Genetic evaluation with autosomal and X-chromosomal inheritance

Summary

At present, genetic evaluation in livestock using best linear unbiased prediction (BLUP) assumes autosomal inheritance. There is evidence, however, of X-chromosomal inheritance for some traits of economic importance. BLUP can accommodate models that include X-chromosomal in addition to autosomal inheritance. To obtain BLUP with autosomal and X-chromosomal additive inheritance for a population in which allelic frequency is equal in the sexes, and that is in gametic equilibrium, we write y _i = x′_iβ + a_i + s_i + e_i, where y _i is the phenotypic value for individual i, x′_i, is a vector of constants relating y _i to fixed effects, β is a vector of fixed effects, a _i is the additive genetic effect for autosomal loci, S _i is the additive genetic effect for X-chromosomal loci, and e _i is random error. The covariance matrix of a _i's is Aσ ²_A , where A is the matrix of twice the co-ancestries between relatives for autosomal loci, and σ ²_A is the variance of additive genetic effects for autosomal loci. The covariance matrix of s _i's is Sσ ²_F , where S is a matrix of functions of co-ancestries between relatives for X-chromosomal loci and σ ²_F is the variance of additive genetic effects for X-chromosomal loci for noninbred females. Given the covariance matrices of random effects a _i, s_i, and e _i, BLUPs of autosomal and of X-chromosomal additive effects can be obtained using mixed model equations. Recursive rules to construct S and an efficient algorithm to compute its inverse are given.