The Wilcoxon Rank-Sum test is a non-parametric analysis designed to determine whether two independent samples come from populations with identical or different distributions. This test is essentially identical to the Mann-Whitney U Test. The Wilcoxon Rank-Sum test serves as an alternative to the T-Test for two independent samples, especially when the assumption of normality cannot be met. In the use 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 assess the efficacy of a new antidepressant, ten patients suffering from clinical depression are randomly placed into one of two groups. Five patients are placed in Group 1, who are administered the antidepressant for a six-month period. The other five patients are placed in Group 2, who are administered a placebo for the same six-month period. Assume that before initiating the experimental treatment, the researcher ensures that the depression levels in both groups are identical. After six months have elapsed, all ten subjects are evaluated by a psychiatrist (who is blind to the experimental conditions of the subjects) regarding their depression levels. The psychiatrist's depression scores for the five subjects in each group are as follows (higher scores indicate greater depression): Group 1: 11, 1, 0, 2, 0; Group 2: 11, 11, 5, 8, 4. Do the data suggest that the antidepressant is effective?
Analysis Steps
The following are the steps for conducting the Wilcoxon Rank-Sum test and the post hoc test using SmartstatXL, an Excel Add-in:
- Activate the worksheet (Sheet) to be analyzed.
- Place the cursor on the dataset (for dataset creation, see Data Preparation method). The dataset can be organized in two layouts:
- Grouped based on level/tier (comparison between level)
- Grouped based on variable (comparison between variables)
- If the active cell is not on the dataset, SmartstatXL will automatically attempt to determine the dataset.
- Activate the SmartstatXL Tab
- Click on 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).
- If it is correct, click the Next Button
- Next, the Non-Parametric Test dialog box will appear:
- If the data layout used is based on comparison between variables, the following dialog box will appear:
- Finally, press the "OK" button
Analysis Results
The following are the Analysis Output of the Wilcoxon Rank-Sum Test:
Statistical Summary
From the Wilcoxon Rank-Sum Test results, we aim to determine whether the new antidepressant has a significant effect in reducing depression levels compared to a placebo.
Let's interpret these analysis results:
- 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 the placebo.
- H1 (Alternative Hypothesis): Both groups have different medians.
- Statistical Results:
- The rank sum R₁ is 19.000.
- Normal approximation indicates that R₁ approximates a normal distribution with a mean (μ) of 27.50 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 1.803.
- The p-value (probability) is 0.071.
- The critical value for this test is ±1.960.
- Interpretation:
- Because the Z-value (1.803) is smaller than the critical value (1.960), and the p-value (0.071) is greater than 0.05, we fail to reject the null hypothesis (H0). This means there is insufficient evidence to state that there is a significant difference between the group given the antidepressant and the group given the placebo in terms of depression levels.
Conclusion:
Based on the Wilcoxon Rank-Sum Test analysis, there is insufficient evidence to state that the new antidepressant is effective in reducing depression levels compared to a placebo at the 0.05 significance level.
Mean Comparison
Table of mean values, Mean Rank, Sum Rank, and Post Hoc Test
From the table, we can observe the comparison between the two groups based on their mean values, mean ranks, and sum ranks:
- Data Description:
- Group 1: With a mean of 2.80, mean rank of 3.80, and sum rank of 19.00.
- Group 2: With a mean of 7.80, mean rank of 7.20, and sum rank of 36.00.
- 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 mean and mean rank comparison, both groups do not show a significant difference in depression levels.
- Comparison with Mann-Whitney Test and Kolmogorov–Smirnov Test:
- Mann-Whitney Test: Although there is a difference in mean values between the two groups, the statistical test results indicate that the difference is not significant at the 0.05 level.
- Kolmogorov–Smirnov Test: The results of this test indicate that there is a significant difference between the two groups based on their cumulative distribution.
- Wilcoxon Rank-Sum Test: The results of this test, indicated by the same subset label ("a"), show that there is no significant difference between the two groups based on their mean ranks.
Considering all three tests, we can conclude that although there is a difference in mean values between the two groups, this difference may not be statistically significant. However, it is important to note that each test has different assumptions and calculation methods, which can yield different conclusions. Therefore, the appropriate test selection should be based on the data's nature and research objectives.
