Rules for assortative mating in relation to selection for linear merit functions.

This study examined how assortative mating (without selection) based on linear combinations of two traits could be used to change genetic parameters so as to increase efficiency of selection. The efficiency of the Smith-Hazel index for improvement of multiple traits is a function of phenotypic and genetic variances and covariances, and of the relative economic values of the traits involved. Assortative mating is known to change genetic variances and covariances. Recursive formulae were derived to obtain these variances and covariances after t generations of assortative mating on linear combinations (mating rules) of phenotypic values for two traits, with a given correlation between mates. Selection efficiency after t generations of assortative mating without selection was expressed as a function of random mating genetic parameters, economic values, the mating rule, and the correlation between mates. Selection efficiency was maximized with respect to the coefficients in the mating rule. Because the objective function was nonlinear, a computer routine was used for maximizing it. Two cases were considered. When random mating heritabilities for the two traits were h X (2) =0.25 and h Y (2) =0.50, the genetic correlation rXY=-0.60, and the economic values were aX=3 and aY=1, continued assortative mating based on the optimal mating rule for 31 generations (with a correlation of 0.80 between mates) increased selection efficiency by 29%. Heritabilities changed to 0.38 and 0.66, respectively, and the genetic correlation became - 0.79. When h X (2) =0.60, h Y (2) =0.60, rXY=- 0.20, a1=1 and a2=1, 36 generations of continued assortative mating with the optimal mating rule increased the efficiency of selection by 17%, heritabilities became h X (2) = h Y (2) =0.71, and the genetic correlation changed to 0.25. Only three generations of assortative mating were required to change the sign of the genetic correlation.