ABSTRACT.This study aimed to analyze the robustness of mixed models for the study of genotype-environment interactions (G x E). Simulated unbalancing of real data was used to determine if the method could predict missing genotypes and select stable genotypes. Data from multienvironment trials containing 55 maize hybrids, collected during the 2005-2006 harvest season, were used in this study. Analyses were performed in two steps: the variance components were estimated by restricted maximum likelihood, using the expectation-maximization (EM) algorithm, and factor analysis (FA) was used to calculate the factor scores and relative position of each genotype in the biplot. Random unbalancing of the data was performed by removing 10, 30, and 50% of the plots; the scores were then re-estimated using the FA model. It was observed that 10, 30, and 50% unbalancing exhibited mean correlation values of 0.7, 0.6, and 0.56, respectively. Overall, the genotypes classified as stable in the biplot had smaller prediction error sum of squares (PRESS) value and prediction 14263 Factor analysis using mixed models ©FUNPEC-RP www.funpecrp.com.br Genetics and Molecular Research 14 (4): 14262-14278 (2015) amplitude of ellipses. Therefore, our results revealed the applicability of the PRESS statistic to evaluate the performance of stable genotypes in the biplot. This result was confirmed by the sizes of the prediction ellipses, which were smaller for the stable genotypes. Therefore, mixed models can confidently be used to evaluate stability in plant breeding programs, even with highly unbalanced data.

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.

ABSTRACT. In this study, we identified simple sequence repeat, amplified fragment length polymorphism, and sequence-related amplified polymorphism markers linked to quantitative trait loci (QTLs) for resistance to white mold disease in common bean progenies derived from a cross between lines CNFC 9506 and RP-2, evaluated using the oxalic acid test and using Bayesian analysis. DNA was extracted from 186 F 2 plants and their parental lines for molecular analysis. Fifteen experiments were carried out for phenotypic analysis, which included 186 F 2:4 progenies, the F 1 generation, the F 2 generation, and the lines CNFC 9506, RP-2, and G122 as common treatments. A completely randomized experimental design with 3 replications was used in controlled environments. The adjusted means for the F 2:4 generation were to identify QTLs by Bayesian shrinkage analysis. Significant differences were observed among the progenies for the reaction to white mold. The moving away method under the Bayes- Identification of QTLs for resistance to white mold ian approach was effective for identifying QTLs when it was not possible to obtain a genetic map because of low marker density. Using the Wald test, 25 markers identified QTLs for resistance to white mold, as well as 16 simple sequence repeats, 7 amplified fragment length polymorphisms, and 2 sequence-related amplified polymorphisms. The markers BM184, BM211, and PV-gaat001 showed low distances from QTLs related white mold resistance. In addition, these markers showed, signal effects with increasing resistance to white mold and high heritability in the analysis with oxalic acid, and thus, are promising for marker-assisted selection.

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