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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