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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 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]} 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.
History
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^{[]}.
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 smallest 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.^{[1]}
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 $|z|>z_{critical}$.
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 original test using the T statistic is applied.
Denoted by Siegel as the T statistic, it is the smaller of the two sums of ranks of given sign; in the example given below, therefore, T would equal 3+4+5+6=18. Low values of T are required for significance. As will be obvious from the example below, T is easier to calculate by hand than W and the test is equivalent to the
Low two-sided test described above; however, the distribution of the statistic under $H_{0}$ has to be adjusted.
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.
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.^{[4]}^{[5]} This modification is more robust for data on an ordinal scale.^{[5]}
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.
^{[6]}
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.^{[6]} 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.
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.^{[7]}
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.
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
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