Is Fisher's exact test the most powerful test for 2x2 contingency table? I don't find the powerfulness of Fisher's exact test on Wikipedia. Are there any results about the power of FET vs other methods on 2x2 contingency tables?
 A: No Fisher's Exact Test is not the most powerful because it doesn't condition on the sufficient statistic, the odds ratio, of the likelihood. Pearson's chi-square test, on the other hand, is the score test which is asymptotically equivalent to the likelihood ratio test, and the LRT is uniformly most powerful (among unbiased tests). You can even show that the $\alpha$-level Fisher's Exact test results in sub-alpha false positive rates (it does not attain its nominal size).
This has been known for a long time. Bennett and Hsu (1960) is one of many papers discussing the point.  There are many reasons to like the FET, but power is not one of them.
A: A computation of the power always depends on your assumptions. I recently uploaded a preprint to arxiv about the implementation of an unconditional test for contingency tables tentatively called m-test (link). It is work in progress, but we used a Monte Carlo simulation to compare the power of the m-test, FET and Barnard's test in R at different values of $\alpha$. The results with different marginals suggest that Barnard's test is more powerful than FET in most cases. The m-test was consistently the most powerful test of these three. There is one example here (the figures are at the end of the file, the Barnard package is verbose). The algorithm for the m-test is packaged here.
