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.

The coefficient of multiple correlation takes values between 0 and 1. Higher values indicate higher 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.

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

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.

Computation

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

is the transpose is the inverse of the matrix

is the identity 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 squares of residualsthat is, the sum of the squares of the prediction errorsdivided by the sum of squares of deviations of the values of the dependent variable from its expected value.

Properties

is uncorrelated providing no explanatory power it will be exactly as large.

References

Further reading

• Allison, Paul D. (1998). Multiple Regression: A Primer. London: Sage Publications.

• Cohen, Jacob, et al. (2002). Applied Multiple Regression: Correlation Analysis for the Behavioral Sciences.

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

• Edwards, Allen Louis (1985). Multiple Regression and the Analysis of Variance and Covariance.

• 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.

• Stanton, Jeffrey M. (2001). "Galton, Pearson, and the Peas: A Brief History of Linear Regression for Statistics Instructors", Journal of Statistics Education, 9 (3).