Wilcoxon Signed-Rank Test : Overview and Requirements
The Wilcoxon signed-rank test, also known as the Wilcoxon matched-pairs signed-rank test, is a non-parametric statistical test used to compare two paired samples. It is a powerful alternative to the paired t-test when the data may not be normally distributed or the sample size is small.
Purpose:
- Compares two dependent samples, meaning each data point in one group is paired with a corresponding data point in the other group.
- Useful for analyzing repeated measures or before-after data.
- Tests whether the medians of the two paired samples are different.
Requirements:
- Paired Samples: The data must consist of paired observations, where each data point in one group has a corresponding data point in the other group.
- Ordinal or Continuous Data: The data can be either ordinal (ranked data) or continuous (numerical data).
- Independence: The differences between the paired observations should be independent.
Steps to Conduct a Wilcoxon Signed-Rank Test:
- Formulate Hypotheses:
- Null Hypothesis (H₀): There is no significant difference between the medians of the two paired samples.
- Alternative Hypothesis (H₁): There is a significant difference between the medians of the two paired samples.
- Calculate the Differences:
- For each pair of observations, calculate the difference between the two values.
- Rank the Differences:
- Combine all the differences and rank them from smallest to largest, ignoring the signs (positive or negative).
- If there are tied ranks, assign the average rank to each tied value.
- Assign Signs:
- Assign a positive sign (+) to the rank if the corresponding difference is positive.
- Assign a negative sign (-) to the rank if the corresponding difference is negative.
- Calculate the Wilcoxon Signed-Rank Statistic (T):
- Sum the ranks with positive signs (T+).
- Sum the ranks with negative signs (T-).
- The Wilcoxon signed-rank statistic (T) is the smaller of T+ and T-.
- Determine the Critical Value:
- Consult a Wilcoxon signed-rank table or use statistical software to find the critical value based on the number of paired samples (n) and chosen significance level (α).
- Compare the T-Statistic to the Critical Value:
- If the absolute value of the T-statistic is greater than the critical value, reject the null hypothesis (H₀). This suggests a statistically significant difference between the medians of the two paired samples.
- If the absolute value of the T-statistic is less than or equal to the critical value, fail to reject the null hypothesis (H₀). There is not enough evidence to claim a significant difference.
- Interpret the Results:
- A rejected null hypothesis suggests a significant difference, but it doesn’t tell you the direction of the difference (which group has a higher median).
- Consider the magnitude of the T-statistic and the context of your research question for a more comprehensive interpretation.
Formulas:
- Wilcoxon Signed-Rank Statistic (T):
- T = min(T+, T-)
- Where:
- T+ = Sum of ranks with positive signs
- T- = Sum of ranks with negative signs
Example:
Suppose we want to compare the anxiety levels of participants before and after a relaxation technique. We measure their anxiety levels using a scale and obtain the following paired data:
| Participant | Before | After | Difference | Rank | Sign |
|---|---|---|---|---|---|
| 1 | 80 | 70 | 10 | 4 | + |
| 2 | 75 | 65 | 10 | 4 | + |
| 3 | 60 | 50 | 10 | 4 | + |
| 4 | 85 | 75 | 10 | 4 | + |
| 5 | 90 | 80 | 10 | 4 | + |
Steps:
- Formulate Hypotheses:
- H₀: There is no significant difference in anxiety levels before and after the relaxation technique.
- H₁: There is a significant difference in anxiety levels before and after the relaxation technique.
- Calculate the Differences:
- All differences are 10.
- Rank the Differences:
- All differences are ranked 4.
- Assign Signs:
- All signs are +.
- Calculate the T-Statistic:
- T = min(T+, T-) = min(5 * 4, 0) = 0
- Determine the Critical Value:
- Using a Wilcoxon signed-rank table with n = 5 and α = 0.05, the critical value is 2.
- Compare T-Statistic to Critical Value:
- The absolute value of T (0) is less than the critical value (2).
- Interpret the Results:
- We fail to reject the null hypothesis. There is not enough evidence to conclude that the relaxation technique significantly reduces anxiety levels.
Note:
- Statistical software can automate most of these calculations, making the Wilcoxon signed-rank test easier to perform.
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