I have a dataset within which I have a particular response variable that I'm interested in and numerous predictor variables. All variables are nominal and have as many as 15 possible values. When I cross tabulate any given predictor variable with the response variable, I get many cells with 0 counts, making performing a chi-squared test of independence inappropriate. That's fine, because I can use Fisher's exact test, but it's a problem in terms of calculating effect size since Cramer's V and every other method I've found that works for nominal data like mine seems to rely on chi-squared. Are there any alternatives to Cramer's V that don't have this problem? Or, if I'm misunderstanding something, is it still valid to user Cramer's V even if a chi-squared test is inappropriate?
The effect sizes I assume you are considering — Cramer's V, (phi), Contingency coefficient C, and Cohen's w — can all be calculated with the chi-square value. But the chi-square is simply calculated from the difference of observed values from expected values. This is way Cohen defines his w in Cohen (1988).
I assume that because there's no inference with these statistics, that it is fine to report them even if some test using the chi-square statistic would not be appropriate. It's like saying the difference between two means is some value, without addressing whether or not you could use a t-test or not in this case.