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.
Efficient analysis of datasets from multi-environment trials (MET) is of paramount importance in plant breeding programs. Several methods have been proposed for this purpose, each of them having advantages and disadvantages, depending on the objectives of the study. We examined the robustness in the predictive power of models that have been widely used in the study of genotype-byenvironment interaction such as AMMI (additive main-effects and multiplicative interaction) models via EM algorithm, Bayesian AMMI models with homogeneity (BAMMI), heterogeneity of variances (BAMMI-H) and the Analytical Factorial model (FA). To check the efficiency of these methods, genotype and genotype-byenvironment interaction effects were simulated and further unbalances were included at levels of 10, 33 and 50% loss of genotypes in the environments. To evaluate the predictive power of the proposed models, the PRESS (prediction error sum square) statistics and the Cor (correlation between predicted and observed ©FUNPEC-RP www.funpecrp.com.br Genetics and Molecular Research 18 (3): gmr18176 R.F. Romão et al. 2value) were used. The genotype-environment interaction models had low sensitivity to missing data since all models showed correlations above 0.5 in all scenarios -even with high unbalance levels (50%). In general, there were differences in predictive accuracy among the models in different scenarios, with a slight advantage for the Bayesian models in the correlation among observed and predicted data ranging from 0.79 to 0.855 compared to 0.591 to 0.853 obtained from the competing models. Similar results were observed for the PRESS (4.988 to 8.027) in Bayesian models compared to competing models (5.411 to 23,361). Overall, there was slight advantage of the Bayesian models in unbalanced scenarios.
Rice is one of the world’s most important crops. The search for genotypes that are more productive and have wide adaptation to different environments is paramount. One of the major breeder’s obstacles faced is identification of superior strains is the presence of Genotype × Environment Interaction (GEI), which motivated the development of countless statistical procedures aiming to offer more efficient studies. In this work we analysed adaptability and stability of 13 upland rice lineages as part of a genetic improvement program in nine different environments, resulting from local combination and years of agriculture. The experiment was conducted in a completely randomized block design, with three replicates. The main variable is the grain storage in kg/ha. The model applied is the Bayesian Main Additive Effects and Multiplicative Interaction (Bayesian-AMMI). Our implementation implies an extra assumption of random effects from genotypes coming from a single population as opposed to previous works in the literature. Credibility regions with maximum posteriori density allowed identification of cultivars with higher average yield. Stable genotypes showed an initial evidence of adaptation to an environment in this rice breeding program. Bayesian-AMMI is flexible, and starts to be more widely used, but our suggestion is promising in making it a more powerful tool
The dissection of genotype × environment interaction (GEI) is a crucial aspect of the final stages of plant breeding pipelines and recommendation of cultivars. Linearbilinear models used to analyze this interaction, such as the additive main effects and multiplicative interaction (AMMI) and genotype plus GEI (GGE), often assume homogeneity of the residual variances across environments which affects the estimates and therefore, interpretations and conclusions. Our main objective was to propose a GGE model that considers heteroscedasticity across environments using Bayesian inference and to evaluate its implications in the interpretation of real and simulated data. The GGE model assuming common variance was also fitted for comparison purposes. The great flexibility of the Bayesian inference is transferred to the biplots, allowing the construction of credible regions for genotypic and environmental scores. The inference on the stability and adaptability of genotypes might change when heteroscedasticity is ignored. When real data are used, different patterns of correlations between environments also affect the representativeness and discrimination of the target environment. The modeling of heteroscedasticity allowed the clustering of environments into subgroups, with similar effects for GEI. The proposed GGE model was more adequate and realistic to deal with scenarios of heterogeneous variance in multienvironment trials, which can be useful for exploiting the GEI.Abbreviations: AE, average environment; AEA, axis of the average environment; AEC, average environment coordination; AMMI, additive main effects and multiplicative interaction model; BGGE, Bayesian GGE model under homogeneity of residual variances across sites; BGGEH, Bayesian GGE model under heterogeneity of residual variances across sites; FA, Factor-Analytic Model; G, Genotype; GEI, Genotype × environment interaction; GGE, Genotype main effect + genotype × environment interaction; HPD, Highest posterior density; IG, ideal genotype; MCMC, Markov chain Monte Carlo; MET, Multienvironmental trial; MLM, mixed linear model; SS GGE , total sum of squares for G + GEI; SVD, Singular value decomposition.
O objetivo foi avaliar os efeitos dos recipientes e substratos no crescimento e na qualidade de mudas de cafeeiro em diferentes cultivares. O experimento foi conduzido no setor de produção de mudas do Ifsuldeminas - Campus Inconfidentes, Inconfidentes/MG, em um delineamento de blocos casualizados com três repetições por tratamento, sendo as parcelas constituídas por 30 plantas. O esquema fatorial foi de 2 x 2 x 5 sendo constituídos por dois recipientes, dois tipos de substratos e cinco cultivares de cafeeiro. Foram avaliados os parâmetros de crescimento e qualidade das mudas. Os resultados obtidos permitiram verificar que para a produção de mudas, os recipientes e os substratos interfere no crescimento das mudas cafeeiras, apresentando melhor qualidade as mudas no recipiente tubete com a utilização do substrato comercial, sendo a cultivar Acauã e a cultivar Topázio 1190, produzida em sacolas com substrato comercial com melhores resultados. Pela análise multivariada verificou-se que o maior crescimento das mudas em altura e menor diâmetro de coleto foi responsável pela menor qualidade das mesmas. Diante disso, conclui-se que o saquinho preenchidos com substrato comercial, proporcionou mudas com maior crescimento. O tubete, preenchido com substrato comercial, proporcionou mudas de melhor qualidade; a cultivar que apresentou um maior desequilíbrio foi a Catuaí 144, no saquinho; as cultivares que apresentam melhores IQD são a Acauã com uso de tubete em substrato comercial e a Topázio 1190 na sacolinha; a variável que mais contribui para o IQD foi a MSSR; o substrato padrão limitou o crescimento das raízes.
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