Coefficient of multiple correlation: Difference between revisions

Revision as of 09:40, 18 November 2010

Revision as of 04:08, 20 November 2010

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The coefficient of multiple determination ''R''² (a [[scalar (mathematics)|scalar]]), is computed using the [[Euclidean space|vector]] ''c'' of cross-correlations between the predictor variables and the criterion variable, its [[transpose]] ''c''', and the [[Matrix (mathematics)|matrix]] ''R''_''xx'' of inter-correlations between predictor variables. The fundamental equation of multiple regression analysis is

::''R''² = ''c''' ''R''_''xx''^−1 ''c''.

The expression on the left side denotes the coefficient of multiple determination (the squared coefficient of multiple correlation). The terms on the right side are the transposed vector ''c'' ' of cross-correlations, the [[Matrix inversion|inverse]] of the matrix ''R''_''xx'' of inter-correlations, and the vector ''c'' of cross-correlations. The inverted matrix of the inter-correlations removes the redundant variance that results from the inter-correlations of the predictor variables. The square root of the resulting coefficient of multiple determination is the coefficient of multiple correlation ''R''. Note that if all the predictor variables are uncorrelated, the matrix ''R''_''xx'' is the identity matrix and ''R''² simply equals ''c''' ''c'', the sum of the squared cross-correlations.

{{nofootnotes}}

==References==

* Paul D. Allison. ''Multiple Regression: A Primer'' (1998)

* Cohen, Jacob, et al. ''Applied Multiple Regression - Correlation Analysis for the Behavioral Sciences'' (2002) (ISBN: 0805822232)

* Crown, William H. ''Statistical Models for the Social and Behavioral Sciences: Multiple Regression and Limited-Dependent Variable Models'' (1998) (ISBN: 0275953165)

* Edwards, Allen Louis. ''Multiple regression and the analysis of variance and covariance'' (1985)(ISBN: 0716710811)

* Timothy Z. Keith. '' Multiple Regression and Beyond'' (2005)

* Fred N. Kerlinger, Elazar J. Pedhazur, ''Multiple Regression in Behavioral Research.'' (1973)

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* [http://www.visualstatistics.net/Visual%20Statistics%20Multimedia/multiple_regression_analysis.htm A Guide To Computing <math>R^2</math> For Multiple Correlation]

* [http://www.docstoc.com/docs/3530187/A-Derivation-of-the-Sample-Multiple-Corelation-Formula-for-Standard-Scores, "Derivations"]

{{DEFAULTSORT:Multiple Correlation}}

[[Category:Regression analysis]]