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'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. Other names may include the 't-test for matched pairs' or the 't-test for dependent samples'.

Assumptions

1. Data is paired and comes from the same population.
2. Each pair is chosen randomly and independent.
3. The data is at least ordinal.

Test procedure

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.

$H_{0}:{\text{median difference between the pairs is zero}}$    $H_{1}:{\text{median difference is not zero}}$.

1. For i = 1, ..., N, calculate $|x_{2,i}-x_{1,i}|$ and $\operatorname {sgn}(x_{2,i}-x_{1,i})$, where sgn is the sign function.
   
   
2. Exclude pairs with $|x_{2,i}-x_{1,i}|=0$. Let $N_{r}$ be the reduced sample size.
   
   
3. Order the remaining $m$ pairs from smallest absolute difference to largest absolute difference, $|x_{2,i}-x_{1,i}|$.
   
   
4. 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.
   
   
5. Calculate the test statistic W.
   
   $W=|\sum _{i=1}^{n}[\operatorname {sgn}(x_{2,i}-x_{1,i})\cdot R_{i}]|$, the absolute value of the sum of the signed ranks.
   
   
6. As $N_{r}$ increases, the sampling distribution of W converges to a normal distribution. Thus,
   
   For $N_{r}\geq 10$, 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, reject H₀.
   
   For $N_{r}<10$, $W$ is compared to a critical value from a reference table. If $W\geq W_{critical,N_{r}}$, reject H₀. Alternatively, a p-value can be calculated from enumeration of all possible combinations of $W$ given $N_{r}$.
   
   

Example

$x_{2,i}-x_{1,i}$ $i_{}$ $x_{2,i}$ $x_{1,i}$ $\operatorname {sgn}$ ${\text{abs}}$ 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

$x_{2,i}-x_{1,i}$ $i_{}$ $x_{2,i}$ $x_{1,i}$ $\operatorname {sgn}$ ${\text{abs}}$ $R_{i}$ $\operatorname {sgn} \cdot R_{i}$ 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.

$N_{r}=10-1=9,W=|1.5+1.5-3-4-5-6+7+8+9|=9.$

$W<W_{\alpha =0.05,9}=35\therefore {\text{fail to reject}}H_{0}$

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