If I have a $2 \times 2$ contingency table for treatment/no treatment in the rows and response/no response, would Fisher's exact test break down due to assumptions of fixed marginals? In other words, suppose the table looked like:

                Cancer    No Cancer
Drug Given           a            b
No Drug Given        c            d

where the total number of trials is $a+b+c+d = N$.

In this case, the normal interpretation would be to see if a drug tends to induce cancer. Fisher's exact test assumes that $(a+b)$, $(c+d)$, $(a+c)$, and $(b+d)$ are fixed.

From the perspective of an experiment, we normally can decide ahead of time and fix how many people are given a drug or not. However, we cannot fix how many people are going to get Cancer. For this, does that mean Fisher's test is not applicable here in that we cannot fix column sizes? Or am I interpreting something wrongly? Thanks!



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