How to Analyze Mann-Whitney U Test - Non Parametric

The Mann-Whitney test, also known as the Mann-Whitney U test or Wilcoxon–Mann–Whitney test, is a non-parametric analysis designed to compare the median differences between two independent groups. This test is suitable for use when the scale of the dependent variable is either ordinal or continuous. Developed by H.B. Mann and D.R. Whitney in 1947, the Mann-Whitney test serves as an alternative to the Independent Samples T-Test, particularly when the assumption of normality is not met. In the context of SmartstatXL, this test provides a reliable statistical solution for data analysis under such conditions.

Case Example

Sourced from: Handbook of Parametric and Nonparametric Statistical Procedures, Fifth Edition by David J. Sheskin

Example 12.1

To evaluate the efficacy of a new antidepressant medication, ten patients suffering from clinical depression are randomly assigned to one of two groups. Five patients are placed in Group 1, who are administered the antidepressant medication for a six-month period. The other five patients are placed in Group 2, who are given a placebo for the same six-month period. Assume that before commencing the experimental treatment, researchers ensure that the levels of depression in both groups are the same. After six months have passed, all ten subjects are assessed by a psychiatrist (who is unaware of the subjects' experimental conditions) regarding their levels of depression. The depression ratings by the psychiatrist for the five subjects in each group are as follows (the higher the rating, the more depressed the subject): Group 1: 11, 1, 0, 2, 0; Group 2: 11, 11, 5, 8, 4. Do the data suggest that the antidepressant medication is effective?

Analysis Steps

The following are the steps for the Mann–Whitney U test and Post hoc Test using SmartstatXL, Excel Add-in:

1. Activate the worksheet (Sheet) that will be analyzed.
2. Place the cursor on the dataset (for creating a dataset, see the Data Preparation method). The dataset can be organized in two layouts:
3. Grouped based on levels (comparison between levels)
4. Grouped based on variables (comparison between variables)
   
5. If the active cell is not on the dataset, SmartstatXL will automatically attempt to determine the dataset.
6. Activate the SmartstatXL Tab
7. Click the Non-Parametric Menu. SmartstatXL will display a dialog box to confirm whether the dataset is correct or not (usually, the cell address of the dataset is automatically selected correctly).
   
8. If it is correct, click the Next Button
9. Next, the following Non-Parametric Test Dialog Box will appear:
   
10. If the data layout is based on a comparison between variables, the following dialog box will appear:
    
11. Finally, press the "OK" button

Analysis Results

The following is the Output Analysis for the Mann-Whitney U Test:

Statistical Summary

Based on the analysis results, we aim to determine whether the new antidepressant medication has a significant effect in reducing depression levels compared to a placebo. To ascertain this, we use the Mann-Whitney Test.

Let's interpret these analysis results:

1. Proposed Hypotheses:
   • H0 (Null Hypothesis): Both groups have the same median. This means there is no difference between the group given the antidepressant and the group given a placebo.
   • H1 (Alternative Hypothesis): Both groups have different medians.
2. Statistical Results:
   • The U value is 4.000.
   • Normal approximation shows that R₁ approaches a normal distribution with a mean (μ) of 12.50 and a standard deviation (σ) of 4.71.
   • The Z-value (without adjustment for ties) is 1.776, while the Z-value (with adjustment for ties) is 1.798.
   • The p-value (probability) is 0.072.
   • The critical value for this test is ±1.960.
3. Interpretation:
   • Because the Z-value (1.798) is smaller than the critical value (1.960), and the p-value (0.072) is greater than 0.05, we fail to reject the null hypothesis (H0). This means that there is insufficient evidence to state that there is a significant difference between the group given the antidepressant and the group given a placebo in terms of depression levels.

Conclusion:

Based on the Mann-Whitney Test analysis, there is insufficient evidence to claim that the new antidepressant medication is effective in reducing depression levels compared to a placebo at the 0.05 significance level.

Average Comparisons

Table of average values, Mean Rank, Sum Rank, and Post hoc Test

From the table, we can observe the comparison between the two groups based on average values, mean ranks, and sum of ranks.

Let's interpret these comparison results:

1. Data Description:
   • Group 1: With an average of 2.80, mean rank of 3.80, and sum of ranks 19.00.
   • Group 2: With an average of 7.80, mean rank of 7.20, and sum of ranks 36.00.
2. Subset Interpretation:
   • Both groups have the same subset label ("a"). This indicates that there is no significant difference between the two groups based on their mean ranks.

Conclusion:

Based on the comparison of average values and mean ranks, both groups do not show a significant difference in depression levels. Although Group 2 has higher average and mean rank values compared to Group 1, this difference is not statistically significant.