Wilcoxon signed-rank test

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 Sidney Siegel (1956) in his influential textbook on non-parametric statistics.[3] Siegel used the symbol T for a value related to, but not the same as, ${\displaystyle W}$. In consequence, the test is sometimes referred to as the Wilcoxon T test, and the test statistic is reported as a value of T.

1. Data are paired and come from the same population.
2. Each pair is chosen randomly and independently.
3. The data are measured on at least an interval scale when, as is usual, within-pair differences are calculated to perform the test (though it does suffice that within-pair comparisons are on an ordinal scale).

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

H₀: difference between the pairs follows a symmetric distribution around zero

H₁: difference between the pairs does not follow a symmetric distribution around zero.

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

   ${\displaystyle W=\sum _{i=1}^{N_{r}}[\operatorname {sgn}(x_{2,i}-x_{1,i})\cdot R_{i}]}$, the sum of the signed ranks.

6. Under null hypothesis, ${\displaystyle W}$ follows a specific distribution with no simple expression. This distribution has an expected value of 0 and a variance of ${\displaystyle {\frac {N_{r}(N_{r}+1)(2N_{r}+1)}{6}}}$.

   ${\displaystyle W}$ can be compared to a critical value from a reference table.[4]

   The two-sided test consists in rejecting ${\displaystyle H_{0}}$ if ${\displaystyle |W|>W_{critical,N_{r}}}$.

7. As ${\displaystyle N_{r}}$ increases, the sampling distribution of ${\displaystyle W}$ converges to a normal distribution. Thus,

   For ${\displaystyle N_{r}\geq 20}$, a z-score can be calculated as ${\displaystyle z={\frac {W}{\sigma _{W}}}}$, where ${\displaystyle \sigma _{W}={\sqrt {\frac {N_{r}(N_{r}+1)(2N_{r}+1)}{6}}}}$.

   To perform a two-sided test, reject ${\displaystyle H_{0}}$ if ${\displaystyle z_{critical}<|z|}$.

   Alternatively, one-sided tests can be performed with either the exact or the approximate distribution. p-values can also be calculated.

8. For ${\displaystyle N_{r}<20}$ the exact distribution needs to be used.

Example

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

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

${\displaystyle \operatorname {sgn} }$ is the sign function, ${\displaystyle {\text{abs}}}$ is the absolute value, and ${\displaystyle R_{i}}$ 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.

${\displaystyle W=1.5+1.5-3-4-5-6+7+8+9=9}$

${\displaystyle |W|<W_{crit(\alpha =0.05,\ 9{\text{, two-sided}})}=33}$

${\displaystyle \therefore {\text{failed to reject }}H_{0}}$ that the two medians are the same.

The ${\displaystyle p}$-value for this result is ${\displaystyle 0.6113}$

As demonstrated in the example, when the difference between the groups is zero, the observations are discarded. This is of particular concern if the samples are taken from a discrete distribution. In these scenarios the modification to the Wilcoxon test by Pratt 1959, provides an alternative which incorporates the zero differences.[5][6] This modification is more robust for data on an ordinal scale.[6]

To compute an effect size for the signed-rank test, one can use the rank-biserial correlation.

If the test statistic W is reported, the rank correlation r is equal to the test statistic W divided by the total rank sum S, or r = W/S. [7] Using the above example, the test statistic is W = 9. The sample size of 9 has a total rank sum of S = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) = 45. Hence, the rank correlation is 9/45, so r = 0.20.

If the test statistic T is reported, an equivalent way to compute the rank correlation is with the difference in proportion between the two rank sums, which is the Kerby (2014) simple difference formula.[7] To continue with the current example, the sample size is 9, so the total rank sum is 45. T is the smaller of the two rank sums, so T is 3 + 4 + 5 + 6 = 18. From this information alone, the remaining rank sum can be computed, because it is the total sum S minus T, or in this case 45 - 18 = 27. Next, the two rank-sum proportions are 27/45 = 60% and 18/45 = 40%. Finally, the rank correlation is the difference between the two proportions (.60 minus .40), hence r = .20.

• R includes an implementation of the test as wilcox.test(x,y, paired=TRUE), where x and y are vectors of equal length.[8]
• ALGLIB includes implementation of the Wilcoxon signed-rank test in C++, C#, Delphi, Visual Basic, etc.
• 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
• Accord.NET includes an implementation of the Wilcoxon signed-rank test in C# for .NET applications
• MATLAB implements this test using "Wilcoxon rank sum test" as [p,h] = signrank(x,y) also returns a logical value indicating the test decision. The result h = 1 indicates a rejection of the null hypothesis, and h = 0 indicates a failure to reject the null hypothesis at the 5% significance level
• Julia HypothesisTests package includes the wilkoxian Signed Rank test as "value(SignedRankTest(x, y))

• 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)

• Wilcoxon Signed-Rank Test in R
• 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
• Brief guide by experimental psychologist Karl L. Weunsch - Nonparametric effect size estimators (Copyright 2015 by Karl L. Weunsch)
• Kerby, D. S. (2014). The simple difference formula: An approach to teaching nonparametric correlation. Comprehensive Psychology, volume 3, article 1. doi:10.2466/11.IT.3.1. link to article