The Wilcoxon Signed Rank Test is a widely used non-parametric statistical test designed to compare two related or paired samples. It is an alternative to the paired t-test when the data is not normally distributed, when sample size is small, or when the data is ordinal or ranked. Unlike the Sign Test—which considers only the direction of change—the Wilcoxon Signed Rank Test incorporates both direction and magnitude of differences, making it more powerful.
When to Use the Wilcoxon Signed Rank Test?
- When comparing paired or matched observations.
- When differences between pairs are not normally distributed.
- When data is ordinal, ranked, or non-normally distributed interval data.
- When sample size is small.
- When evaluating improvement or change in the same subjects.
Assumptions
- Data consists of paired observations.
- Differences between pairs can be ranked.
- Distribution of differences is symmetric (not necessarily normal).
- No ties in differences; ties should be handled appropriately.
Principle of the Test
The test evaluates whether the median difference between paired observations is zero. It ranks the absolute differences, assigns signs, and uses the sum of ranks to determine whether the observed change is statistically significant.
Steps in Performing the Wilcoxon Signed Rank Test
- List all paired observations (Before vs. After).
- Calculate the difference (D = After − Before).
- Ignore zero differences (ties).
- Take the absolute value of each difference.
- Rank the absolute differences (1 = smallest difference).
- Assign positive or negative signs based on original difference.
- Sum all positive ranks (T+) and negative ranks (T−).
- Use the smaller of T+ or T− as the test statistic (T).
- Compare the test statistic with the critical value from Wilcoxon tables.
Test Statistic
The Wilcoxon test statistic is:
T = min(T+, T−)
For large samples (n > 25), a normal approximation may be used:
Z = (T − μ) / σ
Hypotheses
- H₀: Median difference = 0 (no change).
- H₁: Median difference ≠ 0 (significant change).
Example (Simple Illustration)
A researcher measures pain scores before and after treatment in the same patients. The differences in scores are ranked, and the signed ranks are used to compute the test statistic. If the calculated T is less than the critical T from the table, the treatment effect is significant.
Advantages of the Wilcoxon Signed Rank Test
- More powerful than the Sign Test.
- Does not require normal distribution.
- Suitable for ordinal and non-normal interval data.
- Useful for small samples.
- Accounts for magnitude and direction of change.
Limitations
- Not suitable for nominal data.
- Less powerful than parametric tests when data is normally distributed.
- Ties and zero differences complicate calculations.
Applications
- Before-and-after treatment comparisons.
- Matched pair studies in clinical trials.
- Analyzing patient satisfaction or pain scales.
- Non-normal physiological measurements.
Detailed Notes:
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