Fisher's Exact Test#

Fisher's exact test is a statistical test for examining independence in a 2×2 contingency table. Unlike the chi-square test, it calculates the exact p-value and is therefore particularly suitable for small samples where the chi-square approximation is unreliable.

When to Use#

Use Fisher's exact test when you want to:

Examine the association between two dichotomous variables
You have a 2×2 contingency table
The expected frequencies in one or more cells are less than 5
The sample is small (typically $N < 30$ )
You need an exact p-value without approximation

Assumptions#

2×2 contingency table (two dichotomous variables)
Independent observations
Fixed marginal totals or random sampling
Random assignment of observations

Formula#

The probability of a particular arrangement in the 2×2 table is calculated using the hypergeometric distribution:

p = \frac{\binom{a+b}{a} \binom{c+d}{c}}{\binom{N}{a+c}} = \frac{(a+b)! \cdot (c+d)! \cdot (a+c)! \cdot (b+d)!}{N! \cdot a! \cdot b! \cdot c! \cdot d!}

for a table of the form:

	Column 1	Column 2
Row 1	a	b
Row 2	c	d

The p-value is the sum of all probabilities that are equal to or more extreme than the observed distribution.

Example#

Practical Example: Drug Side Effect

A small clinical trial examines whether a drug causes a particular side effect. 20 patients are studied:

	Side Effect	No Side Effect	Total
Drug	4	6	10
Placebo	1	9	10
Total	5	15	20

Since the expected frequency for "Placebo/Side Effect" is only $\frac{10 \cdot 5}{20} = 2.5$ , the chi-square test is unreliable. Fisher's exact test provides the exact p-value.

Effect Size#

The odds ratio (OR) as effect size:

OR = \frac{a \cdot d}{b \cdot c}

Interpretation	Odds Ratio
No effect	1.0
Small effect	1.5
Medium effect	2.5
Large effect	4.3

Alternatively, Cramer's V or the phi coefficient can be used:

\phi = \frac{ad - bc}{\sqrt{(a+b)(c+d)(a+c)(b+d)}}