Wilcoxon signed-rank test

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 $$x_i$$ and second observation be $$y_i$$.

Assumptions

 * 1) Let Zi = Yi - Xi for i = 1, ..., n. The differences Zi are assumed to be independent.
 * 2) Each Zi comes from a continuous population (they must be identical) and is symmetric about a common median &theta;.

Test Procedure
The null hypothesis tested is H0 : &theta; = 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 &phi;i = I(Zi>0) where I''(.) is an indicator function. The Wilcoxon signed ranked statistic W+ is defined as,

$$W^+ = \sum_{i=1}^n \phi_i R_i\,\!$$

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.

Implementations

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