- Check the conditions for Fisher’s Exact Test
- Explain the relationship of two qualitative binary variables using Fisher’s Exact Test
[latex]\chi^2[/latex] Test for Independence vs. Fisher’s Exact Test
We use the [latex]\chi^2[/latex] test for independence to decide whether two variables (factors) are independent or dependent. In this case, there will be two qualitative survey questions or experiments, and a contingency table will be constructed. The goal is to see if the two variables are unrelated (independent) or related (dependent).
The null and alternative hypotheses are:
- [latex]H_0[/latex]: The variables (factors) are independent.
- [latex]H_A[/latex]: The variables (factors) are dependent.
Next, the conditions for the test of independence must be checked. It is important to know when you can or can not perform the [latex]\chi^2[/latex] test of independence.
- Condition # 1: Independence/Randomness Condition: The [latex]\chi^2[/latex] test assumes that observations are independent. This means that the outcome for one observation is not associated with the outcome of any other observation.
- Condition # 2: Large Sample Sizes Condition: The sample sizes need to be large enough so that the expected count in each cell is at least five.
Question: So, what happens if one or more of the expected counts is less than 5? This means that the large sample sizes condition is violated, and therefore [latex]\chi^2[/latex] test of independence cannot be used in this case.
This is when Fisher’s Exact Test comes into play. You would then want to consolidate the contingency table to a [latex]2 \times 2[/latex] contingency table and check the condition again.
So, the null and alternative hypotheses for Fisher’s Exact Test are:
- [latex]H_0[/latex]: The two variables (factors) are independent.
- [latex]H_A[/latex]: The two variables (factors) are dependent.
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