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I have a $m \times n$ contingency table (with $m, n > 2)$ . The experiment yields ~15% cells with expected frequencies lower 5 and also zero counts in the empirical data. The prerequisite for the asymptotic $\chi^2$-Test do not hold. Thus, I want to use an exact test (or its simulation variants). Also, one margin is conditional (fixed by design) and the other margin is undconditional (e.g. fixed counts in row margin and random counts in column margin).

Which test is the right one to choose here? Fisher's exact test is conditional on both margins, so the requirements are violated. Barnard's test is unconditional on both margins but AFAIK only applicable to $2 \times 2$ tables. I could not find any hints to a $m \times n$ generalization of Barnard's test.

Which test is applicable to my situation?

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@Mark, not sure which tool you are considering to do the test with but refer to this link https://mrnoutahi.com/2016/01/03/Fisher-exac-test-for-mxn-table/ where Fischer exact test was done for the unconditional likelihood estimate and the code is written in python.

Hope this is helpful.

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  • $\begingroup$ I am an R user but Python works as well for me :), thanks $\endgroup$ – Mark Heckmann Apr 26 '18 at 8:01
  • $\begingroup$ As far as I understand, the implementation in the link is the one for fixed margins. The OP is about one margin fixed, the other not. $\endgroup$ – Michael M Apr 26 '18 at 8:04
  • $\begingroup$ Actually, it is mentioned nowhere that it is for fixed margins. Also, could someone post the thread where there is the explanation of fixed margins. I have gone through the wikipedia page, things are not clear for me. $\endgroup$ – Suffer Surfer Apr 26 '18 at 11:41

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