Wilcoxon Signed Ranks Test

The Wilcoxon signed ranks test may be used as a one-sample test of location or as a test of difference in location between two dependent samples. The underlying assumptions are that the distribution is continuous and symmetric.

The Wilcoxon signed ranks procedure is primarily used as a test for location, such as the median. In the one-sample case, the one-sided test of the null hypothesis is H0: φ = median, which is tested against the alternative Ha: φ> median or Ha: φ< median. The alternative hypothesis for the two-sided test is Ha: φ≠ median. In the two-sample case, the Wilcoxon signed ranks test is applied to matched or paired data, such as an examination of pretest versus posttest scores. Although the Wilcoxon test may also be used as a test of symmetry, there are often more powerful procedures.

The Wilcoxon test is nonparametric. This means it preserves the Type I error rate (i.e., the false positive rate) to nominal alpha regardless of the population shape. This is a fundamental advantage over its parametric counterparts, the one-mean Student t test and the two-dependent-samples Student t test, which rely on the normality distribution assumption.

Theoretical power comparisons indicate that the asymptotic relative efficiency of the Wilcoxon signed ranks test with the Student t test is .955 for the normal distribution, but it can be as high as ∞ for nonnormal distributions. Monte Carlo simulations indicate that the spectacular power gains achieved by the independent-samples version of this test (i.e., the Wilcoxon rank-sum test; see the entry on Mann-Whitney U Test) over the Student t test are not realized although the Wilcoxon signed ranks test is often at least slightly more powerful than the t test for departures from population normality. The explanation is that in the paired-samples case, tests are conducted on the distribution of the difference scores. Subtracting a pretest score from the posttest score, for example, is a mildly normalizing procedure, producing a distribution that is less deviant than the parent population and more similar to the normal distribution.

To conduct the two-sample matched-pairs Wilcoxon signed ranks test, compute the difference (d) between each pair of scores, ignoring the signs. (This is achieved by taking the absolute value |d|.) Next, denote the ranks of the positive differences with a + sign and the negative differences with a − sign. Finally, sum the ranks associated with the lesser-occurring sign.