No abstract

The additive main effects and multiplicative interaction (AMMI) model is frequently applied in plant breeding for studying the genotype × environment (G × E) interaction. One of the main difficulties related to this method of analysis is the incorporation of inference to the bilinear terms that compose the biplot representation. This study aimed to incorporate credible intervals for the genotypic and environmental scores in the AMMI model by using an informative prior for the genotype effect. This approach differs from the Bayesian methods that have been presented so far, which assume the same restrictions as the fixed effects model. The method was exemplified by using data from a study with 55 maize hybrids in nine different environments for which variable being studied was the yield of unhusked ears. Our results demonstrated that the credible intervals allowed for the identification of genotypes and environments that did not contribute to the G × E interaction. In addition, it facilitated recognition of homogeneous subgroups of genotypes and environments (with respect to the effect of the interaction) and the adaptability of genotypes to specific environments of great interest to breeders. The posterior distributions of singular vectors were bimodal but with the same density peaks in absolute value. This reflects the arbitrary choice of signs of the main component that was used in different mathematical algorithms. Although our data set was based on unrelated single cross hybrids, the choice of genotypes as random effects enabled the Bayesian AMMI to accommodate the additive and nonadditive relationship matrices. Additionally, the flexibility of the analysis facilitated working with unbalanced data and the incorporation of heterogeneity of variances.

Linear-bilinear models, especially the additive main effects and multiplicative interaction (AMMI) model, are widely applicable to genotype-by-environment interaction (GEI) studies in plant breeding programs. These models allow a parsimonious modeling of GE interactions, retaining a small number of principal components in the analysis. However, one aspect of the AMMI model that is still debated is the selection criteria for determining the number of multiplicative terms required to describe the GE interaction pattern. Shrinkage estimators have been proposed as selection criteria for the GE interaction components. In this study, a Bayesian approach was combined with the AMMI model with shrinkage estimators for the principal components. A total of 55 maize genotypes were evaluated in nine different environments using a complete blocks design with three replicates. The results show that the traditional Bayesian AMMI model produces low shrinkage of singular values but avoids the usual pitfalls in determining the credible intervals in the biplot. On the other hand, Bayesian shrinkage AMMI models have difficulty with the credible interval for model parameters, but produce stronger shrinkage of the principal components, converging to GE matrices that have more shrinkage than those obtained using mixed models. This characteristic allowed more parsimonious models to be chosen, and resulted in models being selected that were similar to those obtained by the Cornelius F-test (α = 0.05) in traditional AMMI models and cross validation based on leave-one-out. This characteristic allowed more parsimonious models to be chosen and more GEI pattern retained on the first two components. The resulting model chosen by posterior distribution of singular value was also similar to those produced by the cross-validation approach in traditional AMMI models. Our method enables the estimation of credible interval for AMMI biplot plus the choice of AMMI model based on direct posterior distribution retaining more GEI pattern in the first components and discarding noise without Gaussian assumption as requested in F-based tests or deal with parametric problems as observed in traditional AMMI shrinkage method.

In analyses of multienvironment trials, it is common to assume homogeneity of variances in additive main effect and multiplicative interaction (AMMI) models for further inferences about the genotypes × environment interaction (GEI). However, it is not always reasonable to adopt such an assumption because it could mislead the evaluation and selection of the best genotypes. In this context, modeling the heterogeneity of variance jointly with GEI models has been of particular interest in plant breeding, since the experimental accuracy may float across the trial network. In this study, we used the Bayesian AMMI model in real and simulated frameworks to study GEI effects in the presence of heterogeneous variances (BAMMI‐H) across environments, highlighting the differences that can arise when this scenario is neglected. The findings indicate that neglecting the differences among the experimental variances across environments can influence the biplot precision and the conclusions regarding adaptability and stability. The main differences observed between the naive AMMI (assuming homogeneity) and BAMMI‐H biplots were related to the biplot precision for the genotypic and environment scores and the ability to recover information about experimental differences among the trials in the biplot. The results observed in this study suggest the importance of taking the heterogeneity of variance into account in the AMMI analysis to select genotypes for stability and adaptability.

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.