Genetic evaluation with autosomal and X-chromosomal inheritance

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Abstract
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 + ai + si + ei, 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 A 2 , where A is the matrix of twice the co-ancestries between relatives for autosomal loci, and σ A 2 is the variance of additive genetic effects for autosomal loci. The covariance matrix of s i's is Sσ F 2 , where S is a matrix of functions of co-ancestries between relatives for X-chromosomal loci and σ F 2 is the variance of additive genetic effects for X-chromosomal loci for noninbred females. Given the covariance matrices of random effects a i, si, 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.