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Wilcoxon Matched-Pairs Signed-Ranks Test

Overview​

When comparing related measurements from the same participants — pre-test vs. post-test, two matched groups, or repeated measures — and the data violates the assumptions of the dependent-samples t-test, the Wilcoxon Matched-Pairs Signed-Ranks (MPSR) test is the appropriate non-parametric alternative. Developed by Wilcoxon (1945, 1949), this test takes into account both the direction and magnitude of differences between pairs, making it more powerful than the simpler Sign Test.

This Unit Covers
  1. Definition and logic of the Wilcoxon Matched-Pairs Signed-Ranks test
  2. Assumptions and background
  3. Step-by-step procedure for small samples (n ≤ 25)
  4. Large-sample Z approximation
  5. SPSS procedure
  6. Comparison of Mann-Whitney U, Wilcoxon, and t-tests

2.8 The Wilcoxon Matched-Pairs Signed-Ranks Test: Core Logic​

The Wilcoxon MPSR test works on difference scores between matched pairs, then ranks those differences by absolute value. It examines whether the population of differences has a median of zero.

Step-by-Step Logic​

  1. Compute difference scores: For each pair, di=Scorecondition1−Scorecondition2d_i = \text{Score}_{condition1} - \text{Score}_{condition2}
  2. Rank differences by absolute value (ignoring sign), giving rank 1 to the smallest absolute difference
  3. Restore signs: Assign original + or − sign to each rank
  4. Sum positive ranks: T+T^+
  5. Sum negative ranks: T−T^-
  6. Test statistic T = the smaller of T+T^+ and T−T^-
  7. Compare T to critical value from table (significant if T ≤ table value)
Why Differences of Zero Are Excluded

Pairs with zero difference (no change) cannot be ranked because they provide no directional information. These pairs are excluded from analysis, reducing the effective N.

What the Test Assesses​

The null hypothesis:

H₀: In the underlying population, the median difference score equals zero.

If H₀ is rejected, it indicates the two conditions likely represent populations with different median values — i.e., the intervention or condition had a real effect.

2.9 Assumptions and Background​

The Wilcoxon MPSR test was developed as a more sensitive alternative to the Sign Test — unlike the Sign Test, it uses the magnitude of differences, not just their direction.

Formal Assumptions​

  1. Random sampling: The n subjects are randomly selected from the population
  2. Interval/ratio data: The original scores should be at interval or ratio level (the differences are then ranked)
  3. Symmetric difference distribution: The distribution of difference scores in the population is symmetric about the median
Key Distinction from Mann-Whitney

The Mann-Whitney U test assumes independence between the two samples. The Wilcoxon MPSR test applies when samples are dependent (same subjects tested twice, or matched pairs). These tests are not interchangeable!

Categorisation Debate​

Most sources categorise the Wilcoxon MPSR test as an ordinal-level test because it involves a ranking step. However, some sources consider it an interval-level test since it ranks difference scores (which carry magnitude information), not the original data directly.

Design Contexts​

The Wilcoxon MPSR test is applicable to:

Design TypeExample
Pre-test / Post-test (same subjects)Anxiety scores before and after CBT intervention
Matched pairs (different subjects)Twins randomly assigned to different treatments
Counterbalanced repeated measuresSame participants completing two different tasks

2.10 Step-by-Step: Wilcoxon Test for Small Samples (n ≤ 25)​

Use when: n ≤ 25 pairs AND:

  • Differences between conditions can only be ranked in size, OR
  • Data is obviously non-normal, OR
  • Obvious variance differences between conditions

Procedure​

STEP 1: Compute the difference between each pair: d=Score1−Score2d = \text{Score}_1 - \text{Score}_2 (retain sign)

STEP 2: Rank the absolute values of differences from smallest to largest. If differences are tied, assign the mean of the tied ranks.

STEP 3: Calculate T = sum of ranks for differences with the less frequent sign

STEP 4: Look up T in Table J. Significant if observed T ≤ table value.

STEP 5: Interpret the result in terms of the research question.

Worked Example: Reaction Time and Alcohol​

Eight pairs of twins were tested on complex reaction time. One member of each pair drank 3 double whiskies; the other was sober.

PairSoberWhiskyDifference (Sober − Whisky)Absolute DiffRank
1310300−10101
2340320−20202
3290360+70705
4270320+50504
5370540+1701706
6330360+30303
7320680+3603607
83201120+8008008

STEP 3: Less frequent sign = negative (2 negative differences)

T=Rank1+Rank2=1+2=3T = \text{Rank}_1 + \text{Rank}_2 = 1 + 2 = 3

STEP 4: From Table J, for N = 8: critical value = 4.

Since T = 3 ≤ 4 (table value), the result is significant.

STEP 5: Complex reaction time scores are significantly higher after consuming alcohol than when sober. The Wilcoxon MPSR test confirms the alcohol-induced impairment in reaction time.

Handling Ties in Ranks

When two or more differences have the same absolute value, assign each the mean of the ranks they would have occupied. For example, if two values would occupy ranks 3 and 4, each receives rank 3.5.

2.11 Step-by-Step: Wilcoxon Test for Large Samples (n > 25)​

For larger samples, T is approximately normally distributed:

Z=T−N(N+1)4N(N+1)(2N+1)24Z = \frac{T - \frac{N(N+1)}{4}}{\sqrt{\frac{N(N+1)(2N+1)}{24}}}

Significance Criteria​

Test Typeα = .05 threshold
Two-tailed|Z| > 1.96
One-tailed|Z| > 1.64

As with Mann-Whitney, for large samples the Wilcoxon test is still valid but often unnecessary — large samples meet parametric assumptions well enough that the t-test is preferred for its greater power.

2.12 Computing Wilcoxon Signed-Ranks in SPSS​

Analyse → Non-Parametric Tests → Legacy Dialogs → 2 Related Samples

Detailed Steps​

  1. Analyse → Non-Parametric Tests → 2 Related Samples
  2. Click on both variables that form the pair → click the arrow to move them into Test Pair(s) List
  3. Under Test Type, select Wilcoxon
  4. For exact probabilities: click Exact → Exact → Continue
  5. Click OK

Interpreting SPSS Output​

SPSS provides:

  • Negative Ranks: n pairs where condition 1 < condition 2 (and their mean rank, sum of ranks)
  • Positive Ranks: n pairs where condition 1 > condition 2 (and their mean rank, sum of ranks)
  • Ties: n pairs with equal scores
  • Z statistic and Asymptotic Significance (2-tailed) p-value

Self-Assessment Questions​

Level 1 — Recall​

  1. What is the key difference between the Wilcoxon MPSR test and the Sign Test?

  2. Why are pairs with a difference score of zero excluded from the Wilcoxon analysis?

  3. In the Wilcoxon test, if you obtain T = 6 and the critical table value for N = 10 at α = .05 is 8, what is your conclusion?

Level 2 — Application​

  1. A therapist measures depression scores (BDI) for 12 clients before and after a 10-week CBT programme. Scores are continuous but the distribution of differences is skewed. Which test is appropriate? Why not the paired t-test?

  2. Compute the Wilcoxon T statistic for the following 5 pairs: differences = +4, −1, +7, +2, −3 (absolute values ranked: 1=1, 2=2, 3=3, 4=4, 5=7). Which sign is less frequent? What is T?

Level 3 — Critical Thinking​

  1. A researcher has matched pairs (n = 15) and is debating between the Wilcoxon MPSR test and the dependent-samples t-test. The data are roughly normally distributed. Which would you recommend, and why? What is "lost" by choosing the non-parametric option?

  2. The Wilcoxon MPSR test is sometimes described as an interval-level test rather than ordinal. Construct an argument for each position. Does this classification affect when you would use the test?

🧠 Memory Aid​

"DRIED Ranks" — Wilcoxon MPSR procedure:

  • Differences: compute d=Score1−Score2d = \text{Score}_1 - \text{Score}_2
  • Remove zeros (exclude ties from analysis)
  • Ignore signs; rank absolute differences
  • Endorse signs back onto ranks
  • Decide: T = sum of less-frequent-sign ranks; compare to table

Tip: The smaller T is, the more significant the result. T approaching zero means one direction of difference is nearly absent — strong evidence of an effect.

📚 External Resources​

Foundational Reading​

Tutorials and Guides​

Video Resources​

Research Articles​

  • 📄 Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics Bulletin, 1(6), 80–83. (Original paper)
  • 📄 Conover, W. J., & Iman, R. L. (1981). Rank transformations as a bridge between parametric and nonparametric statistics. The American Statistician, 35(3), 124–129.
  • 📄 Rosner, B., Glynn, R. J., & Lee, M. T. (2006). The Wilcoxon signed rank test for paired comparisons of clustered summary measures. Biometrics, 62(1), 185–192.

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