Fisher’s exact test is used for small sample sizes and contingency tables to calculate statistical significance exactly. It determines the probability of a significant relationship between two categorical variables, such as gender and pet ownership, by calculating the p-value using factorials. The test is accurate for tables with a small sample size or large discrepancies between cell numbers.
Fisher’s exact test is a test of statistical significance used for small sample sizes. It is one of several tests used to analyze contingency tables, which show the interaction of two or more variables. This test was invented by English scientist Ronald Fisher and is called exact because it calculates statistical significance exactly, rather than using an approximation.
To understand how Fisher’s exact test works, it is essential to understand what a contingency table is and how it is used. In the simplest example, there are only two variables to compare in a contingency table. Usually, these are categorical variables. For example, imagine you’re conducting a study on whether gender correlates with pet ownership. There are two categorical variables in this study: gender, male or female, and pet ownership.
A contingency table is set up with one variable at the top and one to the left, so that there is one box for each combination of variables. Totals are shown at the bottom and far right. Here’s what a contingency table for the example study would look like, assuming a survey of 24 individuals:
Pet owner
Not a pet owner
Total
Man
1
9
10
Donna
11
3
14
Total
12
12
24
Fisher’s exact test calculates the deviation from the null hypothesis, which argues that there are no biases in the data or that the two categorical variables have no correlation with each other. In the case of the present example, the null hypothesis is that men and women are equally likely to own pets. Fisher’s exact test was designed for contingency tables with a small sample size or large discrepancies between cell numbers, like the one shown above. For contingency tables with a large sample size and well-balanced numbers in each cell of the table, Fisher’s exact test is not accurate and the chi-squared test is preferred.
In analyzing the data in the table above, Fisher’s exact test is used to determine the probability that pet ownership is unevenly distributed among men and women in the sample. We know that ten of the 24 people interviewed have pets and that 12 of the 24 are female. The probability that ten people randomly selected from the sample consist of nine women and one man will suggest the statistical significance of the distribution of pet owners in the sample.
The probability is denoted by the letter p. Fisher’s exact test determines the p-value for the above data by multiplying the factorials of each marginal total — in the table above, 10, 14, 12, and 12 — and dividing the result by the product of the factorials of each cell number and of the grand total. A factorial is the product of all positive integers less than or equal to a given number. 10!, pronounced “ten factorials,” is therefore equal to 10X9X8X7X6X5X4X3X2X1, or 3,628,800.
For the above table, therefore, p= (10!)(14!)(12!)(12!)/(1!)(9!)(11!)(3!)(24!). Using a calculator, it can be determined that the probability of getting the numbers in the table above is less than 2%, well below the probability, if the null hypothesis is true. Therefore, it is very unlikely that there is no contingency, or significant relationship, between gender and pet ownership in the study sample.
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