The inequality between the coefficient of determination and the sum of squared simple correlation coefficients

Abstract

The inequality between the coefficient of determination and the sum of two squared simple correlation coefficients in a two-variable regression model is reexamined through two relative measures. They are the relative coefficient of determination and the relative simple correlation, which are the ratio of the coefficient of determination to the sum of squares of the two simple correlations and the ratio of two simple correlations, respectively. This approach not only permits new insights into their relationship hut also allows clear and informative visual representations of various aspects of the counterintuitive condition. We consider the occurrence and corresponding magnitude? probability and expected magnitude of the enhancement-synergism situation. Numerical examples are presented to illustrate these phenomena.