Wilcoxon signed-rank test

The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used when comparing two related samples, matched samples, or repeated measurements on a single sample to assess whether their population mean ranks differ (i.e. it is a paired difference test). It can be used as an alternative to the paired Student's t-test, t-test for matched pairs, or the t-test for dependent samples when the population cannot be assumed to be normally distributed.^[1]

1 History
2 Assumptions
3 Test procedure
4 Example
5 See also
6 References
7 External links
- 7.1 Implementations

History[edit]

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).^[2] The test was popularized by Siegel (1956)^[3] in his influential text book on non-parametric statistics. Siegel used the symbol T for the value defined below as $W$ . In consequence, the test is sometimes referred to as the Wilcoxon T test, and the test statistic is reported as a value of T. Other names may include the "t-test for matched pairs" or the "t-test for dependent samples".

Assumptions[edit]

Data are paired and come from the same population.
Each pair is chosen randomly and independent.
The data are measured at least on an ordinal scale, but need not be normal.

Test procedure[edit]

Let $N$ be the sample size, the number of pairs. Thus, there are a total of 2N data points. For $i = 1, ..., N$ , let $x_{1,i}$ and $x_{2,i}$ denote the measurements.

H₀: median difference between the pairs is zero

H₁: median difference is not zero.

For $i = 1, ..., N$ , calculate $|x_{2,i} - x_{1,i}|$ and $\sgn(x_{2,i} - x_{1,i})$ , where $\sgn$ is the sign function.
Exclude pairs with $|x_{2,i} - x_{1,i}| = 0$ . Let $N_r$ be the reduced sample size.
Order the remaining $N_r$ pairs from smallest absolute difference to largest absolute difference, $|x_{2,i} - x_{1,i}|$ .
Rank the pairs, starting with the smallest as 1. Ties receive a rank equal to the average of the ranks they span. Let $R_i$ denote the rank.
Calculate the test statistic $W$

$W = |\sum_{i=1}^{N_r} [\sgn(x_{2,i} - x_{1,i}) \cdot R_i]|$ , the absolute value of the sum of the signed ranks.
As $N_r$ increases, the sampling distribution of $W$ converges to a normal distribution. Thus,

For $N_r \ge 10$ , a z-score can be calculated as $z = \frac{W - 0.5}{\sigma_W}, \sigma_W = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{6}}$ .

If $z > z_{critical}$ then reject $H_0$

For $N_r < 10$ , $W$ is compared to a critical value from a reference table.^[1]

If $W \ge W_{critical, N_r}$ then reject $H_0$

Alternatively, a p-value can be calculated from enumeration of all possible combinations of $W$ given $N_r$ .

Example[edit]

			$x_{2,i} - x_{1,i}$
$i_{}$	$x_{2,i}$	$x_{1,i}$	$\sgn$	$\text{abs}$
1	125	110	1	15
2	115	122	–1	7
3	130	125	1	5
4	140	120	1	20
5	140	140		0
6	115	124	–1	9
7	140	123	1	17
8	125	137	–1	12
9	140	135	1	5
10	135	145	–1	10

order by absolute difference

			$x_{2,i} - x_{1,i}$
$i_{}$	$x_{2,i}$	$x_{1,i}$	$\sgn$	$\text{abs}$	$R_i$	$\sgn \cdot R_i$
5	140	140		0
3	130	125	1	5	1.5	1.5
9	140	135	1	5	1.5	1.5
2	115	122	–1	7	3	–3
6	115	124	–1	9	4	–4
10	135	145	–1	10	5	–5
8	125	137	–1	12	6	–6
1	125	110	1	15	7	7
7	140	123	1	17	8	8
4	140	120	1	20	9	9

$sgn$ is the sign function, $\text{abs}$ is the absolute value, and $R_i$ is the rank. Notice that pairs 3 and 9 are tied in absolute value. They would be ranked 1 and 2, so each gets the average of those ranks, 1.5.

$N_r = 10 - 1 = 9, W = |1.5+1.5-3-4-5-6+7+8+9| = 9.$

$W < W_{\alpha = 0.05, 9} = 39 \therefore \text{fail to reject } H_0.$

References[edit]

^ ^a ^b Lowry, Richard. "Concepts & Applications of Inferential Statistics". Retrieved 24 March 2011.
^ Wilcoxon, Frank (Dec 1945). "Individual comparisons by ranking methods". Biometrics Bulletin 1 (6): 80–83.
^ Siegel, Sidney (1956). Non-parametric statistics for the behavioral sciences. New York: McGraw-Hill. pp. 75–83.

External links[edit]

Wilcoxon Signed-Rank Test in R
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
Discussion and table of critical values for the original Wilcoxon Rank-Sum Test, which uses a slightly different test statistic (pdf)

Implementations[edit]

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.
SciPy includes an implementation of the Wilcoxon signed-rank test in Python

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