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The Wilcoxon signed-rank test is a non-parametric alternative to the paired Student's t-test for the case of two related samples or repeated measurements on a single sample. The test is named for Frank Wilcoxon (1892–1965) who, in a single paper, proposed both it and the rank-sum test for two independent samples (Wilcoxon, 1945).
Like the t-test, the Wilcoxon test involves comparisons of differences between measurements, so it requires that the data are measured at an interval level of measurement. However it does not require assumptions about the form of the distribution of the measurements. It should therefore be used whenever the distributional assumptions that underlie the t-test cannot be satisfied.
Suppose we collect 2n observations, two observations of each of the n subjects. Let i denote the particular subject that is being referred to and the first observation measured on subject i be denoted by and second observation be .
- Let Zi = Yi - Xi for i = 1, ... , n. The differences Zi are assumed to be independent.
- Each Zi comes from a continuous population (they must be identical) and is symmetric about a common median θ.
The null hypothesis tested is H0: θ = 0. The Wilcoxon signed rank statistic W+ is computed by ordering the absolute values |Z1|, ..., |Zn|, the rank of each ordered |Zi| is given a rank of Ri. Denote where I(.) is an indicator function. The Wilcoxon signed ranked statistic W+ is defined as
It is often used to test the difference between scores of data collected before and after an experimental manipulation, in which case the central point would be expected to be zero. Scores exactly equal to the central point are excluded and the absolute values of the deviations from the central point of the remaining scores is ranked such that the smallest deviation has a rank of 1. Tied scores are assigned a mean rank. The sums for the ranks of scores with positive and negative deviations from the central point are then calculated separately. A value S is defined as the smaller of these two rank sums. S is then compared to a table of all possible distributions of ranks to calculate p, the statistical probability of attaining S from a population of scores that is symmetrically distributed around the central point.
As the number of scores used, n, increases, the distribution of all possible ranks S tends towards the normal distribution. So although for n <= 20, exact probabilities would usually be calculated, for n > 20, the Normal approximation is used. The recommended cutoff varies from textbook to textbook - here we use 20 although some put it lower (10) or higher (25).
The Wilcoxon test was popularised by Siegel (1956) in his influential text book on non-parametric statistics. Siegel used the symbol T for the value defined here as S. In consequence, the test is sometimes referred to as the Wilcoxon T test, and the test statistic is reported as a value of T.
- Mann-Whitney-Wilcoxon test (the two-sample variant)
- Siegel, S. (1956). Non-parametric statistics for the behavioral sciences. New York: McGraw-Hill.
- Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics, 1, 80-83.
- Description of how to calculate p for the Wilcoxon signed-ranks test
- Example of using the Wilcoxon signed-rank test
- An online version of the test
- ALGLIB includes implementation of the Wilcoxon signed-rank test in C++, C#, Delphi, Visual Basic, etc.
- Blair, R. C., & Higgins, J. J. (1980). A comparison of the power of Wilcoxon's rank-sum statistic to that of student's t statistic under various nonnormal distributions: Journal of Educational Statistics Vol 5(4) Win 1980, 309-335.
- Blair, R. C., & Higgins, J. J. (1980). The power of t and Wilcoxon statistics: A comparison: Evaluation Review Vol 4(5) Oct 1980, 645-656.
- Blair, R. C., & Higgins, J. J. (1985). A comparison of the power of the paired samples rank transform statistic to that of Wilcoxon's signed ranks statistic: Journal of Educational Statistics Vol 10(4) Win 1985, 368-383.
- Blair, R. C., & Higgins, J. J. (1985). Comparison of the power of the paired samples t test to that of Wilcoxon's signed-ranks test under various population shapes: Psychological Bulletin Vol 97(1) Jan 1985, 119-128.
- Blair, R. C., & Higgins, J. L. (1981). A note on the asymptotic relative efficiency of the Wilcoxon rank-sum test relative to the independent means t test under mixtures of two normal distributions: British Journal of Mathematical and Statistical Psychology Vol 34(1) May 1981, 124-128.
- Cronholm, J. N., & Revusky, S. H. (1965). A sensitive rank test for comparing the effects of two treatments on a single group: Psychometrika 30(4) 1965, 459-467.
- Curtis, D. A., & Marascuilo, L. A. (1992). Point estimates and confidence intervals for the parameters of the two-sample and matched-pair combined tests for ranks and normal scores: Journal of Experimental Education Vol 60(3) Spr 1992, 243-269.
- Guiard, V., & Rasch, D. (2004). The robustness of two sample tests for means--A reply on von Eye's comment: Psychology Science Vol 46(4) 2004, 549-554.
- Kornbrot, D. E. (1990). The rank difference test: A new and meaningful alternative to the Wilcoxon signed ranks test for ordinal data: British Journal of Mathematical and Statistical Psychology Vol 43(2) Nov 1990, 241-264.
- Mahfoud, Z. R., & Randles, R. H. (2005). Practical tests for randomized complete block designs: Journal of Multivariate Analysis Vol 96(1) Sep 2005, 73-92.
- Mielke, P. W., & Berry, K. J. (1976). An extended class of matched pairs tests based on powers of ranks: Psychometrika Vol 41(1) Mar 1976, 89-100.
- Nanna, M. J., & Sawilowsky, S. S. (1998). Analysis of Likert scale data in disability and medical rehabilitation research: Psychological Methods Vol 3(1) Mar 1998, 55-67.
- Rasmussen, J. L. (1985). The power of student's t and Wilcoxon W statistics: A comparison: Evaluation Review Vol 9(4) Aug 1985, 505-510.
- Van den Brink, W. P., & Van den Brink, S. G. (1989). A comparison of the power of the t test, Wilcoxon's test, and the approximate permutation test for the two-sample location problem: British Journal of Mathematical and Statistical Psychology Vol 42(2) Nov 1989, 183-189.
- Von Eye, A. (2004). Robustness is parameter-specific a comment on Rasch and Guiard's robustness study: Psychology Science Vol 46(4) 2004, 544-548.
- Wike, E. L., & Church, J. D. (1977). Further comments on nonparametric multiple-comparison tests: Perceptual and Motor Skills Vol 45(3, Pt 1) Dec 1977, 917-918.
- Wike, E. L., & Church, J. D. (1978). A Monte Carlo investigation of four nonparametric multiple-comparison tests for k independent groups: Bulletin of the Psychonomic Society Vol 11(1) Jan 1978, 25-28.
- Zimmerman, D. W. (1996). An efficient alternative to the Wilcoxon signed-ranks test for paired nonnormal data: Journal of General Psychology Vol 123(1) Jan 1996, 29-40.
- Zimmerman, D. W., & Zumbo, B. D. (1990). The relative power of the Wilcoxon-Mann-Whitney test and Student t test under simple bounded transformations: Journal of General Psychology Vol 117(4) Oct 1990, 425-436.
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