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; a higher value indicates a better 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 computed 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 ${\displaystyle \mathbf {c} ={(r_{x_{1}y},r_{x_{2}y},\dots ,r_{x_{N}y})}^{\top }}$ of correlations ${\displaystyle r_{x_{n}y}}$ between the predictor variables ${\displaystyle x_{n}}$ (independent variables) and the target variable ${\displaystyle y}$ (dependent variable), and the correlation matrix ${\displaystyle R_{xx}}$ of correlations between predictor variables. It is given by

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