# Interpreting Fisher's exact test OR, CI and p-values

In the context of Fisher's exact test results, it seems that a bunch of online tutorials mainly focus on interpreting the p-value exclusively, without considering the confidence intervals or the odds ratio.
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Based on those results, I cannot state that there is a significant association between the variant and the phenotype, because:

• the confidence interval does not include a 1, indicating no significance?

Thanks a lot.

• You split your question into two posts, so I answered your other post instead of this one. My answer is directly related to this post. Next time please don’t split posts. Commented Jan 21 at 13:46

If you take a look at the wikipedia article on odds ratio, the value of an odds ratio simply means the following:

An odds ratio greater than 1 indicates that the condition or event is more likely to occur in the first group. And an odds ratio less than 1 indicates that the condition or event is less likely to occur in the first group.

You'll notice that this quote says nothing about p-values, confidence intervals, or statistical significance.

So on its own, the value of an odds-ratio observed in a sample won't tell you if you can infer something about your population of interest, relative to the true odds ratio. It means that you can't say just from the odds ratio if it is "statistically significant" or not. For that, you have to look at the p-value (is it smaller than your alpha level?) or at the confidence interval (does it exclude the null hypothesis? in your case, does it exclude the value 1?).

You say:

the confidence interval does not include a 1, indicating no significance

Your statement is incorrect. Think about a confidence interval including 1: it shows uncertainty about whether the true value of the odds ratio in your population of interest is larger or smaller than 1 (i.e. uncertainty on whether the event is less or more likely to occur in the first group).

So a confidence interval excluding 1 is consistent with having a small p-value. In your case, there does not seem to be an problem relative to that (the p-value is quite small, and the confidence interval excludes 1).

As a side note, the sample contingency table from your question gives the following output in R, which is not in line with the OR, p-value, and confidence interval you mention (this is incidentally an example of a confidence interval including 1, with a large p-value):

tab = rbind(c(15835, 923),
c(1867425, 111832))
fisher.test(tab)

Fisher's Exact Test for Count Data

data:  tab
p-value = 0.4396
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
0.9611067 1.0993676
sample estimates:
odds ratio
1.0274

• Thanks a lot. This brings more clarity into my thoughts. I want to address the output you posted and test my understanding in the interpretation: Since the confidence interval includes a 1 (includes also my H0) and the p value is not small enough to claim significance, and having the OR>1 stating that the association is more likely to occur in the first group (first group being miRNA correlating with phenotype), we conclude that there is no significant association between the variant of interest and phenotype. Is this correct? Commented Jan 19 at 19:11
• @mango This seems correct, however your statement about "the association is more likely to occur in the first group" is a bit ambiguous. The odds ratio is > 1 in the sample, but as the confidence interval includes 1 (which also implies a large p-value), we can't be very confident that the odds ratio is really > 1 in the population on which you're trying to infer. In other words, it's certainly difficult to draw conclusions from this sample about the real direction of the odds ratio (it could be > 1, it could be < 1). Commented Jan 19 at 19:53