Regression of Genotypic on Phenotypic Value of a Ratio-Defined Character

An exact expression for the joint probability density function of the phenotypic and genotypic values is examined for the ratio-defined character in which two component characters are assumed to follow a bivariate normal law and to have positive values. An approximation to the function is derived in terms of parameters of two component characters. A nonlinear approximation to the true regression function of the genotypic value on the phenotypic value is obtained. Similar approximations to the regression functions of the genotypic values of two component characters on the phenotypic value of the ratio-defined character are also given. An example is used to discuss the validity of the given approximations and to illustrate the geometric shapes of the regression functions.