Carlos Alberto Martínez

We addressed genomic prediction accounting for partial correlation of marker effects, which entails the estimation of the partial correlation network/graph (PCN) and the precision matrix of an unobservable m‐dimensional random variable. To this end, we developed a set of statistical models and methods by extending the canonical model selection problem in Gaussian concentration, and directed acyclic graph models. Our frequentist formulations combined existing methods with the EM algorithm and were termed Glasso‐EM, Concord‐EM and CSCS‐EM, whereas our Bayesian formulations corresponded to hierarchical models termed Bayes G‐Sel and Bayes DAG‐Sel. We implemented our methods in a real bull fertility dataset and then carried out gene annotation of seven markers having the highest degrees in the estimated PCN. Our findings brought biological evidence supporting the usefulness of identifying genomic regions that are highly connected in the inferred PCN. Moreover, a simulation study showed that some of our methods can accurately recover the PCN (accuracy up to 0.98 using Concord‐EM), estimate the precision matrix (Concord‐EM yielded the best results) and predict breeding values (the best reliability was 0.85 for a trait with heritability of 0.5 using Glasso‐EM).

Summary: The aim of this paper was to develop statistical models to estimate individual breed composition based on the previously proposed idea of regressing discrete random variables corresponding to counts of reference alleles of biallelic molecular markers located across the genome on the allele frequencies of each marker in the pure (base) breeds. Some of the existing regression‐based methods do not guarantee that estimators of breed composition will lie in the appropriate parameter space, and none of them account for uncertainty about allele frequencies in the pure breeds, that is, uncertainty about the design matrix. To overcome these limitations, we proposed two Bayesian generalized linear models. For each individual, both models assume that the counts of the reference allele at each marker locus follow independent Binomial distributions, use the logit link and pose a Dirichlet prior over the vector of regression coefficients (which corresponds to breed composition). This prior guarantees that point estimators of breed composition such as the posterior mean pertain to the appropriate space. The difference between these models is that model termed BIBI does not account for uncertainty about the design matrix, while model termed BIBI2 accounts for such an uncertainty by assigning independent Beta priors to the entries of this matrix. We implemented these models in a data set from the University of Florida's multibreed Angus‐Brahman population. Posterior means were used as point estimators of breed composition. In addition, the ordinary least squares estimator proposed by Kuehn et al. () (OLSK) was also computed. BIBI and BIBI2 estimated breed composition more accurately than OLSK, and BIBI2 had a 7.69% improvement in accuracy as compared to BIBI.

The aim of this paper was to develop statistical models to estimate individual breed composition based on the previously proposed idea of regressing discrete random variables corresponding to counts of reference alleles of biallelic molecular markers located across the genome on the allele frequencies of each marker in the pure (base) breeds. Some of the existing regression-based methods do not guarantee that estimators of breed composition will lie in the appropriate parameter space and none of them account for uncertainty about allele frequencies in the pure breeds, that is, uncertainty about the design matrix. In order to overcome these limitations, we proposed two Bayesian generalized linear models. For each individual, both models assume that the counts of the reference allele at each marker locus follow independent Binomial distributions, use the logit link, and pose a Dirichlet prior over the vector of regression coefficients (which corresponds to breed composition). This prior guarantees that point estimators of breed composition like the posterior mean pertain to the appropriate space. The difference between these models is that model termed BIBI does not account for uncertainty about the design matrix, while model termed BIBI2 accounts for such an uncertainty by assigning independent Beta priors to the entries of this matrix. We implemented these models in a multibreed Angus-Brahman population. Posterior means were used as point estimators of breed composition. In addition, the ordinary least squares estimator proposed by Kuehn et al. (2011) (OLSK) was also computed. BIBI and BIBI2 estimated breed composition more accurately than OLSK, and BIBI2 had an 8.3% improvement in accuracy as compared to BIBI.

In genome-wide prediction, independence of marker allele substitution effects is typically assumed; however, since early stages of this technology it has been known that nature points to correlated effects. In statistics, graphical models have been identified as a useful and powerful tool for covariance estimation in high dimensional problems and it is an area that has recently experienced a great expansion. In particular, Gaussian concentration graph models (GCGM) have been widely studied. These are models in which the distribution of a set of random variables, the marker effects in this case, is assumed to be Markov with respect to an undirected graph G. In this paper, Bayesian (Bayes G and Bayes G-D) and frequentist (GML-BLUP) methods adapting the theory of GCGM to genome-wide prediction were developed. Different approaches to define the graph G based on domain-specific knowledge were proposed, and two propositions and a corollary establishing conditions to find decomposable graphs were proven. These methods were implemented in small simulated and real datasets. In our simulations, scenarios where correlations among allelic substitution effects were expected to arise due to various causes were considered, and graphs were defined on the basis of physical marker positions. Results showed improvements in correlation between phenotypes and predicted breeding values and accuracies of predicted breeding values when accounting for partially correlated allele substitution effects. Extensions to the multiallelic loci case were described and some possible refinements incorporating more flexible priors in the Bayesian setting were discussed. Our models are promising because they allow incorporation of biological information in the prediction process, and because they are more flexible and general than other models accounting for correlated marker effects that have been proposed previously.

Summary: Several statistical models used in genome‐wide prediction assume uncorrelated marker allele substitution effects, but it is known that these effects may be correlated. In statistics, graphical models have been identified as a useful tool for covariance estimation in high‐dimensional problems and it is an area that has recently experienced a great expansion. In Gaussian covariance graph models (GCovGM), the joint distribution of a set of random variables is assumed to be Gaussian and the pattern of zeros of the covariance matrix is encoded in terms of an undirected graphG. In this study, methods adapting the theory of GCovGM to genome‐wide prediction were developed (Bayes GCov, Bayes GCov‐KR and Bayes GCov‐H). In simulated data sets, improvements in correlation between phenotypes and predicted breeding values and accuracies of predicted breeding values were found. Our models account for correlation of marker effects and permit to accommodate general structures as opposed to models proposed in previous studies, which consider spatial correlation only. In addition, they allow incorporation of biological information in the prediction process through its use when constructing graphG, and their extension to the multi‐allelic loci case is straightforward.

Several statistical models used in genome-wide prediction assume independence of marker allele substitution effects, but it is known that these effects might be correlated. In statistics, graphical models have been identified as a useful tool for covariance estimation in high dimensional problems and it is an area that has recently experienced a great expansion. In Gaussian covariance graph models (GCovGM), the joint distribution of a set of random variables is assumed to be Gaussian and the pattern of zeros of the covariance matrix is encoded in terms of an undirected graph G. In this study, methods adapting the theory of GCovGM to genome-wide prediction were developed (Bayes GCov, Bayes GCov-KR and Bayes GCov-H). In simulated and real datasets, improvements in correlation between phenotypes and predicted breeding values and accuracies of predicted breeding values were found. Our models account for correlation of marker effects and permit to accommodate general structures as opposed to models proposed in previous studies which consider spatial correlation only. In addition, they allow incorporation of biological information in the prediction process through its use when constructing graph G, and their extension to the multiallelic loci case is straightforward.