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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).## Setup

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 $x\_i$ and second observation be $y\_i$.
## Assumptions

## Test procedure

The null hypothesis tested is H_{0}: θ = 0. The Wilcoxon signed rank statistic W_{+} is computed by ordering the absolute values |Z_{1}|, ..., |Z_{n}|, the rank of each ordered |Z_{i}| is given a rank of R_{i}. Denote $varphi\_i\; =\; I(Z\_i\; >\; 0),\; ,$ where I(.) is an indicator function. The Wilcoxon signed ranked statistic W_{+} is defined as## See also

## References

## External links

## Implementations

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.

- Let Z
_{i}= Y_{i}- X_{i}for i = 1, ... , n. The differences Z_{i}are assumed to be independent. - Each Z
_{i}comes from a continuous population (they must be identical) and is symmetric about a common median θ.

- $W\_+\; =\; sum\_\{i=1\}^n\; varphi\_i\; R\_i.,!$

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 variant for two independent samples)

- 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.
- The free statistical software R includes an implementation of the test as
`wilcox.test(x,y, paired=TRUE)`

, where x and y are vectors of equal length.

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Last updated on Thursday October 02, 2008 at 11:55:37 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday October 02, 2008 at 11:55:37 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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