Covariance Analysis with Unequal Subclass Numbers: Component-Estimation in Quantitative Genetics

Abstract

SUMMARY Analysis of covariance for the hierarchical design with unequal subclass numbers is presented in this expository paper. The derivations of the expectations of the sums of cross products of two covariables and of the coefficients of the covariance components are given. The covariance components are estimated and their genetic interpretation is shown. The estimation of phenotypic, genetic, and environmental correlations employing covariance components is discussed. A numerical example demonstrating the use of covariance analysis in genetic studies is offered. In the application of quantitative genetics, phenotypic, genetic, and environmental correlation coefficients describing the relationship between two characters are useful statistics in the evaluation of correlated responses during selection. Moreover, they aid in the formulation of breeding programs designed to achieve maximum improvement in total productivity. The covariance components required in estimating these correlations can be obtained from an analysis of covariance; the latter is an extension of the procedures used in analysis of variance. King and Henderson [1954] gave the details of the analysis of variance for variance component estimation from data with unequal subclass numbers. Hazel et al. [1943] first demonstrated the extension of variance component analysis to the estimation of covariance components assumning equal numbers. Considerable statistical knowledge would be required to carry out the analysis with non-orthogonal data (Henderson [1953]). The recent appearance of computer programs designed to process analyses of covariance for varying types of data (see e.g. Bogyo [1965]) requires that some effort be made to present the theory in an acceptable form if practical applications are to be as efficient as possible.