I'd like to see an extension of this discussion of the age-old chi-sq vs. Fisher's exact test debate, broadening the scope a bit. There are many many tests for interactions in a contingency table, enough to make my head spin. I'm hoping to get an explanation of what test I should use and when, and of course an explanation as to why one test should be preferred over another.
My current problem is the classic $n \times m$ case, but answers regarding higher dimensionality are welcome, as are tips for implementing the various solutions in R, at least, in cases where it is non-obvious how to proceed.
Below I've listed all the tests I'm aware of; I hope by exposing my errors they can be corrected.
$\chi^2$. The old standby. There are three major options here:
- The correction built into R for 2x2 tables: "one half is subtracted from all $|O-E|$ differences." Should I always be doing this?
- "$N-1$" $\chi^2$ Test, not sure how to do this in R.
- Monte Carlo simulation. Is this always best? Why does R not give me df when I do this?
- Traditionally advised when any cell is expected to be <4, but apparently some dispute this advice.
- Is the (usually false) assumption that the marginals are fixed really the biggest problem with this test?
- Another exact test, except I've never heard of it.
- One thing that always confuses me about glms is exactly how to do this significance tests so help on that would be appreciated. Is it best to do nested model comparison? What about a Wald test for a particular predictor?
- Should I really just always be doing Poisson regression? What's the practical difference between this and a $\chi^2$ test?