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I have a question for using Fisher's exact test for subgroups.

I have 39 subjects that was divided into 4 groups( n = 9, 5, 10, 15, respectively).

There were 3 conditions dividing the groups. Table below shows the summary.

Conditions Group1 Group 2 Group 3 Group 4
Condition A O X X X
Condition B O O X X
Condition C O O O X

Each subjects chose between 2 options (A or B) after the treatment. So I got 4x2 contingency data as below.

Choice Group1 Group 2 Group 3 Group 4
Choose A 6 1 2 3
Choose B 3 4 8 12

My initial hypothesis was that, Condition A increases % choosing option A. So I designed and conducted an experiment with only group 1 and group 4. I used Fisher's exact Test to compare them and found that there was significant difference (p<0.05).

However, I later came out with an alternative theory that, maybe condition B, C could also influence % choosing A. So I did the same experiment with group 2 and 3, and found that the result seemed similar to group 4 rather then group 1.

How can I show that both condition B and C does not influence the "% of A-choice" significantly?

ps: I am currently thinking of using the following methods:

(a) Fisher's Exact test to compare (Group 1) vs (Group 2+3+4 combined) to show the significant difference of % choosing exists in condition A.

(b) Fisher's Exact test to compare (Group 1+2 combined) vs (Group 3+4 combined) to show significant difference in doesn't exist in condition B.

(c) Fisher's Exact test to compare (Group 1+2+3 combined) vs (group 4) to show significant difference in doesn't exist in condition C.

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    $\begingroup$ Fisher's exact test is not very accurate for comparing two groups, so I wouldn't use it for four. $\endgroup$
    – Frank Harrell
    Commented Apr 28, 2022 at 11:25
  • $\begingroup$ Thanks for the comment. It feels like this is going to be a dumb question, but do you mean that it would be better to avoid using Fisher's exact test for comparing small number of groups? $\endgroup$
    – Roas Clack
    Commented Apr 28, 2022 at 11:55
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    $\begingroup$ I would avoid using Fisher's "exact" test for any use. Its p-values are too large, e.g., if testing at $\alpha=0.05$ the probability of rejecting $H_0$ when it is true is $< 0.05$. Ordinary Pearson $\chi^2$ works quite well. For your particular needs I'd embed this into a model such as the polytomous (multinomial) logistic model which allows you to compute any contrast you need, and even get simultaneous coverage confidence intervals over multiple contrasts. $\endgroup$
    – Frank Harrell
    Commented Apr 28, 2022 at 12:16
  • $\begingroup$ Thanks a lot. I'll check it out. $\endgroup$
    – Roas Clack
    Commented Apr 28, 2022 at 12:27
  • $\begingroup$ Would it be possible to use multinomial logistic model? Because independent variables (condition A, B, C) might be conflicted with the assumption of multicollinearity, since All subjects with condition A also have condition B and C. $\endgroup$
    – Roas Clack
    Commented May 10, 2022 at 4:12

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