# Cox & Snell $R^2$ rule of thumb threshold

• Like p value is usually compared to 0.05,
• What is the magic number that is considered a good fit for a logistic regression Cox & Snell pseudo $$R^2$$ result?
• >0.1? >0.5? something else?

The analogy suggested in this question--between a p-value statistical significance threshold and a Cox-Snell logistic regression pseudo-$$R^2$$ that represents a "good fit"--isn't apt.
Think about ordinary least squares multiple linear regression. That produces both a p-value for the model as a whole and an $$R^2$$ value that is the fraction of variance in the outcome variable that is explained by the model. Those represent two different things.
The linear multiple regression $$R^2$$ provides that information about the fraction of variance explained by the model. But there is no single $$R^2$$ value that represents a "good fit"; that depends on the underlying subject matter. What's considered a "good fit" $$R^2$$ in a biomedical study might be considered woefully inadequate for a "good fit" in a physical-science study, even if a large number of observations provided a "statistically significant" p-value of p < 0.05 in the latter study.
It's similar in logistic regression: the p-value tells you how likely you are to have found an apparent relationship even if there isn't really one. The value of a Cox-Snell pseudo-$$R^2$$ that represents a "good fit" will depend on what is being studied.
Finally, a good argument can be made that pseudo-$$R^2$$ values shouldn't even be reported for logistic regression. Other measures more closely related to the purpose of the study are much more important.