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Bayesian Modeling of Heterogeneous Error and Genotype x Environment Interaction Variances

^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

View larger version (40K): [in a new window]   Fig. 1. Posterior distributions of variances of heterogeneity parameters, ag2, ae2, bg2, and be2. A value of zero for one of these parameters would indicate homogeneity of variance across the corresponding factor of genotype or environment.

View larger version (22K): [in a new window]   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.

View larger version (29K): [in a new window]   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.

View larger version (40K): [in a new window]   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.

View larger version (53K): [in a new window]   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.

View larger version (24K): [in a new window]   Fig. 6. Comparison of point estimators of genotype effects, 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.

View larger version (25K): [in a new window]   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.

View larger version (15K): [in a new window]   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).

View larger version (25K): [in a new window]   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.

View larger version (21K): [in a new window]   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.

View larger version (20K): [in a new window]   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.