# 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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