The McNemar test is a statistical test used to analyze paired nominal (categorical) data. It is applied when we have two dependent (paired) samples and want to determine if there is a significant change or difference between the two related conditions.
When to Use the McNemar Test?
- The data is categorical (e.g., Yes/No, Success/Failure, Present/Absent).
- The samples are paired (i.e., the same subjects are measured twice).
- The goal is to test if there is a significant change in proportions between two related conditions.
Applications
- Medical studies: To check if a new treatment is more effective than the old one by comparing the same patients before and after treatment.
- Marketing research: To assess if a customer changes their preference after a promotional campaign.
- Psychology & behavioral studies: To evaluate whether an intervention changes behavior.
Contingency Table for McNemar Test
The test is based on a 2×2 contingency table:
| Condition B: Yes | Condition B: No | Row Total | |
|---|---|---|---|
| Condition A: Yes | a | b | a + b |
| Condition A: No | c | d | c + d |
| Column Total | a + c | b + d | N |
- a = Number of cases where both conditions are “Yes.”
- b = Number of cases where the first condition is “Yes” but the second is “No.”
- c = Number of cases where the first condition is “No” but the second is “Yes.”
- d = Number of cases where both conditions are “No.”
The McNemar test focuses on the discordant pairs (b and c).
Hypothesis of the McNemar Test
- Null Hypothesis (H₀): The proportions of the two related groups are equal (no significant change).
- Alternative Hypothesis (H₁): There is a significant difference between the two related proportions.
Test Statistics
\(\chi^2 = \frac{(b-c)^2}{b+c}\)
For small sample size use Edwards continuity correction
\(\chi^2 = \frac{(|b-c|-1)^2}{b+c}\)
The test statistic follows a Chi-square \(\chi^2\) distribution with 1 degree of freedom.
Decision Rule
- Compare the computed \(\chi^2\) value with the critical value from the Chi-square table at α = 0.05.
- Alternatively, calculate the p-value:
- If p ≤ 0.05, reject H0H_0H0 (significant difference).
- If p > 0.05, fail to reject H0H_0H0 (no significant difference).