replications-within-environmental effects in the RANDOM statement.

While the solution options in the MODEL statement and in the RANDOM statement provide EBLUE of ß and EBLUP of u, respectively, it is the use of the ESTIMATE statement that allows estimating and testing for any linear combinations of fixed and random effects, thereby testing for COI. The usual and familiar use of the ESTIMATE statement is for constructing estimable functions (linear combinations of fixed effects). When random effects are included as well, the ESTIMATE statement uses a vertical bar (|) to separate fixed effects (before the bar) from random effects (after the bar). Thus the coefficients given in the ESTIMATE statement constitute the elements of vector K for the fixed effects and those of vector M for random effects. Given that multiple ESTIMATE statements are allowed under one PROC MIXED session (so long as they all appear after the MODEL and RANDOM statements), the SAS MACRO facility is used to create different ESTIMATE statements for comparing pairs of genotypes at each and every random environment or comparing pairs of environments for each and every random genotype. This works well for a moderate number of comparisons, but is problematic for a very large number of comparisons. For example, for g = 30 genotypes (fixed) and e = 50 environments (random), one needs to create a total of e x [g(g – 1)/2] = 21750 ESTIMATE statements for all possible comparisons, which is too numerous to be handled by a computer with a modest amount of memory!

An alternative strategy is to use the estimated C matrix (cf. Eq. [4]) and the EBLUE of ß and EBLUP of u produced by the MMEQSOL option of PROC MIXED to calculate all possible comparisons, their associated standard errors, and associated t statistics, following Eq. [6] and [11]. As a partial check on the calculations, the results from comparing the first two genotypes (or environments) at each and every random environment (or genotype) are confirmed with outputs from a subset of ESTIMATE statements for corresponding comparisons. These calculations are implemented in a SAS program called "mixed_COI.sas." The core part of the program is listed and explained in the Appendix for the case of the mixed-effect model involving fixed genotypes and random environments. Modifications of SAS codes can be readily made to accommodate the case of the mixed-effect model involving random genotypes and fixed environments and the case of the fixed- or random-effect model. The complete program is available from me on request.

Data Sets
Two data sets were used for the COI assessment under the mixed-effect model. The first data set used for the mixed-effect model, involving fixed genotypic effects and random environmental effects, was taken from the Canadian Prairies Barley Trials as described in Yang et al. (2006). While these trials included a large number of cultivars and advanced breeding lines, a focus has been on the following six two-row barley cultivars (Helm et al., 2004): three eligible for feed grades—‘CDC Dolly’, ‘Seebe’, and ‘Xena’—and three others eligible for malting grades—‘AC Metcalfe’, ‘Harrington’, and ‘Merit’. Details of cultivar development, yield performance, and agronomic and quality characteristics are given in Field Crop Development Centre (2006). I analyzed the yield data of these six cultivars evaluated in 2003 at 18 sites across the Province of Alberta including the two neighboring sites in the Province of British Columbia (Table 1). Two sites in Lethbridge, representing rainfed (dryland) and irrigated conditions, are within ≤10 km and are indistinguishable at the scale of the map. The cultivar trials across all sites were conducted using a RCBD with three or four replications. Cultural practices such as fertility, tillage, and pest control varied from site to site but were considered to be the most appropriate for the individual sites.


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  		Table 1. Yields of six barley (Hordeum vulgare L.) cultivars at 18 sites across Alberta, Canada, in 2003.

 
The second data set for the mixed-effect model, involving random genotypic effects and fixed environmental effects, was taken from a field pea cultivar trials conducted at 21 sites across Alberta in 2001 (Yang et al., 2005). Specifically, I analyzed the yield data of 33 registered cultivars or advanced breeding lines at four selected sites—Brooks, Vegreville, Namao, and High Prairie—representing four regions of definite geography and soil characteristics: southern Alberta, east-central Alberta, west-central Alberta, and the Peace River region (Yang et al., 2005). The Brooks site was under irrigation, whereas the other three sites were under a dryland condition. These four sites also belong to four distinctive isoyield groups based on clustering analysis (Yang et al., 2005). The experimental design for the trials at all four sites was a RCBD with four replications.


   RESULTS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
DISCUSSION
 APPENDIX
 REFERENCES
 
Crossover Interactions in the Barley Data Set
Table 1 is a two-way table giving the average yields of 108 combinations between six barley cultivars and 18 sites, based on four replications at seven sites—Big Lake, CDC North, Dawson Creek, Ft. Kent, Ft. St. John, Northern Sunrise, and St. Paul—and three replications at the remaining 11 sites. These cell means based on the fixed-effect model (i.e., both cultivar and site effects are fixed) serve as reference points in subsequent analyses under mixed- or random-effect models. The least square means and simple means for individual sites are the same because the data set is balanced within each site. The number of replications is not constant across sites, however, so the best estimates of genotype means are least square means, which differ from the simple means.

Given in Table 2 are the estimates of variance components of random effects for sites, replications (sites), cultivar x site interactions, and errors using the four estimation methods: SAS Type 1, SAS Type 3, REML, and ML. The REML estimates differ from those by Type 1 and Type 3, as expected for an unbalanced data set like the barley MET data; they would be identical if the data were balanced. The ML estimates are only slightly smaller than the estimators of the other three methods because the deduction of the number of fixed (cultivar) effects (six) from the total observations of 366 has a negligible effect. The same estimates by Type 1 and Type 3 are not surprising because of the hierarchical structure of different effects.


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  		Table 2. Estimates of variance components (±SE) by four estimation methods—Type 1, Type 3, restricted maximum likelihood (REML), and maximum likelihood (ML)—in the mixed-model analysis of barley (Hordeum vulgare L.) multiple-environment trial data.

 
While the significance of cultivar x site interaction can be assessed directly and quickly by determining if the cultivar x site variance component is greater than zero by more than twice its standard error (Wald's Z test), this test is not particularly reliable for small sample sizes and for the variance components with a skewed or bounded sampling distribution (Littell et al., 2006). Thus, I carried out a LR test (Yang, 2002) by comparing –2(Res)log likelihoods for two models, one with the interaction term (full model) as in Eq. [1] and the other without it (reduced model). The LR tests based on all four estimation methods show significant GEI variability (Table 3). For example, the –2(Res)log likelihoods by the REML method for the full and reduced models are 691.8 and 737.4, respectively, and LR = 737.4 – 691.8 = 45.6. Under the null hypothesis that the GEI variance component is zero, LR follows a chi-square distribution with one degree of freedom ({chi}12), leading to a probability of 1.45 x 10–11. However, this probability needs to be halved (7.25 x 10–12) because the asymptotic distribution of the LR statistic with the boundary value of the covariance parameter (zero GEI variance) in the reduced model is actually a 50:50 mixture of {chi}02 and {chi}12 distributions (Self and Liang 1987). Given that the {chi}02 distribution takes the value 0 with probability 1, the mixture distribution takes the value 0 with probability 1/2 and takes a value drawn from a {chi}12 distribution with probability 1/2. The small probability obviously indicates that the null hypothesis of zero GEI variance is unlikely to be true and that interactions between cultivars and sites are significant.


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  		Table 3. Likelihood ratio (LR) tests for cultivar x site interactions under three linear models in the mixed-model analysis of barley (Hordeum vulgare L.) multiple-environment trial data. The methods of estimating covariance parameters are: Type 1, Type 3, restricted maximum likelihood (REML), and maximum likelihood (ML).

 
Of 2295 [6(6 – 1)18(18 – 1)/4] possible quadruples evaluated for COI, the number of significant COI varies with different models (fixed, mixed, and random), estimation methods (Type1, Type 3, REML, and ML), and test criteria (experiment-wise, comparison-wise, and interaction-wise) as shown in Table 4. While different numbers of significant COI are observed for combinations of estimation methods, models, and test criteria, the least and most sensitive of these combinations share the same quadruples that display significant COI. Three points are quite obvious from the table.


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  		Table 4. Number of significant crossover interactions with three different linear models (fixed, mixed, and random effects), four estimation methods (Type1, Type 3, restricted maximum likelihood, and maximum likelihood), and three test criteria (experiment-wise, comparison-wise, and interaction-wise) in the mixed-model analysis of barley (Hordeum vulgare L.) multiple-environment trial data. The total of quadruples is 2295.

 
First, treating random site and GEI effects as fixed under the fixed-effect model would overestimate the power of detecting COI, as the numbers of significant COI are higher under the fixed-effect model than under mixed- or random-effect models. The comparison under the fixed-effect model is based on unshrunken differences between the marginal means of a pair of cultivars across all sites and the cell means at a particular site. In contrast, the comparison under the mixed- or random-effect models is based on shrunken differences between the means of a pair of cultivars across all sites or at a particular site. Consequently, the difference between cultivars based on the mixed- or random-effect models is smaller than those based on the fixed effect model, as the random effects are scaled toward their true expectation of zero. The miniscule difference in detecting COI between the mixed- and random-effect models is somewhat expected because the cultivar difference at a random environment under both models involves the shrunken GEI effects with similar but not identical shrinkage factor (the variance components differ slightly under the two models). The shrinkage effect is illustrated in Fig. 1, in which line plots are made of differences for two cultivar pairs, CDC Dolly–Harrington and AC Metcalfe–Harrington, at all 18 sites under the three linear models. In Fig. 1, the dashed lines represent upper and lower bounds (based on the interaction-wise criterion) beyond which significant cultivar differences are indicated. It is hardly surprising that the number of sites showing significant cultivar differences is not equal on positive and negative sides, as yields of CDC Dolly and AC Metcalfe are higher than those of Harrington at most sites. Thus, the total number of significant COI for each cultivar pair is the number of sites with significant cultivar differences on the positive side multiplied by that on the negative side. For example, in the top graph of Fig. 1, the number of sites showing significant positive cultivar differences (CDC Dolly–Harrington, above the upper dashed line) is 7 under the fixed-effect model and 6 under the mixed- and random-effect models, whereas the number of sites showing significant negative cultivar differences (CDC Dolly–Harrington, below the lower dashed line) is 3 under the fixed-effect model and 2 under the mixed- and random-effect models. Thus, the number of significant COI for this cultivar pair is 21 (7 x 3) under the fixed-effect model and 12 (6 x 2) under the mixed- and random-effect models. It is evident from the same argument that there are zero COI for AC Metcalfe–Harrington under all three models.


Figure 1
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  		Figure 1. Differences in yields (Mg ha–1) for two barley (Hordeum vulgare L.) cultivar pairs, CDC Dolly–Harrington and AC Metcalfe–Harrington, evaluated at 18 sites across Alberta under three linear models (fixed-, mixed-, and random-effect models). The dashed lines represent upper and lower bounds (based on interaction-wise criterion) beyond which significant cultivar differences are indicated.

 
Second, the number of significant COI is larger if the ML method is used to estimate variance components than if any of the other three methods (Type 1, Type 3, and REML) is used under the fixed-effect model (Table 4). The latter three methods give identical numbers of significant COI even though the REML estimates of variance components are slightly different from those by the Type 1 and Type 3 methods (Table 2). The ML estimates of variance components are slightly biased downward, thereby leading to either smaller standard errors of estimable functions or smaller shrinkages of differences in means. Thus, higher frequencies of significant COI result under the ML estimation method for the fixed-effect model.

Third, the percentages of significant COI evaluated by the interaction-wise test criterion ranged from 4% (97) to 12% (279), whereas those evaluated by the experiment-wise test criterion ranged from 0% (0) to 0.7% (15). Obviously, the ranking for sensitivities of the three test criteria is: interaction-wise > comparison-wise > experiment-wise. Under the null hypothesis that there are no significant COI with an error rate of {alpha} = 0.05, the test would find 115 significant COI by chance alone. Judging from the observed numbers of significant COI, no COI is present in all cases except for the interaction-wise test criterion under the fixed-effect model.

Crossover Interactions in the Field Pea Data Set
The analysis of field pea data with random genotypes (cultivars or breeding lines) and fixed sites shows that the numbers of significant COI are all >159 (5% of 3168, the total number of quadruples for 33 genotypes and four environments) as expected by chance alone, regardless of different linear models (fixed, mixed, and random), estimation methods (Type1, Type 3, REML, and ML), and test criteria (experiment-wise, comparison-wise, and interaction-wise) (Table 5). The extent of significant COI is obviously much higher in this field pea data than in the barley data; however, the patterns of significant COI are similar for both data sets. First, treating random genotypic and GEI effects as fixed under the fixed-effect model would overestimate the power of detecting COI, as the numbers of significant COI are higher under the fixed-effect model than under mixed- or random-effect models. Second, there are higher frequencies of significant COI under the ML estimation method than the other three estimation methods for the fixed-effect model. Third, the ranking for sensitivities of the three test criteria is: interaction-wise > comparison-wise > experiment-wise. As in the barley data set, the same quadruples that display significant COI occur throughout different estimation methods, models, and test criteria.


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  		Table 5. Number of significant crossover interactions with three different linear models (fixed, mixed, and random), four estimation methods (Type1, Type 3, restricted maximum likelihood, and maximum likelihood) and three test criteria (experiment-wise, comparison-wise, and interaction-wise) in the mixed-model analysis of field pea (Pisum sativum L.) multiple-environment data. The total number of quadruples is 3168.

 
High levels of significant COI in the field pea data are somewhat expected given marked and inconsistent differences in yield among 33 genotypes across the four selected sites. The range of yields across 33 genotypes is 2.38 to 6.41 Mg ha–1 for Brooks, 1.52 to 5.64 Mg ha–1 for High Prairie, 2.69 to 6.63 Mg ha–1 for Namao, and 1.21 to 4.56 Mg ha–1 for Vegreville. The presence of significant COI across the four sites is also indicated by imperfect correlations between pairs of sites. The estimated correlations between genotypic performances at all six pairs of sites vary from 0.066 for Brooks–High Prairie to 0.763 for High Prairie–Namao, but all these correlations are significantly less than unity according to Fisher's z transformation (SAS Institute, 2004). In this field pea example, the assumption of fixed site and random genotypic effects appears to be particularly reasonable. For nontraditional crops such as field pea in western Canada, one of the major objectives of cultivar testing programs is to test a large number of introduced cultivars or advanced breeding lines and to identify which sites are suitable for optimal crop production. In addition, the four selected sites are more regularly used for cultivar and line testing across years than the other sites that are chosen yearly on an ad hoc basis. The four sites differ markedly in their geography and performance (Yang et al., 2005). While the across-site combined analysis was performed to illustrate the flexibility of the new test for COI, it needs to be pointed out that in reality, if environments are considered fixed, then results for cultivar performances would typically be reported separately for each environment.


   DISCUSSION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
APPENDIX
 REFERENCES
 
The proposed test for COI based on the mixed-model theory is quite general as it is applicable to all the models regardless of whether genotypic, environmental, and GEI effects are fixed or random. If both genotypic and environmental effects are fixed, the new test reduces to the conventional Azzalini–Cox test for COI (Baker, 1988; Cornelius et al., 1992). In the past, the Azzalini–Cox test has often been used for assessing COI but its use is based on the limiting assumption that all effects are fixed. In reality, either genotypes or environments or both should be random, as argued in Baker (1996) and Smith et al. (2005). The mixture of fixed and random effects is now accommodated in the new mixed-model test, which uses linear combinations of BLUPs of random effects and BLUEs of fixed effects for detecting COI. It is also the responsibility of the users of the new test, however, to ensure that the choice of model should be made a priori and be based on the inference space and sampling of cultivars or environments for a MET. In both examples analyzed in this study, I used a selected set of cultivars (barley data set) and environments (field pea data set) for clearer illustration of the mixed-model test for COI. Nevertheless, the new test also works with the whole set of cultivars or environments. In fact, similar patterns and extent of COI are observed when analyzing all 41 cultivars and breeding lines (barley data) and all sites (field pea data). For example, the mixed-model analysis of the complete barley data set involves the evaluation of 125460 [41(41 – 1)18(18 – 1)/4] possible quadruples of fixed cultivars and random sites; the numbers of significant COI are 7 (experiment-wise), 770 (comparison-wise), and 3738 (interaction-wise) if the estimation methods of REML, Type 1, or Type 3 are used. All these numbers are obviously <5% of the total number of quadruples (125460). Thus, the new test is applicable to any number of genotypes and environments if the nature of each effect (fixed or random) is clearly identified.

It is clear from Table 4 that the choice of which model is used to test for COI would lead to a different conclusion about the nature of cultivar x site interactions. Looking at the test results based on the interaction-wise error rate, the fixed-effect model analysis would identify significant COI in this barley data set, whereas the mixed- and random-effect model analyses would not. An important consequence is that false claims may be made if either genotypes or environments (and thus GEI) should be random but are treated as fixed. In the barley example, if both cultivars and sites are considered to be fixed effects, then the claim under the fixed-effect model is appropriate. While the 18 sites for the barley trials are known to belong to different soil zones or eco-regions in Alberta with "fixed" agroclimatic characteristics, however, the effects of soil zones or eco-regions alone account for only a small portion of site-to-site variability (Yang et al., 2006). If the site effects are random, as argued above, there is no evidence of significant COI in this barley data set (the observed 97 significant COI based on the most sensitive interaction-wise test is still <115 [5%] expected by chance alone). A similar conclusion was implied in an earlier investigation on the same six barley cultivars (Helm et al., 2004). In addition, in crop improvement, the adaptability of cultivars needs to be effectively assessed by applying inference beyond the observed sites to the entire population of environments. Thus, site and cultivar x site interaction effects are considered random for most breeding applications (e.g., Baker, 1996; Balzarini, 2002).

It is crucial to correctly determine whether or not the observed significant GEI involves rank changes (COI) because COI are the only type of interactions that impact selection programs. Even in this case, COI may still be irrelevant to a breeder if their occurrence coincides with the situation where one cultivar consistently outperforms other cultivars in the majority of the environments in a MET. In general, the presence of COI would suggest that much of the improvement made in one or one set of environments will not be carried over when the selected genotypes are grown in other environments. In this case, one must select one genotype for one set of environments and a different genotype for other environments. On the other hand, if significant GEI merely reflects differences in scale (i.e., absence of COI), then there is no need to consider any aspect of GEI (Baker, 1996). In this case, the breeding efforts or production system can be greatly simplified because a genotype that is the best in one environment will be the best in all environments. Thus, in the absence of COI, a single genotype would optimize production in all environments.

Despite a growing use of the mixed-model analysis for studying GEI (e.g., Piepho, 1998; Yang, 2002; Crossa et al., 2004, 2006), much of the GEI literature remains focused on the ANOVA-based analysis. One of the major drawbacks of the ANOVA-based analysis is the dichotomy in its applications: statistical inference is based either on estimable functions of fixed effects under the assumed fixed-effect model or on the magnitudes and structures of variances of the different types of GEI under the random-effect model. I propose to study GEI by estimating and testing for appropriate predictable functions (linear combinations of both fixed [genotype] and random [GEI] effects). While estimation of fixed effects and prediction of random effects have been a major focus of the mixed-model theory (Henderson, 1984), the hypothesis testing concerning a mixture of these two types of effects is largely neglected (Kennedy, 1991). Given that the most realistic models for studying GEI are those based on a mixture of fixed and random effects, further research should be directed to estimation, prediction and testing for predictable functions that can characterize the nature and patterns of GEI.

This study can be extended to other areas related to characterization of GEI when the mixed- or random-effect model is considered. For example, in the past, the criteria developed for clustering genotypes or environments into subsets with negligible COI are based on the fixed GEI effect (e.g., Crossa et al., 2004; Navabi et al., 2006). It would be of interest to investigate the effectiveness of the clustering procedures based on the random GEI effect. In addition, this study used simple covariance structure models for error and GEI effects to permit direct comparison with the ANOVA-based analysis. In the future, another area of investigation may be to exploit more complicated error and GEI covariance structures from the mixed-model perspective (e.g., Piepho, 1998; Yang, 2002; Qiao et al., 2004; Casanoves et al., 2005; Crossa et al., 2004, 2006) or from the Bayesian perspective (Cotes et al., 2006; Edwards and Jannink, 2006) to assess their sensitivity to detection of significant COI. Finally, the commonly used additive main effects and multiplicative interaction (AMMI) model (Gauch, 2006) and the genotype main effects and genotype x environment interaction effects (GGE) model (Yan and Tinker, 2006) are fixed-effect models. When some or all effects are random, the present mixed-model analysis may conceivably be extended to provide BLUPs of genotype and environment eigenvectors for biplot characterization of GEI.


   APPENDIX
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
REFERENCES
 
Listed below is the syntax for the PROC MIXED analysis of the barley MET, where cultivar effects (&G_EFF with an ampersand sign [ &] signaling a SAS macro variable) are fixed, whereas site effects (&E_EFF), blocks-within-sites effects [&B_EFF(&E_EFF)] and cultivar x site interaction effects (&E_EFF*&G_EFF) are random. This is the core part of the SAS program mixed_COI.sas described above.

PROC MIXED METHOD = &METH MMEQSOL COVTEST;
CLASS &E_EFF &B_EFF &G_EFF;
MODEL &DVAR = &G_EFF/DDFM = KR SOLUTION;
RANDOM &E_EFF &B_EFF(&E_EFF) &E_EFF*&G_EFF/SOLUTION;
ESTIMATE ‘CULT1-CULT2 AT SITE 1’ &G_EFF 1–1 |&E_EFF*&G_EFF 1–1;
(17 more ESTIMATE statements)...
ODS OUTPUT MMEQSOL = MMESOL(DROP = ROW) ESTIMATES = EVAL;
RUN;

In the PROC MIXED statement, the option of METHOD = &METH specifies which of the four estimation methods (Type 1, Type 3, REML, and ML) is used, where &METH can be "type1," "type3," "reml," or "ml" (two other methods, Type 2 and MIVQUE0 [minimum variance quadratic unbiased estimation] are not included). The option of MMEQSOL requests SAS to output a solution to the mixed-model equations along with the inverted coefficients matrix (cf. Eq. [3]). The option of COVTEST asks SAS to generate asymptotic standard errors and Wald Z tests for the variance parameter estimates.

The MODEL statement includes one dependent variable (&DVAR) and one fixed effect (&G_EFF). The option of DDFM = KR identifies the method of Kenward and Roger (1997) for computing the denominator degrees of freedom for the tests of fixed effects specified in the MODEL statement and for tests of predictable functions specified in the ESTIMATE statements (below); this option involves calculating an inflation factor for the estimated variance–covariance matrix of the fixed and random effects (Prasad and Rao, 1990; Harville and Jeske, 1992) and then computing Satterthwaite-type degrees of freedom on the inflated variance–covariance matrix. The option of SOLUTION is added to request SAS output estimates and tests of fixed effects for checking purposes.

The random statement lists all three random effects [&E_EFF &B_EFF(&E_EFF) &E_EFF*&G_EFF]. The option of SOLUTION is added to demand SAS to output estimates and tests of random effects for checking purposes.

A selected set of ESTIMATE statements is specified to provide desired outputs for evaluating significant COI and to check the calculations of predictable functions and their standard errors using the outputs from the MMESOL option. For example, a series of 18 estimate statements can be given to estimate and test for the differences between AC Metcalfe and CDC Dolly for each of the 18 test sites. In general, a SAS MACRO is available in mixed_COI.sas to allow for automatically creating these ESTIMATE statements.

Finally, the ODS statement is used to request SAS to provide the outputs from the MMESOL option and ESTIMATE statements. A SAS/IML subroutine is available in mixed_COI.sas to perform tests for COI based on these outputs.


   ACKNOWLEDGMENTS
 
I thank Drs. James Holland, Jose Crossa, and two anonymous reviewers for valuable comments. This research was supported in part by Alberta Agriculture, Food, and Rural Development's Industry Development Sector New Initiative Fund and a Natural Sciences and Engineering Research Council of Canada grant.


   NOTES
TOP
 NOTES
ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
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Received for publication September 25, 2006.


   REFERENCES
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 




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