Wilcoxon and Mann-Whitney

Non-Parametric Tests of Difference: Wilcoxon and Mann-Whitney

When data are at ordinal level, substantially non-normal, or contain outliers, the non-parametric tests of difference — the Wilcoxon signed-rank test and the Mann-Whitney U test — are appropriate alternatives to the parametric t-tests. Both work by converting scores to ranks before analysis, making no assumptions about the shape of the underlying distribution.

Wilcoxon Signed-Rank Test (Related Design)

The Wilcoxon signed-rank test is used when: the research question concerns a difference; the design is related (repeated measures or matched pairs); and data are at ordinal level or above but parametric assumptions are not met.

Procedure:

1. Calculate the difference (d) between each pair of scores.
2. Exclude pairs where d = 0 (ties); N = number of remaining pairs.
3. Rank the absolute values of d from smallest to largest (ignoring sign; tied ranks receive average rank).
4. Assign the original sign (+ or −) to each rank.
5. Sum the positive ranks → W+; sum the negative ranks → W−.
6. The test statistic T = the smaller of W+ and W−.
7. Significant if T ≤ critical value from the Wilcoxon table for given N and α.

Mann-Whitney U Test (Unrelated Design)

The Mann-Whitney U test is used when: the research question concerns a difference; the design is unrelated (independent groups); and data are at ordinal level or above but parametric assumptions are not met.

Procedure:

1. Rank all scores from both groups combined (lowest = rank 1), keeping track of which group each score belongs to. Tied ranks receive average rank.
2. Sum the ranks for Group 1 → R₁; sum the ranks for Group 2 → R₂.
3. Calculate: U₁ = n₁n₂ + [n₁(n₁+1)/2] − R₁, and U₂ = n₁n₂ + [n₂(n₂+1)/2] − R₂.
4. The test statistic U = the smaller of U₁ and U₂.
5. Significant if U ≤ critical value from the Mann-Whitney table for given n₁, n₂ and α.

A useful check: U₁ + U₂ = n₁ × n₂.

Comparison