Coefficient of multiple correlation: Difference between revisions

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In [[statistics]], the coefficient of '''multiple correlation''' is a measure of how well a given variable can be predicted using a linear relationshipfunction amongof morea thanset of twoother variables. It is measured by the [[coefficient of multiple determination]], denotedbut asunder R²,the whichparticular isassumption athat measurethat ofthe best possible linear predictors are used, whereas the fitcoefficient of adetermination [[linearis regression]]defined for more general cases. The Acoefficient regression'sof R²multiple determination fallstakes somewherevalues between zero and one (assuming; a constanthigher termvalue hasindicates beena includedbetter inpredictability of the regression);[[dependent aand higherindependent valuevariables|dependent indicatesvariable]] afrom strongerthe relationship[[dependent amongand theindependent variables|independent variables]], with a value of one indicating that all data points fall exactly on a linethe inpredictions multidimensionalare spaceexact and a value of zero indicating that no relationshiplinear atcombination allof betweendependent variables is better than the independentsimpler variablespredictor collectivelywhich andconsists of mean of the dependenttarget variable.

==Definition==

Unlike the [[coefficient of determination]] in a regression involving just two variables, the coefficient of multiple determination is not computationally [[commutative]]: a regression of ''y'' on ''x'' and ''z'' will in general have a different R² than will a regression of ''z'' on ''x'' and ''y''. For example, suppose that in a particular sample the variable ''z'' is [[Correlation and dependence|uncorrelated]] with both ''x'' and ''y'', while ''x'' and ''y'' are linearly related to each other. Then a regression of ''z'' on ''y'' and ''x'' will yield an R² of zero, while a regression of ''y'' on ''x'' and ''z'' will yield a positive R².

The coefficient of multiple determination ''R''² (a [[scalar (mathematics)|scalar]]), can be computed using the [[Euclidean space|vector]] ''c'' of cross-[[correlation]]s (i.e. [[covariance]]sbetween NO,the THISpredictor IS NOT CORRECTvariables (NOTindependent COVARIANCESvariables). ALSO, THE CITED SOURCE NO LONGER EXISTS.) between the predictor variables and the criteriontarget variable, its(dependent [[transpose]]&nbsp;''c'''variable), and the [[Matrix (mathematics)|matrix]] ''R''_''xx'' of inter-correlations between predictor variables. TheIt "fundamentalis equationgiven of multiple regression analysis"<ref>[http://www.visualstatistics.net/Visual%20Statistics%20Multimedia/multiple_regression_analysis.htm Visualstatistics.net]</ref> isby

::''R''² = ''c''' ''R''_''xx''^&minus;1 ''c'',

==Fundamental equation of multiple regression analysis==

The coefficient of multiple determination ''R''² (a [[scalar (mathematics)|scalar]]), can be computed using the [[Euclidean space|vector]] ''c'' of cross-[[correlation]]s (i.e. [[covariance]]s NO, THIS IS NOT CORRECT (NOT COVARIANCES). ALSO, THE CITED SOURCE NO LONGER EXISTS.) between the predictor variables and the criterion variable, its [[transpose]]&nbsp;''c''', and the [[Matrix (mathematics)|matrix]] ''R''_''xx'' of inter-correlations between predictor variables. The "fundamental equation of multiple regression analysis"<ref>[http://www.visualstatistics.net/Visual%20Statistics%20Multimedia/multiple_regression_analysis.htm Visualstatistics.net]</ref> is

::where ''Rc''² =' is the [[transpose]] of ''c''', and ''R''_''xx''^&minus;1 is [[Matrix inversion|inverse]] of the matrix ''cR''_''xx''.

The expression on the left side denotes the coefficient of multiple determination. 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. Note that ifIf 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. Otherwise,If thethere invertedis matrixcross-correlation ofamong the inter-correlationspredictor removesvariables, the redundantinverse variance that results fromof the intercross-correlationscorrelation ofmatrix theaccounts predictorfor variablesthis.

==Properties==

Unlike the [[coefficient of determination]] in a regression involving just two variables, the coefficient of multiple determination is not computationally [[commutative]]: a regression of ''y'' on ''x'' and ''z'' will in general have a different R² than will a regression of ''z'' on ''x'' and ''y''. For example, suppose that in a particular sample the variable ''z'' is [[Correlation and dependence|uncorrelated]] with both ''x'' and ''y'', while ''x'' and ''y'' are linearly related to each other. Then a regression of ''z'' on ''y'' and ''x'' will yield an R² of zero, while a regression of ''y'' on ''x'' and ''z'' will yield a positive R².

==References==

{{Reflist}}

* Allison, Paul D. Allison.(1998) ''Multiple Regression: A Primer'' (1998){{full}}

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

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

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

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

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

* Stanton, Jeffrey M. (2001) [http://www.amstat.org/publications/jse/v9n3/stanton.html "Galton, Pearson, and the Peas: A Brief History of Linear Regression Analysisfor Statistics Instructors"], ''Journal of Statistics Education'', 9 (3)

==External links==

* [http://www.amstat.org/publications/jse/v9n3/stanton.html A Brief History of Linear Regression Analysis]

{{DEFAULTSORT:Multiple Correlation}}