Coefficient of multiple correlation: Difference between revisions

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{{Short description|Statistical concept}}

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{{More footnotes|date=November 2010}}

In [[statistics]], the '''coefficient of multiple correlation''' is a measure of how well a given variable can be predicted using a [[linear function]] of a set of other variables. It is the [[Pearson correlation|correlation]] between the variable's values and the best predictions that can be computed [[linear equation|linearly]] from the predictive variables.<ref>[http://onlinestatbook.com/2/regression/multiple_regression.html Introduction to Multiple Regression] </ref>

In [[statistics]], '''multiple correlation''' is a linear relationship among more than two variables. It is measured by the [[Coefficient of determination|coefficient of multiple determination]], denoted as R², which is a measure of the fit of a [[linear regression]]. A regression's R² falls somewhere between zero and one (assuming a constant term has been included in the regression); a higher value indicates a stronger relationship among the variables, with a value of one indicating that all data points fall exactly on a line in multidimensional space and a value of zero indicating no relationship at all between the independent variables collectively and the dependent variable.

The coefficient of multiple correlation takes values between 0 and 1. Higher values indicate higher predictability of the [[dependent and independent variables|dependent variable]] from the [[dependent and independent variables|independent variables]], with a value of 1 indicating that the predictions are exactly correct and a value of 0 indicating that no linear combination of the independent variables is a better predictor than is the fixed [[mean]] of the dependent variable.<ref>[http://mtweb.mtsu.edu/stats/regression/level3/multicorrel/multicorrcoef.htm Multiple correlation coefficient]</ref>

==Fundamental equation of multiple regression analysis==

{| class="wikitable"

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]]&nbsp;''c''', and the [[Matrix (mathematics)|matrix]] ''R''_''xx'' of inter-correlations between predictor variables. The "fundamental equation of multiple regression analysis"<ref>Visualstatistics.net [http://www.visualstatistics.net/Visual%20Statistics%20Multimedia/multiple_regression_analysis.htm]</ref> is

|Correlation Coefficient (r)

|Direction and Strength of Correlation

|-

|1

|Perfectly positive

|-

|0.8

|Strongly positive

|-

|0.5

|Moderately positive

|-

|0.2

|Weakly positive

|-

|0

|No association

|-

| -0.2

|Weakly negative

|-

| -0.5

|Moderately negative

|-

| -0.8

|Strongly negative

|-

| -1

|Perfectly negative

|}

The coefficient of multiple correlation is known as the square root of the [[coefficient of determination]], but under the particular assumptions that an intercept is included and that the best possible linear predictors are used, whereas the coefficient of determination is defined for more general cases, including those of nonlinear prediction and those in which the predicted values have not been derived from a model-fitting procedure.

==Definition==

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

The coefficient of multiple correlation, denoted ''R'', is a [[scalar (mathematics)|scalar]] that is defined as the [[Pearson correlation coefficient]] between the predicted and the actual values of the dependent variable in a linear regression model that includes an [[Y-intercept|intercept]].

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.

==Computation==

The square of the coefficient of multiple correlation can be computed using the [[Euclidean space|vector]] <math>\mathbf{c} = {(r_{x_1 y}, r_{x_2 y},\dots,r_{x_N y})}^\top</math> of [[correlation]]s <math>r_{x_n y}</math> between the predictor variables <math>x_n</math> (independent variables) and the target variable <math>y</math> (dependent variable), and the [[correlation matrix]] <math>R_{xx}</math> of correlations between predictor variables. It is given by

==References==

{{reflist}}

::<math>R^2 = \mathbf{c}^\top R_{xx}^{-1}\, \mathbf{c},</math>

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

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

where <math>\mathbf{c}^\top</math> is the [[transpose]] of <math>\mathbf{c}</math>, and <math>R_{xx}^{-1}</math> is the [[Matrix inversion|inverse]] of the matrix

* 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)

::<math>R_{xx} = \left(\begin{array}{cccc}

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

r_{x_1 x_1} & r_{x_1 x_2} & \dots & r_{x_1 x_N} \\

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

r_{x_2 x_1} & \ddots & & \vdots \\

\vdots & & \ddots & \\

r_{x_N x_1} & \dots & & r_{x_N x_N}

\end{array}\right).</math>

If all the predictor variables are uncorrelated, the matrix <math>R_{xx}</math> is the identity matrix and <math>R^2</math> simply equals <math>\mathbf{c}^\top\, \mathbf{c}</math>, the sum of the squared correlations with the dependent variable. If the predictor variables are correlated among themselves, the inverse of the correlation matrix <math>R_{xx}</math> accounts for this.

The squared coefficient of multiple correlation can also be computed as the fraction of variance of the dependent variable that is explained by the independent variables, which in turn is 1 minus the unexplained fraction. The unexplained fraction can be computed as the [[sum of squares of residuals]]&mdash;that is, the sum of the squares of the prediction errors&mdash;divided by the [[Total sum of squares|sum of squares of deviations of the values of the dependent variable]] from its [[expected value]].

==Properties==

With more than two variables being related to each other, the value of the coefficient of multiple correlation depends on the choice of dependent variable: a regression of <math>y</math> on <math>x</math> and <math>z</math> will in general have a different <math>R</math> than will a regression of <math>z</math> on <math>x</math> and <math>y</math>. For example, suppose that in a particular sample the variable <math>z</math> is [[Correlation and dependence|uncorrelated]] with both <math>x</math> and <math>y</math>, while <math>x</math> and <math>y</math> are linearly related to each other. Then a regression of <math>z</math> on <math>y</math> and <math>x</math> will yield an <math>R</math> of zero, while a regression of <math>y</math> on <math>x</math> and <math>z</math> will yield a strictly positive <math>R</math>. This follows since the correlation of <math>y</math> with its best predictor based on <math>x</math> and <math>z</math> is in all cases at least as large as the correlation of <math>y</math> with its best predictor based on <math>x</math> alone, and in this case with <math>z</math> providing no explanatory power it will be exactly as large.

==References==

{{Reflist}}

==ExternalFurther linksreading==

* Allison, Paul D. Allison(1998). ''Multiple Regression: A Primer''. (1998)London: Sage Publications. {{ISBN|9780761985334}}

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

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

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

* 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 regressionRegression and the analysisAnalysis of varianceVariance and covarianceCovariance''. (1985)({{ISBN |0716710811)}}

* Keith, Timothy Z. Keith(2006). '' Multiple Regression and Beyond''. (2005)Boston: Pearson Education.

* Fred N. Kerlinger, Elazar J. Pedhazur, (1973). ''Multiple Regression in Behavioral Research.'' (1973)New York: Holt Rinehart Winston. {{ISBN|9780030862113}}

* 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).

{{DEFAULTSORT:Multiple Correlation}}

[[Category:Correlation indicators]]

[[Category:Regression analysis]]