21. WILCOXON RANK SUM TEST

The Wilcoxon Rank Sum Test is an important non-parametric statistical test used to compare two independent groups when the data does not meet the assumptions of parametric tests such as the independent t-test. It evaluates whether the two populations have the same distribution by ranking all observations and comparing the sum of ranks between the two groups.

This test is commonly used in clinical and pharmaceutical research where sample sizes are small, data is skewed, or measurements are ordinal. It is also known as the Wilcoxon-Mann-Whitney Test or simply the Rank Sum Test.

When to Use the Wilcoxon Rank Sum Test?

When comparing two independent samples.
When data is non-normally distributed or ordinal.
When sample sizes are small (n < 30).
When values represent ranks, scores, or non-parametric observations.
When independent t-test assumptions (normality, equal variance) are violated.

Hypotheses for the Wilcoxon Rank Sum Test

H₀: The two populations have the same distribution (no difference).
H₁: The two populations differ in distribution (there is a difference).

Principle of the Test

The test combines the data from both groups, ranks them from lowest to highest, and then calculates the sum of ranks for each group. A significant difference in rank sums suggests that the two groups differ in central tendency.

Step-by-Step Procedure

Combine data from both groups into a single dataset.
Rank all the observations from smallest to largest.
If ties exist, assign average ranks.
Calculate the sum of ranks for each group (R₁ and R₂).
Compute the test statistic:
W = smaller rank sum
Determine expected mean and variance of W (for large samples).
Compare the calculated value with the critical value or convert to a Z score.
Make a conclusion based on the significance level (α).

Test Statistic (Large Sample Approximation)

Z = (W − μ_W) / σ_W

Where:

μ_W = n₁(n₁ + n₂ + 1) / 2
σ_W = √[n₁n₂(n₁ + n₂ + 1) / 12]

Example (Illustration)

Suppose two pain relief drugs are compared by measuring pain score reduction in two independent groups of patients. All scores are ranked and the rank sums are computed. If the rank sum difference is large enough, it indicates a significant difference between the two drugs.

Assumptions

Two samples are independent.
Data is ordinal or continuous but non-normally distributed.
Shapes of the distributions are similar.
Observations are mutually independent.

Advantages

Does not require normal distribution.
Simple and easy to compute.
Useful for small sample sizes.
Works with ordinal, ranked, or skewed data.

Limitations

Less powerful than the t-test when data is normal.
Cannot be used for paired data (use Wilcoxon Signed-Rank Test instead).
Assumes similar distribution shapes in both groups.

Applications

Comparing two treatment groups in clinical studies.
Evaluating patient outcomes between independent samples.
Comparing scores, symptom severity, or rankings.
Pharmacy and biomedical research with non-normal datasets.

Detailed Notes:

For PDF style full-color notes, open the complete study material below: