Fisher's Exact Test Calculator

Fisher's exact test is based on the hypergeometric distribution. Given fixed row and column marginals, the probability of observing a specific table configuration is:

$$P = \frac{\binom{a+b}{a} \binom{c+d}{c}}{\binom{n}{a+c}} = \frac{(a+b)!(c+d)!(a+c)!(b+d)!}{n! \cdot a! \cdot b! \cdot c! \cdot d!}$$

The p-value is computed by summing probabilities of all tables as extreme or more extreme than the observed table, while keeping marginals fixed. The odds ratio provides a measure of effect size:

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

The odds ratio compares the odds of the outcome in one group versus the other. OR = 1 indicates no association, OR > 1 indicates higher odds in the first group, and OR < 1 indicates lower odds. The risk ratio (relative risk) is \(RR = (a/(a+b)) / (c/(c+d))\), comparing proportions directly.

The log odds ratio \(\ln(OR)\) is useful because its sampling distribution is approximately normal for moderate sample sizes, facilitating confidence interval construction: \(\ln(OR) \pm z_{\alpha/2} \sqrt{1/a + 1/b + 1/c + 1/d}\).

Interpreting the results from Fisher's exact test:

• Odds Ratio (OR): OR = 1 means no association. OR > 1 suggests the outcome is more likely in Row 1 relative to Row 2. OR < 1 suggests the opposite. The magnitude indicates strength of association.
• Risk Ratio (RR): Directly interpretable as the ratio of probabilities. RR = 2 means the event is twice as likely in group 1. RR is preferred in prospective studies; OR is used in case-control studies.
• Marginals: Row and column totals provide context for the cell frequencies and indicate the constraint structure of the test.
• When to use: Fisher's test is especially important when any expected cell frequency would be less than 5, making the chi-square approximation unreliable.