# Significant Fisher's exact test, post hoc analysis for subgroup comparisons?

I'm analysing 2 groups of patients with 2 different DISEASE_STAGES: MILD disease and MODERATE disease, as defined by a complex clinical diagnosis. The sample size is relatively small: a total of 80 patients characterised by SMOKING_STATUS with 3 levels: active smoker, ex-smoker and never smoked.

I've performed a Fisher exact test because one cell has a frequency of 1

fisher.test(matrix(c(1, 5, 14, 3, 33, 22), nrow=2, ncol=3, byrow=TRUE))

Fisher's Exact Test for Count Data
p-value = 0.03039
alternative hypothesis: two.sided


I reject the null hypothesis that the disease is not affected by smoking status.

My question: Is it possible and how can I perform a post-hoc analysis with pairwise comparisons of the proportions for a Fisher exact test? How should I correct p-values to account for the multiple testing (what kind of statistical significance should I accept for these subgroup comparisons)?

• You could just do a test of proportions between each group and use a Bonferonni correction, but I think there might be better things to do. Would it be reasonable to assume there is a trend in disease stages as a function of smoking status (e.g. more people are moderately diseased if they smoke more)? There is also the issue of your first group having only 4 patients, which is extremely small. – Demetri Pananos Nov 13 '18 at 16:14
• I was a bit worried about doing a Bonferroni correction for a Fisher with such low counts in 3 cells. I've done it for a Chi-squared with higher cell counts and I'm not sure it can be applied the same way to a Fisher. Yes, I assume there is an association between disease and smoking but not sure how to express this with categorical variables – Mia Nov 13 '18 at 16:20
• I think testing for a relationship between the ordinal variables and the outcome would be more informative than doing multiple comparisons (and would likely have better statistical power). Most tests are asymptotic tests though, so that first group is a real problem. I'd have to know more about the problem and your goal in order to recommend something. Is this homework or something, or is this a clinical investigation? – Demetri Pananos Nov 13 '18 at 16:28
• Yes the first group is very small because it is rare for elderly diseased patients to be active smokers. I'm not sure if collapsing to only 2 levels: exposed to smoke (active smokers and ex-smokers) and not exposed to smoke would help or make things worse – Mia Nov 13 '18 at 16:30
• It's a clinical investigation – Mia Nov 13 '18 at 16:31