Wilcoxon Signed-Rank Test Examples

Understanding the Wilcoxon Signed-Rank Test

The Wilcoxon Signed-Rank Test is a non-parametric statistical test used to determine whether there is a significant difference between the medians of two related groups. This test is particularly useful when the data does not meet the assumptions required for a parametric test, such as the paired t-test. It is commonly used in situations where you have paired observations, such as before-and-after measurements, or matched subjects. Below are three practical examples that illustrate the application of the Wilcoxon Signed-Rank Test in various contexts.

Example 1: Effectiveness of a Weight Loss Program

In a study aimed at evaluating the effectiveness of a specific weight loss program, researchers collected data from 10 participants. Each participant’s weight was measured before and after the program was implemented. The objective was to determine if there was a significant reduction in weight after the program.

The weight measurements (in kilograms) are as follows:

• Before: [82, 76, 91, 85, 78, 90, 84, 79, 88, 92]
• After: [80, 75, 89, 83, 76, 88, 82, 78, 87, 90]

To apply the Wilcoxon Signed-Rank Test:

1. Calculate the differences for each pair (Before - After):

• [2, 1, 2, 2, 2, 2, 2, 1, 1, 2]

  2. Rank the absolute differences:

• [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

  3. Assign signs to the ranks based on the direction of the difference (positive or negative):

• All ranks receive a positive sign.

  4. Calculate the test statistic W, which is the sum of the ranks of the positive differences.
  5. Compare the test statistic to the critical value from the Wilcoxon Signed-Rank distribution to determine significance.

Notes: This example shows how the Wilcoxon Signed-Rank Test can be used to assess the effectiveness of interventions such as weight loss programs. Additionally, the test does not assume normality, making it suitable for weight data that can be skewed.

Example 2: Pre- and Post-Test Scores in Education

An educational researcher wants to assess the impact of a new teaching method on student performance. A group of 15 students took a pre-test before the new method was introduced and a post-test afterward. The aim is to determine if the new teaching method significantly improved students’ test scores.

The scores (out of 100) are as follows:

• Pre-Test: [65, 70, 75, 80, 68, 72, 74, 76, 78, 81, 67, 73, 79, 77, 69]
• Post-Test: [75, 80, 82, 85, 78, 76, 81, 83, 84, 89, 74, 78, 80, 82, 76]

Steps to conduct the Wilcoxon Signed-Rank Test:

1. Calculate the differences:

• [10, 10, 7, 5, 10, 4, 7, 7, 6, 8, 7, 5, 1, 5, 7]

  2. Rank the absolute differences:

• Ranks would be assigned based on the magnitude of the differences.

  3. Compute W, the sum of the ranks for positive differences.
  4. Use the Wilcoxon Signed-Rank distribution table to find the p-value.

Notes: This example illustrates the application of the Wilcoxon Signed-Rank Test in educational settings to measure the effectiveness of teaching methods. The non-parametric nature of the test is beneficial when dealing with small sample sizes or non-normally distributed data.

Example 3: Measuring Patient Satisfaction Before and After Treatment

In a healthcare study, researchers aimed to measure the impact of a new treatment protocol on patient satisfaction. A survey was administered to 20 patients before and after the treatment, assessing their satisfaction on a scale from 1 (very dissatisfied) to 10 (very satisfied). The goal is to determine if patient satisfaction improved following the new treatment.

Satisfaction scores are as follows:

• Before: [5, 6, 7, 4, 5, 6, 7, 5, 6, 4, 3, 5, 6, 7, 4, 5, 6, 5, 6, 7]
• After: [7, 8, 9, 7, 6, 9, 8, 7, 8, 8, 5, 7, 9, 9, 6, 8, 8, 9, 9, 9]

Steps to perform the Wilcoxon Signed-Rank Test:

1. Calculate the differences:

• [2, 2, 2, 3, 1, 3, 1, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 4, 3, 2]

  2. Rank the absolute differences:

• Assign ranks based on the ordered differences.

  3. Compute the W statistic based on the ranks.
  4. Compare W to the critical value for significance.

Notes: This example highlights the use of the Wilcoxon Signed-Rank Test in the healthcare field, where patient satisfaction is a critical metric. It demonstrates the test’s ability to handle ordinal data effectively, making it ideal for survey responses.

By understanding these practical examples of the Wilcoxon Signed-Rank Test, researchers and practitioners can confidently apply this non-parametric test in their own studies and analyses.