Generalized Coefficient of Determination

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:

$R^2 = 1 - \biggl\{\frac{L(0)}{L(\hat{{\beta}})}\biggr\}^ {\frac{2}n}$

where L(0) is the likelihood of the intercept-only model, $L(\hat{{\beta}})$ is the likelihood of the specified model, and n is the sample size. The quantity R² achieves a maximum of less than one for discrete models, where the maximum is given by

R_max² = 1 - {L(0)}^[2/n]

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

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

Properties and interpretation of R² and ${\tilde{R}^2}$ are provided in Nagelkerke (1991). In the "Testing Global Null Hypothesis: BETA=0" table, R² is labeled as "RSquare" and ${\tilde{R}^2}$ is labeled as "Max-rescaled RSquare." Use the RSQUARE option to request R² and ${\tilde{R}^2}$ .

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