Use of the reversible jump Markov chain Monte Carlo algorithm to select multiplicative terms in the AMMI-Bayesian model.

The model selection stage has become a central theme in applying the additive main effects and multiplicative interaction (AMMI) model to determine the optimal number of bilinear components to be retained to describe the genotype-by-environment interaction (GEI). In the Bayesian context, this problem has been addressed by using information criteria and the Bayes factor. However, these procedures are computationally intensive, making their application unfeasible when the model's parametric space is large. A Bayesian analysis of the AMMI model was conducted using the Reversible Jump algorithm (RJMCMC) to determine the number of multiplicative terms needed to explain the GEI pattern. Three a priori distributions were assigned for the singular value scale parameter under different justifications, namely: i) the insufficient reason principle (uniform); ii) the invariance principle (Jeffreys' prior) and iii) the maximum entropy principle. Simulated and real data were used to exemplify the method. An evaluation of the predictive ability of models for simulated data was conducted and indicated that the AMMI analysis, in general, was robust, and models adjusted by the Reversible Jump method were superior to those in which sampling was performed only by the Gibbs sampler. In addition, the RJMCMC showed greater feasibility since the selection and estimation of parameters are carried out concurrently in the same sampling algorithm, being more attractive in terms of computational time. The use of the maximum entropy principle makes the analysis more flexible, avoiding the use of procedures for correcting prior degrees of freedom and obtaining improper posterior marginal distributions.

Shrinkage in the Bayesian analysis of the GGE model: A case study with simulation.

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.

Functional models in genome-wide selection.

The development of sequencing technologies has enabled the discovery of markers that are abundantly distributed over the whole genome. Knowledge about the marker locations in reference genomes provides further insights in the search for causal regions and the prediction of genomic values. The present study proposes a Bayesian functional approach for incorporating the marker locations into genomic analysis using stochastic methods to search causal regions and predict genotypic values. For this, three scenarios were analyzed: F2 population with 300 individuals and three different heritability levels (0.2, 0.5, and 0.8), along with 12,150 SNP markers that were distributed through ten linkage groups; F∞ populations with 320 individuals and three different heritability levels (0.2, 0.5, and 0.8), along with 10,020 SNP markers that were distributed through ten linkage groups; and data related to Eucalyptus spp. to measure the model performance in a real LD setting, with 611 individuals whose phenotypes were simulated from QTLs distributed through a panel of 36,812 SNPs with known positions. The performance of the proposed method was compared with those of other genome selection models, namely, RR-BLUP, Bayes B and Bayesian Lasso. The Bayesian functional model presented higher or similar predictive ability when compared with those classical regressions methods in simulated and real scenarios on different LD structures. In general, the Bayesian functional model also achieved higher computational efficiency, using 12 SNPs per MCMC round. The model was efficient in the identification of causal regions and showed high flexibility of analysis, as it is easily adaptable to any genomic selection model.

Bayesian factor analytic model: An approach in multiple environment trials.

One of the main challenges in plant breeding programs is the efficient quantification of the genotype-by-environment interaction (GEI). The presence of significant GEI may create difficulties for breeders in the selection and recommendation of superior genotypes for a wide environmental network. Among the diverse statistical procedures developed for this purpose, we highlight those based on mixed models and factor analysis that are called factor analytic (FA) models. However, some inferential issues are related to the factor analytic model, such as Heywood cases that make the model non-identifiable. Moreover, the representation of the loads and factors in the conventional biplot does not involve any measurement of uncertainty. In this work, we propose dealing with the FA model using the Bayesian framework with direct sampling of factor loadings via spectral decomposition; this guarantees identifiability in the estimation process and eliminates the need for the rotationality of factor loadings or imposition of any ad hoc constraints. We used simulated and real data to illustrate the method's application in multi-environment trials (MET) and to compare it with traditional FA mixed models on controlled unbalancing. In general, the Bayesian FA model was robust under different simulated unbalanced levels, presenting the superior predictive ability of missing data when compared to competing models, such as those based on FA mixed models. In addition, for some scenarios, the classical FA mixed model failed in estimating the full FA model, illustrating the parametric problems of convergence in these models. Our results suggest that Bayesian factorial models might be successfully used in plant breeding for MET analysis.

Genome-wide association analysis of ear rot resistance caused by Fusarium verticillioides in maize.

The identification of causal regions associated with resistance to Fusarium verticillioides can be useful to understand resistance mechanisms and further be used in breeding programs. In this study, a genome-wide association study (GWAS) was conducted to identify candidate markers associated with resistance to the ear rot caused by the fungus F. verticillioides. A total of 242 maize inbred lines were genotyped with 23,153 DArT-seq markers. A total of 12 DArTs were associated with ear rot resistance. Some DArTs were localized close to genes with functions directly related to ear rot resistance, such as a gene responsible for the innate immune response that belongs to the class of NBS-LRR receptors. Some markers were also found to be closely associated with genes that synthesize transcription factors (nactf11 and nactf61), genes responsible for the oxidation-reduction process and peroxidase activity. These results are encouraging since some candidate markers can present functional relationship with ear rot resistance in maize.

Bayesian reversible-jump for epistasis analysis in genomic studies.

BACKGROUND: The large amount of data used in genomic analysis has allowed geneticists to achieve some understanding of the genetic architecture of complex traits. Although the information gathered by molecular markers has permitted gains in predictive accuracy and gene discovery, epistatic effects have been ignored based on exhaustive searches requesting estimates of its effects on the whole genome. In this work, we propose the reversible-jump technique to estimate epistasis in the genome without drastically altering the model dimension. To this end, we used a real maize dataset based on 256 F2:3 progenies plus a simulation data set based on 300 F2 individuals. In the simulation scenario, six QTL presenting main effects (additive and dominance) were combined with seven other epistatic effects totaling 13 QTL controlling the trait. RESULTS: Our model explored 18,624 candidate epistases, but even in this vast space, only one spurious interaction was found. The three epistases selected by our model, named here as 18x26, 56x68 and 59x93, were very close to simulated ones (19x25, 54x72, 59x91 and 59x94). In the real dataset, we estimate 33,024 epistatic effects, and several minor epistatic combinations were found to explain a significant proportion of the genetic variance. The broad participation of epistasis in the real dataset may indicate the presence of pervasive epistasis acting on maize grain yield. CONCLUSIONS: The power of selecting true epistasis in thousands of possible combinations suggests the attractiveness of our model to handle genomic data.

Genomic selection to resistance to Stenocarpella maydis in maize lines using DArTseq markers.

BACKGROUND: The identification of lines resistant to ear diseases is of great importance in maize breeding because such diseases directly interfere with kernel quality and yield. Among these diseases, ear rot disease is widely relevant due to significant decrease in grain yield. Ear rot may be caused by the fungus Stenocarpella maydi; however, little information about genetic resistance to this pathogen is available in maize, mainly related to candidate genes in genome. In order to exploit this genome information we used 23.154 Dart-seq markers in 238 lines and apply genome-wide selection to select resistance genotypes. We divide the lines into clusters to identify groups related to resistance to Stenocarpella maydi and use Bayesian stochastic search variable approach and rr-BLUP methods to comparate their selection results. RESULTS: Through a principal component analysis (PCA) and hierarchical clustering, it was observed that the three main genetic groups (Stiff Stalk Synthetic, Non-Stiff Stalk Synthetic and Tropical) were clustered in a consistent manner, and information on the resistance sources could be obtained according to the line of origin where populations derived from genetic subgroup Suwan presenting higher levels of resistance. The ridge regression best linear unbiased prediction (rr-BLUP) and Bayesian stochastic search variable (BSSV) models presented equivalent abilities regarding predictive processes. CONCLUSION: Our work showed that is possible to select maize lines presenting a high resistance to Stenocarpella maydis. This claim is based on the acceptable level of predictive accuracy obtained by Genome-wide Selection (GWS) using different models. Furthermore, the lines related to background Suwan present a higher level of resistance than lines related to other groups.

Inclusion of Dominance Effects in the Multivariate GBLUP Model.

New proposals for models and applications of prediction processes with data on molecular markers may help reduce the financial costs of and identify superior genotypes in maize breeding programs. Studies evaluating Genomic Best Linear Unbiased Prediction (GBLUP) models including dominance effects have not been performed in the univariate and multivariate context in the data analysis of this crop. A single cross hybrid construction procedure was performed in this study using phenotypic data and actual molecular markers of 4,091 maize lines from the public database Panzea. A total of 400 simple hybrids resulting from this process were analyzed using the univariate and multivariate GBLUP model considering only additive effects additive plus dominance effects. Historic heritability scenarios of five traits and other genetic architecture settings were used to compare models, evaluating the predictive ability and estimation of variance components. Marginal differences were detected between the multivariate and univariate models. The main explanation for the small discrepancy between models is the low- to moderate-magnitude correlations between the traits studied and moderate heritabilities. These conditions do not favor the advantages of multivariate analysis. The inclusion of dominance effects in the models was an efficient strategy to improve the predictive ability and estimation quality of variance components.

A Bayesian Shrinkage Approach for AMMI Models.

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.

Prediction of maize single cross hybrids using the total effects of associated markers approach assessed by cross-validation and regional trials.

The present study aimed to predict the performance of maize hybrids and assess whether the total effects of associated markers (TEAM) method can correctly predict hybrids using cross-validation and regional trials. The training was performed in 7 locations of Southern Brazil during the 2010/11 harvest. The regional assays were conducted in 6 different South Brazilian locations during the 2011/12 harvest. In the training trial, 51 lines from different backgrounds were used to create 58 single cross hybrids. Seventy-nine microsatellite markers were used to genotype these 51 lines. In the cross-validation method the predictive accuracy ranged from 0.10 to 0.96, depending on the sample size. Furthermore, the accuracy was 0.30 when the values of hybrids that were not used in the training population (119) were predicted for the regional assays. Regarding selective loss, the TEAM method correctly predicted 50% of the hybrids selected in the regional assays. There was also loss in only 33% of cases; that is, only 33% of the materials predicted to be good in training trial were considered to be bad in regional assays. Our results show that the predictive validation of different crop conditions is possible, and the cross-validation results strikingly represented the field performance.

Inheritance of nitrogen use efficiency in inbred progenies of tropical maize based on multivariate diallel analysis.

The objective of our study was to characterize and determine the patterns of genetic control in relation to tolerance and efficiency of nitrogen use by means of a complete diallel cross involving contrasting inbred progenies of tropical maize based on a univariate approach within the perspective of a multivariate mixed model. Eleven progenies, previously classified regarding the tolerance and responsiveness to nitrogen, were crossed in a complete diallel cross. Fifty-five hybrids were obtained. The hybrids and the progenies were evaluated at two different nitrogen levels, in two locations. The grain yield was measured as well as its yield components. The heritability values between the higher and lower nitrogen input environment did not differ among themselves. It was observed that the general combining ability values were similar for both approaches univariate and multivariate, when it was analyzed within each location and nitrogen level. The estimate of variance of the specific combining ability was higher than general combining ability estimate and the ratio between them was 0.54. The univariate and multivariate approaches are equivalent in experiments with good precision and high heritability. The nonadditive genetic effects exhibit greater quantities than the additive genetic effects for the genetic control of nitrogen use efficiency.

Identification of QTLs for Resistance to Sclerotinia sclerotiorum in Carioca Common Bean by the Moving Away Method.

The aim of this study was to use multiple DNA markers for detection of QTLs related to resistance to white mold in an F2 population of common bean evaluated by the straw test method. The DNA from 186 F2 plants and from the parents was extracted for genotypic evaluation using SSR, AFLP, and SRAP markers. For phenotypic analysis, 186 F2:4 progenies and ten lines were evaluated, in a 14 × 14 triple lattice experimental design. The adjusted mean values of the F2:4 progenies were used for identification of QTLs by Bayesian shrinkage analysis. Significant differences were observed among the progenies for reaction to white mold. In identification of QTLs, 17 markers identified QTLs for resistance-13 SSRs and 4 AFLPs. The moving away method under the Bayesian approach proved to be efficient in the identification of QTLs when a genetic map is not used due to the low density of markers. The ME1 and BM211 markers are near the QTLs, with the effect of increasing resistance to white mold, and they have high heritability. They are thus promising for marker-assisted selection.

Bayesian mapping of multiple traits in maize: the importance of pleiotropic effects in studying the inheritance of quantitative traits.

Pleiotropy has played an important role in understanding quantitative traits. However, the extensiveness of this effect in the genome and its consequences for plant improvement have not been fully elucidated. The aim of this study was to identify pleiotropic quantitative trait loci (QTLs) in maize using Bayesian multiple interval mapping. Additionally, we sought to obtain a better understanding of the inheritance, extent and distribution of pleiotropic effects of several components in maize production. The design III procedure was used from a population derived from the cross of the inbred lines L-14-04B and L-08-05F. Two hundred and fifty plants were genotyped with 177 microsatellite markers and backcrossed to both parents giving rise to 500 backcrossed progenies, which were evaluated in six environments for grain yield and its components. The results of this study suggest that mapping isolated traits limits our understanding of the genetic architecture of quantitative traits. This architecture can be better understood by using pleiotropic networks that facilitate the visualization of the complexity of quantitative inheritance, and this characterization will help to develop new selection strategies. It was also possible to confront the idea that it is feasible to identify QTLs for complex traits such as grain yield, as pleiotropy acts prominently on its subtraits and as this "trait" can be broken down and predicted almost completely by the QTLs of its components. Additionally, pleiotropic QTLs do not necessarily signify pleiotropy of allelic interactions, and this indicates that the pervasive pleiotropy does not limit the genetic adaptability of plants.