Coefficient of 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.
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 between the predictor variables and the target 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 squares of residuals—that is, the sum of the squares of the prediction errors—divided by the 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 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.