Correlated Responses in Two Traits when Selection is Practised on their Ratio

In many fields of application in plant breeding and crop science, ratios of two component traits X and Y are of interest (harvest index in cereals, leaf-to-stem ratio in forage legumes, height-to-diameter ratio in forest trees etc.). When selection is practised on the ratio X/Y of two traits X and Y, the experimenter may be interested in the resulting changes of both trait means. Based on improved approximations for the covariances between X and X/Y and between Y and X/Y and for the variance of X/Y the changes in the means of X and Y can be predicted by applying the regression approach from conventional selection theory. Explicit expressions for these correlated responses in X and Y when selection is practised on their ratio X/Y are derived and discussed. The different outcomes (decrease, zero change or increase) for the selection pressures on X and Y are characterized by phenotypic coefficients of variation of X and Y, phenotypic and genotypic correlations between X and Y and heritabilities of X and Y.