Abstract
The genotype main effects plus the genotype × environment interaction effects model has been widely used to analyze multi-environmental trials data, especially using a graphical biplot considering the first two principal components of the singular value decomposition of the interaction matrix. Many authors have noted the advantages of applying Bayesian inference in these classes of models to replace the frequentist approach. This results in parsimonious models, and eliminates parameters that would be present in a traditional analysis of bilinear components (frequentist form). This work aims to extend shrinkage methods to estimators of those parameters that composes the multiplicative part of the model, using the maximum entropy principle for prior justification. A Bayesian version (non-shrinkage prior, using conjugacy and large variance) was also used for comparison. The simulated data set had 20 genotypes evaluated across seven environments, in a complete randomized block design with three replications. Cross-validation procedures were conducted to assess the predictive ability of the model and information criteria were used for model selection. A better predictive capacity was found for the model with a shrinkage effect, especially for unorthogonal scenarios in which more genotypes were removed at random. In these cases, however, the best fitted models, as measured by information criteria, were the conjugate flat prior. In addition, the flexibility of the Bayesian method was found, in general, to attribute inference to the parameters of the models which related to the biplot representation. Maximum entropy prior was the more parsimonious, and estimates singular values with a greater contribution to the sum of squares of the genotype + genotype × environmental interaction. Hence, this method enabled the best discrimination of parameters responsible for the existing patterns and the best discarding of the noise than the model assuming non-informative priors for multiplicative parameters.
Highlights
In multi-environmental trials (MET), it is common to observe differential genotype responses in different environments, and this phenomenon is called genotype × environment interaction (GEI)
The bilinear parameters of these models are obtained by Singular Value Decomposition (SVD) of the GEI or G+GEI matrix, which makes it possible to separate noise patterns, thereby approaching high dimensional matrices by matrices with reduced dimensions [2, 6, 7]
As observed in our results, BGGE produces some shrinkage if compared to the method of ordinary least squares estimators (OLS)
Summary
In multi-environmental trials (MET), it is common to observe differential genotype responses in different environments, and this phenomenon is called genotype × environment interaction (GEI). In the presence of GEI, genotypes with good performance in one environment may perform poorly in another, which imposes difficulties in terms of a wide selection and broad recommendation of superior cultivars. In these situations, it is prudent to group locations (or environments) into relatively homogeneous subgroups. Multiplicative (or linear-bilinear) models for fixed effects are widely applicable in the study of patterns of genotype responses between environments [4, 5] Examples of such models are the main additive effects and multiplicative interaction model (AMMI) and the genotype main effects (G) plus GEI model (G+GEI, or GGE), known as the site regression model (SREG). The bilinear parameters of these models are obtained by Singular Value Decomposition (SVD) of the GEI or G+GEI matrix, which makes it possible to separate noise patterns, thereby approaching high dimensional matrices by matrices with reduced dimensions [2, 6, 7]
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