# Fisher's exact test in 3x2 contingency table

I have two groups of patients (A and B) with a congenital malformation which might present itself in 3 forms (a or b or c). Sample sizes are small as you can see, so I think the best test to check whether there's a statistically significant difference between the 3 forms in the 2 groups is a Fisher's exact test in a following 3x2 contingency table:

group A: 2 a, 12 b, 1 c
group B: 5 a,  3 b, 1 c


My question is: how should I interpret the p value? I don't understand what is that referred to. Having the p value, how can I say that one of the three forms is statistically significantly more represented than the others (if true)?

• On the final question, I'd just look at the Pearson residuals or the (signed square roots of) the contribution to chi-square to decide where the main 'causes' of the difference come from. Commented Aug 14, 2013 at 22:42
• My comment here: Think carefully about what you are actually trying to determine. The Fisher exact test on the contingency table determines e.g. if the distribution of Forms differ between the Groups. The way you wrote the question, I wonder if you want to know if the frequency is different across Forms. Either within each Group. Or perhaps pooling the Groups. Commented Oct 13, 2022 at 14:27

## 2 Answers

It sounds like you are asking a lot of different questions here.

My question is: how should I interpret the p value? I don't understand what is that referred to.

The null hypothesis for Fisher's Exact test is that the groups do not affect the outcome, i.e. that they are independent. Rejection of the null hypothesis indicates the outcome (a, b, or c) is dependent on group.

fisher.test(matrix(c(2, 12, 1, 5, 3, 1),
nrow=2, ncol=3, byrow=TRUE))
Fisher's Exact Test for Count Data

data:  dta
p-value = 0.05082
alternative hypothesis: two.sided


In this case your $$p$$ value is approximately 0.05082. I will let you decide whether to reject the null.

Having the p value, how can I say that one of the three forms is statistically significant more represented than the others (if true)?

This is a separate question and I'm not sure what you are trying to ask.

• Hi Ellis, thanks, you right there were 2 questions. Let's say i consider the difference significant, it means the null hypothesis is not true and the outcomes depend on the group, but then how can i say something more about the 3 forms a,b,c? They depend on the group ok, but in which way? How can I see for instance if the form b is statistically more reported in the group A? Commented Aug 14, 2013 at 14:05
• @Gabriele look at this in a table form, calculate proportions and see if that helps. Commented Aug 14, 2013 at 14:14
• Make sure you really want to use this test as it is less powerful than the Pearson $\chi^2$ test. Commented Aug 14, 2013 at 16:12
• Pearson $\chi^2$ is now known to work well when expected cell frequencies exceed 1.0, not the 5.0 originally uttered by Pearson without checking. Commented Aug 14, 2013 at 21:03
• @Frank Harrell +1. A rule of thumb that frequencies be >1 was cited by Harold Jeffreys, Theory of probability, 1939, pp.88-9, anticipating the modern consensus. Jeffreys did a lot of calculating as well as being a first-rate mathematician. Commented Aug 14, 2013 at 21:58

Totally agree with Ellis, the null hypothesis of the Fisher's exact test is that the outcome is independent from the groups. If rejected, it might be helpful looking at the observed frequencies and discuss such result in a broader context by integrating other sources of information. This latter could be your best shot as drawing solid conclusion with such a small sample size can be misleading.