Wilcoxon Signed-Rank Test Explained

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:

1. 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.
2. Calculate the Differences:
   • For each pair of observations, calculate the difference between the two values.
3. 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.
4. 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.
5. 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-.
6. 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).
7. 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.
8. 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:

1. 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.
2. Calculate the Differences:
   • All differences are 10.
3. Rank the Differences:
   • All differences are ranked 4.
4. Assign Signs:
   • All signs are +.
5. Calculate the T-Statistic:
   • T = min(T+, T-) = min(5 * 4, 0) = 0
6. Determine the Critical Value:
   • Using a Wilcoxon signed-rank table with n = 5 and α = 0.05, the critical value is 2.
7. Compare T-Statistic to Critical Value:
   • The absolute value of T (0) is less than the critical value (2).
8. 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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Also Read:

• Mann-Whitney U Test Explained
• Chi-Square Test Explained
• ANOVA Analysis of Variance Explained
• T-Test Explained
• Z-Test Explained
• Wilcoxon Signed-Rank Test Explained