Fisher’s Exact Test

1. Testing the association between two nominal variables

When measuring the association between two nominal variables, one can conduct a Chi-square test. However, the Chi-square test is conducted under the assumption that it allowed maximum of 20% of the cells to have expected count <5. If this assumption is violated, one can consider an alternative of the test which is the Fisher’s Exact test.

Fisher’s Exact test

Fisher’s exact test is normally being conducted when there are more than 20% of the cells have expected counts that is less than 5.

2. Formula for Fisher’s Exact test

p-value = [(a+b)!(c+d)!(a+c)!(b+d)!]/(a!b!c!d!n!)

	Group 1	Group 2
Group A	a	b
Group B	c	d

3. Decision rules for Fisher’s Exact test

For such test, one can determine whether or not to reject the null hypothesis by comparing the p-value they calcuated from the Fisher’s Exact Test with the level of significance. The null hypothesis will be rejected if the calculated p-value is less than the level of significance (usually 0.05).

Exercise

Determine whether to reject the null hypothesis of the Fisher’s Exact Test for each for the following pairs of data (Level of Significance = 0.10).

Question 1

Var 1	Var 2
1	1
1	2
1	1
2	1
1	2
1	1
1	2
1	1
2	2
2	1
2	2
1	1
1	2
1	2
1	1
1	2
1	1
1	1
1	2
1	2
1	1
1	1
2	2
2	1

Question 2

Var 3	Var 4
1	2
1	2
2	1
1	2
2	2
1	2
2	2
1	1
2	2
2	1
1	2
2	1
1	2
1	2
1	2
1	2
2	1
2	2
2	1
2	1
1	1
1	2
1	1
2	1

Question 3

Var 5	Var 6
2	1
1	1
1	1
2	1
2	2
2	1
1	1
2	2
2	1
2	1
2	1
1	2
1	1
2	2
1	2
2	1
2	2
1	2
2	2
2	2
2	2
2	2
2	1
1	1

Question 4

Var 7	Var 8
1	1
1	2
1	1
2	2
1	2
1	1
2	1
1	2
1	1
1	1
1	2
1	1
2	2
2	1
2	1
2	2
1	2
2	1
1	1
2	1
1	1
2	2
2	1
2	2

Question 5

Var 9	Var 10
1	2
1	2
1	1
2	1
1	2
1	1
2	2
2	1
1	2
1	2
2	2
1	2
2	2
1	1
2	1
2	1
2	2
1	1
2	2
1	2
1	1
1	1
2	1
2	1

Question 6

Var 11	Var 12
1	2
1	1
2	1
1	2
1	1
1	2
1	1
2	2
1	2
1	2
1	1
2	1
1	1
1	1
2	1
1	2
2	2
2	1
2	1
2	1
2	2
2	1
1	2
2	2

Answer

Do not reject (p-value = 1)
Reject (p-value = 0.095)
Do not reject (p-value = 0.679)
Do not reject (p-value = 1)
Do not reject (p-value = 1)
Do not reject (p-value = 0.444)

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