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I know that uniformly most powerful tests have to be based on the likelihood ratios as test statistic, which is not the case for the Fisher exact test. Nevertheless couldn't I use the G2 test metric, but calculate & tabulate its test statistic numerically for all relevant small numbers instead of approximating it by $\chi_1$?

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    $\begingroup$ It would help if you explained your setup properly. Are you talking about the independence of two categorical variables? $\endgroup$ Dec 8, 2022 at 10:56
  • $\begingroup$ yes. updated question $\endgroup$ Dec 8, 2022 at 18:12
  • $\begingroup$ For small sample sizes, you generally shouldn't rely on asymptotic approximations so you shouldn't really care about what test, asymptotically has the most power. Simulations are better. $\endgroup$
    – num_39
    Dec 8, 2022 at 20:55
  • $\begingroup$ It's something that has been discussed for 2 x 2 tables. For example, with Fisher's, Barnard's, Boschloo's tests. $\endgroup$ Dec 9, 2022 at 1:03
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    $\begingroup$ Beyond 2x2 things can be complicated, especially with small samples. For example, you don't have one-sided tests (no Karlin-Rubin theorem for example), you're dealing with omnibus alternatives and generally there's no UMP test. $\endgroup$
    – Glen_b
    Dec 9, 2022 at 1:35

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