Decomposing the Coefficient of Determination in Multiple Regression

Abstract

In multiple regression, the coefficient of determination (or R-square) has a very useful interpretation. The statistic is the ratio of the variation that is explained by the regression equation to the total variation of the dependent variable. For example, a coefficient of
determination equal to .45 indicates that the independent variables can explain 45 percent of the variation in the dependent variable.

It follows immediately that a person's next instinct is to want to allocate among the several Independent variables the explained variation In the dependent variable. For example, many people would like to say that if the regression of a dependent variable on three independent variables explains 45 percent of the variance, that (say) 25 percent was due to the first independent variable, 15 percent to the second, and 5 percent to the third. While this interpretation is tempting, it should be avoided. The reason for avoiding it is that there is no unique way of decomposing the explained variance, and If there is no unique way of doing so, then there is no meaningful way of doing so.