**ERIC Number:**EJ945678

**Record Type:**Journal

**Publication Date:**2011-Oct

**Pages:**16

**Abstractor:**As Provided

**Reference Count:**49

**ISBN:**N/A

**ISSN:**ISSN-0033-3123

The Geometry of Enhancement in Multiple Regression

Waller, Niels G.

Psychometrika, v76 n4 p634-649 Oct 2011

In linear multiple regression, "enhancement" is said to occur when R[superscript 2] = b[prime]r greater than r[prime]r, where b is a p x 1 vector of standardized regression coefficients and r is a p x 1 vector of correlations between a criterion y and a set of standardized regressors, x. When p = 1 then b [is congruent to] r and enhancement cannot occur. When p = 2, for all full-rank R[subscript xx] is not equal to I, R[subscript xx] = E[xx[prime]] = V[image omitted]V[prime] (where V[image omitted]V[prime] denotes the eigen decomposition of R[subscript xx]; [lambda][superscript 1] greater than [lambda][superscript 2]), the set B[subscript 1] := {b[subscript i] : R[superscript 2] = b[prime][subscript i]r[subscript i] = r[prime][subscript i]r[subscript i]; 0 less than R[superscript 2] less than or equal to 1} contains four vectors; the set B[subscript 2] := {b[subscript i] : R[superscript 2] = b[prime][subscript i]r[subscript i] greater than r[prime][subscript i]r[subscript i]; 0 less than R[superscript 2] less than or equal to 1; R[superscript 2][lambda][subscript p] less than or equal to r[prime][subscript i]r[subscript i] less than R[superscript 2]} contains an infinite number of vectors. When p greater than or equal to 3 (and [lambda][subscript 1] greater than [lambda][subscript 2] greater than [image omitted] greater than [lambda][subscript p]), both sets contain an uncountably infinite number of vectors. Geometrical arguments demonstrate that B[subscript 1] occurs at the intersection of two hyper-ellipsoids in [set of real numbers][superscript P]. Equations are provided for populating the sets B[subscript 1] and B[subscript 2] and for demonstrating that maximum enhancement occurs when b is collinear with the eigenvector that is associated with [lambda][subscript 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. (Contains 2 figures.

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**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**N/A

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A