Coefficient of multiple correlation: Difference between revisions

Revision as of 04:32, 2 December 2014

Revision as of 09:32, 3 February 2015

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==Computation==

The square of the coefficient of multiple correlation can be computed using the [[Euclidean space|vector]] ''<math>\mathbf{c''}</math> of cross-[[correlation]]s between the predictor variables <math>x_n</math> (independent variables) and the target variable <math>y</math> (dependent variable), and the [[correlation matrix]] ''R''<submath>''R_{xx''}</submath> of inter-correlations between predictor variables. It is given by

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

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

where ''<math>\mathbf{c'' '}^\top</math> is the [[transpose]] of ''<math>\mathbf{c''}</math>, and ''R''<submath>''R_{xx''</sub><sup>&minus;}^{-1}</supmath> is the [[Matrix inversion|inverse]] of the matrix ''R''<submath>''R_{xx''}</submath>.

If all the predictor variables are uncorrelated, the matrix ''R''<submath>''R_{xx''}</submath> is the identity matrix and ''R''<supmath>R^2</supmath> simply equals ''<math>\mathbf{c'''}^\top\, ''\mathbf{c''}</math>, the sum of the squared cross-correlations with the dependent variable. If there is cross-correlation among the predictor variables are correlated among themselves, the inverse of the cross-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 squared residuals]]&mdash;that is, the sum of the squares of the prediction errors&mdash;divided by the [[Total sum of squares|sum of the squared deviations of the values of the dependent variable]] from its [[expected value]].