Covariance between breeding values in the ancestral regression Gaussian model

CANTET, R. J. C.; MUNILLA, S.; GARCÍA-BACCINO, C. A.

Tallinn

Congreso; 68th Annual Meeting of the European Association for Animal Production; 2017

European Association for Animal Production

We have recentlyintroduced the ancestral regression (AR) as a conditional model that generalizesthe parental regression in the animal model. The breeding value (BV) of ananimal in AR is regressed to the average of parental BV plus to a linearfunction of the BV of all four grandparents. This function is orthogonal to parentalBV in the absence of inbreeding. Hence, AR reduces Mendelian residual varianceand increases accuracy of prediction. The AR model includes two linear parametersper animal evaluated, which are path coefficients expressing the relativedifference of contributions of grandsires over granddams (or viceversa) in thegenome and BV of the individual. Segmental inheritance (non-independent geneticeffects) is accounted for if probabilities of IBD at two loci are used toestimate the linear parameters. Whole genomic relationships between BV are thenfunctions of these parameters and are also the elements of the covariancestructure S.  Asymptotic normality isemployed to obtain the distribution of BV, and the resulting causal distributionis Gaussian and Markovian. This latter property allows inverting S as itis done with the additive relationship matrix because, conditionally on the BVof grandparents and parents, the BV of any animal is independent of anythingelse. Another plus of the AR is that covariances between BV are not limited toindependent genetic effects but also to linked genes at the same gamete, andthis allows including non-additive genetic effects that are transmitted as apart of the BV. Our goal here is to introduce the general expression for thecovariance between relatives in the AR, and to present some important covariancessuch as parent-offspring, grandparent grand-offspring, full and half sibs. Whereasin the general case 36 elements of S are involved whencalculating any covariance, the formula reduces to simple expressions in theabsence of inbreeding because many of the elements are zero. When there is nogenomic information available to identify the linear parameters, all of them arezero and covariances between relatives turn into the classical expressions.