Incorporation of economic values into the component traits of a ratio: feed efficiency.

Direct selection on a ratio (R) of 2 traits (x1/x2) does not have a mechanism to accommodate the relative economic values (a1 and a2) between x1 and x2 because selection criteria x1/x2 and a1x1/a2x2 rank animals in the same order. This study presented a procedure to incorporate the economic weights into ratio traits through linear transformation. The partial derivatives of a nonlinear profit function evaluated at the means were widely taken as economic weights in the literature. This study showed that the economic weights derived in this manner were erroneous because they actually contain a mixture of actual economic weights and transformation effects. The ratios 1/2 and 2/4 are considered equal by selection on R, but are treated differently by the linear index. In addition, this study presented a unified approach to compare 4 different selection strategies for genetic improvement of ratio traits: linear index (I), selection on the ratio (R), selection on difference between x1 and x2 (D), and selection on x1 alone. This study considered 3 levels of heritability each for variables x1 and x2 and 2 levels of genetic correlations (γG), 2 ratios of means (µ1/µ2), and 4 ratios of phenotypic variances giving a total of 96 scenarios. Linear index I was the most efficient of the 4 criteria compared in all 96 scenarios studied. The superiority of index I over R, D, and selection on x1 alone are particularly remarkable when x1 and x2 have a large difference in heritability and are highly correlated. Selection on x1 alone is an economically viable alternative to criterion I or R for the improvement of ratio traits particularly when x1 is more heritable than x2 and when x2 is costly to measure. Selection on D is more efficient than direct selection on R or selection on x1 alone when x1 is less heritable than x2 and the difference between µ1 and µ2 is small.