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The '''Wilcoxon signed-rank test''' is a [[Non-parametric statistics|non-parametric]] [[statistical hypothesis testing|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 [[Student's t-test|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]].<ref name=lowry>{{cite web|last=Lowry|first=Richard|title=Concepts & Applications of Inferential Statistics|url=http://faculty.vassar.edu/lowry/ch12a.html|accessdate=24 March 2011}}</ref>
==History==
The test is named for [[Frank Wilcoxon]] (1892–1965) who, in a single paper, proposed both it and the [[Mann-Whitney-Wilcoxon test|rank-sum test]] for two independent samples (Wilcoxon, 1945).<ref>{{cite journal|last=Wilcoxon|first=Frank|title=Individual comparisons by ranking methods|journal=Biometrics Bulletin|date=Dec 1945|volume=1|issue=6|pages=80–83|url=http://sci2s.ugr.es/keel/pdf/algorithm/articulo/wilcoxon1945.pdf}}</ref> The test was popularized by [[Sidney Siegel|Siegel]] (1956)<ref>{{cite book|last=Siegel|first=Sidney|title=Non-parametric statistics for the behavioral sciences|year=1956|publisher=McGraw-Hill|location=New York|pages=75–83|url=http://books.google.com/books?ei=9cWLTfaTIcmEOs_NuM0L&ct=result&id=ebfRAAAAMAAJ&dq=Wilcoxon+statistics+for+the+behavioral+sciences+Non-parametric&q=Wilcoxon#search_anchor}}</ref> in his influential text book on non-parametric statistics. Siegel used the symbol ''T'' for a value related to, but not the same as, <math>W</math>.  In consequence, the test is sometimes referred to as the '''Wilcoxon ''T'' test''', and the test statistic is reported as a value of ''T''.


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==Assumptions==
 
# Data are paired and come from the same population.
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# Each pair is chosen randomly and independent.
 
# The data are measured at least on an [[ordinal scale]], but need not be normal.
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# The distribution of the differences is symmetric around the median.{{citation needed|date=July 2013}}
 
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</ul>


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==Test procedure==
Let <math>N</math> be the sample size, the number of pairs. Thus, there are a total of ''2N'' data points. For <math>i = 1, ..., N</math>, let <math>x_{1,i}</math> and <math>x_{2,i}</math> denote the measurements.


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: H<sub>0</sub>: median difference between the pairs is zero
 
: H<sub>1</sub>: median difference is not zero.
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# For <math>i = 1, ..., N</math>, calculate <math>|x_{2,i} - x_{1,i}|</math> and <math>\sgn(x_{2,i} - x_{1,i})</math>, where <math>\sgn</math> is the [[sign function]].
# Exclude pairs with <math>|x_{2,i} - x_{1,i}| = 0</math>. Let <math>N_r</math> be the reduced sample size.
# Order the remaining <math>N_r</math> pairs from smallest absolute difference to largest absolute difference, <math>|x_{2,i} - x_{1,i}|</math>.
# [[Ranking#Ranking_in_statistics|Rank]] the pairs, starting with the smallest as 1. Ties receive a rank equal to the average of the ranks they span. Let <math>R_i</math> denote the rank.
# Calculate the test statistic <math>W</math>
#: <math>W = |\sum_{i=1}^{N_r} [\sgn(x_{2,i} - x_{1,i}) \cdot R_i]|</math>, the absolute value of the sum of the signed ranks.
# As <math>N_r</math> increases, the sampling distribution of <math>W</math> converges to a normal distribution. Thus,
#: For <math>N_r \ge 10</math>, a z-score can be calculated as <math>z = \frac{W - 0.5}{\sigma_W}, \sigma_W = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{6}}</math>.
#: If <math>z > z_{critical}</math> then reject <math>H_0</math>
#: {{-}}
#: For <math>N_r < 10</math>, <math>W</math> is compared to a critical value from a reference table.<ref name=lowry></ref>
#: {{-}}
#: If <math>W \ge W_{critical, N_r}</math> then reject <math>H_0</math>
#: {{-}}
#: Alternatively, a ''p''-value can be calculated from enumeration of all possible combinations of <math>W</math> given <math>N_r</math>.


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The ''T'' statistic used by Siegel is the smaller of 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''.  
 
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== Longchamp Fiyatları ==
==Example==
{| border="0" | style="text-align: right;"
|
{| class="wikitable"
|-
| &nbsp;
| &nbsp;
| &nbsp;
| colspan="2" | <math>x_{2,i} - x_{1,i}</math>
|-
| <math>i_{}</math>
| <math>x_{2,i}</math>
| <math>x_{1,i}</math>
| <math>\sgn</math>
| <math>\text{abs}</math>
|-
| 1
| 125
| 110
| 1
| 15
|-
| 2
| 115
| 122
|&nbsp;–1
| 7
|-
| 3
| 130
| 125
| 1
| 5
|-
| 4
| 140
| 120
| 1
| 20
|-
| 5
| 140
| 140
| &nbsp;
| 0
|-
| 6
| 115
| 124
|&nbsp;–1
| 9
|-
| 7
| 140
| 123
| 1
| 17
|-
| 8
| 125
| 137
|&nbsp;–1
| 12
|-
| 9
| 140
| 135
| 1
| 5
|-
| 10
| 135
| 145
|&nbsp;–1
| 10
|}
| style="vertical-align:center;" | order by absolute difference
|
{| class="wikitable"
|-
| &nbsp;
| &nbsp;
| &nbsp;
| style="text-align: center;" colspan="4" | <math>x_{2,i} - x_{1,i}</math>
|-
| <math>i_{}</math>
| <math>x_{2,i}</math>
| <math>x_{1,i}</math>
| <math>\sgn</math>
| <math>\text{abs}</math>
| <math>R_i</math>
| <math>\sgn \cdot R_i</math>
|-
| 5
| 140
| 140
| &nbsp;
| 0
| &nbsp;
| &nbsp;
|-
| 3
| 130
| 125
| 1
| 5
| 1.5
| 1.5
|-
| 9
| 140
| 135
| 1
| 5
| 1.5
| 1.5
|-
| 2
| 115
| 122
|&nbsp;–1
| 7
| 3
|&nbsp;–3
|-
| 6
| 115
| 124
|&nbsp;–1
| 9
| 4
|&nbsp;–4
|-
| 10
| 135
| 145
|&nbsp;–1
| 10
| 5
|&nbsp;–5
|-
| 8
| 125
| 137
|&nbsp;–1
| 12
| 6
|&nbsp;–6
|-
| 1
| 125
| 110
| 1
| 15
| 7
| 7
|-
| 7
| 140
| 123
| 1
| 17
| 8
| 8
|-
| 4
| 140
| 120
| 1
| 20
| 9
| 9
|}
|}


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:<math>sgn</math> is the sign function, <math>\text{abs}</math> is the absolute value, and <math>R_i</math> 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.
 
:<math>N_r = 10 - 1 = 9, W = |1.5+1.5-3-4-5-6+7+8+9| = 9.</math><br><br>
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:<math>W < W_{\alpha = 0.05, 9} = 39
 
  \therefore \text{fail to reject } H_0.</math>
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== Christian Louboutin Malaysia  1913 ==
==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)


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==References==
 
{{Reflist}}
  <li>[http://agent.yunhosting.com/stop/hold.htm?mod=spacecp&ac=blog&blogid= http://agent.yunhosting.com/stop/hold.htm?mod=spacecp&ac=blog&blogid=]</li>
 
 
==External links==
  <li>[http://ks35439.kimsufi.com/spip.php?article171/ http://ks35439.kimsufi.com/spip.php?article171/]</li>
*[http://www.r-tutor.com/elementary-statistics/non-parametric-methods/wilcoxon-signed-rank-test Wilcoxon Signed-Rank Test in R]
 
*[http://comp9.psych.cornell.edu/Darlington/wilcoxon/wilcox0.htm Description of how to calculate ''p'' for the Wilcoxon signed-ranks test]
  <li>[http://ukvotism.com/activity/p/462692/ http://ukvotism.com/activity/p/462692/]</li>
*[http://faculty.vassar.edu/lowry/ch12a.html Example of using the Wilcoxon signed-rank test]
 
*[http://faculty.vassar.edu/lowry/wilcoxon.html An online version of the test]
  <li>[http://enseignement-lsf.com/spip.php?article64#forum18665494 http://enseignement-lsf.com/spip.php?article64#forum18665494]</li>
*[http://www.sussex.ac.uk/Users/grahamh/RM1web/WilcoxonTable2005.pdf A table of critical values for the Wilcoxon signed-rank test]
 
 
  <li>[http://verdamilio.net/tonio/spip.php?article1970/ http://verdamilio.net/tonio/spip.php?article1970/]</li>
 
 
===Implementations===
</ul>
* [http://www.alglib.net/statistics/hypothesistesting/wilcoxonsignedrank.php ALGLIB] includes implementation of the Wilcoxon signed-rank test in C++, C#, Delphi, Visual Basic, etc.
* The free statistical software [[R (programming language)|R]] includes an implementation of the test as <code>wilcox.test(x,y, paired=TRUE)</code>, where x and y are vectors of equal length.
* [[GNU Octave]] implements various one-tailed and two-tailed versions of the test in the <code>wilcoxon_test</code> function.
* [http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.wilcoxon.html SciPy] includes an implementation of the Wilcoxon signed-rank test in Python
 
{{statistics|inference}}
 
{{DEFAULTSORT:Wilcoxon Signed-Rank Test}}
[[Category:Statistical tests]]
[[Category:Non-parametric statistics]]
[[Category:U-statistics]]

Revision as of 18:31, 2 October 2013

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.[1]

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 Siegel (1956)[3] in his influential text book on non-parametric statistics. Siegel used the symbol T for a value related to, but not the same as, . 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

  1. Data are paired and come from the same population.
  2. Each pair is chosen randomly and independent.
  3. The data are measured at least on an ordinal scale, but need not be normal.
  4. The distribution of the differences is symmetric around the median.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

Test procedure

Let be the sample size, the number of pairs. Thus, there are a total of 2N data points. For , let and denote the measurements.

H0: median difference between the pairs is zero
H1: median difference is not zero.
  1. For , calculate and , where is the sign function.
  2. Exclude pairs with . Let be the reduced sample size.
  3. Order the remaining pairs from smallest absolute difference to largest absolute difference, .
  4. Rank the pairs, starting with the smallest as 1. Ties receive a rank equal to the average of the ranks they span. Let denote the rank.
  5. Calculate the test statistic
    , the absolute value of the sum of the signed ranks.
  6. As increases, the sampling distribution of converges to a normal distribution. Thus,
    For , a z-score can be calculated as .
    If then reject
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    For , is compared to a critical value from a reference table.[1]
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    If then reject
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    Alternatively, a p-value can be calculated from enumeration of all possible combinations of given .

The T statistic used by Siegel is the smaller of 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.

Example

     
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
     
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
is the sign function, is the absolute value, and 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.


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)

References

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External links


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

Template:Statistics

  1. ↑ 1.0 1.1 Template:Cite web
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