R-Squared Statistics

Figure 1. Model Summary

In the linear regression model, the coefficient of determination, R ², summarizes the proportion of variance in the dependent variable associated with the predictor (independent) variables, with larger R ² values indicating that more of the variation is explained by the model, to a maximum of 1. For regression models with a categorical dependent variable, it is not possible to compute a single R ² statistic that has all of the characteristics of R ² in the linear regression model, so these approximations are computed instead. The following methods are used to estimate the coefficient of determination.

Cox and Snell's R ² ¹ is based on the log likelihood for the model compared to the log likelihood for a baseline model. However, with categorical outcomes, it has a theoretical maximum value of less than 1, even for a "perfect" model.
Nagelkerke's R ² ² is an adjusted version of the Cox & Snell R-square that adjusts the scale of the statistic to cover the full range from 0 to 1.
McFadden's R ² ³ is another version, based on the log-likelihood kernels for the intercept-only model and the full estimated model.

What constitutes a “good” R ² value varies between different areas of application. While these statistics can be suggestive on their own, they are most useful when comparing competing models for the same data. The model with the largest R ² statistic is “best” according to this measure.

¹ Cox, D. R., and E. J. Snell. 1989. The Analysis of Binary Data, 2nd ed. London: Chapman and Hall.

² Nagelkerke, N. J. D. 1991. A note on the general definition of the coefficient of determination. Biometrika, 78:3, 691-692.

³ McFadden, D. 1974. Conditional logit analysis of qualitative choice behavior. In: Frontiers in Economics, P. Zarembka, eds. New York: Academic Press.