Selection Differentials of Component Traits in Truncation Selection for a Proportion-Defined Trait

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Abstract

In this paper, for one specific type of ratio traits, or a proportion-defined trait (denoted simply as proportion trait), whose numerator trait is contained also in the denominator as its one component, formulae to predict selection differentials for the component traits are derived, assuming upward and downward truncation selection based on phenotypic value of the proportion trait. Two component traits are considered, supposing that these traits follow bivariate normal law and their coefficients of variation are small. The phenotypic changes of component traits are described in terms of a given proportion of population selected, the corresponding truncation point on the proportion trait and phenotypic parameters for the component traits. Using the formulae derived, a deterministic investigation of the selection differentials is conducted for three selected sets of phenotypic parameters for component traits with which the shapes of the distribution of the proportion trait are very nearly symmetric, skewed to the right and skewed to the left. In addition, the signs of the selection differentials are theoretically determined by examining the related inequality including a function of truncation point and the phenotypic parameters. As a consequence, it is revealed that for each of upward and downward selection, there occur three kinds of possible combinations of the signs according to the situations of the phenotypic parameters.

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