Statistica Sinica

Abraham Genizi

Abstract:Let correlated regressors and a dependent variable have a joint distribution, and assume that a suitable regression model has been found. A decomposition ofR^{2}into components corresponding to the regressors is proposed. The components are to serve as descriptive intuitive statistics indicating the relative importance of each regressor with respect to its overall effect on the dependent variable. When it is possible to partition the set of regressors into mutually orthogonal subsets, the sum of components of this decomposition in any subset is equal to the multipleR^{2}due to that subset. Each component consists of a subcomponent ``specific'' to that regressor, and of ``common'' subcomponents with each of the other regressors. A measure of deviation of the set of regressors from orthogonality is given to help in assessing the amount o f approximation used in the decomposition.

Key words and phrases:Multiple correlation coefficient, Decomposition of explained variance, Relative importance of regressors, Geometrical representation, Measure of deviation from orthogonality.