Wilcoxon signed-rank test

WikiDoc Resources for Wilcoxon signed-rank test
Articles
Most recent articles on Wilcoxon signed-rank test Most cited articles on Wilcoxon signed-rank test Review articles on Wilcoxon signed-rank test Articles on Wilcoxon signed-rank test in N Eng J Med, Lancet, BMJ
Media
Powerpoint slides on Wilcoxon signed-rank test Images of Wilcoxon signed-rank test Photos of Wilcoxon signed-rank test Podcasts & MP3s on Wilcoxon signed-rank test Videos on Wilcoxon signed-rank test
Evidence Based Medicine
Cochrane Collaboration on Wilcoxon signed-rank test Bandolier on Wilcoxon signed-rank test TRIP on Wilcoxon signed-rank test
Clinical Trials
Ongoing Trials on Wilcoxon signed-rank test at Clinical Trials.gov Trial results on Wilcoxon signed-rank test Clinical Trials on Wilcoxon signed-rank test at Google
Guidelines / Policies / Govt
US National Guidelines Clearinghouse on Wilcoxon signed-rank test NICE Guidance on Wilcoxon signed-rank test NHS PRODIGY Guidance FDA on Wilcoxon signed-rank test CDC on Wilcoxon signed-rank test
Books
Books on Wilcoxon signed-rank test
News
Wilcoxon signed-rank test in the news Be alerted to news on Wilcoxon signed-rank test News trends on Wilcoxon signed-rank test
Commentary
Blogs on Wilcoxon signed-rank test
Definitions
Definitions of Wilcoxon signed-rank test
Patient Resources / Community
Patient resources on Wilcoxon signed-rank test Discussion groups on Wilcoxon signed-rank test Patient Handouts on Wilcoxon signed-rank test Directions to Hospitals Treating Wilcoxon signed-rank test Risk calculators and risk factors for Wilcoxon signed-rank test
Healthcare Provider Resources
Symptoms of Wilcoxon signed-rank test Causes & Risk Factors for Wilcoxon signed-rank test Diagnostic studies for Wilcoxon signed-rank test Treatment of Wilcoxon signed-rank test
Continuing Medical Education (CME)
CME Programs on Wilcoxon signed-rank test
International
Wilcoxon signed-rank test en Espanol Wilcoxon signed-rank test en Francais
Business
Wilcoxon signed-rank test in the Marketplace Patents on Wilcoxon signed-rank test
Experimental / Informatics
List of terms related to Wilcoxon signed-rank test

Please Take Over This Page and Apply to be Editor-In-Chief for this topic: There can be one or more than one Editor-In-Chief. You may also apply to be an Associate Editor-In-Chief of one of the subtopics below. Please mail us [1] to indicate your interest in serving either as an Editor-In-Chief of the entire topic or as an Associate Editor-In-Chief for a subtopic. Please be sure to attach your CV and or biographical sketch.

Overview

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.

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 <math>x_i</math> and second observation be <math>y_i</math>.

Assumptions

1. Let Z_i = Y_i - X_i for i = 1, ... , n. The differences Z_i are assumed to be independent.
2. 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 H₀: θ = 0. The Wilcoxon signed rank statistic W₊ is computed by ordering the absolute values |Z₁|, ..., |Z_n|, the rank of each ordered |Z_i| is given a rank of R_i. Denote φ_i = I(Z_i>0) where I(.) is an indicator function. The Wilcoxon signed ranked statistic W₊ is defined as,

<math>W^+ = \sum_{i=1}^n \phi_i R_i\,\!</math>

It is often used to test difference 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, n, used increases, the distribution of all possible ranks S tends towards the normal distribution, so for an n of less than 10 this distribution is often used to calculate p.

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.

See also

• Mann-Whitney-Wilcoxon test (the two-sample variant)

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

• 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

Implementations

• ALGLIB includes implementation of the Wilcoxon signed-rank test in C++, C#, Delphi, Visual Basic, etc.