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Wilcoxon signed-rank test
The Wilcoxon signed-rank test is a non-parametricstatistical hypothesis test used to compare 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 use as an alternative to the paired Student's t-test (also known as "t-test for matched pairs" or "t-test for dependent samples") when the distribution of the differences between the two samples cannot be assumed to be normally distributed^{[1]}. A Wilcoxon signed-rank test is a nonparametric test that can be used to determine whether two dependent samples were selected from populations having the same distribution.
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, $W$. In consequence, the test is sometimes referred to as the Wilcoxon T test, and the test statistic is reported as a value of T.
Assumptions
Data are paired and come from the same population.
Each pair is chosen randomly and independently^{[citation needed]}.
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).
Test procedure
Let $N$ be the sample size, i.e., the number of pairs. Thus, there are a total of 2N data points. For pairs $i=1,...,N$, let $x_{1,i}$ and $x_{2,i}$ denote the measurements.
H_{0}: difference between the pairs follows a symmetric distribution around zero
H_{1}: difference between the pairs does not follow a symmetric distribution around zero.
For $i=1,...,N$, calculate $|x_{2,i}-x_{1,i}|$ and $\operatorname {sgn} (x_{2,i}-x_{1,i})$, where $\operatorname {sgn}$ 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 pair with the smallest non-zero absolute difference as 1. Ties receive a rank equal to the average of the ranks they span. Let $R_{i}$ denote the rank.
$W=\sum _{i=1}^{N_{r}}[\operatorname {sgn}(x_{2,i}-x_{1,i})\cdot R_{i}]$, the sum of the signed ranks.
Under null hypothesis, $W$ follows a specific distribution with no simple expression. This distribution has an expected value of 0 and a variance of ${\frac {N_{r}(N_{r}+1)(2N_{r}+1)}{6}}$.
$W$ can be compared to a critical value from a reference table.^{[4]}
The two-sided test consists in rejecting $H_{0}$ if $|W|>W_{critical,N_{r}}$.
As $N_{r}$ increases, the sampling distribution of $W$ converges to a normal distribution. Thus,
For $N_{r}\geq 20$, a z-score can be calculated as $z={\frac {W}{\sigma _{W}}}$, where $\sigma _{W}={\sqrt {\frac {N_{r}(N_{r}+1)(2N_{r}+1)}{6}}}$.
To perform a two-sided test, reject $H_{0}$ if $z_{critical}<|z|$.
Alternatively, one-sided tests can be performed with either the exact or the approximate distribution. p-values can also be calculated.
For $N_{r}<20$ the exact distribution needs to be used.
Example
$i$
$x_{2,i}$
$x_{1,i}$
$x_{2,i}-x_{1,i}$
$\operatorname {sgn}$
${\text{abs}}$
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
$i$
$x_{2,i}$
$x_{1,i}$
$x_{2,i}-x_{1,i}$
$\operatorname {sgn}$
${\text{abs}}$
$R_{i}$
$\operatorname {sgn} \cdot R_{i}$
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
$\operatorname {sgn}$ 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.
$\therefore {\text{failed to reject }}H_{0}$ that the two medians are the same.
The $p$-value for this result is $0.6113$
Historical T statistic
In historical sources a different statistic, denoted by Siegel as the T statistic, was used. The T statistic is the smaller of the two sums of ranks of given sign; in the example, therefore, T would equal 3+4+5+6=18. Low values of T are required for significance. T is easier to calculate by hand than W and the test is equivalent to the two-sided test described above; however, the distribution of the statistic under $H_{0}$ has to be adjusted.
$\therefore {\text{failed to reject }}H_{0}$ that the two medians are the same.
Note: Critical T values ($T_{crit}$) by values of $N_{r}$ can be found in appendices of statistics textbooks, for example in Table B-3 of Nonparametric Statistics: A Step-by-Step Approach, 2nd Edition by Dale I. Foreman and Gregory W. Corder
([www.oreilly.com]).
Limitation
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]}
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.
Software implementations
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
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)
^Pratt, J (1959). "Remarks on zeros and ties in the Wilcoxon signed rank procedures". Journal of the American Statistical Association. 54 (287): 655–667. doi:10.1080/01621459.1959.10501526.
^ ^{a}^{b}Derrick, B; White, P (2017). "Comparing Two Samples from an Individual Likert Question". International Journal of Mathematics and Statistics. 18 (3): 1–13.
^ ^{a}^{b}Kerby, Dave S. (2014), "The simple difference formula: An approach to teaching nonparametric correlation.", Comprehensive Psychology, 3: 11.IT.3.1, doi:10.2466/11.IT.3.1
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