Mann-Whitney U Test
The Mann-Whitney U test, also known as the Wilcoxon rank-sum test, is a non-parametric statistical test used to compare two independent groups. It helps assess whether the two groups are likely to come from the same population or have different medians.
What it does:
- Compares the medians of two independent groups.
- Useful when data is not normally distributed, or sample sizes are small.
- Often considered the non-parametric alternative to the independent t-test.
- Ranks the data points instead of using their raw values, making it robust to outliers.
Requirements:
- Independent Groups: The two groups being compared must be independent, meaning the selection of a data point in one group doesn’t influence the selection of a point in the other group.
- Ordinal or Continuous Data: The data can be either ordinal (ranked data) or continuous (numerical data). The test is particularly useful when the assumptions of a parametric test (like the t-test) are not met.
Steps to Conduct a Mann-Whitney U Test:
- Define Your Hypothesis:
- Null Hypothesis (H₀): The medians of the two groups are the same.
- Alternative Hypothesis (H₁): The medians of the two groups are different.
- Rank the Data: Combine all data points from both groups and assign ranks from 1 (smallest) to the total number of data points. If there are tied ranks, assign the average rank to each tied value.
- Calculate the Mann-Whitney U Statistic:
- Calculate the sum of the ranks for each group, R1 and R2.
- Use the formulas:
- U1 = n1n2 + n1(n1 + 1) / 2 – R1
- U2 = n1n2 + n2(n2 + 1) / 2 – R2
- Where n1 and n2 are the sample sizes of the two groups, and R1 and R2 are the sums of the ranks for each group.
- The Mann-Whitney U statistic is the smaller of U1 and U2.
- Determine the Critical Value:
- Consult a Mann-Whitney U table based on the sample sizes (n1 and n2) and chosen significance level (e.g., 0.05).
- Compare the U Statistic to the Critical Value:
- If the U statistic is less than or equal to the critical value, reject the null hypothesis (H₀). This suggests a statistically significant difference between the medians of the two groups.
- If the U statistic is greater than the critical value, fail to reject the null hypothesis (H₀). There’s not enough evidence to claim a significant difference.
- Interpret the Results:
- A rejected null hypothesis suggests a statistically significant difference in medians between the groups. However, it does not indicate the direction of the difference (which group has a higher median).
- Consider the magnitude of the U statistic and the context of your research question for a more comprehensive interpretation.
Additional Considerations:
- Using Statistical Software: Statistical software like SPSS, R, and Python can automate most of these calculations, making the Mann-Whitney U test easier to perform.
- Effect Size: Consider calculating the effect size to understand the practical significance of the results.
- Robustness to Outliers: The Mann-Whitney U test is less sensitive to outliers than parametric tests, making it a good choice when the data might have extreme values.
Example Calculation:
Suppose we have two groups with the following data:
Group 1: [2, 4, 6]
Group 2: [1, 3, 5]
Combine and rank the data: [1, 2, 3, 4, 5, 6] -> Ranks: [1, 2, 3, 4, 5, 6]
Sum of ranks for Group 1 (R1): 2 + 4 + 6 = 12
Sum of ranks for Group 2 (R2): 1 + 3 + 5 = 9
Calculate U statistics:
U1 = 3 × 3 + 3(3 + 1) / 2 – 12 = 9 + 6 – 12 = 3
U2 = 3 × 3 + 3(3 + 1) / 2 – 9 = 9 + 6 – 9 = 6
Use the smaller U value: U = 3
Compare U = 3 to the critical value from the Mann-Whitney U table. If the critical value at α = 0.05 is greater than 3, reject the null hypothesis.
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