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I see that in R, there is an example of running Fisher's exact test (see ?fisher.test) with an $r \times c$ contingency table with $r, c > 2$.

I could not find any such material in Agresti's Categorical Data Analysis, 3rd ed. Is there a standard reference for Fisher's exact test for such contingency tables? Textbook and/or journal article would be fine; however, I don't want non-peer-reviewed websites.

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    $\begingroup$ In the 2nd edition, Agresti lists "Fisher's exact text" in the index and references extensive material around pp 91-104 and "software" around pp 623-635. That's section 3.5, Small-sample tests of independence, and section 16.2, R. A. Fisher's Contributions. On Amazon's "Look Inside" page for the 3rd edition, the test is listed right there on p. 708 and refers to pp 90-96. $\endgroup$
    – whuber
    May 26 '20 at 17:27
  • $\begingroup$ @whuber If I've read the text correctly, only the 2 x 2 case is discussed in Agresti. $\endgroup$ May 26 '20 at 18:01
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    $\begingroup$ In the 2nd edition, at 3.5.7 "Derivation of Exact Conditional Distribution," Agresti writes "We do this for $I\times J$ tables, since we next discuss extensions of Fisher's exact test for them." This he does in section 3.5.8, "Exact Tests of Independence for $I\times J$ Tables." It's hard to imagine Agresti dropped all that material in the 3rd edition, but I can't find any reference to it in the contents or index for the 3rd edition. So, maybe "use the 2nd edition" is the answer. $\endgroup$
    – whuber
    May 30 '20 at 12:58
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    $\begingroup$ @whuber I've confirmed that it is in the 2nd edition, but not in the 3rd. Thank you for that. $\endgroup$ Jun 13 '20 at 21:17
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I think the original reference is by Freeman and Halton in an article entitled "A note on an exact treatment of contingency, goodness of fit and other problems of significance" published in Biometrika in 1951.

Indeed it is sometimes referred to as the Freeman-Halton method, soemtimes with the addition of Fisher's name.

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  • $\begingroup$ +1 This is the reference Agresti used in the second edition of his book (at p. 97). $\endgroup$
    – whuber
    May 30 '20 at 13:01

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