## Abstract

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′]=VΛV′ (where VΛV′ denotes the eigen decomposition of R_{xx}; λ_{1} > λ_{2}), the set contains four vectors; the set 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.

Original language | English (US) |
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Pages (from-to) | 634-649 |

Number of pages | 16 |

Journal | Psychometrika |

Volume | 76 |

Issue number | 4 |

DOIs | |

State | Published - Oct 1 2011 |

## Keywords

- multiple regression
- suppression
- suppressor variable