Belcy Karine Angarita Barajas

Genomic relationships can be computed with dense genome-wide genotypes through different methods, either based on identity-by-state (IBS) or identity-by-descent (IBD). The latter has been shown to increase the accuracy of both estimated relationships and predicted breeding values. However, it is not clear whether an IBD approach would achieve greater heritability (h2$$ {\boldsymbol{h}}<^>{\mathbf{2}} $$) and predictive ability (ry,y$$ {\hat{r}}_{y,\hat{y}} $$) than its IBS counterpart for data with low-depth pedigrees. Here, we compare both approaches in terms of the estimated of h2$$ {\boldsymbol{h}}<^>{\mathbf{2}} $$ and ry,y$$ {\hat{r}}_{y,\hat{y}} $$, using data on meat quality and carcass traits recorded in experimental crossbred pigs, with a pedigree constrained to only three generations. Three animal models were fitted which differed on the relationship matrix: an IBS model (GIBS$$ {\boldsymbol{G}}_{\mathbf{IBS}} $$), an IBD (defined within the known pedigree) model (GIBD$$ {\boldsymbol{G}}_{\mathbf{IBD}} $$), and a pedigree model (A22$$ {\boldsymbol{A}}_{\mathbf{22}} $$). In 9 of 20 traits, the range of increase for the estimates of sigma u2$$ {\sigma}_{\mathrm{u}}<^>2 $$ and h2$$ {\boldsymbol{h}}<^>{\mathbf{2}} $$ was 1.2-2.9 times greater with GIBS$$ {\boldsymbol{G}}_{\mathbf{IBS}} $$ and GIBD$$ {\boldsymbol{G}}_{\mathbf{IBD}} $$ models than with A22$$ {\boldsymbol{A}}_{\mathbf{22}} $$. Whereas for all traits, both parameters were similar between genomic models. The ry,y$$ {\hat{r}}_{y,\hat{y}} $$ of the genomic models was higher compared to A22$$ {\boldsymbol{A}}_{\mathbf{22}} $$. A scarce increment in ry,y$$ {\hat{r}}_{y,\hat{y}} $$ was found with GIBS$$ {\boldsymbol{G}}_{\mathbf{IBS}} $$ when compared to GIBD$$ {\boldsymbol{G}}_{\mathbf{IBD}} $$, most likely due to the former recovering sizeable relationships among founder F-0 animals.

Background: The covariance matrix of breeding values is at the heart of prediction methods. Prediction of breeding values can be formulated using either an “observed” or a theoretical covariance matrix, and a major argument for choosing one or the other is the reduction of the computational burden for inverting such a matrix. In this regard, covariance matrices that are derived from Markov causal models possess properties that deliver sparse inverses. Results: By using causal Markov models, we express the breeding value of an individual as a linear regression on ancestral breeding values, plus a residual term, which we call residual breeding value (RBV). The latter is a noise term that accounts for the uncertainty in prediction due to lack of fit of the linear regression. A notable property of these models is the parental Markov condition, through which the multivariate distribution of breeding values is uniquely determined by the distribution of the mutually independent RBV. Animal breeders have long been relying on a causal Markov model, while using the additive relationship matrix as the covariance matrix structure of breeding values, which is calculated assuming gametic equilibrium. However, additional covariances among breeding values arise due to identity disequilibrium, which is defined as the difference between the covariance matrix under the multi-loci probability of identity-by-descent ( $$\varvec{\Sigma}$$ Σ ) and its expectation under gametic phase equilibrium, i.e., A . The disequilibrium term $$\varvec{\Sigma}$$ Σ − A is considered in the model for predicting breeding values called the “ancestral regression” (AR), a causal Markov model. Here, we introduce the “ancestral regression to parents” (PAR) causal Markov model, which reduces the computational burden of the AR approach. By taking advantage of the conditional independence property of the PAR Markov model, we derive covariances between the breeding values of grandparents and grand-offspring and between parents and offspring. In addition, we obtain analytical expressions for the covariance between collateral relatives under the PAR model, as well as for the inbreeding coefficient. Conclusions: We introduced the causal PAR Markov model that captures identity disequilibrium in the covariances among breeding values and produces a sparse inverse covariance matrix to build and solve a set of mixed model equations.