# McFadden’s Pseudo-R2 – Interpretation and Meaning

logisticregressionself-study

I have a binary logistic regression model with a McFadden's pseudo R-squared of 0.192 with a dependent variable called payment (1 = payment and 0 = no payment). What is the interpretation of this pseudo R-squared?

Is it a relative comparison for nested models (e.g. a 6 variable model has a McFadden's pseudo R-squared of 0.192, whereas a 5 variable model (after removing one variable from the aforementioned 6 variable model), this 5 variable model has a pseudo R-squared of 0.131. Would we would want to keep that 6th variable in the model?) or is it an absolute quantity (e.g. a given model that has a McFadden's pseudo R-squared of 0.192 is better than any existing model with a McFadden's pseudo R-squared of 0.180 (for even non-nested models)? These are just possible ways to look at McFadden’s pseudo R-squared; however, I assume these two views are way off, thus the reason why I am asking this question here.

I have done a great deal of research on this topic, and I have yet to find the answer that I am looking for in terms of being able to interpret a McFadden's pseudo R-squared of 0.192. Any insight and/or references are greatly appreciated! Before answering this question, I am aware that this isn't the best measure to describe a logistic regression model, but I would like to have a greater understanding of this statistic regardless!

So I figured I'd sum up what I've learned about McFadden's pseudo $$R^2$$ as a proper answer.

The seminal reference that I can see for McFadden's pseudo $$R^2$$ is: McFadden, D. (1974) “Conditional logit analysis of qualitative choice behavior.” Pp. 105-142 in P. Zarembka (ed.), Frontiers in Econometrics. Academic Press. http://eml.berkeley.edu/~mcfadden/travel.html Figure 5.5 shows the relationship between $$\rho^2$$ and traditional $$R^2$$ measures from OLS. My interpretation is that larger values of $$\rho^2$$ (McFadden's pseudo $$R^2$$) are better than smaller ones.

The interpretation of McFadden's pseudo $$R^2$$ between 0.2-0.4 comes from a book chapter he contributed to: Bahvioural Travel Modelling. Edited by David Hensher and Peter Stopher. 1979. McFadden contributed Ch. 15 "Quantitative Methods for Analyzing Travel Behaviour on Individuals: Some Recent Developments". Discussion of model evaluation (in the context of multinomial logit models) begins on page 306 where he introduces $$\rho^2$$ (McFadden's pseudo $$R^2$$). McFadden states "while the $$R^2$$ index is a more familiar concept to planner who are experienced in OLS, it is not as well behaved as the $$\rho^2$$ measure, for ML estimation. Those unfamiliar with $$\rho^2$$ should be forewarned that its values tend to be considerably lower than those of the $$R^2$$ index...For example, values of 0.2 to 0.4 for $$\rho^2$$ represent EXCELLENT fit."

So basically, $$\rho^2$$ can be interpreted like $$R^2$$, but don't expect it to be as big. And values from 0.2-0.4 indicate (in McFadden's words) excellent model fit.