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Published online 24 February 2006
Published in Crop Sci 46:820-833 (2006)
© 2006 Crop Science Society of America
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CROP BREEDING, GENETICS & CYTOLOGY

Bayesian Modeling of Heterogeneous Error and Genotype x Environment Interaction Variances

Jode W. Edwardsa,* and Jean-Luc Janninkb

a USDA Agricultural Research Service, Corn Insects and Crop Genetics Research Unit, Department of Agronomy, Iowa State University, Ames, IA 50011
b Department of Agronomy, Iowa State University, Ames, IA 50011

* Corresponding author (jode{at}iastate.edu)


   ABSTRACT
TOP
 NOTES
 ABSTRACT
INTRODUCTION
 METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
An important assumption in the analysis of multienvironment cultivar trials is homogeneity of error and genotype x environment interaction variances. When variances are heterogeneous, the best estimators of performance are obtained by weighting inversely to variance components. However, because variances are almost never known and must be estimated, the additional error introduced into the model from estimating many variances may cause weighted estimators to perform poorly. Our objective was to test a Bayesian approach to estimating heterogeneous error and genotype x environment interaction variances. A Bayesian model for multienvironment yield trials that includes a linear model for error and genotype x environment interaction variances was applied to yield data from the Iowa State University Oat Variety Trial for the years 1997 to 2003. The Bayesian approach revealed that error variances were highly heterogeneous among environments and that genotype x environment interaction variances were heterogeneous among environments and genotypes. Incorporation of heterogeneity of variances significantly decreased estimates of marginal error, genotypic, and genotype x environment variance components, with the largest change being a reduction in the marginal genotype x environment interaction variance. Repeatabilities were higher in the heterogeneous variance model but not at a high level of statistical significance. Genotype-specific estimates of genotype x environment interaction variances were correlated with estimated genotypic yields and heading dates, providing biological validity to our estimates of genotype-specific estimators of genotype x environment interaction variances as stability estimators.


   INTRODUCTION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
YIELD TRIALS conducted in multiple years and locations are central to plant breeding efforts to evaluate and improve crops. The typical analysis of such trials assumes homogeneity of microenvironment error variances and genotype x environment interaction variances across environments and across genotypes. In their classic review of genotype x environment interaction, Comstock and Moll (1963) state "we know from experience that the plot error variance is variable from one experiment to another, ... and there is nothing that compels the variances of the GE interaction effects to be homogeneous." Though they do not present definitive methods to assess variance heterogeneity, they conclude from an example data set "that the observed variation in estimates ... is sufficient to cast doubt on the assumptions that effect covariances and interaction effect variances are uniform from one macro-environment to another." Comstock and Moll (1963) appeared primarily concerned with the adequacy of the ANOVA analysis of the yield trial under heterogeneous variances. It has been well established in statistics that the minimum variance estimator of a linear function, such as a combined mean across environments, is weighted by the inverses of variances of individual means or values. Yates and Cochran (1938) provided detailed analyses for combining data across multiple experiments with possibly heterogeneous variances. However, the persistent problem with weighting individual means by the inverses of variances is that these variances are almost never known, and in fact, must be estimated. Because the exact sampling distributions for variance components do not exist except in the very simplest of models, the error of estimating the weights (i.e., the variances of individual variances) cannot be properly accounted for in the computation of the weighted means (Harville, 1977; Jeske and Harville, 1988; Kackar and Harville, 1984; Searle et al., 1992). There are two costs of estimating the variances that are used in computing weighted means: (i) the error incurred in estimating the variances is added to the error of estimating individual means so that despite the reduction in error due to weighting, there is also a penalty of increased error from estimating the weights, (ii) sampling distributions of the weighted means do not exist, and therefore, exact interval procedures (e.g., to obtain upper and lower confidence intervals or exact hypothesis tests) cannot be obtained for weighted means estimates.

Parallel to the desire to use weighted estimates of combined means to achieve lower variances of multienvironment genotypic means, other researchers have focused on heterogeneity of genotype x environment interaction variances as a measure of cultivar stability. Genotype-specific microenvironment and genotype x environment variances represent a measure of the stability of the genotype in the face of unpredictable conditions, with low variance being desirable. Finlay and Wilkinson (1963) and Eberhart and Russell (1966) developed regression approaches to assess stability. Shukla showed how to estimate the genotype-specific error variance (Shukla, 1972b) and showed how to test the null hypothesis that all genotype-specific error variances were equal, i.e., homoscedastic (Shukla, 1972a). Shukla (1972b) also remarked that, considering observations assembled into a genotype x environment matrix, estimation and hypothesis testing procedures performed on the row (genotypic) dimension of the matrix could also be performed on the column (environmental) dimension of the matrix. More recently, Piepho (1999; 1998) has shed further light on the statistical underpinnings of genotype stability parameters by casting their estimation in the context of fitting mixed models to yield trial data. In particular, considering genotypes (Piepho, 1998) as fixed effects and environments as random effects leads to the estimation of a variance–covariance matrix of genotypic effects across environments. For example, testing five genotypes results in estimating a 5 x 5 genotype-mean variance–covariance matrix, where the diagonal elements represent genotype-specific variances and off-diagonal elements represent covariances among genotypes. Different conceptions of stability then require different structures of the genotype-mean variance–covariance matrix. A highly useful feature of these methods based on mixed models estimated using maximum likelihood, or preferably restricted maximum likelihood, is that they allow unbalanced datasets (Piepho, 1999) to be analyzed for variance heterogeneity.

It should be evident from the preceding discussion that there is value in considering variance heterogeneity in both genotypic and environmental dimensions of the genotype x environment matrix. Furthermore, assuming the existence of heterogeneity in both dimensions, appropriate estimation of heterogeneity in either dimension requires estimation of heterogeneity in both dimensions jointly, particularly for unbalanced datasets. In such datasets, variance heterogeneity among genotypes will not be orthogonal to variance heterogeneity among environments. Consequently, assessments of variance heterogeneity among genotypes will be affected by variance heterogeneity among environments and conversely, unless a joint estimation is performed. To give a simple example, assume that among a set of environments, microenvironment variance is greater in Environment A than Environment B. In the balanced case where all genotypes are evaluated in all environments, this difference across environments will not differentially affect estimates of genotype-specific microenvironment variance. In the unbalanced case, however, genotype-specific microenvironment variances estimated for genotypes evaluated in an environment with a larger microenvironment variance will tend to be biased upward relative to microenvironment variances estimated for genotypes evaluated only in environments with lower microenvironment variances. The bias in estimation of genotype-specific microenvironment variance will adversely affect estimates of genotypic main effects as well, as the weights placed on means from individual environments will be biased as well. Thus, best estimates, meaning estimates with minimum variance, of genotypic main effects require estimating variance heterogeneity in both genotypic and environmental dimensions of the genotype environment effect matrix.

In light of these issues, our objective was to develop a model for estimating heterogeneity of variances in both dimensions of the genotype x environment matrix that would also distinguish between heterogeneity in microenvironmental variance from heterogeneity in genotype x environment variance. A problem attending the estimation of genotype- or environment-specific variances is that the estimates are based on fewer observations, that is, only those observations on the given genotype or environment. Variances may then be poorly estimated and extreme estimates might arise. Following up on suggestions from Henderson (1984), Gianola (1986) suggested using the Bayesian solution of pooling prior and data information to regress estimated variances toward their average for estimating best linear unbiased predictors in animal breeding data. Numerous authors have proposed modeling variances in a generalized linear model with a natural logarithmic link function (Nair and Pregibon, 1988; Verbyla, 1993; Smyth, 2002; Smyth, 1989; Cook and Weisberg, 1983; Aitkin, 1987; Sorensen and Waagepetersen, 2003; Jaffrezic et al., 2000; Foulley et al., 1992; Gianola, 1986; Leonard, 1975) in both the frequentist and Bayesian paradigms. In all cases, the basic idea is that the natural logarithms of variances are modeled using a linear model to account for heterogeneity of the variances (on a logarithmic scale) in terms of covariates and factor levels. In the Bayesian paradigm, hierarchical models (Gelman et al. (2004), chapter 5) can be used to model heterogeneity of variances on the log scale, as outlined in great detail by Leonard (1975). In such a model, each genotype- or environment-specific variance parameter is considered to be an independent draw from a distribution common to that variance type. The model generally specifies only the type of distribution from which the variances are sampled (e.g., normal or gamma), but the parameters of the distribution, called hyperparameters, (e.g., µ and {sigma}2 if a normal distribution is assumed) are estimated from the data. The complete model then represents a hierarchy: hyperparameters determine the distribution of variance parameters which in turn determine the distributions of error deviations specific to each genotype or each environment. The hierarchical model provides the advantage that each variance estimate constrains the other variance estimates because each estimate contributes to determining the distribution of variances. This approach leads to information sharing between genotypes and between environments in that, for example, each environment-specific variance estimate is influenced both by the properties of the full set of environments and by the particulars of the individual environment.

Incorporating the notion of a Bayesian hierarchical model, our objective was to test a Bayesian approach to modeling heterogeneity of error variances and genotype x environment interaction variances among oat (Avena sativa L.) cultivars. Our specific objectives to test the approach were (i) to determine if our Bayesian model provided convincing evidence for heterogeneity of error variances and genotype x environment interaction variances among genotypes and among environments, (ii) to determine the effects of modeling heterogeneous variances in an oat cultivar trial, and (iii) to determine if genotype-specific estimates of the variance of genotype x environment interactions were correlated to other performance indicators.


   METHODS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 METHODS
RESULTS
 DISCUSSION
 REFERENCES
 
Data
Data analyzed in this report were obtained from the Iowa State University Oat Variety Trial for the years 1997 to 2003. The trial contained 40 genotypes each year with on average 30 released cultivars and 10 experimental lines. The trial was grown in five Iowa locations: Ames (central Iowa), Nashua (northeastern Iowa), Crawfordsville (southeastern Iowa), Lewis (southwestern Iowa), and Sutherland (northwestern Iowa). Soils at each location were: Nicollet silty loam (fine-loamy, mixed, superactive, mesic Aquic Hapludoll) at Ames, Readlyn loam (fine-loamy, mixed, mesic Aquic Hapludoll) at Nashua, Mahaska silty clay loam (fine, smectitic, mesic Aquic Argiudoll) at Crawfordsville, Marshall silty clay loam (fine-silty, mixed, superactive, mesic Typic Hapludoll) at Lewis, and Galva silty clay-loam (fine-silty, mixed, mesic Typic Hapludoll) at Sutherland. In 1998, the trial was not grown in Sutherland, so that the data represent a total of 34 environments. For identification, environments were numbered (Table 1). The trial was replicated three times in each environment, in a randomized complete block design. Individual plots were 1.5 x 2.4 m in size. The trial contained three long-term check cultivars, Cherokee, Richland, and Multiline E77. Over the 7 yr of the trial analyzed here, a total of 85 genotypes were evaluated. We removed from this analysis five genotypes because their low yields made them outliers. Those genotypes were the three long-term checks, the hull-less oat ‘Paul’, and one experimental line. The number of environments in which individual genotypes appeared ranged from three to all 34 of the environments analyzed. Yield was measured at all locations, heading date was measured only at the Ames location.


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  		Table 1. Identification numbers used in figures for environments included in the analysis.

 
Statistical Model
Observations were modeled with field-block effects, genotype effects, and genotype x environment interactions according to the linear model:

Formula 1[1]
where ßjk = effect of kth block in environment j, {gamma}i = effect of ith genotype, {delta}ij = interaction of ith genotype and jth environment, and {varepsilon}ijk = residual error for the ith genotype in the jth environment in the kth block.

We defined a hierarchical Bayesian model for the problem to model both means (ßjk, {gamma}i, and {delta}ij) and variances ({sigma}{delta}(ij)2 and {sigma}{varepsilon}(ij)2) in terms of explanatory variables (block, location, and genotype; note that variances are subscripted according to location and genotype, indicating unique variance parameters for each level of these variables). At the first level of the Bayesian hierarchy, observations were assumed to be exchangeable samples (implying they can be modeled as independent samples from some probability distribution) from a normal distribution:

Formula 1

The second level of the Bayesian hierarchy includes prior distributions for location parameters (i.e., means) ßjk, {gamma}i, {delta}ij, and observational variances {sigma}ij2. Priors on all location parameters were normal with means zero and variances defined to condition the desired level of information sharing among levels of the factor. For block means, ßjk, the prior was defined with a very large variance to make the prior noninformative:

Formula 1

With a variance of 107, the likelihood of a block mean of 0 is 1.262, compared a likelihood of 1.259 for a block mean of 200, making the prior distribution very flat in the entire range of values that would reasonably be expected. Hence, the prior imparts essentially no information to the model and estimates of each block mean are determined almost entirely by the data. This prior construction was used so that estimates of block effects would be data driven since we had no prior information as to what particular block means should be. The flat and independent priors used on the block effects is a Bayesian equivalent to defining the block effects as fixed effects in classical (frequentist) linear models. In classical linear model theory, if individual environment effects and block effects are jointly fit in a linear model as fixed effects, these effects are unestimable because unique estimates of all parameters (environments and blocks) do not exist. Solutions to models are obtained with generalized inverses and/or arbitrary restrictions (such as restricting block effects to sum to zero). However, such tools do not exist in conjunction with modern Bayesian computing tools, and thus, models need to be defined with all parameters identifiable (or estimable). We have excluded environment effects from our model to maintain the identifiability of all parameters in the model. However, the exclusion of environment effects from the model does not effect our ability to estimate the mean response in an environment. It could be estimated from the posterior mean of the three block effects within the environment of interest, Formula 1j. Likewise, the grand mean of the experiment could be estimated from the posterior mean of all block effects for the entire data set (all environments).

Genotype effects, {gamma}i, and genotype x environment interaction effects, {delta}ij, were modeled with priors that treated these effects like random effects in classical mixed linear models. Genotype effects, i.e., the response of genotypes across all environments, were modeled as exchangeable samples from a normal distribution with variance {sigma}{gamma}2:

Formula 1

Genotype x environment interaction effects, {delta}ij, were defined as exchangeable samples from a normal distribution with the variance of genotype x environment interaction effects as a function of environment and genotype:

Formula 1

The subscripted notation on the variance of genotype x environment interactions indicates that every genotype x environment combination had an unique variance of the interaction, just as the variance of observations (i.e., the error variance) had an unique value for each genotype x environment combination. The observational, or error, variance is the last remaining parameter in the first level of the Bayesian hierarchy requiring specification of a prior. Residual variances, {sigma}{varepsilon}(ij)2, were modeled with a generalized linear model using a natural-log link function as suggested in numerous contexts in past studies (Leonard, 1975; Cook and Weisberg, 1983; Nair and Pregibon, 1988; Leonard and Hsu, 1992; Verbyla, 1993; Aitkin, 1987; Smyth, 1989; Foulley et al., 1992):

Formula 1

The quantity a + agi + aej is a linear additive model for the natural logarithm of the residual variance that provides the expected value of the variance (through the natural log link function) as a function of the parameters where a = average natural logarithm of residual variances, agi = genotype effect on the natural logarithm of residual variances, and aej = environment effect on the natural logarithm of residual variances.

The parameter a conditions a sort of average variance across all genotypes and environments. The parameter agi describes the degree to which the residual variance tends to be higher for observations on genotype i, (positive values of agi for an increased logvariance) or to which the residual variance tends to be lower for observations on genotype i (negative values of agi for reduced logvariance). Likewise, values of the parameters aej describe the degree to which observations in environment j tend to have larger (positive aej) or smaller (negative) residual variances. The marginal, or "mean" residual variance across all genotypes and environments, was predicted by exp(a) (Table 2). On taking expectations with respect to environment j, the marginal error variance for that environment, averaging over a random sample of many genotypes, was exp(a + aej). Likewise, the marginal error variance across a random sample of environments for genotype i was exp(a + agi). The predicted error variance for a specific genotype i observed in environment j was exp(a + agi + aej).


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  		Table 2. Tabular representation of individual and marginal error variances given the Bayesian prior specification in the materials and methods section.

 
To complete the Bayesian model specification for residual variances, the parameters (a, agi, and aej) in the linear model of residual logvariances require prior distributions, which have been specified as follows: a ~ N(0,107), agi | {sigma}ag2 ~ N(0, {sigma}ag2), and aej | {sigma}ae2 ~ N(0, {sigma}ae2).

The parameter a was given a normal prior with large variance to make it a noninformative prior so that the natural logarithm of residual variances is almost entirely data driven. Genotype effects, agi, and environment effects, aej, on log-variances were each assumed to be exchangeable samples from normal distributions with variances {sigma}ag2 and {sigma}ae2, respectively. Variances {sigma}ae2 and {sigma}ag2 express the degree of heterogeneity of residual variances. Large heterogeneity of residual variances among environments would be reflected in a large variance, {sigma}ae2, of environment effects. Homogeneous residual variances among environments would correspond to {sigma}ae2 = 0, and all aej parameters having values of zero. Thus, a test of {sigma}ae2 > 0 is a test for heterogeneity of residual variances among environments (and likewise for {sigma}ag2 with genotypes). To complete the hierarchical model for the residual variances, variances of genotype and environment effects on log-variances, {sigma}ae2 and {sigma}ag2, were given noninformative priors: {sigma}ae2 ~ IG(0.001,0.001), and {sigma}ag2 ~ IG(0.001,0.001).

The inverse gamma distribution with shape and scale parameters of 0.001 was chosen for two reasons. First, it is a conjugate prior, which has computational advantages for complex models, which will be described in slightly more detail when computation is described later. Second, the inverse gamma prior with shape and scale parameters of 0.001 is relatively flat, so that it is a noninformative prior. Thus, noninformative priors were assigned to the three hyperparameters a, {sigma}ag2, and {sigma}ae2 so that the expected natural logarithm of the residual variance (a) and the degree of heterogeneity x genotype and x environment are data driven (no information provided a priori).

Genotype effects, {gamma}i, and genotype x environment interaction effects, {delta}ij, defined in the second level of the Bayesian hierarchy had priors that were conditioned on variance parameters, {sigma}{gamma}2 and {sigma}{delta}(ij)2, respectively. Variance among genotypes was modeled by a single variance, {sigma}{gamma}2, which had a noninformative prior of the form: {sigma}{gamma}2 ~ IG(0.001,0.001).

Variances of genotype x environment interactions were modeled as a function of genotype and environment effects in a linear additive model as in the case of the residual variance:

Formula 1
where b = intercept, bgi = effect of ith genotype on the log-variance, and bej = effect of the jth environment on the log-variance.

The prior construction for effects b, bgi, and bej was the same structure as the prior on the model effects for the residual variances: b ~ N(0,107), bgi | {sigma}bg2 ~ N(0, {sigma}bg2), and bej | {sigma}be2 ~ N(0, {sigma}be2).

Priors on the variances of genotype and environment effects on variances were chosen to be noninformative as in the case of corresponding effects on residual variances: {sigma}be2 ~ IG(0.001,0.001), and {sigma}bg2 ~ IG(0.001,0.001).

The model and prior distributions described so far represent the full model for heterogeneous variances. The entire prior construction for the heterogeneous model was designed to represent a completely objective Bayesian analysis, in that no prior data is used to inform the prior distributions. Rather, the prior distributions for parameters of interest were defined using hyperparameters which themselves had noninformative prior distributions so that the specific location and scale of the prior distributions were completely data driven. After estimating parameters in this model for the oat data set, it was determined that the parameter {sigma}ag2 for the variance of genotype effects on the error variance was very nearly zero, and thus was dropped from the model to form a reduced model. Results from the reduced model and the full model were quite similar. Both the full model and the reduced model were compared with a model with homogeneous variances which corresponds to the classical model commonly used. Effects in the homogeneous model had normal priors with constant variances: {gamma}i | {sigma}{gamma}2 ~ N(0, {sigma}{gamma}2), {delta}ij | {sigma}{delta}2 ~ N(0, {sigma}{delta}2), and {varepsilon}ij(k) | {sigma}{varepsilon}2 ~ N(0, {sigma}{varepsilon}2).

At the final level of the hierarchy in the homogeneous model, the variance components had noninformative inverse gamma priors: {sigma}{gamma}2 ~ IG(0.001,0.001), {sigma}{delta}2 ~ IG(0.001,0.001), and {sigma}{varepsilon}2 ~ IG(0.001,0.001).

All parameters were obtained via Markov Chain Monte Carlo simulation using the Bayesian Gibbs Sampling software WinBUGS (Spiegelhalter et al., 2004). WinBUGS uses Gibbs sampling steps for those parameters for which analytical forms exist for the full conditional posterior distribution. Some of the parameters in our model will have analytically obtainable conditionals as they were constructed with conjugate priors. However, the generalized linear model approach to modeling variances is a nonconjugate construction, meaning that an analytical form does not exist for the full conditional posterior distribution. For such parameters, a Metropolis-within-Gibbs approach is used in WinBUGS to generate random samples from the posterior. Hence, where it seemed biologically reasonable, we tried to use conjugate prior constructions, as directly sampling from an analytically obtained posterior is more efficient than simulating random samples using Metropolis approaches.

In our evaluation of all three models, we used estimates of some parameters generated in SAS (SAS Institute, Cary, NC) Proc Mixed (genotypic effects, block means, and some variances) as starting values while many parameters were set to zero. Two Markov chains were obtained for each model with a burn-in period of 1000 iterations. After the burn-in period, parameter values were saved from every 20th iteration until 5000 parameter values per chain had been saved (100 000 total iterations) for a total of 10 000 quasi-independent parameter values. Inspection of autocorrelation plots for saved parameter values (every 20th iteration) suggested some correlation between successive values, but almost no correlation between a parameter value with the third successive parameter value in the chain.


   RESULTS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS
DISCUSSION
 REFERENCES
 
Heterogeneity of Variances
The hyperprior parameters {sigma}ae2 and {sigma}be2 were the variances of the effects aej and bej of individual environments on the logarithms of error variances and genotype x environment interaction variances, respectively. Thus, the greater the values of {sigma}ae2 and {sigma}be2, the greater the heterogeneity of variances among environments. The posterior distributions of both of these parameters had essentially zero probability mass at or near zero (Fig. 1 ), indicating that we can reject the null hypothesis of homogeneous variances among environments for either error or genotype x environment interaction or error. The heterogeneity of error variances in these data can also be seen by examining posteriors for individual environments. For example, Environments 10 and 15 had substantially higher error variances than most other environments (Fig. 2 ). The 95% posterior support intervals for Environments 10 and 15 overlap with few other environments (Fig. 2). Conversely, some environments, 16 and 31 in particular, had substantially lower error variances than most environments included in the analysis. Separation among environments was not as clear with respect to variances of genotype x environment interactions (Fig. 3 ), but it was still apparent that some environments tended to have larger than average variability in genotype x environment interactions, Environments 6, 15, 28, and 33 in particular. Genotypes also had a marked impact on variances of genotype x environment interactions (Fig. 1 and Fig. 4 ), with four or five genotypes appearing to have greater variability in their interactions than other genotypes. However, genotypes had almost no impact on the error variance. The posterior support interval for the variance, {sigma}ag2, of effects of individual genotypes effects, agi, on the error variance, was very close to zero (Fig. 1). In addition, very little variation could be seen among posterior support intervals of marginal error variances among genotypes (Fig. 5 ).


Figure 1
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  		Fig. 1. Posterior distributions of variances of heterogeneity parameters, {sigma}ag2, {sigma}ae2, {sigma}bg2, and {sigma}be2. A value of zero for one of these parameters would indicate homogeneity of variance across the corresponding factor of genotype or environment.

 

Figure 2
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  		Fig. 2. Boxplots of marginal posterior support intervals on error variances for individual environments. Boxes represent central 50% intervals for variances, whiskers represent central 95% support intervals, and the lines at the centers of the boxes represent the medians of the marginal posterior parameter distributions. See Table 1 for locations and years of specific environments.

 

Figure 3
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  		Fig. 3. Boxplots of marginal posterior support intervals on genotype x environment interaction variance estimates for individual environments. Boxes represent central 50% intervals for variances, whiskers represent central 95% support intervals, and the lines at the centers of the boxes represent the medians of the marginal posterior parameter distributions. See Table 1 for locations and years of specific environments.

 

Figure 4
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  		Fig. 4. Boxplots of marginal posterior support intervals on genotype x environment interaction variances for individual genotypes. Boxes represent central 50% intervals for variances, whiskers represent central 95% support intervals, and the lines at the centers of the boxes represent the medians of the marginal posterior parameter distributions.

 

Figure 5
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  		Fig. 5. Boxplots of marginal posterior support intervals on error variance estimates for genotypes. Boxes represent central 50% intervals for variances, whiskers represent central 95% support intervals, and the lines at the centers of the boxes represent the medians of the marginal posterior parameter distributions. See Table 1 for locations and years of specific environments.

 
Model Comparison
Variance components estimated in the homogeneous variance model were numerically larger than marginal variance estimates in the heterogeneous variance model for all three random sources of variability, genetic variance, genotype x environment interaction variance, and error variance (Table 3). The larger values of marginal variances in the homogeneous model than the variance estimates in the heterogeneous model was a surprising result to the authors. Bayesian posterior probabilities for these comparisons were computed by a direct comparison of individual values in the Monte Carlo Markov chains between models. The ith parameter value in the homogeneous model chain was compared to the ith parameter value in the heterogeneous chain. The proportion of times that the parameter value for the homogeneous model was greater than the parameter value for the heterogeneous model provided an estimate of Bayesian posterior probability that the parameter was greater in the homogeneous model than in the heterogeneous model. The probability that marginal variance of genotype x environment interactions was larger under the homogeneous variance model than the heterogeneous variance model was 0.986 (Table 3). Thus, there was a 98.6% probability that the marginal genotype x environment variances (i.e., the expected variance of genotype x environment variance) was larger in the homogeneous model than the corresponding component in the heterogeneous model. The corresponding probabilities for error variance and genotypic variance were smaller, 0.861 and 0.688, respectively. In addition to variances, we computed repeatabilities under both models. Because of the larger variance estimates in the homogeneous variance model, the larger estimate of variance of genotype x environment interactions in particular, marginal repeatabilities tended to be larger under the heterogeneous model (Table 4). However, the probability that the true repeatability was larger under the heterogeneous model than under the homogeneous model was less compelling than for variance components themselves.


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  		Table 3. Estimates and 95% posterior support interval bounds of genetic variance, marginal genotype x environment interaction variance (G x E), and Error variance under models with homogeneous versus heterogeneous variances among environments and genotypes. The Probability indicates the probability that the variance obtained under the heterogeneous model was smaller than that obtained under the homogeneous model.

 

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  		Table 4. Estimates and 95% posterior support intervals of Repeatability on a single plot basis, an environment-mean basis, and on a five-environment mean basis. The Probability indicates the probability that the repeatability obtained under the heterogeneous model was larger than that obtained under the homogeneous model.

 
Bayesian point estimators of genotype effects, i.e., cultivar performance, {gamma}i, were computed as the mean of the {gamma}i effects across the 10 000 random draws from the marginal posterior distributions in both the homogeneous variance model and the heterogeneous variance model. Comparison of the genotype effects between the two models revealed that the two models provided highly correlated but nevertheless distinct estimators of genotype effects (Fig. 6 ).


Figure 6
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  		Fig. 6. Comparison of point estimators of genotype effects, {gamma}i, under the homogeneous variance model and the heterogeneous variance model. Point estimators of genotypic effects were obtained from the means of 10 000 samples from the Bayesian Posterior.

 
Variances of Genotype x Environment Interaction and Performance
Estimated genotypic effects had a negative correlation with estimated square roots of genotype x environment interaction variances of –0.33 (Fig. 7 ). The correlation was tested by a permutation test with 100 000 random permutations of the data with 321 permutations having a correlation greater than 0.33 or less than –0.33 for an estimated p value of 0.00321. The relationship shows that cultivars with higher levels of performance also had lower variances of genotype x environment interactions, i.e., were more stable across environments. As another indication of the relationship of the variance of genotype x environment interactions with performance, it was observed that the genotypes with the largest estimated variance of genotype x environment interactions tended to have the earliest and latest heading dates (Fig. 8 ). The relationship between variance of genotype x environment interaction and heading dates suggests the cultivars with heading dates closest to the center of the distribution tended to be the most stable, whereas the earliest and latest genotypes tended to have larger variances of genotype x environment interaction variance. In this case, it appeared that heading date was predictive of the genotype x environment interaction variance, or stability, of cultivars.


Figure 7
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  		Fig. 7. Bayesian point estimates of genotypic value versus the square roots of Bayesian point estimators of square roots of genotype x environment interaction variances. Bayesian point estimators of variances were taken as the medians of marginal posterior distributions.

 

Figure 8
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  		Fig. 8. Square roots of Bayesian point estimates of genotype x environment interaction variances versus heading dates for individual genotypes. Bayesian point estimates of variances were the medians of the marginal posterior distributions. The curve is a fitted quadratic regression equation for the Bayesian point estimators of the variance regressed on heading date (regression was performed on the original variance scale and then the curve was transformed for plotting purposes).

 

   DISCUSSION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS
 DISCUSSION
REFERENCES
 
Heterogeneity of variances of error and genotype x environment interactions has long been recognized as an important assumption violation in the analysis of plant cultivar evaluations (Comstock and Moll, 1963). Despite decades of work, there is still debate on how to deal with this violation. Some recent authors advocate retaining the assumption of homogeneity by using unweighted means (Bernardo, 1992; Huhn, 1997), while others recommend using weighted approaches to account for heterogeneity of variances (Frensham et al., 1997). The decision as to whether to ignore heterogeneity of variances is a choice between the lesser of two evils: (i) the error incurred by assuming homogeneous variances when they are in fact heterogeneous or (ii) the error incurred from estimating many variances, each from comparatively few observations, to account for heterogeneity. We propose that hierarchical models and Bayesian estimation offer a potential solution to this dilemma. Bayesian estimation methods have two advantages over frequentist estimation methods of particular importance to this problem: (i) incorporating shrinkage estimation into complex models and (ii) accounting for error of estimation of all parameters in a model regardless of model complexity. A key point to both advantages is that computational methods can be used to fit Bayesian models even in cases when analytical solutions do not exist.

Shrinkage estimation is a commonly used term to refer to estimation of random effects, as in best linear unbiased prediction (BLUP) of multivariate normal means in classical mixed linear models (Henderson, 1973; Henderson, 1984; Robinson, 1991). In the Bayesian (or empirical Bayesian) context, shrinkage estimation is described with the term "information borrowing" (Louis and Shen, 1999; Morris, 1983) to emphasize the "borrowing strength" among levels of a random factor to improve all estimators. The hierarchical model proposed in this paper takes advantage of shrinkage estimation in the estimators of both means and variances, in exactly the fashion outlined in a one-way classification by Leonard (1975). Leonard (1975) described a Bayesian hierarchical model with multivariate normal structure for means and for natural logarithms of variances that clearly demonstrates important properties of shrinkage estimators when applied to both means and variances. In particular, estimators of individual levels of a random factor are weighted averages of the data on a particular level and the expected value, or mean, of all levels. Leonard (1975) considered a one-way random classification in which each level of the classification had a random mean and a random variance that were modeled with exchangeable random variables, i.e.: yij | {theta}i, {sigma}i2 ~ N({theta}i, {sigma}i2), where {theta}i | {sigma}i2 ~ N(0, {sigma}i2) and {sigma}i2 = e{alpha}i, with {alpha}i | {sigma}{alpha}2 ~ N(0, {sigma}{alpha}2).

Each level in this one-way classification is characterized by a random location parameter (or mean, where the mean is a parameter), {theta}i, and a random within-level error variance, {sigma}i2. The random within-level error variances were normally distributed on the logarithmic scale (as in our model), i.e., {sigma}i2 = e{alpha}i implies that ln({sigma}i2) = {alpha}i, with the {alpha}i being normally distributed random variables with variance {sigma}{alpha}2.

Leonard's (1975) model illustrates the inherent shrinkage estimation of means and variances as in our model. Leonard (1975) obtained relatively straightforward conditional estimators of the true level means, {theta}i:

Formula 1
where Formula 1i = sample mean of level i, ni = sample size of level i, and

Formula 1
and of the natural logarithms, {alpha}i, of the variances, {sigma}i2:

Formula 1
where si2 = Formula 1, and Formula 1 = Formula 1 with {rho}{alpha}(i) = Formula 1.

Shrinkage estimators are "weighted averages" of a "prior" estimator, which in this case is an estimator of the mean of all levels (Formula 1 and Formula 1), combined with the data on a particular level in the form of either of the sample mean of level i, Formula 1i, or the natural logarithm of the sample variance, ln(si2). The term "prior estimator" is used here to mean an estimator derived from the prior probability distribution. Weights on the prior estimators (Formula 1 and Formula 1) and on the data [Formula 1i and ln(si2)] are the relative precisions of the prior estimator versus the data as estimators of an individual level. For example, {sigma}{theta}2 is the relative precision of the prior estimator, Formula 1, as an estimator of {theta}i, while ni/{sigma}i2 is the relative precision of the sample mean, Formula 1i, as an estimator of {theta}i. Neither the prior estimator (e.g., Formula 1) nor the data (e.g., Formula 1i) are perfect estimators of true mean of an individual level (e.g., of {theta}i). The best compromise between the two estimators is the weighted combination of the prior estimator and the data. This weighted combination of prior estimator and data provides the best available estimator, given the prior distribution and the data, meaning the combined estimator has the lowest variance among possible estimators. With respect to estimation of within-level variances, 1/2{sigma}{alpha}2 and 1/2ni are relative precisions for the prior estimator and the data as estimators of {alpha}i, the natural logarithm of the true sampling variance for the ith level, {sigma}i2. These weights provide an optimal combination of the prior estimator, Formula 1, and the data, ln(si2), in obtaining estimators of true variances for individual levels.

The weight, {rho}{theta}(i), placed on the sample mean, Formula 1i, in the Bayesian estimator of {theta}i can be expressed as {rho}{theta}(i) = Formula 1, which is a function of two quantities, the sample size, ni, and the repeatability of individual observations collected from the ith level, ri2 = Formula 1. The weights {rho}{theta}(i) decrease with decreasing sample size and decreasing repeatability, and approach unity as repeatability approaches one or sample size gets very large (Fig. 9 ). At low repeatability (when the error variance within levels is high relative to the variance among true level means, {theta}i) and low sample size, the values of {rho}{theta}(i) can become quite small. For example, with r2 = 0.25 and sample size of 2, the weight on the sample mean is 0.4 and the weight placed on the prior estimator is 0.6, i.e., the best estimator of the true value of {theta}i derives only 40% of its value from data on the ith sample, Formula 1i, (Fig. 9). The term "shrinkage estimator" comes directly from the weighting of the prior estimator and the data. As the weight on the data decreases and the weight on the prior estimator increases, the combined estimator gradually "shrinks" toward the prior estimator.


Figure 9
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  		Fig. 9. Weight placed on individual sample means in combined shrinkage estimators versus repeatability on an observation basis. Weights are shown at three different sample sizes for individual levels.

 
Similarly, weights on the data for level specific variances, si2, can be expressed as a function of sample size and the variance of the {alpha}i:

Formula 1

The quantities 1/2ni and 1/{sigma}{alpha}2 are the relative precisions of the data, ln(si2), and of the prior estimator, Formula 1, as estimators of {alpha}i. The weight, {rho}{alpha}(i), on the natural logarithm of the sample variance, increases with increasing sample size and increasing heterogeneity (as 1/{sigma}{alpha}2} gets smaller) (Fig. 10 ). With decreasing heterogeneity, i.e., as {sigma}{alpha}2 approaches zero, weights on individual levels, {rho}{alpha}(i), approach zero so that the best estimator of individual sample variances approaches a pooled error variance estimated by eFormula 1 (Fig. 10). Such a result has intuitive appeal because with decreasing heterogeneity (as {sigma}{alpha}2 approaches zero) we expect that all individual levels have nearly the same sampling variance and we desire a single pooled estimator of the sample variance. Whereas, with increasing heterogeneity of sample variances, we increasingly place more weight on individual levels through si2 and less weight on a pooled estimator through Formula 1.


Figure 10
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  		Fig. 10. Weight placed on natural logarithms of individual-level samples variances in combined shrinkage estimators of within-level variances versus the variance of natural logarithms of within-level variances. Weights are shown at three different sample sizes for individual levels.

 
The variance of the {alpha}i, {sigma}{alpha}2, quantifies the degree of heterogeneity and has a simple interpretation because of the logarithmic scale. Approximately 95% of true within-level sample variances, {sigma}i2, will be contained in the interval (eFormula 1 + 1.96{sigma}{alpha},eFormula 1–1.96{sigma}{alpha}) = ({sigma}Formula 12e1.96{sigma}{alpha},{sigma}Formula 12e – 1.96{sigma}{alpha}), where {sigma}Formula 12 = eFormula 1. In other words, the quantities e1.96{sigma}{alpha} and e–1.96{sigma}{alpha} provide approximate bounds for 95% of true within-level variances, {sigma}i2, as proportions of the prior estimate, {sigma}Formula 12. Bounds for within-level variances corresponding to values of {sigma}{alpha}2 ranging from 0.015625 to 4.0 are shown in Table 5. In our study, variances of bgi and bej for heterogeneity of genotype x environment interaction variances, each had posterior modes around 0.4 suggesting that 95% of true variances of genotype x environment interaction variances for a given genotype or a given environment ranged from approximately 0.53 to 1.88 times the posterior mode of genotype x environment interaction variance. In our study, little shrinkage was expected for estimators of error variances among environments because of large sample sizes. However, with respect to genotype x environment interaction variances, some genotypes were observed in only 5 environments (1 yr). For a genotype observed in only five environments, five is the effective sample size for that genotype that determines the amount of shrinkage of the genotype-specific variance estimator, and thus fairly substantial shrinkage could be expected for the genotype-specific variance of genotype x environment interactions.


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  		Table 5. Proportional 95% upper and lower bounds for variance components as a function of the variance of natural logarithms of the component. Bounds are expressed as a proportion of the marginal (expected) variance.

 
In Bayesian terms, the parameter {sigma}{alpha}2 is a hyperprior parameter that quantifies the amount of heterogeneity of variances and subsequently conditions how level-specific sample variances are estimated. Large heterogeneity (large {sigma}{alpha}2) results in more weight on data from individual levels in the estimates of the {sigma}i2; decreasing heterogeneity (and smaller {sigma}{alpha}2) results in more weight placed on the pooled variance estimator. Hence, these estimators are data-driven compromises between complete-pooling and level-specific variances. The problem with using weighted means historically has been that the weights, computed from individual variances, were unknown, had to be estimated, and had unknown sampling distributions (Cochran, 1954; Harville, 1985; Kackar and Harville, 1984; Yates and Cochran, 1938). Estimating the variances introduces additional (unquantifiable) error to the estimation of means that may offset any advantage incurred by weighting the means. With the Bayesian compromise estimators, if there is little information in the data to support heterogeneity, i.e., {sigma}{alpha}2 is small and or sample sizes within individual levels are small, individual variances are all very close to a pooled estimate and means across environments or genotypes are nearly unweighted. Whereas, with larger heterogeneity and larger sample sizes within individual levels, individual variance estimators are more specific to individual levels (environments or genotypes) and there is greater weighting of combined estimators of means according to level-specific variances. Hence, in the Bayesian hierarchy, estimators of means are neither "weighted" nor "unweighted" in a strict sense, but can be thought of as an optimal compromise between "weighted" and "unweighted" with the degree of weighting proportional to the information in the data to support estimation of heterogeneous variances.

Models of heterogeneous variances using a logarithmic link function have been proposed in the frequentist framework in numerous contexts (Verbyla, 1993; Smyth, 1989; Aitkin, 1987; Cook and Weisberg, 1983; Smyth, 2002; Foulley et al., 1992). Thus, heterogeneity of variances can easily be modeled and frequentist point estimators can be obtained. However, because of the complexity of these models, exact sampling distributions of parameter estimators do not exist. Hence, it is very difficult to obtain exact sampling distributions for any of the parameters in these models. Bayesian estimation approaches actually suffer the same problem, in that analytical solutions do not exist for posterior distributions. However, the introduction of Markov Chain Monte Carlo methods to Bayesian analysis has provided a way to deal with complex models (Gilks et al., 1996; Tierney, 1994). Markov Chain methods have provided a way to make random draws from the marginal posterior distributions of parameters regardless of whether analytical forms for the density function exist or not. Rather than requiring exact solutions for the means, modes, and quantiles of the posterior, we can simulate a large number of random draws from the posterior and summarize the simulated parameter values. The summarization of the simulated posterior correctly accounts for the error of estimation of all parameters; each simulated posterior distribution represents an "average" over the joint posterior distributions of all other parameters in the model so that any uncertainty of estimation of other parameters is fully accounted for in both the mean or the mode of simulated posteriors and in the dispersion of the posterior. This "Bayesian averaging" over the uncertainty of estimation is a very desirable property of Bayesian analysis. If the data have very poor information on particular parameters, and they are poorly estimated, the remaining parameters in the models are "averages" over the full range of plausible values conditioned by the prior distributions. If any parameters depend strongly on poorly estimated parameters, their posteriors will be relatively flat, i.e., the posterior will be a direct indication that the available precision on the parameter is very poor. Such a property does not exist in Maximum Likelihood (including Restricted Maximum Likelihood as a subclass of Maximum Likelihood) approaches because very often Likelihood-based estimators are those estimators that maximize part of the likelihood conditional on estimators of other parameters. When estimators are plugged into the likelihood as known values, no account is taken of the fact that estimators, and not known values, were used.

The primary goal of our work was to provide weighted estimates of performance as advocated by early workers for situations with heterogeneous variances (Cochran, 1954; Shukla, 1972b; Yates and Cochran, 1938). The statistical advantage, in a practical sense, to modeling heterogeneity of error variances is that greater weight is placed on those environments with higher quality data as defined by lower error variance. In modeling the heterogeneity of genotype x environment interaction variances among environments and genotypes, we can account for differences in both environments and genotypes in the variability of responses to environments. Those environments that are unique, and thus have high variance of genotype x environment interaction, will have lower weight placed on them in the estimation of average genotypic values. Greater weight will be placed on environments in which cultivars respond most like the norm of response across all environments as defined by the low variance of genotype x environment interaction. The weight, i.e., the degree of shrinkage, for a particular genotype also depends on the variance of genotype x environment interactions for that particular genotype. Of particular interest is the impact on unstable genotypes, i.e., those with large variances of genotype x environment interactions. For genotypes with high genotype x environment interaction variance (relative to other cultivars) we have much less certainty of their performance across environments because their relative performance is highly variable across environments. In our model, the estimates of average genotypic value have smaller weights on the data for particular cultivars (because of their larger genotype x environment interaction variance) and thus estimators of genotypic value tend to have greater shrinkage toward the mean. In other words, the more unstable a cultivar is, the less certain we are that it is any different from any other genotypes (different from the mean of other genotypes).

Shrinkage can be examined under alternative models by examining the absolute values of genotype effects, {gamma}i: greater shrinkage will result in smaller absolute values. By comparing the absolute values of genotype effects between the homogeneous variance model and the heterogeneous variance model, we can examine the degree of differential amounts of shrinkage under homogeneous versus heterogeneous assumptions. We computed these absolute values under the homogeneous variance model and the heterogeneous variance model. We then subtracted the absolute value of the estimator under the homogeneous model from the absolute value of the estimator under the heterogeneous model to obtain a shrinkage "deviation." We have taken the shrinkage deviation as an estimate of the difference in shrinkage between the two models. Negative values of the shrinkage deviations would correspond to greater shrinkage in the heterogeneous model, whereas positive values would correspond to greater shrinkage in the homogeneous model. The deviations were plotted against square roots of genotype x environment interaction variances estimated under the heterogeneous model to determine if there was a relationship between variance and shrinkage in these data (Fig. 11 ). As the genotype-specific variance of genotype x environment interaction variance (estimated in the heterogeneous model) increased, the difference between absolute values became more negative, suggesting greater shrinkage in the heterogeneous model (Fig. 11). Likewise, for genotypes with smaller genotype x environment interaction variances, there appeared to be less shrinkage in the heterogeneous model, i.e., there is greater apparent certainty of the performance of these genotypes across environments.


Figure 11
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  		Fig. 11. Deviations versus the square root of the Bayesian point estimates of square roots of genotype x environment interaction variance. Bayesian point estimates were the medians of the marginal posterior distributions of the parameters. Deviations were defined as the absolute values of the genotype effects in the homogeneous model minus the absolute values of genotype effects in the heterogeneous model.

 
The weighting of cultivar performance estimates by the inverse of the genotype x environment interaction variance, i.e., stability, provided an automatic penalty to unstable genotypes, because their estimators were shrunken closer to the mean more than stable cultivars. We have used the term "penalty" to refer to a performance estimate that is shrunken toward the mean; for low-yielding cultivars, the penalty is negative in that unstable cultivars would also be shrunken toward the mean and thus have increased estimators of performance. The penalty for unstable cultivars with high performance values is desirable for breeders and growers because a cultivar that performs well in some environments but not others may still have high estimated performance. However, because of the instability of its performance, its high estimated performance may be a poor indicator of its true performance relative to other cultivars. The larger shrinkage applied to unstable cultivars than to stable cultivars adjusts for the greater uncertainty in performance estimates of unstable cultivars.

We have described an application of Bayesian analysis to arrive at a more parsimonious answer to the question of whether to use weighted or unweighted means in multienvironment yield trials. Our analysis provides a data-driven compromise between weighted and unweighted analysis in which the relative degree of weighting by individual variances versus pooling of variances is determined by the information content in the data to support estimation of separate variances. The results of our analyses suggested that the heterogeneous model may be advantageous for predicting future cultivar performance. In our comparison between models, we found that the heterogeneous model had a slightly higher estimated repeatability than the model with homogeneous variances. This model-based comparison suggests the heterogeneous model produces better, i.e., more repeatable, predictions. This observation does not, however, guarantee that the heterogeneous variance model is truly the better model. The true parameter values and their true distributions cannot be known; that one model happens to have higher estimated repeatability in this situation does not prove it is a better model. However, we found very strong evidence for heterogeneity of both error and genotype x environment interaction variances. The heterogeneity of variances revealed in the heterogeneous model demonstrates that there is a strong assumption violation in the homogeneous model. Nonetheless, additional work is needed. First, cross validation approaches would be valuable to provide a better test of the improvement in predictive value of the heterogeneous model. Second, analyses of additional data sets are needed to determine if the apparent superiority of the approach would be found in other datasets. Third, alternative forms of the prior distribution of heterogeneous variances need to be evaluated to determine how best to represent the distribution of true variances. Finally, simulation would be valuable to test the model under known conditions to determine under what parameter values and with what samples sizes the model will perform well and when it will not perform well. Thus, while the analyses presented here do not allow us to recommend the best approach, they do show very strong evidence for heterogeneity of variances. Furthermore, we have shown that modeling heterogeneity can illuminate interesting biological relationships (e.g., between stability and heading date), which may have useful properties for the estimation of marginal genotypic values (e.g., the relationship between stability and shrinkage). In addition to more thoroughly testing the Bayesian approach to modeling heterogeneous variances, it is noteworthy that Bayesian approaches have been advocated for addressing other issues in multienvironment yield trials such as use of environment specific covariates (Theobald et al., 2002), modeling of multiplicative genotype x environment interactions (Viele and Srinivasan, 2000) and spatial analysis (Besag and Higdon, 1999). Comparisons of our approach to these other applications would also be quite valuable.


   NOTES
TOP
 NOTES
ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
Mention of trade names or commercial products in this article is solely for the purpose of providing scientific information and does not imply recommendation or endorsement by the U.S. Department of Agriculture or Iowa State University.

Received for publication February 23, 2005.


   REFERENCES
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 




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