The Geometry of Enhancement in Multiple Regression

In linear multiple regression, "enhancement" is said to occur when R²=b′r > r′r, where b is a p×1 vector of standardized regression coefficients and r is a p×1 vector of correlations between a criterion y and a set of standardized regressors, x. When p=1 then b≡r and enhancement cannot occur. When p=2, for all full-rank R_xx ≠ I, R_xx=E[xx′]=VΛV′ (where VΛV′ denotes the eigen decomposition of R_xx; λ₁ > λ₂), the set contains four vectors; the set contains an infinite number of vectors. When p ≥ 3 (and λ₁ > λ₂ >... > λ_p), both sets contain an uncountably infinite number of vectors. Geometrical arguments demonstrate that B₁ occurs at the intersection of two hyper-ellipsoids in ℝ^p. Equations are provided for populating the sets B₁ and B₂ and for demonstrating that maximum enhancement occurs when b is collinear with the eigenvector that is associated with λ_p (the smallest eigenvalue of the predictor correlation matrix). These equations are used to illustrate the logic and the underlying geometry of enhancement in population, multiple-regression models. R code for simulating population regression models that exhibit enhancement of any degree and any number of predictors is included in Appendices A and B.