Fisher's exact test

The Fisher Exact test ( Fisher -Yates test, exact chi-square test) is a significance test of independence in the contingency table. However, in contrast to the chi-square test, it does not provide conditions on the sample size and also provides for a small number of observations reliable results. He goes back to the British statistician Ronald Aylmer Fisher. It was originally developed for two dichotomous variables, ie 2x2 contingency tables, but it can also be extended to larger contingency tables.

Idea

Fisher's exact test is an alternative to a test based on a 2x2 Kontigenztafel. The upper right contingency table contains the observed frequencies, and for the four combinations of features, while the upper left contingency table contains the expected frequencies under the null hypothesis. The test statistic would result in an -based test to

And would then approximate distributed with one degree of freedom. Thus, the approximation is valid, but must be valid, and.

Are the four marginal frequencies, and tight, then it is sufficient, however, to consider one of the cells. Once, for example, the value of is fixed, are due to the fixed marginal frequencies, the values ​​for, and finally fixed.

Fisher showed that the number of observations in the upper left corners of a hypergeometric distribution follows:

The unknown marginal frequencies are estimated from the sample by means of their marginal frequencies, so that follows:

And the probability is given by

Alternatively, after Bortz, Lienert and Boehnke (1990), the probability can be written as

If the value of the sample is too small or too large, then the null hypothesis should be rejected.

Method

The independence of student performance by gender can not be tested using the chi -square test or the four-field test on its statistical significance in the example. The exact test of Fisher disagrees even with a few observations the required level.

If one chooses, for example, a significance level, the result is the critical values ​​as 2 or 3, ie the null hypothesis of independence of student achievement of gender can not be discarded when or. If or, then the null hypothesis can be rejected. In the example, that is, the null hypothesis of independence of student achievement of gender can not be discarded.

In addition, there are three other tables (see below), for the rule is that the sum of the column and row frequencies are equal to the observed values.

This example also shows that the Fisher exact test is a conservative test. Because the probability that you mistakenly accepts the alternative hypothesis ( type 1 error ) is given by

Thus smaller than the pre- bene significance level.