- Wilcoxon signed-rank test
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 forFrank 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.**etup**Suppose we collect 2"n" 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**# 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".**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 anindicator function . The Wilcoxon signed ranked statistic "W"_{+}is defined as:$W\_+\; =\; sum\_\{i=1\}^n\; varphi\_i\; R\_i.,!$

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 value s 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 amean rank. Thesum s 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 statisticalprobability 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".

**ee also***

Mann-Whitney-Wilcoxon test (the variant for two independent samples)**References***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.**External links*** [

*http://comp9.psych.cornell.edu/Darlington/wilcoxon/wilcox0.htm Description of how to calculate "p" for the Wilcoxon signed-ranks test*]

* [*http://faculty.vassar.edu/lowry/ch12a.html Example of using the Wilcoxon signed-rank test*]

* [*http://faculty.vassar.edu/lowry/wilcoxon.html An online version of the test*]**Implementations*** [

*http://www.alglib.net/statistics/hypothesistesting/wilcoxonsignedrank.php 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.

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Wilcoxon signed rank test**— signed rank t … Medical dictionary**Wilcoxon signed-rank Test**— Der Wilcoxon Vorzeichen Rang Test ist ein statistischer Test für die Häufigkeitsverteilung gepaarter Stichproben. Im Anwendungsbereich ergänzt er den Vorzeichentest, da er nicht nur die Richtung der Differenzen, sondern auch die Stärke der… … Deutsch Wikipedia**Wilcoxon rank sum test signed rank test**— Wil·cox·on rank sum test, signed rank test (wil kokґsən) [Frank Wilcoxon, American chemist and statistician, 1892â€“1962] see rank sum test and signed rank test, under test … Medical dictionary**signed rank test**— a nonparametric statistical test for ordinal data, comparing two populations of data by examining the differences between matched pairs in the two populations. It is based on the signed rank statistic, calculated by arranging all samples in order … Medical dictionary**Rank test**— In statistics, a rank test is any test involving ranks. Examples include: *Wilcoxon signed rank test *Kruskal Wallis one way analysis of variance **Mann Whitney U (special case) *Page s trend test *Friedman test *Rank products … Wikipedia**Wilcoxon**— is a surname, and may refer to: * Henry Wilcoxon, an actor * Frank Wilcoxon, chemist and statistician, inventor of two non parametric tests for statistical significance: ** The Wilcoxon signed rank test ** The Wilcoxon rank sum test (also known… … Wikipedia**Wilcoxon Test**— The Wilcoxon test, which refers to either the Rank Sum test or the Signed Rank test, is a nonparametric test that compares two paired groups. The test essentially calculates the difference between each set of pairs and analyzes these differences … Investment dictionary**Student's t-test**— A t test is any statistical hypothesis test in which the test statistic follows a Student s t distribution if the null hypothesis is supported. It is most commonly applied when the test statistic would follow a normal distribution if the value of … Wikipedia**Frank Wilcoxon**— (1892–1965) was a chemist and statistician, known for the development of statistical tests. Frank Wilcoxon was born to American parents on 2 September 1892 in County Cork, IrelandBradley, R.A. (1966) Obituary: Frank Wilcoxon. Biometrics 22(1):… … Wikipedia**Nicht-parametrischer Test**— Der Zweig der Statistik, der als parameterfreie Statistik bekannt ist, beschäftigt sich mit parameterfreien statistischen Modellen und parameterfreien statistischen Tests. Andere gebräuchliche Bezeichnungen sind nicht parametrische Statistik oder … Deutsch Wikipedia