Generalized Coefficient of Determination

Cox and Snell (1989, pp. 208–209) propose the following generalization of the coefficient of determination to a more general linear model:

\[  R^2 = 1 - \biggl \{ \frac{L(\mb {0})}{L(\hat{\btheta })}\biggr \} ^{\frac{2}{n}}  \]

where $L(\mb {0})$ is the likelihood of the intercept-only model, $L(\hat{\btheta })$ is the likelihood of the specified model, and n is the sample size. The quantity $R^2$ achieves a maximum of less than 1 for discrete models, where the maximum is given by

\[  R_{\max }^2 = 1 - \{ L(\mb {0})\} ^{\frac{2}{n}}  \]

Nagelkerke (1991) proposes the following adjusted coefficient, which can achieve a maximum value of 1:

\[  \tilde{R}^2 = \frac{R^2}{R_{\max }^2}  \]

Properties and interpretation of $R^2$ and $\tilde{R}^2$ are provided in Nagelkerke (1991). In the Testing Global Null Hypothesis: BETA=0 table, $R^2$ is labeled as RSquare and $\tilde{R}^2$ is labeled as Max-rescaled RSquare.  Use the RSQUARE option to request $R^2$ and $\tilde{R}^2$.