The Geometry of Enhancement in Multiple Regression.

In linear multiple regression, "enhancement" is said to occur when R (2)=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']=V Λ V' (where V Λ V' denotes the eigen decomposition of R xx; λ 1>λ 2), the set [Formula: see text] contains four vectors; the set [Formula: see text]; [Formula: see text] contains an infinite number of vectors. When p≥3 (and λ 1>λ 2>⋯>λ p ), both sets contain an uncountably infinite number of vectors. Geometrical arguments demonstrate that B 1 occurs at the intersection of two hyper-ellipsoids in ℝ (p) . Equations are provided for populating the sets B 1 and B 2 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.