Qualitatively, I want to answer the question:
Given that the effect of a covariate is statistically significant (as assessed by the p-value / confidence interval of its coefficient), how important is it in explaining the response? A little bit? A lot?"
We can compare the magnitude of coefficients for continuous variables but doing the same for categorical variables with $n > 2$ values (the motivation for my question) is not so straightforward since there are $n-1 \geq 2$ parameters.
$R^2$, in the case of an identity link, and pseudo-$R^2$ in the case of a logit link, quantifies how much variation is explained by a model. So one way to answer the question would be to compute the differences in $R^2$ between models that include vs. exclude a particular covariate. These differences could then, at least in the case of an identity link, be interpreted as the differences in "how much" (of the variability) of the response is explained by a covariate. How to interpret the result is not as straightforward in the case of pseudo-$R^2$.
My questions are:
- Is the method of comparing $R^2$ between models that include / exclude covariates one that is used in practice?
- Are there any caveats to consider with this method of comparing $R^2$?
- Are there other well-known methods that quantify the "contribution" of a categorical covariate that can take on more than 2 values?