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It is known that Fisher exact test assumes that all 4 margins of 2 by 2 table are fixed. $\chi^2$ test does not impose restriction on sampling design beforehand in general where I mean $\chi^2\sim\sum |Observed-Expected|^2/Expected$ or derived from likelihood.

Q: Should sampling/experiment design fix 4 margins in advance for Fisher exact test? This seems unachievable in non experimental setting. I thought 4 margins are fixed due to testing assumption for independence and in this case Fisher exact applies. Other than independence testing, Fisher exact does not apply. There is Boschloo's test which fixes either row margins or column margins. It also tests association. Boschloo's test seems to be more adapted to prospective/retrospective design.

Q': In Boschloo's test, it seems that this test imposes restriction on sampling design already?

user45765
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  • I would say that the sampling design restricts the appropriate statistical methods, not the other way around. – wzbillings Sep 29 '22 at 15:20
  • @wzbillings Of course sampling design restricts the methods. However, people were told you should do chi square testing and if entries<5, do fisher. That is the reason I am having confusion on when Fisher to should be applied. I can see it should be applied only when testing for independence. Otherwise, no. It does not seems to have sampling design requirement as well here. – user45765 Sep 29 '22 at 15:24
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    Generally, statistical model assumptions are formal and reality is not. There is no such thing as a "truly random" number distinguished from fixed. Fisher's test interprets the margins as fixed, but one could also interpret the test as being conditional on the margins. Ultimately whether Fisher's test is good or not in a situation with marginals that are not fixed by design can only be found out by simulating (or elaborating) its performance in situations of interest without fixed margins. I haven't tried it out but I'd expect it to be fine in most if not all such cases. – Christian Hennig Sep 29 '22 at 15:30
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    Fisher's exact test conditions on the margins. That's not the same thing as requiring the margins to be fixed. – Glen_b Sep 29 '22 at 15:54
  • e.g. see here: https://stats.stackexchange.com/questions/441139/what-does-the-assumption-of-the-fisher-test-that-the-row-and-column-totals-shou/441146#441146 – Glen_b Sep 29 '22 at 16:02
  • @Glen_b Thanks for clarification and the related post. – user45765 Sep 29 '22 at 16:20

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