I have SPSS output for a logistic regression model. The output reports two measures for the model fit, Cox & Snell and Nagelkerke.

So as a rule of thumb, which of these $R^²$ measures would you report as the model fit?

Or, which of these fit indices is the one that is usually reported in journals?



Some Background: The regression tries to predict the presence or absence of a bird (capercaillie) from some environmental variables (e.g., steepness, vegetation cover, ...). Unfortunately, the bird did not appear very often (35 hits to 468 misses) so the regression performs rather poorly. Cox & Snell is .09, Nagelkerke, .23.  

The subject is environmental sciences or ecology.

Normally I wouldn't report $R^2$ at all.  Hosmer and Lemeshow, in their textbook Applied Logistic Regression (2nd Ed.), explain why:


  In general, [$R^2$ measures] are based on various comparisons of the predicted values from the fitted model to those from [the base model], the no data or intercept only model and, as a result, do not assess goodness-of-fit.  We think that a true measure of fit is one based strictly on a comparison of observed to predicted values from the fitted model.


[At p. 164.]

Concerning various ML versions of $R^2$, the "pseudo $R^2$" stat, they mention that it is not "recommended for routine use, as it is not as intuitively easy to explain," but they feel obliged to describe it because various software packages report it.

They conclude this discussion by writing,


  ...low $R^2$ values in logistic regression are the norm and this presents a problem when reporting their values to an audience accustomed to seeing linear regression values. ... Thus [arguing by reference to running examples in the text] we do not recommend routine publishing of $R^2$ values with results from fitted logistic models.  However, they may be helpful in the model building state as a statistic to evaluate competing models.


[At p. 167.]

My experience with some large logistic models (100k to 300k records, 100 - 300 explanatory variables) has been exactly as H & L describe.  I could achieve relatively high $R^2$ with my data, up to about 0.40.  These corresponded to classification error rates between 3% and 15% (false negatives and false positives, balanced, as confirmed using 50% hold-out datasets).  As H & L hinted, I had to spend a lot of time disabusing the client (a sophisticated consultant himself, who was familiar with $R^2$) concerning $R^2$ and getting him to focus on what mattered in the analysis (the classification error rates).  I can warmly recommend describing the results of your analysis without reference to $R^2$, which is more likely to mislead than not.
