Wilcoxon signed-rank test

The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used when comparing two related samples or repeated measurements on a single sample to assess whether their population mean ranks differ (i.e. it's a paired difference test).

It can be used as an alternative to the paired Student's t-test when the population cannot be assumed to be normally distributed or the data is on the ordinal scale.

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 Siegel (1956) in his influential text book on non-parametric statistics. Siegel used the symbol T for the value defined below 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.

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. For each i in the observations, x_i and y_i should be paired together.

Assumptions

Let Z_i = X_i – Y_i for i = 1, ... , n .

1. The differences Z_i are assumed to be independent.
2. Each Z_i comes from the same continuous population.
3. The values which X_i and Y_i represent are ordered (at least the ordinal level of measurement), so the comparisons "greater than", "less than", and "equal to" are useful. X and Y need to be on scales that make differences orderable, which can mean that X and Y are interval scaled.
4. If we wish to make an inference about the mean (or about the median) difference, then we assume the distribution of the differences is symmetric. If we only want to test the hypothesis that the probability that the sum of a randomly chosen pair of differences exceeds zero is 0.5 then no distributional assumption is needed.

Test procedure

The null hypothesis tested is H₀: θ = 0.

1. Exclude observations with Z_i = 0. Let m be the reduced sample size. (But see the note on #Excluding zero differences below.)
2. Order the absolute values |Z₁|, ..., |Z_n| in ascending sequence, and let the rank of each non-zero |Z_i| be R_i (the smallest positive |Z_i| gets the rank of 1, and a mean rank is assigned to tied scores).
3. Denote the positive Z_i values with φ_i = I(Z_i > 0), where I(.) is an indicator function: φ_i = 1 for Z_i > 0, otherwise φ_i = 0.
4. The Wilcoxon signed ranked statistic W₊ is defined as
   $W_+ = \sum_{i=1}^n \phi_i R_i.\,\!$
5. Define W₋ similarly by summing ranks of the negative differences Z_i.
6. Calculate S as the smaller of these two rank sums: S = min(W₊, W₋).
7. Find the critical value for the given sample size n (or m?^{[citation needed]}), and the wanted confidence level.
   • For samples of a small size the critical value is obtained from a table (which is calculated by considering 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).
8. Compare S to the critical value, and reject H₀ if S is less than or is equal to the critical value.

Confidence interval for the Wilcoxon signed-rank test

A median confidence interval can be constructed based on Wilcoxon signed-rank test for matched pairs. To create the confidence interval, all possible pairs (X_i,X_j) are used to compute the differences D_i=X_i-X_j; then, compute all of the averages, υ_ij use:

• υ_ij=(D_i+D_j)/2

There would be [n(n-1)/2]+n averages. Then arrange all the averages from smallest to largest, and the median of ordered averages gives a point estimate of the population.

Example

Subject (i)	Xi	Yi	Sign of Xi – Yi	Xi – Yi	Absolute Xi – Yi	Rank of Absolute	Signed Rank
1	125	110	+	15	15	7	7
2	115	122	–	–7	7	3	–3
3	130	125	+	5	5	1.5	1.5
4	140	120	+	20	20	9	9
5	140	140		0	0
6	115	124	–	–9	9	4	–4
7	140	123	+	17	17	8	8
8	125	137	–	–12	12	6	–6
9	140	135	+	5	5	1.5	1.5
10	135	145	–	–10	10	5	–5

1. The sign of X_i – Y_i is denoted in the Sign column by either (+) or (–). If X_i and Y_i are equal, then the value is thrown out.
2. The values of X_i – Y_i are given in the next two columns.
3. The last two columns are the ranks. The absolute rank column has no signs, and the signed rank column gives the ranks along with their signs.
4. The data is ranked from the smallest value to the largest value. In the case of a tie, ranks are added together and divided by the number of ties. For example, in this data, there were two instances of the value 5. The ranks corresponding to 5 are 1 and 2. The sum of these ranks is 3. After dividing by the number of ties, you get a mean rank of 1.5, and this value is assigned to both instances of 5.
5. The test statistic, W₊, is given by the sum of all of the positive values in the Signed Rank column. The test statistic, W_–, is given by the sum of all of the negative values in the Signed Rank column. For this example, W₊ = 27 and W_–=18. The minimum of these is 18.
6. Lastly, this test statistic is analyzed using a table of critical values. If the test statistic is less than or equal to the critical value based on the number of observations n, then the null hypothesis is rejected for the alternative hypothesis. Otherwise, the null hypothesis is not rejected. See table here.

In this case the test statistic is W = 18 and the critical value is 8 for a two-tailed p-value of 0.05. The test statistic must be less than this to be significant at this level, so in this case the null hypothesis is not rejected.

See also

• 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)

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
• A table of critical values for the Wilcoxon signed-rank test

Implementations

• 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.
• GNU Octave implements various one-tailed and two-tailed versions of the test in the wilcoxon_test function.