Fisher's Exact Test

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A Fisher's Exact Test is a statistical hypothesis test that ...



    • Fisher's exact test[1][2][3] is a statistical significance test used in the analysis of contingency tables where sample sizes are small. It is named after its inventor, R. A. Fisher, and is one of a class of exact tests, so called because the significance of the deviation from a null hypothesis can be calculated exactly, rather than relying on an approximation that becomes exact in the limit as the sample size grows to infinity, as with many statistical tests. Fisher is said to have devised the test following a comment from Dr Muriel Bristol, who claimed to be able to detect whether the tea or the milk was added first to her cup; see lady tasting tea.

      The test is useful for categorical data that result from classifying objects in two different ways; it is used to examine the significance of the association (contingency) between the two kinds of classification. So in Fisher's original example, one criterion of classification could be whether milk or tea was put in the cup first; the other could be whether Dr Bristol thinks that the milk or tea was put in first. We want to know whether these two classifications are associated – that is, whether Dr Bristol really can tell whether milk or tea was poured in first. Most uses of the Fisher test involve, like this example, a 2 × 2 contingency table. The p-value from the test is computed as if the margins of the table are fixed, i.e. as if, in the tea-tasting example, Dr Bristol knows the number of cups with each treatment (milk or tea first) and will therefore provide guesses with the correct number in each category. As pointed out by Fisher, this leads under a null hypothesis of independence to a hypergeometric distribution of the numbers in the cells of the table.

      With large samples, a chi-squared test can be used in this situation. However, the significance value it provides is only an approximation, because the sampling distribution of the test statistic that is calculated is only approximately equal to the theoretical chi-squared distribution. The approximation is inadequate when sample sizes are small, or the data are very unequally distributed among the cells of the table, resulting in the cell counts predicted on the null hypothesis (the "expected values") being low. The usual rule of thumb for deciding whether the chi-squared approximation is good enough is that the chi-squared test is not suitable when the expected values in any of the cells of a contingency table are below 5, or below 10 when there is only one degree of freedom (this rule is now known to be overly conservative.

  1. Fisher, R. A. (1922). "On the interpretation of χ2 from contingency tables, and the calculation of P". Journal of the Royal Statistical Society 85 (1): 87–94. doi:10.2307/2340521. JSTOR 2340521. 
  2. Fisher, R.A. (1954). Statistical Methods for Research Workers. Oliver and Boyd. ISBN 0050021702. 
  3. Agresti, Alan (1992). "A Survey of Exact Inference for Contingency Tables". Statistical Science 7 (1): 131–153. doi:10.1214/ss/1177011454. JSTOR 2246001.