What are potential advantages of chi-square test of independence over fisher's exact test? Alternatively, does fisher's have limitations?
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1$\begingroup$ Fisher's exact test has been criticized as being too conservative (i.e. that its actual rejection rate is below the nominal significance level). This is the primary limitation I am aware of, and as sample sizes increase the chi-square and fisher should be equivalent., $\endgroup$– Demetri PananosCommented Apr 14 at 23:12
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1$\begingroup$ Lachin, John M. Biostatistical Methods: The Assessment of Relative Risks. John Wiley & Sons, 2014. Pg 34 of the second edition mentions its conservative nature. $\endgroup$– Demetri PananosCommented Apr 14 at 23:45
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1$\begingroup$ When some marginal totals are small any test that conditions on the margins will have a limited selection of available significance levels, particularly for the 2x2 cases. If you accept Barnard's 1949 argument (that his unconditional test was mistaken and that you should in fact condition on the margins), then there's not a great deal to be done about it. You can mitigate very slightly by breaking ties with another statistic but all the most natural candidates for doing so are very closely related to each other in the 2x2 case and ... ctd $\endgroup$– Glen_bCommented Apr 15 at 4:09
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1$\begingroup$ ctd ... so it usually only adds a few more levels well below the typical overall significance levels (unless you're testing a lot of 2x2 tables and adjusting for familywise error in which case it does sometimes help). There's more to gain in slightly larger tables particularly if applying a correction for multiple tests (whether across tables or doing contrasts or post hoc comparisons within the current one) but also somewhat less need there as there are more available levels to start with. $\endgroup$– Glen_bCommented Apr 15 at 4:09
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1$\begingroup$ Of course one can always do a randomized test if there's at least one significance level below your nominal one, but that may run into difficulties getting people to accept it (due to the feature that two different people using the same test on the same data may arrive at different conclusions) $\endgroup$– Glen_bCommented Apr 15 at 4:16
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