Parameter estimation in the ancestral regression with missing parental or grandparental genotypes

CANTET, R. J. C.; FORNERIS, N.S.

Ghent

Congreso; 70th Annual Meeting of the European Federation of Animal Science; 2019

European Federation of Animal Science (EAAP)

The ancestral regression (AR) is a genetic evaluation model that generalizes the Gaussian animal model by reducing the variance of the Mendelian residuals. Fitting of the AR requires two linear parameters (βS and βD) per genotyped animal. These partial regression coefficients represent the excess (or defect) with respect to 0.25, from the genome shared IBD between the individual and any grandsire over the IBD shared between the individual and the respective granddam, and it is due to recombination. The original formulation of AR was presented on the assumption that all four grandparents are genotyped, although for most data sets this principle is not fulfilled. The objective of this research was to provide with a Bayesian algorithm for estimating βS and βD when one or more of the four grandparents, plus any parent, are not genotyped. The algorithm is based on the property of the covariance structure to be written as Σ = (I ? B)−1 D (I ? B´)−1. Matrix D is diagonal containing the standardized variances of the Mendelian errors in AR, Var(ϕ). In addition, each row of the triangular matrix B contains βS and βD, with positive sign in the columns of both grandsires and negative signs in the columns pertaining to both granddams, plus the two 0.5 values in the paternal columns. The algorithm uses the Cholesky root free decomposition of the covariance structure for a Gaussian vector of BV, and Σ−1 follows a Wishart density of small dimension, i.e. it includes the BV of all four grandparents, both parents and the individual. However, due to the restrictions on the positive definitiveness of Σ, a more efficient approach is to sample individual elements of this 7 × 7 matrix. Sufficient statistics are the differences between the covariance between the grandsire and X, minus the covariance between the granddam and X. The diagonal elements in D are the variance of the Mendelian residuals and are sampled from a truncated chi-square density. The beta parameters are sampled directly from a truncated normal in the interval −0.25 to 0.25. It was observed in simulated data that if either grandparent is not genotyped, the estimate of any beta goes to zero. The missing information can be supplied by ancestors and half-sibs of the grandparent, but at the expense of a higher standard error. Hence, the breed association that adopted AR for genomic evaluation is asking commercial breeders to genotype the calf and its dam, whereas the association takes care of genotyping the bulls. An example is given with actual data on beef cattle to illustrate the method.