Fisher's exact test

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Fisher's Exact Test is a statistical significance test used in the analysis of categorical data where sample sizes are small. It is a non-parametric test that is used to determine if there are nonrandom associations between two categorical variables. Developed by Ronald Fisher in the early 20th century, the test is an alternative to the chi-squared test when the data does not meet the chi-squared test's large sample size assumptions.

Overview

Fisher's Exact Test is primarily used when dealing with 2x2 contingency tables. A contingency table in this context displays the distribution of two categorical variables and their respective frequencies. The test calculates the exact probability of observing the data assuming that there is no association between the variables (null hypothesis). It is particularly useful when the dataset is small, and the chi-squared approximation may be unreliable.

Calculation

The calculation of Fisher's Exact Test involves the use of the hypergeometric distribution. The test considers all possible distributions of the data that could occur under the null hypothesis and calculates the probability of each. The sum of these probabilities for all distributions as extreme or more extreme than the observed distribution gives the p-value, which indicates the likelihood of observing the given data if the null hypothesis were true.

Application

Fisher's Exact Test is widely used in medical research, especially in studies with small sample sizes, such as case-control studies. It is also applied in other fields such as biology, psychology, and social sciences, where researchers deal with categorical data and need to test for independence or association between two variables.

Advantages and Limitations

One of the main advantages of Fisher's Exact Test is its ability to provide an exact p-value, which is not based on any approximation. This makes it particularly reliable for small sample sizes. However, the test has limitations, including its computational intensity for larger datasets and its restriction to 2x2 tables, although extensions to larger tables exist.

Comparison with Chi-squared Test

While both Fisher's Exact Test and the chi-squared test are used to analyze categorical data, they have different applications and assumptions. The chi-squared test is suitable for larger sample sizes and relies on a distribution approximation, which can be inaccurate for small samples. Fisher's Exact Test, on the other hand, does not rely on these approximations and is thus preferred for smaller samples.

Extensions

Extensions of Fisher's Exact Test for larger contingency tables have been developed. These extensions often involve more complex calculations or simulations to obtain the exact p-value for tables larger than 2x2.

Conclusion

Fisher's Exact Test remains a fundamental tool in the analysis of categorical data, especially in situations where sample sizes are small, and the assumptions of other tests like the chi-squared test are not met. Its exact nature provides a reliable method for testing the association between categorical variables.

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