Fisher's Exact Test

Fisher's exact test (FET) is a nonparametric version of the chi-square test. The most common use of FET is with small data sets, in particular when at least one cell in a cross-tabulation table has an expected frequency of less than 5. Small data sets and sparse cells may render the results reached by a chi-square test invalid, because it is based on asymptotic normality. Because the FET is a nonparametric test, this assumption does not apply to it. The FET calculates the exact probability (p value) of observing the table containing the data or more extreme distribution of the data. This probability is calculated by looking at all possible rearrangements of the table (in the direction of the alternative). The row and column totals are held constant when determining all possible more extreme rearrangements. Instructions on how to calculate the FET with 2 × 2 tables are given below. If one has another table of other dimensions, it is best to use a computer package such as SAS.

Finding the Exact Probability of a 2 × 2 Table

For a 2 × 2 table, the probability of the table occurring is

This probability can be calculated using either combinatorial functions (the left-hand side of the equation) or factorials (the right-hand side of the equation). In the first expression, the numerator counts the number of ways that the first row can occur (expressed by the combinatorial function describing how many ways the b-cell total can be chosen from the Row 1 total) times the number of ways that the second row can occur (expressed as the combinatorial function describing how many ways the d-cell total can be chosen from the Row 2 total) divided by the number of ways that the row and cell totals can be determined with n items (expressed by the combinatorial functions describing how many ways the first column total can be chosen from the sample size). In the second expression, the relevant factorials are simply multiplied together.

2 × 2 Example

Table 2 contains a 2 × 2 table of categorical data, representing the number of times two groups of people answered ‘yes’ to a question. The null hypothesis [Page 394]is that the proportion of yes is the same for the two column groups and the alternative hypothesis is that the proportion of yes is greater for Group 1 than for Group 2. This is a directional alternative hypothesis.

Table 1 General 2 × 2 Table
	Column 1	Column 2	Total
Row 1	a	b	r1
Row 2	c	d	r2
Total	s1	s2	n

Table 2 Original Data
Group 1	Group 2	Total
Yes	13	3	16
No	20	28	48
Total	33	31	64

The exact probability of Table 2, given our assumptions, is

However, what we want to calculate is the probability of getting a result at least as extreme as this distribution, if the null hypothesis is true. Therefore, we add to this probability the probability of more extreme tables under the alternative hypothesis; more extreme tables would have the (1, 1) cell—that is, cell ‘a’ in Table 2—equal to 14, 15, or 16. These more extreme tables are given in Table 3 with their respective probabilities. Note that these tables are found by keeping the row and column totals the same.

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