Multiple correlation

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 correlation between the variable's values and the best predictions that can be computed linearly from the predictive variables.[1]

The coefficient of multiple correlation takes values between .00 and 1.00; a higher value indicates a high predictability of the dependent variable from the 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.[2]

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.


The coefficient of multiple correlation, denoted R, is a 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 intercept.


The square of the coefficient of multiple correlation can be computed using the vector   of correlations   between the predictor variables   (independent variables) and the target variable   (dependent variable), and the correlation matrix   of correlations between predictor variables. It is given by


where   is the transpose of  , and   is the inverse of the matrix


If all the predictor variables are uncorrelated, the matrix   is the identity matrix and   simply equals  , the sum of the squared correlations with the dependent variable. If the predictor variables are correlated among themselves, the inverse of the correlation matrix   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—that is, the sum of the squares of the prediction errors—divided by the sum of the squared deviations of the values of the dependent variable from its expected value.


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   on   and   will in general have a different   than will a regression of   on   and  . For example, suppose that in a particular sample the variable   is uncorrelated with both   and  , while   and   are linearly related to each other. Then a regression of   on   and   will yield an   of zero, while a regression of   on   and   will yield a strictly positive  . This follows since the correlation of   with its best predictor based on   and   is in all cases at least as large as the correlation of   with its best predictor based on   alone, and in this case with   providing no explanatory power it will be exactly as large.


Further readingEdit

  • Allison, Paul D. (1998). Multiple Regression: A Primer. London: Sage Publications. ISBN 9780761985334
  • Cohen, Jacob, et al. (2002). Applied Multiple Regression: Correlation Analysis for the Behavioral Sciences. ISBN 0805822232
  • Crown, William H. (1998). Statistical Models for the Social and Behavioral Sciences: Multiple Regression and Limited-Dependent Variable Models. ISBN 0275953165
  • Edwards, Allen Louis (1985). Multiple Regression and the Analysis of Variance and Covariance. ISBN 0716710811
  • Keith, Timothy (2006). Multiple Regression and Beyond. Boston: Pearson Education.
  • Fred N. Kerlinger, Elazar J. Pedhazur (1973). Multiple Regression in Behavioral Research. New York: Holt Rinehart Winston. ISBN 9780030862113
  • Stanton, Jeffrey M. (2001). "Galton, Pearson, and the Peas: A Brief History of Linear Regression for Statistics Instructors", Journal of Statistics Education, 9 (3).