Does anyone know of any $\chi^2$ tests to compare the fit of logistic models which factor out the sample size? I'm dealing with a very large sample and I fear the significant $\chi^2$ test I get when adding a single variable to the model is simply the result of the sample size (>200,000 cases). I'm doing what is known as differential item functioning analysis with logistic regression. Basically it's as if I'm checking whether giving the right answer to a question (dependent variable) depends on your ethnicity when controlling for the total exam score.
I'm basically using a chi-squared test to compare model1 to model2. The coefficient significance is not that important but $\chi^2$ and sometimes $R^2$ are generally recommended to check differential item functioning. My problem is that my sample is very large. In theory (for the question I'm considering) there should be no real difference across groups, so I suspect it's simply the sensitivity of the $\chi^2$ to sample size.
I'd rather use the whole dataset instead of taking (small) random samples as it is highly skewed. I've seen things like Phi and Cramer's V for crosstabs but I'm not sure whether they have been used before on logistic regression, if there are better ones and if there are any packages (I generally use Spss, Mplus, Stata, R).