# Wilcoxon Signed-Rank Test

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A Wilcoxon Signed-Rank Test is a non-parametric test equivalent to the one-sample t-test and matched-pair t-test.

**AKA:**Wilcoxon T Test, Signed Rank Test, Wilcoxon Test- It is based on the paired difference test and Sign Test.
- It is used in statistical experiments that have one measurement variable and two nominal variables and , when only one observation for each pair of nominal variables is available (related samples).
- It replaces the Paired Samples t-Test when paired differences distribution are not normally distributed.
- It is replaced by the Wilcoxon–Mann–Whitney Test when the samples are independent.
- It is described by a Wilcoxon Signed-Rank Test Task solved by Wilcoxon Signed-Rank Test System which requires the calculation of a Wilcoxon Signed-Rank Test Statistic.

**Example(s)****Counter-Example(s)****See:**Paired Difference Test, Sign Test, Normally Distributed, Non-Parametric Statistics, Statistical Hypothesis Testing, Paired Difference Test, Student's t-Test.

## References

### 2017

- (ITL-SED, 2017) ⇒ Retrieved 2017-01-08 from NIST (National Intitute of Standards and Technology, US) website http://www.itl.nist.gov/div898//software/dataplot/refman1/auxillar/signrank.htm
- The t-test is the standard test for testing that the difference between population means for two paired samples are equal. If the populations are non-normal, particularly for small samples, then the t-test may not be valid. The signed rank test is an alternative that can be applied when distributional assumptions are suspect. However, it is not as powerful as the t-test when the distributional assumptions are in fact valid.

- The signed rank test is also commonly called the Wilcoxon signed rank test or simply the Wilcoxon test.
- To form the signed rank test, compute [math]d_i = X_i - Y_i[/math] where [math]X[/math] and [math]Y[/math] are the two samples. Rank the [math]d_i[/math] without regard to sign. Tied values are not included in the Wilcoxon test. After ranking, restore the sign (plus or minus) to the ranks. Then compute W+ and W- as the sums of the positive and negative ranks respectively. If the two population means are in fact equal, then the sums of the ranks should also be nearly equal. If the difference between the sum of the ranks is too great, we reject the null hypothesis that the population means are equal.
- Significance levels are based on the fact that if there is no difference in the population means, then there are [math]2^n[/math] equally likely ways for the n ranks to recieve signs.
- More formally, the hypothesis test is defined as follows.
- [math]H_0:\quad \mu_1=\mu_2[/math]
- [math]Ha:\quad \mu_1 \ne \mu_2[/math]

- Test Statistic: W=MIN(W-,W+) where the computation of W- and W+ is discussed above.
- Significance Level: [math]\alpha[/math] (typically set to .05). Due to the discreteness of the ranks, the actual significance level will not in most cases be exact.
- Critical Region: For small samples (N ≤ 30), the critical regions have been tabulated. For N > 30, the test statistic W approaches a normal distribution with a mean of
- [math]\mu_w=n(n+1)/4[/math]

- and a standard deviation of
- [math]\sigma_w=\sqrt{n(n+1)(2n+1)/24}[/math]

- The critical regions are thus based on the normal percent point function. That is, for a 2-sided test,
- [math]\mu_w−\sigma_w\phi^{−1}(\alpha/2) \lt W \lt \mu_w+\sigma_w\phi^{−1}(\alpha/2)[/math]

- where [math]\mu_w[/math] and [math]\sigma_w[/math] are the mean and standard deviation of W as described above and [math]\phi^{−1}[/math] is the normal percent point function.
- Conclusion: Reject null hypothesis if test statistic is in critical region
- Although the above discussion was in terms of a paired two sample test, it can easily be adapted to the following additional cases:

### 2016a

- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test Retrieved:2016-12-17.
- 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.

- The

### 2016b

- (science.psu.edu, 2016) ⇒ STAT 414/415, Lesson 48: The Wilcoxon Tests, https://onlinecourses.science.psu.edu/stat414/node/319
**The Wilcoxon Signed Rank Test for a Median**Developed in 1945 by the statistician Frank Wilcoxon, the signed rank test was one of the first "nonparametric" procedures developed. It is considered a nonparametric procedure, because we make only two simple assumptions about the underlying distribution of the data, namely that:

- (1) the random variable X is continuous
- (2) the probablility density function of X is symmetric

- Then, upon taking a random sample [math]X1, X2, \cdots , Xn,[/math] we are interested in testing the null hypothesis: [math]H_0: m=m_0 [/math] against any of the possible alternative hypotheses: [math]H_A: \;\; m \;\gt \;m_0 \text{ or } H_A:\;\;m\;\lt \;m_0 \text{ or } H_A:\;\;m\;\ne\;m_0[/math]

### 2014

- (McDonald, 2014) ⇒ "Wilcoxon signed-rank test" retrieved from http://www.biostathandbook.com/wilcoxonsignedrank.html . This web page contains the content of pages 186-189 from McDonald, J.H. (2014). Handbook of Biological Statistics (3rd ed.). Sparky House Publishing, Baltimore, Maryland.
**Summary**- Use the Wilcoxon signed-rank test when you'd like to use the paired t–test, but the differences are severely non-normally distributed. (...)

**Null hypothesis**- The null hypothesis is that the median difference between pairs of observations is zero. Note that this is different from the null hypothesis of the paired t–test, which is that the mean difference between pairs is zero, or the null hypothesis of the sign test, which is that the numbers of differences in each direction are equal.

### 2004

- (Rosie Shier, 2004) ⇒ Statistics: 2.2 The Wilcoxon signed rank sum test http://www.statstutor.ac.uk/resources/uploaded/wilcoxonsignedranktest.pdf
- The Wilcoxon signed rank sum test is another example of a non-parametric or distribution free test (see 2.1 The Sign Test). As for the sign test, the Wilcoxon signed rank sum test is used is used to test the null hypothesis that the median of a distribution is equal to some value. It can be used a) in place of a one-sample t-test b) in place of a paired t-test or c) for ordered categorial data where a numerical scale is inappropriate but where it is possible to rank the observations.