Pseudo-R² (McFadden, Nagelkerke, Cox-Snell)

Substitute measures for R² in GLM — quantify improvement of the model over the null model.

Also known as: pseudo-R², pseudo R-squared, McFadden's R², Nagelkerke's R²

GLMs have no true $R^{2}$ because they are fit by maximum likelihood, not least squares. Instead there are several pseudo- $R^{2}$ measures based on the log-likelihood that quantify the improvement of the fitted model over the null model (predictor-free model).

McFadden $R^{2}$ : $1 - L_{M} / L_{0}$ (ratio of log-likelihoods)
Cox-Snell $R^{2}$ : accounts for sample size; has a maximum below $1$
Nagelkerke $R^{2}$ : normalises Cox-Snell to the range $[0, 1]$
Efron $R^{2}$ : simply $1 - RSS / TSS$ on $(y - \overset{p}{^})$ — akin to the classical $R^{2}$

Used in

Regression (Attributive)

In the Algorithm Lab

Sources

McFadden, D. — Conditional Logit Analysis of Qualitative Choice Behavior, in Frontiers in Econometrics, Academic Press, 1974
Nagelkerke, N. J. D. — A Note on a General Definition of the Coefficient of Determination, Biometrika, 78(3), 1991