Wilcoxon Signed Rank Test in Stata|2025

Learn how to perform the Wilcoxon Signed Rank Test in Stata with this step-by-step guide, covering data preparation, test execution, and result interpretation for non-parametric analysis.

The Wilcoxon Signed Rank Test is a non-parametric statistical test that is used to compare paired or related samples to assess whether their population mean ranks differ. It is an alternative to the paired Student’s t-test when the assumptions of normality cannot be met. In this paper, we will explore the Wilcoxon Signed Rank Test, particularly focusing on its application in Stata, a popular statistical software. We will also address its comparison with the Mann-Whitney U test and the Wilcoxon Rank Sum test.

Table of Contents

Understanding the Wilcoxon Signed Rank Test

The Wilcoxon Signed Rank Test, often referred to as the Wilcoxon matched pairs signed rank test, is used when we have two related samples or repeated measurements on a single sample to test if their distributions are the same. Unlike the paired t-test, which assumes normality in the data, the Wilcoxon Signed Rank Test is a non-parametric test and does not require that the data follow a normal distribution.

The test works by calculating the difference between each pair of observations, ranking these differences in absolute value, and then evaluating whether the positive and negative ranks balance each other out. If the sum of the ranks of one sign (positive or negative) is significantly different from the other, it suggests that the samples are from different distributions.

The key assumption in the Wilcoxon Signed Rank Test is that the differences between paired observations come from the same distribution and are symmetrically distributed.

Key Concepts in the Wilcoxon Signed Rank Test

Paired Samples: The Wilcoxon Signed Rank Test is used for paired samples. This can involve before-and-after measurements on the same subjects, or measurements from matched subjects, where each pair has some known relationship.
Ranks: The test works by ranking the absolute differences between the paired values, ignoring the sign. The ranks are then summed for both the positive and negative differences, and the smaller of these two sums is compared to a critical value.
Hypotheses:
- Null Hypothesis (H₀): The median difference between the paired samples is zero.
- Alternative Hypothesis (H₁): The median difference between the paired samples is not zero.

Running the Wilcoxon Signed Rank Test in Stata

Stata is a powerful statistical software that simplifies the implementation of the Wilcoxon Signed Rank Test. To perform this test, the basic syntax is:

Where var1 and var2 are the two variables to be compared.

Example: Wilcoxon Signed Rank Test in Stata

Suppose we are testing whether there is a difference in the blood pressure levels of patients before and after receiving a treatment. We have two variables: bp_before (blood pressure before treatment) and bp_after (blood pressure after treatment). We can run the Wilcoxon Signed Rank Test as follows:

Stata will provide the following output:

In this output, you can see the number of pairs, the sum of ranks, and the test statistics, which are used to compute the significance of the result. Based on the z-score, you can determine whether there is a significant difference between the paired samples.

Mann-Whitney U Test vs. Wilcoxon Signed Rank Test

It is important to note the distinction between the Mann-Whitney U test and the Wilcoxon Signed Rank Test. The Mann-Whitney U test (also called the Wilcoxon Rank Sum Test) is a non-parametric test used to compare two independent groups, while the Wilcoxon Signed Rank Test compares two related or paired samples.

Mann-Whitney U Test: Used to determine if there is a difference between two independent groups. This test ranks all the observations from both groups together and then evaluates the differences between the ranks of the two groups.
Wilcoxon Signed Rank Test: Used to compare two related or paired samples. It ranks the differences between the paired values, not the individual observations.

How to Rank Data for the Wilcoxon Signed Rank Test

Ranking the data is a critical step in the Wilcoxon Signed Rank Test. Here’s a step-by-step process of how to rank the differences:

Calculate the Differences: Subtract one variable from the other. For instance, if you are testing the difference between two variables var1 and var2, calculate difference = var1 - var2.
Rank the Absolute Differences: Rank the absolute values of these differences in ascending order, ignoring the sign of the difference.
Assign the Signs: After ranking the absolute differences, assign the original sign (positive or negative) to each rank.
Sum the Ranks: Calculate the sum of the positive and negative ranks separately.
Test Statistic: The test statistic is based on the smaller of the two summed ranks (positive or negative). If this statistic is significantly small, we reject the null hypothesis.

Wilcoxon Signed Rank Test Interpretation

Interpreting the results of the Wilcoxon Signed Rank Test involves the following steps:

Z-Score: The z-score is a standardized value that reflects the magnitude of the test statistic. If the z-score is large (in absolute value), it suggests a significant difference between the paired groups.
P-Value: The p-value helps determine whether the results are statistically significant. A p-value less than the significance level (typically 0.05) indicates that the null hypothesis can be rejected, meaning there is a significant difference between the paired samples.
Effect Size: The effect size (e.g., r) can be computed to assess the magnitude of the difference between the paired groups. A larger effect size indicates a more significant difference.

For example, if Stata returns a z-score of -3.45 with a p-value of 0.001, this would suggest that there is a statistically significant difference between the two samples.

Reporting the Wilcoxon Signed Rank Test

When reporting the results of a Wilcoxon Signed Rank Test, it is important to include:

A statement of the hypothesis: “The null hypothesis is that the median difference between the two variables is zero.”
Test statistic and p-value: “The Wilcoxon Signed Rank Test yielded a z-score of -3.45 (p = 0.001), indicating a significant difference.”
Effect size: If calculated, report the effect size, such as “The effect size (r) was 0.45, indicating a moderate difference between the groups.”
Context and interpretation: “Based on these results, we reject the null hypothesis and conclude that the treatment has a significant effect on blood pressure.”

Additional Questions and Answers about the Wilcoxon Signed Rank Test

Q: Can I use the Wilcoxon Signed Rank Test for more than two related samples?
- A: No, the Wilcoxon Signed Rank Test is specifically designed for two related samples. For more than two related samples, you should use the Friedman test, which is a non-parametric test for repeated measures.
Q: What should I do if there are ties in the data?
- A: Ties are handled by assigning the average rank to tied values. For example, if two differences have the same absolute value, both are assigned the average of the ranks they would have otherwise received.
Q: How do I check if my data meets the assumptions for the Wilcoxon Signed Rank Test?
- A: The main assumption is that the differences between paired observations come from the same distribution and are symmetrically distributed. You can visually inspect the data using histograms or box plots, or you can test for symmetry using other statistical tests.
Q: Is the Wilcoxon Signed Rank Test always appropriate for non-normal data?
- A: While the Wilcoxon Signed Rank Test is robust to non-normal data, it is most appropriate when you expect the differences between the paired samples to be symmetrically distributed. If symmetry is in doubt, consider using a different approach or conducting a more thorough exploration of your data.

Conclusion

The Wilcoxon Signed Rank Test is an essential tool in non-parametric statistics, allowing researchers to compare two related samples when the assumption of normality is violated. Stata provides a user-friendly platform to conduct this test, and understanding how to run and interpret the results is key to making valid inferences. When using the Wilcoxon Signed Rank Test, it is crucial to understand its assumptions, differences from similar tests like the Mann-Whitney U test, and how to properly report the findings to ensure accurate results and meaningful conclusions.