Bayesian Modeling of Heterogeneous Error and Genotype 3 Environment Interaction Variances

An important assumption in the analysis of multienvironment cultivar trials is homogeneity of error and genotype 3 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 3 environment interaction variances. A Bayesian model for multienvironment yield trials that includes a linear model for error and genotype 3 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 3 environment interaction variances were heterogeneous among environments and genotypes. Incorporation of heterogeneity of variances significantly decreased estimates of marginal error, genotypic, and genotype 3 environment variance components, with the largest change being a reduction in the marginal genotype 3 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 3 environment interaction variances were correlated with estimated genotypic yields and heading dates, providing biological validity to our estimates of genotype-specific estimators of genotype 3 environment interaction variances as stability estimators. 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 3 environment interaction variances across environments and across genotypes. In their classic review of genotype 3 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 macroenvironment 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 3 environment interaction variances as a measure of cultivar stability. Genotype-specific microenvironment and genotype 3 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 3 environment matrix, estimation and hypothesis testing procedures performed on the row (genotypic) dimension of the matrix could J.W. Edwards, USDA Agricultural Research Service, Corn Insects and Crop Genetics Research Unit, Department of Agronomy, Iowa State University, Ames, IA 50011; J.-L. Jannink, Department of Agronomy, Iowa State University, Ames, IA 50011. 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 23 Feb. 2005. *Corresponding author (jode@