The equivalence of two linear methods for the improvement of traits expressed as ratios

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Abstract
Many traits of interest to animal breeders can be expressed as ratios. Yet there remains no uniquely agreed upon method for the genetic evaluation for ratio traits. To generalize, livestock breeders make direct use of ratios (e.g., feed/gain) or linear approximations to ratios. Dairy breeders, on the other hand, tend to use ratios of linear predictors of genetic merit for the evaluation of ratio traits (e.g., fat percent). In the present note, we demonstrate that the two methods are nearly, though not exactly, identical. The proof relies on the expression of the approximate correlation between two ratios.