Wilcoxon Signed Rank Test

The Wilcoxon Signed Rank Test is a non-parametric statistical test used to compare two related groups. It is often applied when the assumptions for the paired t-test (such as normality) are not met. This test evaluates whether there is a significant difference between two paired observations, making it especially useful for non-normally distributed or ordinal data. It is widely used in fields like medicine, psychology and social sciences, where normality assumptions are often violated.

Key Features of the Wilcoxon Signed Rank Test

• Non-parametric Nature: The test does not assume that the data follows any specific distribution, making it ideal for non-normal data or ordinal variables.
• W-Statistic: It calculates the W-statistic which approximates a normal distribution for larger samples (n > 10).

When to Use the Wilcoxon Signed Rank Test?

The test is appropriate when:

• The data consists of paired (dependent) observations.
• The differences between pairs are continuous or ordinal.
• The distribution of differences is not normal but symmetric.

Example

To evaluate the effectiveness of a new teaching method, researchers measure literacy levels of 20 children before and after applying the method. Since the data is non-normal, the Wilcoxon Signed Rank Test is ideal.

Hypotheses

• Null Hypothesis (H₀): The median difference between paired samples is zero (no significant difference).
• Alternative Hypothesis (H₁): The median difference is not zero (there is a significant difference).

Steps to Perform the Wilcoxon Signed Rank Test

1. Formulate Hypotheses:
   • H₀: There is no significant difference between the pairs.
   • H₁: There is a significant difference between the pairs.
2. Find the Difference (D) between paired observations (i.e., D = B − A).
3. Calculate the Absolute Difference (|D|) for each pair.
4. Assign Ranks to the absolute differences from lowest to highest. For tied ranks (duplicate absolute differences), assign them the average rank.
   Example for ties: If two absolute differences are both 3, assign them a rank of \frac{3 + 4}{2} = 3.5.
5. Calculate the Sum of Ranks:
   • T+: Sum of ranks for positive differences.
   • T-: Sum of ranks for negative differences.
6. Find the Wilcoxon Rank Statistic (W_calc): W_calc = min⁡ (T+ ,T−)
7. Compare W_calc to W_table:
   • Use the sample size n and the significance level α (commonly 0.05) to find the critical value W_table from a table of critical values for the Wilcoxon Signed Rank Test.
8. Interpret the Result:
   • If W_calc < W_table ​, reject H₀​.
   • If W_calc > W_table ​, accept H_0​.

Interpretation of Results

• Rejecting H₀​: If W_calc​ is less than the critical value W_table​, it indicates that the two groups are not identically distributed and there is a significant difference.
• Accepting H₀: If W_calc​ is greater than W_table​, it means the two groups are identically distributed and there is no significant difference.

Example: Smog Concentration Comparison

Let's consider an example where we compare smog concentrations in India from May to December. The data is presented for 13 states with measurements in both months.

Step 5: Calculate T+ and T−

• Sum of positive ranks (T+): T+ = 9 + 11 + 2 + 10 + 1 + 7 + 4 + 12 + 8 + 6 + 13 = 82

• Sum of negative ranks (T-): T- = 5 + 3 = 8

Step 6: Calculate W_calc

W_calc = min⁡(T+ ,T−) = min⁡(82 ,8) = 8

Step 7: Find W_table

For n = 13 and α = 0.05 for a two-tailed test, the critical value is 17.

Step 8: Interpret the Result

Since W_calc = 8 is less than W_table = 17, we reject the null hypothesis H_0​. This means that there is a significant difference in the smog concentrations between May and December.

Conclusion

The Wilcoxon Signed Rank Test shows that the smog concentration has significantly changed from May to December.

Code Implementation of Wilcoxon Signed Rank Test

Lets see the implementation of Wilcoxon Signed Rank Test with help of an example in Python. Libraries like NumPy and Wilcoxon from scipy would be used,

import numpy as np
from scipy.stats import wilcoxon

may_smog = np.array([13.3, 10.0, 16.5, 7.9, 9.5, 8.3,
                    12.6, 8.9, 13.6, 8.1, 18.3, 8.1, 13.4])
december_smog = np.array(
    [11.1, 16.2, 15.3, 19.9, 10.5, 15.5, 12.7, 14.2, 15.6, 20.4, 12.7, 11.2, 36.8])

stat, p_value = wilcoxon(may_smog, december_smog)

print("Wilcoxon Signed Rank Test Statistic:", stat)
print("p-value:", p_value)

if p_value < 0.05:
    print("Reject the null hypothesis: There is a significant difference between the two samples.")
else:
    print("Fail to reject the null hypothesis: No significant difference between the two samples.")

Output

The null hypothesis was rejected since there is a significant difference between the two samples.

Difference Between the Wilcoxon Signed Rank Test and Wilcoxon Rank-Sum Test

Lets see the differences between the Wilcoxon Signed Rank Test and Wilcoxon Rank-Sum Test( also known as Mann-Whitney U Test):