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ERIC Number: EJ945678
Record Type: Journal
Publication Date: 2011-Oct
Pages: 16
Abstractor: As Provided
Reference Count: 49
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