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_i + 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, 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 A _A ² , where A is the matrix of twice the co-ancestries between relatives for autosomal loci, and _A ² is the variance of additive genetic effects for autosomal loci. The covariance matrix of s _i's is S _F ² , where S is a matrix of functions of co-ancestries between relatives for X-chromosomal loci and _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, 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.Dedicated to the memory of Dr. C. R. Henderson, whose encouraging comments stimulated the research in this paper. Supported in part by the Illinois Agricultural Experiment Station, Hatch Project 35-0367, Estimation of Genetic Parameters.