Fisher’s Exact Test

 With the Fisher’s exact test we can statistically determine whether there are nonrandom associations between two categorical variables, typically when we have a small sample size. The test is commonly applied to 2x2 matrices (known as contingency tables), where it is assumed that the counts in each row are dependent on which column they belong (or that the rows and columns are independent from one another). For example, we may be interested in whether there are differences in the number of males and females in math and engineering majors, or whether Americans and Canadians prefer cats or dogs as pets. We can phrase the statistical hypotheses for the Fisher’s exact test as follows:

 Null hypothesis: The proportions of the two variables are the equal.

H₀: p₁ = p₂

 Alternative hypothesis: The proportions of the two variables are not equal.

H_A: p₁ ≠ p₂

 Before using Fisher’s exact test to evaluate these hypotheses, we should determine whether the following assumptions are valid.

• The total counts are fixed and not random
• Small sample size
• The samples and observations are independent of each other
• The observations are mutually exclusive and can be classified into only one category
• The expected counts is < 5 but > 1 for at least 20% of the observations (cells in the contingency table)

 While larger contingency tables (2x3, 2x4, etc.) and sample sizes can be used with the Fisher’s exact test, as either increases so does the computation time. This is because unlike some other statistical tests, Fisher’s exact test calculates the exact p-value. Therefore, for larger sample sizes (>1000) or when the expected counts are > 5 for at least 80% of the cells then a χ² test may be preferred.

 After performing the Fisher’s exact test, with an exact p-value less than the chosen statistical threshold (typically α = 0.05) the null hypothesis may be rejected to conclude that there is a statistically significant difference between the two categorical variables.