Bayesian Variable Selection Model Research Articles

Using markers with large effect in genetic and genomic predictions.

The first attempts of applying marker-assisted selection (MAS) in animal breeding were not very successful because the identification of markers closely linked to QTL using low-density microsatellite panels was difficult. More recently, the use of high-density SNP panels in genome-wide association studies (GWAS) have increased the power and precision of identifying markers linked to QTL, which offer new possibilities for MAS. However, when GWAS started to be performed, the focus of many breeders had already shifted from the use of MAS to the application of genomic selection (using all available markers without any preselection of markers linked to QTL). In this study, we aimed to evaluate the prediction accuracy of a MAS approach that accounts for GWAS findings in the prediction models by including the most significant SNP from GWAS as a fixed effect in the marker-assisted BLUP (MA-BLUP) and marker-assisted genomic BLUP (MA-GBLUP) prediction models. A second aim was to compare the prediction accuracies from the marker-assisted models with those obtained from a Bayesian variable selection (BVS) model. To compare the prediction accuracies of traditional BLUP, MA-BLUP, genomic BLUP (GBLUP), MA-GBLUP, and BVS, we applied these models to the trait "number of teats" in 4 distinct pig populations, for validation of the results. The most significant SNP in each population was located at approximately 103.50 Mb on chromosome 7. Applying MAS by accounting for the most significant SNP in the prediction models resulted in improved prediction accuracy for number of teats in all evaluated populations compared with BLUP and GBLUP. Using MA-BLUP instead of BLUP, the increase in prediction accuracy ranged from 0.021 to 0.124, whereas using MA-GBLUP instead of GBLUP, the increase in prediction accuracy ranged from 0.003 to 0.043. The BVS model resulted in similar or higher prediction accuracies than MA-GBLUP. For the trait number of teats, BLUP resulted in the lowest prediction accuracies whereas the highest were observed when applying MA-GBLUP or BVS. In the same data set, MA-BLUP can yield similar or superior accuracies compared with GBLUP. The superiority of MA-GBLUP over traditional GBLUP is more pronounced when training populations are smaller and when relationships between training and validation populations are smaller. Marker-assisted GBLUP did not outperform BVS but does have implementation advantages in large-scale evaluations.

Efficient genomic prediction based on whole-genome sequence data using split-and-merge Bayesian variable selection.

BackgroundUse of whole-genome sequence data is expected to increase persistency of genomic prediction across generations and breeds but affects model performance and requires increased computing time. In this study, we investigated whether the split-and-merge Bayesian stochastic search variable selection (BSSVS) model could overcome these issues. BSSVS is performed first on subsets of sequence-based variants and then on a merged dataset containing variants selected in the first step.ResultsWe used a dataset that included 4,154,064 variants after editing and de-regressed proofs for 3415 reference and 2138 validation bulls for somatic cell score, protein yield and interval first to last insemination. In the first step, BSSVS was performed on 106 subsets each containing ~39,189 variants. In the second step, 1060 up to 472,492 variants, selected from the first step, were included to estimate the accuracy of genomic prediction. Accuracies were at best equal to those achieved with the commonly used Bovine 50k-SNP chip, although the number of variants within a few well-known quantitative trait loci regions was considerably enriched. When variant selection and the final genomic prediction were performed on the same data, predictions were biased. Predictions computed as the average of the predictions computed for each subset achieved the highest accuracies, i.e. 0.5 to 1.1 % higher than the accuracies obtained with the 50k-SNP chip, and yielded the least biased predictions. Finally, the accuracy of genomic predictions obtained when all sequence-based variants were included was similar or up to 1.4 % lower compared to that based on the average predictions across the subsets. By applying parallelization, the split-and-merge procedure was completed in 5 days, while the standard analysis including all sequence-based variants took more than three months.ConclusionsThe split-and-merge approach splits one large computational task into many much smaller ones, which allows the use of parallel processing and thus efficient genomic prediction based on whole-genome sequence data. The split-and-merge approach did not improve prediction accuracy, probably because we used data on a single breed for which relationships between individuals were high. Nevertheless, the split-and-merge approach may have potential for applications on data from multiple breeds.Electronic supplementary materialThe online version of this article (doi:10.1186/s12711-016-0225-x) contains supplementary material, which is available to authorized users.

Genomewide association analysis of sow lactation performance traits in lines of Yorkshire pigs divergently selected for residual feed intake during grow-finish phase.

Lactation is an economically and biologically important phase in the life cycle of sows. Short generation intervals in nucleus herds and low heritability of traits associated with lactation along with challenges associated with collecting accurate lactation performance phenotypes emphasize the importance of using genomic tools to examine the underlying genetics of these traits. We report the first genomewide association study (GWAS) on traits associated with lactation and efficiency in 2 lines of Yorkshire pigs that were divergently selected for residual feed intake during grow-finish phase. A total of 862 farrowing records from 2 parities were analyzed using a Bayesian whole genome variable selection model (Bayes B) to locate 1-Mb regions that were most strongly associated with each trait. The GWAS was conducted separately for parity 1 and 2 records. Marker-based heritabilities ranged from 0.03 to 0.39 for parity 1 traits and from 0.06 to 0.40 for parity 2 traits. For all traits studied, around 90% of genetic variance came from a large number of genomic regions with small effects, whereas genomic regions with large effects were found to be different for the same trait measured in parity 1 and 2. The highest percentage of genetic variance explained by a 1-Mb window for each trait ranged from 0.4% for feed intake during lactation to 4.2% for back fat measured at farrowing in parity 1 sows and from 0.2% for lactation feed intake to 5.4% for protein mass loss during lactation in parity 2 sows. A total of thirteen 1-Mb nonoverlapping windows were found to explain more than 1.5% of genetic variance for either a single trait or across multiple traits. These 1-Mb windows were on chromosomes 2, 3, 6, 7, 8, 11, 14, 15, 17, and 18. The major positional candidate genes within 1 Mb upstream and downstream of these windows were , (SSC2), (SSC6) (SSC7), (SSC8), (SSC11), (SSC14), (SSC17). Further validation studies on larger populations are required to validate these findings and to improve our understanding of the biology and complex genetic architecture of traits associated with sow lactation performance.

Bayesian inference for high‐dimensional linear regression under mnet priors

AbstractFor regression problems that involve many potential predictors, the Bayesian variable selection (BVS) method is a powerful tool. This method associates each model with its posterior probability and achieves excellent prediction performance through Bayesian model averaging. The main challenges of using such models include specifying a suitable prior and computing posterior quantities for inference. We contribute to the literature of BVS modelling in the following aspects. We first propose a new family of priors, called the mnet prior, which is indexed by a few hyperparameters that allow great flexibility in the prior density. The hyperparameters can also be treated as random, so that their values need not be tuned manually, but will instead adapt to the data. Simulation studies are used to demonstrate good prediction and variable selection performances of these models. Secondly, the analytical expression of the posterior distribution is unavailable for the BVS model under the mnet prior in general, as is the case for most BVS models. We develop an adaptive Markov chain Monte Carlo algorithm that facilitates the computation in high‐dimensional regression problems. We finally showcase various ways to do inference with BVS models, highlighting a new way to visualize the importance of each predictor along with estimation of the coefficients and their uncertainties. These are demonstrated through the analysis of a breast cancer gene expression dataset.The Canadian Journal of Statistics44: 180–197; 2016 © 2016 Statistical Society of Canada

Accuracy of genomic prediction using imputed whole-genome sequence data in white layers.

There is an increasing interest in using whole-genome sequence data in genomic selection breeding programmes. Prediction of breeding values is expected to be more accurate when whole-genome sequence is used, because the causal mutations are assumed to be in the data. We performed genomic prediction for the number of eggs in white layers using imputed whole-genome resequence data including ~4.6 million SNPs. The prediction accuracies based on sequence data were compared with the accuracies from the 60K SNP panel. Predictions were based on genomic best linear unbiased prediction (GBLUP) as well as a Bayesian variable selection model (BayesC). Moreover, the prediction accuracy from using different types of variants (synonymous, non-synonymous and non-coding SNPs) was evaluated. Genomic prediction using the 60K SNP panel resulted in a prediction accuracy of 0.74 when GBLUP was applied. With sequence data, there was a small increase (~1%) in prediction accuracy over the 60K genotypes. With both 60K SNP panel and sequence data, GBLUP slightly outperformed BayesC in predicting the breeding values. Selection of SNPs more likely to affect the phenotype (i.e. non-synonymous SNPs) did not improve the accuracy of genomic prediction. The fact that sequence data were based on imputation from a small number of sequenced animals may have limited the potential to improve the prediction accuracy. A small reference population (n=1004) and possible exclusion of many causal SNPs during quality control can be other possible reasons for limited benefit of sequence data. We expect, however, that the limited improvement is because the 60K SNP panel was already sufficiently dense to accurately determine the relationships between animals in our data.

Across population genomic prediction scenarios in which Bayesian variable selection outperforms GBLUP.

BackgroundThe use of information across populations is an attractive approach to increase the accuracy of genomic prediction for numerically small populations. However, accuracies of across population genomic prediction, in which reference and selection individuals are from different populations, are currently disappointing. It has been shown for within population genomic prediction that Bayesian variable selection models outperform GBLUP models when the number of QTL underlying the trait is low. Therefore, our objective was to identify across population genomic prediction scenarios in which Bayesian variable selection models outperform GBLUP in terms of prediction accuracy. In this study, high density genotype information of 1033 Holstein Friesian, 105 Groningen White Headed, and 147 Meuse-Rhine-Yssel cows were used. Phenotypes were simulated using two changing variables: (1) the number of QTL underlying the trait (3000, 300, 30, 3), and (2) the correlation between allele substitution effects of QTL across populations, i.e. the genetic correlation of the simulated trait between the populations (1.0, 0.8, 0.4).ResultsThe accuracy obtained by the Bayesian variable selection model was depending on the number of QTL underlying the trait, with a higher accuracy when the number of QTL was lower. This trend was more pronounced for across population genomic prediction than for within population genomic prediction. It was shown that Bayesian variable selection models have an advantage over GBLUP when the number of QTL underlying the simulated trait was small. This advantage disappeared when the number of QTL underlying the simulated trait was large. The point where the accuracy of Bayesian variable selection and GBLUP became similar was approximately the point where the number of QTL was equal to the number of independent chromosome segments (Me) across the populations.ConclusionBayesian variable selection models outperform GBLUP when the number of QTL underlying the trait is smaller than Me. Across populations, Me is considerably larger than within populations. So, it is more likely to find a number of QTL underlying a trait smaller than Me across populations than within population. Therefore Bayesian variable selection models can help to improve the accuracy of across population genomic prediction.

Genomic prediction using imputed whole-genome sequence data in Holstein Friesian cattle.

BackgroundIn contrast to currently used single nucleotide polymorphism (SNP) panels, the use of whole-genome sequence data is expected to enable the direct estimation of the effects of causal mutations on a given trait. This could lead to higher reliabilities of genomic predictions compared to those based on SNP genotypes. Also, at each generation of selection, recombination events between a SNP and a mutation can cause decay in reliability of genomic predictions based on markers rather than on the causal variants. Our objective was to investigate the use of imputed whole-genome sequence genotypes versus high-density SNP genotypes on (the persistency of) the reliability of genomic predictions using real cattle data.MethodsHighly accurate phenotypes based on daughter performance and Illumina BovineHD Beadchip genotypes were available for 5503 Holstein Friesian bulls. The BovineHD genotypes (631,428 SNPs) of each bull were used to impute whole-genome sequence genotypes (12,590,056 SNPs) using the Beagle software. Imputation was done using a multi-breed reference panel of 429 sequenced individuals. Genomic estimated breeding values for three traits were predicted using a Bayesian stochastic search variable selection (BSSVS) model and a genome-enabled best linear unbiased prediction model (GBLUP). Reliabilities of predictions were based on 2087 validation bulls, while the other 3416 bulls were used for training.ResultsPrediction reliabilities ranged from 0.37 to 0.52. BSSVS performed better than GBLUP in all cases. Reliabilities of genomic predictions were slightly lower with imputed sequence data than with BovineHD chip data. Also, the reliabilities tended to be lower for both sequence data and BovineHD chip data when relationships between training animals were low. No increase in persistency of prediction reliability using imputed sequence data was observed.ConclusionsCompared to BovineHD genotype data, using imputed sequence data for genomic prediction produced no advantage. To investigate the putative advantage of genomic prediction using (imputed) sequence data, a training set with a larger number of individuals that are distantly related to each other and genomic prediction models that incorporate biological information on the SNPs or that apply stricter SNP pre-selection should be considered.Electronic supplementary materialThe online version of this article (doi:10.1186/s12711-015-0149-x) contains supplementary material, which is available to authorized users.

Quantitative trait loci markers derived from whole genome sequence data increases the reliability of genomic prediction

This study investigated the effect on the reliability of genomic prediction when a small number of significant variants from single marker analysis based on whole genome sequence data were added to the regular 54k single nucleotide polymorphism (SNP) array data. The extra markers were selected with the aim of augmenting the custom low-density Illumina BovineLD SNP chip (San Diego, CA) used in the Nordic countries. The single-marker analysis was done breed-wise on all 16 index traits included in the breeding goals for Nordic Holstein, Danish Jersey, and Nordic Red cattle plus the total merit index itself. Depending on the trait’s economic weight, 15, 10, or 5 quantitative trait loci (QTL) were selected per trait per breed and 3 to 5 markers were selected to tag each QTL. After removing duplicate markers (same marker selected for more than one trait or breed) and filtering for high pairwise linkage disequilibrium and assaying performance on the array, a total of 1,623 QTL markers were selected for inclusion on the custom chip. Genomic prediction analyses were performed for Nordic and French Holstein and Nordic Red animals using either a genomic BLUP or a Bayesian variable selection model. When using the genomic BLUP model including the QTL markers in the analysis, reliability was increased by up to 4 percentage points for production traits in Nordic Holstein animals, up to 3 percentage points for Nordic Reds, and up to 5 percentage points for French Holstein. Smaller gains of up to 1 percentage point was observed for mastitis, but only a 0.5 percentage point increase was seen for fertility. When using a Bayesian model accuracies were generally higher with only 54k data compared with the genomic BLUP approach, but increases in reliability were relatively smaller when QTL markers were included. Results from this study indicate that the reliability of genomic prediction can be increased by including markers significant in genome-wide association studies on whole genome sequence data alongside the 54k SNP set.

Genomic prediction of breeding values using previously estimated SNP variances.

BackgroundGenomic prediction requires estimation of variances of effects of single nucleotide polymorphisms (SNPs), which is computationally demanding, and uses these variances for prediction. We have developed models with separate estimation of SNP variances, which can be applied infrequently, and genomic prediction, which can be applied routinely.MethodsSNP variances were estimated with Bayes Stochastic Search Variable Selection (BSSVS) and BayesC. Genome-enhanced breeding values (GEBV) were estimated with RR-BLUP (ridge regression best linear unbiased prediction), using either variances obtained from BSSVS (BLUP-SSVS) or BayesC (BLUP-C), or assuming equal variances for each SNP. Datasets used to estimate SNP variances comprised (1) all animals, (2) 50% random animals (RAN50), (3) 50% best animals (TOP50), or (4) 50% worst animals (BOT50). Traits analysed were protein yield, udder depth, somatic cell score, interval between first and last insemination, direct longevity, and longevity including information from predictors.ResultsBLUP-SSVS and BLUP-C yielded similar GEBV as the equivalent Bayesian models that simultaneously estimated SNP variances. Reliabilities of these GEBV were consistently higher than from RR-BLUP, although only significantly for direct longevity. Across scenarios that used data subsets to estimate GEBV, observed reliabilities were generally higher for TOP50 than for RAN50, and much higher than for BOT50. Reliabilities of TOP50 were higher because the training data contained more ancestors of selection candidates. Using estimated SNP variances based on random or non-random subsets of the data, while using all data to estimate GEBV, did not affect reliabilities of the BLUP models. A convergence criterion of 10−8 instead of 10−10 for BLUP models yielded similar GEBV, while the required number of iterations decreased by 71 to 90%. Including a separate polygenic effect consistently improved reliabilities of the GEBV, but also substantially increased the required number of iterations to reach convergence with RR-BLUP. SNP variances converged faster for BayesC than for BSSVS.ConclusionsCombining Bayesian variable selection models to re-estimate SNP variances and BLUP models that use those SNP variances, yields GEBV that are similar to those from full Bayesian models. Moreover, these combined models yield predictions with higher reliability and less bias than the commonly used RR-BLUP model.Electronic supplementary materialThe online version of this article (doi:10.1186/s12711-014-0052-x) contains supplementary material, which is available to authorized users.

Spatial Bayesian Variable Selection Models on Functional Magnetic Resonance Imaging Time-Series Data.

A common objective of fMRI (functional magnetic resonance imaging) studies is to determine subject-specific areas of increased blood oxygenation level dependent (BOLD) signal contrast in response to a stimulus or task, and hence to infer regional neuronal activity. We posit and investigate a Bayesian approach that incorporates spatial and temporal dependence and allows for the task-related change in the BOLD signal to change dynamically over the scanning session. In this way, our model accounts for potential learning effects in addition to other mechanisms of temporal drift in task-related signals. We study the properties of the model through its performance on simulated and real data sets.

Comparison of genomic predictions using genomic relationship matrices built with different weighting factors to account for locus-specific variances

Various models have been used for genomic prediction. Bayesian variable selection models often predict more accurate genomic breeding values than genomic BLUP (GBLUP), but GBLUP is generally preferred for routine genomic evaluations because of low computational demand. The objective of this study was to achieve the benefits of both models using results from Bayesian models and genome-wide association studies as weights on single nucleotide polymorphism (SNP) markers when constructing the genomic matrix (G-matrix) for genomic prediction. The data comprised 5,221 progeny-tested bulls from the Nordic Holstein population. The animals were genotyped using the Illumina Bovine SNP50 BeadChip (Illumina Inc., San Diego, CA). Weighting factors in this investigation were the posterior SNP variance, the square of the posterior SNP effect, and the corresponding minus base-10 logarithm of the marker association P-value [−log10(P)] of a t-test obtained from the analysis using a Bayesian mixture model with 4 normal distributions, the square of the estimated SNP effect, and the corresponding −log10(P) of a t-test obtained from the analysis using a classical genome-wide association study model (linear regression model). The weights were derived from the analysis based on data sets that were 0, 1, 3, or 5 yr before performing genomic prediction. In building a G-matrix, the weights were assigned either to each marker (single-marker weighting) or to each group of approximately 5 to 150 markers (group-marker weighting). The analysis was carried out for milk yield, fat yield, protein yield, fertility, and mastitis. Deregressed proofs (DRP) were used as response variables to predict genomic estimated breeding values (GEBV). Averaging over the 5 traits, the Bayesian model led to 2.0% higher reliability of GEBV than the GBLUP model with an original unweighted G-matrix. The superiority of using a GBLUP with weighted G-matrix over GBLUP with an original unweighted G-matrix was the largest when using a weighting factor of posterior variance, resulting in 1.7 percentage points higher reliability. The second best weighting factors were −log10 (P-value) of a t-test corresponding to the square of the posterior SNP effect from the Bayesian model and −log10 (P-value) of a t-test corresponding to the square of the estimated SNP effect from the linear regression model, followed by the square of estimated SNP effect and the square of the posterior SNP effect. In addition, group-marker weighting performed better than single-marker weighting in terms of reducing bias of GEBV, and also slightly increased prediction reliability. The differences between weighting factors and scenarios were larger in prediction bias than in prediction accuracy. Finally, weights derived from a data set having a lag up to 3 yr did not reduce reliability of GEBV. The results indicate that posterior SNP variance estimated from a Bayesian mixture model is a good alternative weighting factor, and common weights on group markers with a size of 30 markers is a good strategy when using markers of the 50,000-marker (50K) chip. In a population with gradually increasing reference data, the weights can be updated once every 3 yr.

Dose–Response Modeling Under Simple Order Restrictions Using Bayesian Variable Selection Methods

Bayesian modeling of dose–response data offers the possibility to establish the relationship between a clinical or a genomic response and increasing doses of a therapeutic compound and to determine the nature of the relationship wherever it exists. In this article, we focus on an order-restricted one-way ANOVA model which can be used to test the null hypothesis of no dose effect against an ordered alternative. Within the framework of the dose–response modeling, a model uncertainty can be addressed using model averaging techniques. In this setting, the uncertainty is related to the number of all possible models that can be fitted to the data and should be taken into account for both inference and estimation. In this article, we propose an order-restricted Bayesian variable selection model that addresses the model uncertainty and can be used for both inference and estimation. The proposed method is applied to two case studies and is compared to the likelihood ratio test and the multiple contrast tests in both the analyses of the case studies and a simulation study. This article has online supplementary material.

Genomic evaluation of cattle in a multi-breed context

In order to obtain accurate genomic breeding values a large number of reference animals with both phenotype and genotype data are needed. This poses a challenge for breeds with small reference populations. One option to overcome this obstacle is to use a multi-breed reference population. However, combining populations across breeds is not straightforward due to differences in linkage disequilibrium structure and weak relationships between breeds. This study offers a review of the available literature on the use of reference populations compiled from different cattle breeds. Results show that the effect of multi-breed reference populations on the accuracy of genomic prediction is highly affected by the genetic distance between breeds. When combining populations of the same breeds from different countries, large increases in accuracy are seen, whereas for admixed populations with some exchange of sires, substantial but smaller gains are found. Little or no benefit is found when combining distantly related breeds such as Holstein and Jersey and using the widely used genomic BLUP model. By using more sophisticated Bayesian variable selection models that put more focus on genomic markers in strong linkage disequilibrium with causative variants in combination with denser markers sets or functional subsets of markers, it is however possible to utilize information across distantly related breeds to increase the accuracy of genomic prediction. The further development of multi-breed genomic prediction models offers not only increases in the accuracy of genomic breeding values for small breeds, but will also give a stronger persistence of the accuracy over generations within larger breeds.

A spatio-temporal nonparametric Bayesian variable selection model of fMRI data for clustering correlated time courses

In this paper we present a novel wavelet-based Bayesian nonparametric regression model for the analysis of functional magnetic resonance imaging (fMRI) data. Our goal is to provide a joint analytical framework that allows to detect regions of the brain which exhibit neuronal activity in response to a stimulus and, simultaneously, infer the association, or clustering, of spatially remote voxels that exhibit fMRI time series with similar characteristics. We start by modeling the data with a hemodynamic response function (HRF) with a voxel-dependent shape parameter. We detect regions of the brain activated in response to a given stimulus by using mixture priors with a spike at zero on the coefficients of the regression model. We account for the complex spatial correlation structure of the brain by using a Markov random field (MRF) prior on the parameters guiding the selection of the activated voxels, therefore capturing correlation among nearby voxels. In order to infer association of the voxel time courses, we assume correlated errors, in particular long memory, and exploit the whitening properties of discrete wavelet transforms. Furthermore, we achieve clustering of the voxels by imposing a Dirichlet process (DP) prior on the parameters of the long memory process. For inference, we use Markov Chain Monte Carlo (MCMC) sampling techniques that combine Metropolis–Hastings schemes employed in Bayesian variable selection with sampling algorithms for nonparametric DP models. We explore the performance of the proposed model on simulated data, with both block- and event-related design, and on real fMRI data.

Right-hand-side updating for fast computing of genomic breeding values

BackgroundSince both the number of SNPs (single nucleotide polymorphisms) used in genomic prediction and the number of individuals used in training datasets are rapidly increasing, there is an increasing need to improve the efficiency of genomic prediction models in terms of computing time and memory (RAM) required.MethodsIn this paper, two alternative algorithms for genomic prediction are presented that replace the originally suggested residual updating algorithm, without affecting the estimates. The first alternative algorithm continues to use residual updating, but takes advantage of the characteristic that the predictor variables in the model (i.e. the SNP genotypes) take only three different values, and is therefore termed “improved residual updating”. The second alternative algorithm, here termed “right-hand-side updating” (RHS-updating), extends the idea of improved residual updating across multiple SNPs. The alternative algorithms can be implemented for a range of different genomic predictions models, including random regression BLUP (best linear unbiased prediction) and most Bayesian genomic prediction models. To test the required computing time and RAM, both alternative algorithms were implemented in a Bayesian stochastic search variable selection model.ResultsCompared to the original algorithm, the improved residual updating algorithm reduced CPU time by 35.3 to 43.3%, without changing memory requirements. The RHS-updating algorithm reduced CPU time by 74.5 to 93.0% and memory requirements by 13.1 to 66.4% compared to the original algorithm.ConclusionsThe presented RHS-updating algorithm provides an interesting alternative to reduce both computing time and memory requirements for a range of genomic prediction models.

Combining cow and bull reference populations to increase accuracy of genomic prediction and genome-wide association studies

Genomic selection holds the promise to be particularly beneficial for traits that are difficult or expensive to measure, such that access to phenotypes on large daughter groups of bulls is limited. Instead, cow reference populations can be generated, potentially supplemented with existing information from the same or (highly) correlated traits available on bull reference populations. The objective of this study, therefore, was to develop a model to perform genomic predictions and genome-wide association studies based on a combined cow and bull reference data set, with the accuracy of the phenotypes differing between the cow and bull genomic selection reference populations. The developed bivariate Bayesian stochastic search variable selection model allowed for an unbalanced design by imputing residuals in the residual updating scheme for all missing records. The performance of this model is demonstrated on a real data example, where the analyzed trait, being milk fat or protein yield, was either measured only on a cow or a bull reference population, or recorded on both. Our results were that the developed bivariate Bayesian stochastic search variable selection model was able to analyze 2 traits, even though animals had measurements on only 1 of 2 traits. The Bayesian stochastic search variable selection model yielded consistently higher accuracy for fat yield compared with a model without variable selection, both for the univariate and bivariate analyses, whereas the accuracy of both models was very similar for protein yield. The bivariate model identified several additional quantitative trait loci peaks compared with the single-trait models on either trait. In addition, the bivariate models showed a marginal increase in accuracy of genomic predictions for the cow traits (0.01–0.05), although a greater increase in accuracy is expected as the size of the bull population increases. Our results emphasize that the chosen value of priors in Bayesian genomic prediction models are especially important in small data sets.

Accuracy of genomic prediction using different models and response variables in theNordicRed cattle population

Breeding animals can be accurately evaluated using appropriate genomic prediction models, based on marker data and phenotype information. In this study, direct genomic values (DGV) were estimated for 16 traits of Nordic Total Merit (NTM) Index in Nordic Red cattle population using three models and two different response variables. The three models were as follows: a linear mixed model (GBLUP), a Bayesian variable selection model similar to BayesA (BayesA*) and a Bayesian least absolute shrinkage and selection operator model (Bayesian Lasso). The response variables were deregressed proofs (DRP) and conventional estimated breeding values (EBV). The reliability of genomic predictions was measured on bulls in the validation data set as the squared correlation between DGV and DRP divided by the reliability of DRP. Using DRP as response variable, the reliabilities of DGV among the 16 traits ranged from 0.151 to 0.569 (average 0.317) for GBLUP, from 0.152 to 0.576 (average 0.318) for BayesA* and from 0.150 to 0.570 (average 0.320) for Bayesian Lasso. Using EBV as response variable, the reliabilities ranged from 0.159 to 0.580 (average 0.322) for GBLUP, from 0.157 to 0.578 (average 0.319) for BayesA* and from 0.159 to 0.582 (average 0.325) for Bayesian Lasso. In summary, Bayesian Lasso performed slightly better than the other two models, and EBV performed slightly better than DRP as response variable, with regard to prediction reliability of DGV. However, these differences were not statistically significant. Moreover, using EBV as response variable would result in problems with the scale of the resulting DGV and potential problem due to double counting.

Bayesian nonparametric variable selection as an exploratory tool for discovering differentially expressed genes

High-throughput scientific studies involving no clear a priori hypothesis are common. For example, a large-scale genomic study of a disease may examine thousands of genes without hypothesizing that any specific gene is responsible for the disease. In these studies, the objective is to explore a large number of possible factors (e.g., genes) in order to identify a small number that will be considered in follow-up studies that tend to be more thorough and on smaller scales. A simple, hierarchical, linear regression model with random coefficients is assumed for case-control data that correspond to each gene. The specific model used will be seen to be related to a standard Bayesian variable selection model. Relatively large regression coefficients correspond to potential differences in responses for cases versus controls and thus to genes that might 'matter'. For large-scale studies, and using a Dirichlet process mixture model for the regression coefficients, we are able to find clusters of regression effects of genes with increasing potential effect or 'relevance', in relation to the outcome of interest. One cluster will always correspond to genes whose coefficients are in a neighborhood that is relatively close to zero and will be deemed least relevant. Other clusters will correspond to increasing magnitudes of the random/latent regression coefficients. Using simulated data, we demonstrate that our approach could be quite effective in finding relevant genes compared with several alternative methods. We apply our model to two large-scale studies. The first study involves transcriptome analysis of infection by human cytomegalovirus. The second study's objective is to identify differentially expressed genes between two types of leukemia.

Model Uncertainty in Characterizing Recreation Demand

A Bayesian variable selection procedure is used to control for uncertainty in the specification of a recreational demand model. Specifically, we propose a model that draws on the Bayesian paradigm to integrate the variable selection process into model estimation and to reflect the accompanying uncertainty about which is the best specification in subsequent counterfactual predictions. The advantage of this procedure over previous non-Bayesian approaches is that it overcomes the problem of pre-testing in specification searches. In our application, evaluating demand for recreational lake usage in Iowa, we find clear evidence that site attributes, such as lakes size, handicap facilities and wake restrictions, do impact lake usage. There is also evidence that water quality matters in household recreation choices. Indeed, contrary to Abidoye et al. (Am J Agricult Econ, 2012), in which only a single functional form is considered, we find clear evidence that water quality matters, with posterior probability of less that 10 % associated with a model without any water quality variables. This suggests that the flexibility that the Bayesian variable selection model affords in capturing the linkage between recreation demand and site characteristics can be important.

Makeup of the genetic correlation between milk production traits using genome-wide single nucleotide polymorphism information

The correlated responses between traits may differ depending on the makeup of genetic covariances, and may differ from the predictions of polygenic covariances Therefore, the objective of the present study was to investigate the makeup of the genetic covariances between the well-studied traits: milk yield, fat yield, protein yield, and their percentages in more detail. Phenotypic records of 1,737 heifers of research farms in 4 different countries were used after homogenizing and adjusting for management effects. All cows had a genotype for 37,590 single nucleotide polymorphisms (SNP). A Bayesian stochastic search variable selection model was used to estimate the SNP effects for each trait. About 0.5 to 1.0% of the SNP had a significant effect on 1 or more traits; however, the SNP without a significant effect explained most of the genetic variances and covariances of the traits. Single nucleotide polymorphism correlations differed from the polygenic correlations, but only 10 regions were found with an effect on multiple traits; in 1 of these regions the DGAT1 gene was previously reported with an effect on multiple traits. This region explained up to 41% of the variances of 4 traits and explained a major part of the correlation between fat yield and fat percentage and contributes to asymmetry in correlated response between fat yield and fat percentage. Overall, for the traits in this study, the infinitesimal model is expected to be sufficient for the estimation of the variances and covariances.