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In statistics, multiple correlation is a linear relationship among more than two variables. It is measured by the coefficient of multiple determination, denoted as R2, which is a measure of the fit of a linear regression. A regression's R2 falls somewhere between zero and one (assuming a constant term has been included in the regression); a higher value indicates a stronger relationship among the variables, with a value of one indicating that all data points fall exactly on a line in multidimensional space and a value of zero indicating no relationship at all between the independent variables collectively and the dependent variable.
Fundamental equation of multiple regression analysis
The coefficient of multiple determination R2 (a scalar), can be computed using the vector c of cross-correlations between the predictor variables and the criterion variable, its transpose c', and the matrix Rxx of inter-correlations between predictor variables. The "fundamental equation of multiple regression analysis"[1] is
- R2 = c' Rxx−1 c.
The expression on the left side denotes the coefficient of multiple determination. The terms on the right side are the transposed vector c ' of cross-correlations, the inverse of the matrix Rxx of inter-correlations, and the vector c of cross-correlations. Note that if all the predictor variables are uncorrelated, the matrix Rxx is the identity matrix and R2 simply equals c' c, the sum of the squared cross-correlations. Otherwise, the inverted matrix of the inter-correlations removes the redundant variance that results from the inter-correlations of the predictor variables.
References
- Paul D. Allison. Multiple Regression: A Primer (1998)
- Cohen, Jacob, et al. Applied Multiple Regression - Correlation Analysis for the Behavioral Sciences (2002) (ISBN 0805822232)
- Crown, William H. Statistical Models for the Social and Behavioral Sciences: Multiple Regression and Limited-Dependent Variable Models (1998) (ISBN 0275953165)
- Edwards, Allen Louis. Multiple regression and the analysis of variance and covariance (1985)(ISBN 0716710811)
- Timothy Z. Keith. Multiple Regression and Beyond (2005)
- Fred N. Kerlinger, Elazar J. Pedhazur, Multiple Regression in Behavioral Research. (1973)