fisher's exact t test

3 min read 18-03-2025

Fisher's exact test is a statistical significance test used to determine if there's an association between two categorical variables. Unlike some other tests, it's particularly useful when dealing with small sample sizes, especially when the assumptions of other tests (like the chi-squared test) aren't met. This article will provide a comprehensive overview of Fisher's exact test, explaining its applications, calculations, interpretations, and limitations.

When to Use Fisher's Exact Test

Fisher's exact test shines when you're analyzing contingency tables—tables that summarize the counts of observations for two categorical variables. The test is especially valuable in situations where:

Sample size is small: Traditional tests, such as the chi-squared test, rely on asymptotic approximations that might not be accurate with small sample sizes. Fisher's exact test doesn't have this limitation. A common rule of thumb is to use Fisher's exact test when the expected cell count in any cell of the contingency table is less than 5.
Expected cell counts are low: Even if the overall sample size isn't tiny, if the expected counts in some cells are low, Fisher's exact test is preferred for increased accuracy.
Analyzing the association between two categorical variables: This is the core function of the test. It helps determine if the observed frequencies are significantly different from what would be expected if the variables were independent.

Understanding the Test's Logic

Fisher's exact test is based on calculating the probability of observing the specific contingency table data, or data more extreme, given that the two variables are independent. It uses the hypergeometric distribution to calculate this probability, considering all possible tables with the same marginal totals (row and column sums).

The hypergeometric distribution is perfectly suited for this because it deals with the probability of drawing a specific number of successes (or 'events') from a population of a specific size without replacement. This aligns well with the categorical data analysis performed by Fisher's exact test.

How to Interpret Results

The output of Fisher's exact test typically includes a p-value. This p-value represents the probability of observing the data (or more extreme data) if there is no association between the two categorical variables (the null hypothesis).

p-value ≤ α (significance level): If the p-value is less than or equal to your chosen significance level (commonly 0.05), you reject the null hypothesis. This means there's strong evidence to suggest a significant association between the two variables.
p-value > α: If the p-value is greater than your significance level, you fail to reject the null hypothesis. This indicates that there's not enough evidence to conclude a significant association.

Remember, failing to reject the null hypothesis doesn't automatically prove independence; it simply means there's insufficient evidence to claim dependence.

Calculating Fisher's Exact Test: A Simple Example

Let's say we're investigating the relationship between smoking and lung cancer. We have a small sample:

	Lung Cancer	No Lung Cancer	Total
Smoker	10	2	12
Non-smoker	2	16	18
Total	12	18	30

Statistical software packages (R, SPSS, Python's SciPy) readily perform Fisher's exact test. You input the contingency table, and the software calculates the p-value. Manual calculation is complex due to the factorial calculations involved in the hypergeometric distribution.

Limitations of Fisher's Exact Test

While powerful, Fisher's exact test has limitations:

Computational intensity: For larger tables, calculations can become computationally expensive.
Limited to 2x2 tables: While extensions exist, the standard Fisher's exact test is primarily designed for 2x2 contingency tables.
Not suitable for large samples: Ironically, for very large samples, Fisher's exact test might become computationally impractical. The Chi-squared test would be more appropriate in these situations.

Fisher's Exact Test vs. Chi-Squared Test

The choice between Fisher's exact test and the chi-squared test often depends on the sample size and expected cell counts. The chi-squared test is generally preferred for larger samples because it's computationally less intensive and provides a good approximation. However, when dealing with small samples or low expected cell counts, Fisher's exact test is the more accurate and reliable option.

Conclusion

Fisher's exact test is a valuable tool for assessing the association between two categorical variables, particularly when dealing with small sample sizes or low expected cell counts. By understanding its principles, interpretation, and limitations, researchers can effectively utilize this test in their statistical analyses. Remember to always choose the appropriate statistical test based on the characteristics of your data and research question.