The LOGISTIC Procedure

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:

$R^2 = 1 - \biggl \{ \frac{L(\bm {0})}{L({\widehat{\bbeta }})}\biggr \} ^{\frac{2}{n}}$

where $L(\bm {0})$ is the likelihood of the intercept-only model, $L({\widehat{\bbeta }})$ is the likelihood of the specified model, $n=\sum _ jf_ jn_ j$ is the sample size, $f_ j$ is the frequency of the jth observation, and $n_ j$ is the number of trials when events/trials syntax is specified or $n_ j=1$ with single-trial syntax.

The quantity $R^2$ achieves a maximum of less than one for discrete models, where the maximum is given by

$R_{\max }^2 = 1 - \{ L(\bm {0})\} ^{\frac{2}{n}}$

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

$\tilde{R}^2 = \frac{R^2}{R_{\max }^2}$

Specifying the NORMALIZE option in the WEIGHT statement makes these coefficients invariant to the scale of the weights.

Like the AIC and SC statistics described in the section Model Fitting Information, $R^2$ and $\tilde{R}^2$ are most useful for comparing competing models that are not necessarily nested—larger values indicate better models. More 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 $\tilde{R}^2$ .