Wilcoxon signed-rank test
The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used to compare two related samples, matched samples, or repeated measurements on a single sample to assess whether their population mean ranks differ (i.e., it is a paired difference test). It can be used as an alternative to the paired Student's t-test (also known as "t-test for matched pairs" or "t-test for dependent samples") when the distribution of the difference between two samples' means cannot be assumed to be normally distributed. A Wilcoxon signed-rank test is a nonparametric test that can be used to determine whether two dependent samples were selected from populations having the same distribution.
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). The test was popularized by Sidney Siegel (1956) in his influential textbook on non-parametric statistics. Siegel used the symbol T for a value related to, but not the same as, . In consequence, the test is sometimes referred to as the Wilcoxon T test, and the test statistic is reported as a value of T.
- Data are paired and come from the same population.
- Each pair is chosen randomly and independently.
- The data are measured on at least an interval scale when, as is usual, within-pair differences are calculated to perform the test (though it does suffice that within-pair comparisons are on an ordinal scale).
Let be the sample size, i.e., the number of pairs. Thus, there are a total of 2N data points. For pairs , let and denote the measurements.
- H0: difference between the pairs follows a symmetric distribution around zero
- H1: difference between the pairs does not follow a symmetric distribution around zero.
- For , calculate and , where is the sign function.
- Exclude pairs with . Let be the reduced sample size.
- Order the remaining pairs from smallest absolute difference to largest absolute difference, .
- Rank the pairs, starting with the pair with the smallest non-zero absolute difference as 1. Ties receive a rank equal to the average of the ranks they span. Let denote the rank.
- Calculate the test statistic
- , the sum of the signed ranks.
- Under null hypothesis, follows a specific distribution with no simple expression. This distribution has an expected value of 0 and a variance of .
- can be compared to a critical value from a reference table.
- The two-sided test consists in rejecting if .
- As increases, the sampling distribution of converges to a normal distribution. Thus,
- For the exact distribution needs to be used.
|order by absolute difference||
- that the median of pairwise differences is different than zero.
- The -value for this result is
Historical T statistic
In historical sources a different statistic, denoted by Siegel as the T statistic, was used. The T statistic is the smaller of the two sums of ranks of given sign; in the example, therefore, T would equal 3+4+5+6=18. Low values of T are required for significance. T is easier to calculate by hand than W and the test is equivalent to the two-sided test described above; however, the distribution of the statistic under has to be adjusted.
- that the two medians are the same.
Note: Critical T values () by values of can be found in appendices of statistics textbooks, for example in Table B-3 of Nonparametric Statistics: A Step-by-Step Approach, 2nd Edition by Dale I. Foreman and Gregory W. Corder (https://www.oreilly.com/library/view/nonparametric-statistics-a/9781118840429/bapp02.xhtml).
Alternatively, if n is sufficiently large, the distribution of T under can be approximated by a normal distribution with mean and variance .
As demonstrated in the example, when the difference between the groups is zero, the observations are discarded. This is of particular concern if the samples are taken from a discrete distribution. In these scenarios the modification to the Wilcoxon test by Pratt 1959, provides an alternative which incorporates the zero differences. This modification is more robust for data on an ordinal scale.
If the test statistic W is reported, the rank correlation r is equal to the test statistic W divided by the total rank sum S, or r = W/S.  Using the above example, the test statistic is W = 9. The sample size of 9 has a total rank sum of S = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) = 45. Hence, the rank correlation is 9/45, so r = 0.20.
If the test statistic T is reported, an equivalent way to compute the rank correlation is with the difference in proportion between the two rank sums, which is the Kerby (2014) simple difference formula. To continue with the current example, the sample size is 9, so the total rank sum is 45. T is the smaller of the two rank sums, so T is 3 + 4 + 5 + 6 = 18. From this information alone, the remaining rank sum can be computed, because it is the total sum S minus T, or in this case 45 − 18 = 27. Next, the two rank-sum proportions are 27/45 = 60% and 18/45 = 40%. Finally, the rank correlation is the difference between the two proportions (.60 minus .40), hence r = .20.
- R includes an implementation of the test as
wilcox.test(x,y, paired=TRUE), where x and y are vectors of equal length.
- ALGLIB includes implementation of the Wilcoxon signed-rank test in C++, C#, Delphi, Visual Basic, etc.
- GNU Octave implements various one-tailed and two-tailed versions of the test in the
- SciPy includes an implementation of the Wilcoxon signed-rank test in Python
- Accord.NET includes an implementation of the Wilcoxon signed-rank test in C# for .NET applications
- MATLAB implements this test using "Wilcoxon rank sum test" as [p,h] = signrank(x,y) also returns a logical value indicating the test decision. The result h = 1 indicates a rejection of the null hypothesis, and h = 0 indicates a failure to reject the null hypothesis at the 5% significance level
- Julia HypothesisTests package includes the Wilcoxon signed-rank test as "value(SignedRankTest(x, y))"
- Mann–Whitney–Wilcoxon test (the variant for two independent samples)
- Sign test (Like Wilcoxon test, but without the assumption of symmetric distribution of the differences around the median, and without using the magnitude of the difference)
- "Paired t–test - Handbook of Biological Statistics". www.biostathandbook.com. Retrieved 2019-11-18.
- Wilcoxon, Frank (Dec 1945). "Individual comparisons by ranking methods" (PDF). Biometrics Bulletin. 1 (6): 80–83. doi:10.2307/3001968. hdl:10338.dmlcz/135688. JSTOR 3001968.
- Siegel, Sidney (1956). Non-parametric statistics for the behavioral sciences. New York: McGraw-Hill. pp. 75–83. ISBN 9780070573482.
- Lowry, Richard. "Concepts & Applications of Inferential Statistics". Retrieved 5 November 2018.
- Lowry, Richard. "Concepts & Applications of Inferential Statistics". Retrieved 2020-12-17.
- Pratt, J (1959). "Remarks on zeros and ties in the Wilcoxon signed rank procedures". Journal of the American Statistical Association. 54 (287): 655–667. doi:10.1080/01621459.1959.10501526.
- Derrick, B; White, P (2017). "Comparing Two Samples from an Individual Likert Question". International Journal of Mathematics and Statistics. 18 (3): 1–13.
- Kerby, Dave S. (2014), "The simple difference formula: An approach to teaching nonparametric correlation.", Comprehensive Psychology, 3: 11.IT.3.1, doi:10.2466/11.IT.3.1
- Dalgaard, Peter (2008). Introductory Statistics with R. Springer Science & Business Media. pp. 99–100. ISBN 978-0-387-79053-4.
- Wilcoxon Signed-Rank Test in R
- Example of using the Wilcoxon signed-rank test
- An online version of the test
- A table of critical values for the Wilcoxon signed-rank test
- Brief guide by experimental psychologist Karl L. Weunsch - Nonparametric effect size estimators (Copyright 2015 by Karl L. Weunsch)
- Kerby, D. S. (2014). The simple difference formula: An approach to teaching nonparametric correlation. Comprehensive Psychology, volume 3, article 1. doi:10.2466/11.IT.3.1. link to article