The availability of genomewide dense markers brings opportunities and challenges to breeding programs. An important question concerns the ways in which dense markers and pedigrees, together with phenotypic records, should be used to arrive at predictions of genetic values for complex traits. If a large number of markers are included in a regression model, marker-specific shrinkage of regression coefficients may be needed. For this reason, the Bayesian least absolute shrinkage and selection operator (LASSO) (BL) appears to be an interesting approach for fitting marker effects in a regression model. This article adapts the BL to arrive at a regression model where markers, pedigrees, and covariates other than markers are considered jointly. Connections between BL and other marker-based regression models are discussed, and the sensitivity of BL with respect to the choice of prior distributions assigned to key parameters is evaluated using simulation. The proposed model was fitted to two data sets from wheat and mouse populations, and evaluated using crossvalidation methods. Results indicate that inclusion of markers in the regression further improved the predictive ability of models. An R program that implements the proposed model is freely available.

Fixed linear models have been used for describing genotype x environment interaction (GE). Previous attempts have been made to assess the predictive ability of some linear mixed models when GE components are treated as random effects and modeled by the factor analytic (FA) model. This study compares the predictive ability of linear mixed models when the GE is modeled by the FA model with that of simple linear mixed models when the GE is not modeled. A cross-validation scheme is used that randomly deletes some genotypes from sites; the values for these genotypes are then predicted by the different models and correlated with their observed values to assess model accuracy. A total of six multienvironment trials (one potato [Solanum tuberosum L] trial, three maize [Zea mays L.] trials, and two wheat [Triticum aestivum L.] trials) with GE of varying complexity were used in the evaluation. Results show that for data sets with complex GE, modeling GE using the FA model improved the predictability of the model up to 6%. When GE is not complex, most models (with and without FA) gave high predictability, and models with FA did not seem to lose much predictive ability. Therefore, we concluded that modeling GE with the FA model is a good thing.

Much research has been conducted using least squares estimates of the linear-bilinear model additive main effects and multiplicative interaction (AMMI). The main difficulty with the standard linear-bilinear models is that statistical inference on the bilinear effects of genotype X environment interaction cannot be incorporated easily into the biplot of the first two components. This research proposes a Bayesian approach for the inference on the parameters of the AMMI model using a Gibbs sampler that saves computing time and makes the algorithm stable. Data from one maize {Zea mays L.) multi-environment trial (MET) was used for illustration. Vague but proper prior distributions were introduced. Results show that the various Markov chain Monte Carlo convergence criteria were met for all parameters. Bivariate highest posterior density (HPD) regions for the Bayesian-AMMI interactions are shown in the biplot of the first two bilinear components; these regions offer a statistical inference on the bilinear parameters and allow visualizing homogeneous groups of environments and genotypes.Abbreviations: AIC, Akaike's information criterion; AMMI, additive main effects and multiplicative interaction; BLUP, best linear unbiased prediction; GE, genotype x environment interaction; GGE, genotype plus genotype x environment interaction; HPD, highest posterior density; MET, multi-environment trial; MCMC, Markov Chain Monte Carlo; SREG, sites regression. I n plant breeding, the main purposes of multi-environment trials (METs) are to (i) study genotype x environment interaction (GE), (ii) assess genotypic adaptability and stability, (iii) establish relationships among testing environments, among genotypes, and among genotypes and environments (or sites) simultaneously, and (iv) make predictions of the genotypes' breeding value that will allow making an accurate selection of parents for the next breeding cycle. The presence of GE complicates this process and is usually expressed either as inconsistent responses of some genotypes relative to others, due to genotypic rank change, or as changes in

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