[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wilcoxon-signed-rank-test-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wilcoxon-signed-rank-test-wikipedia\/","headline":"Wilcoxon signed-rank test – Wikipedia","name":"Wilcoxon signed-rank test – Wikipedia","description":"before-content-x4 Statistical hypothesis test after-content-x4 The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used either to test the","datePublished":"2019-09-09","dateModified":"2019-09-09","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/271de587d12c2bd6cc92676dbbd7cc1af68a3629","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/271de587d12c2bd6cc92676dbbd7cc1af68a3629","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wilcoxon-signed-rank-test-wikipedia\/","wordCount":39797,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4Statistical hypothesis test       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used either to test the location of a population based on a sample of data, or to compare the locations of two populations using two matched samples.[1] The one-sample version serves a purpose similar to that of the one-sample Student’s t-test.[2] For two matched samples, it is a paired difference test like the paired Student’s t-test (also known as the “t-test for matched pairs” or “t-test for dependent samples”). The Wilcoxon test can be a good alternative to the t-test when population means are not of interest; for example, when one wishes to test whether a population’s median is nonzero, or whether there is a better than 50% chance that a sample from one population is greater than a sample from another population.Table of Contents       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4History[edit]Test procedure[edit]Null and alternative hypotheses[edit]One-sample test[edit]Paired data test[edit]Zeros and ties[edit]Zeros[edit]Ties[edit]Computing the null distribution[edit]Alternative statistics[edit]Example[edit]Effect size[edit]Software implementations[edit]See also[edit]References[edit]External links[edit]History[edit]The test is named for Frank Wilcoxon (1892\u20131965) who, in a single paper, proposed both it and the rank-sum test for two independent samples.[3] The test was popularized by Sidney Siegel (1956) in his influential textbook on non-parametric statistics.[4] Siegel used the symbol T for the test statistic, and consequently, the test is sometimes referred to as the Wilcoxon T-test.Test procedure[edit]There are two variants of the signed-rank test. From a theoretical point of view, the one-sample test is more fundamental because the paired sample test is performed by converting the data to the situation of the one-sample test. However, most practical applications of the signed-rank test arise from paired data.For a paired sample test, the data consists of samples (X1,Y1),\u2026,(Xn,Yn){displaystyle (X_{1},Y_{1}),dots ,(X_{n},Y_{n})}. Each sample is a pair of measurements. In the simplest case, the measurements are on an interval scale. Then they may be converted to real numbers, and the paired sample test is converted to a one-sample test by replacing each pair of numbers        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4(Xi,Yi){displaystyle (X_{i},Y_{i})} by its difference Xi\u2212Yi{displaystyle X_{i}-Y_{i}}.[5] In general, it must be possible to rank the differences between the pairs. This requires that the data be on an ordered metric scale, a type of scale that carries more information than an ordinal scale but may have less than an interval scale.[6]The data for a one-sample test is a set of real number samples X1,\u2026,Xn{displaystyle X_{1},dots ,X_{n}}. Assume for simplicity that the samples have distinct absolute values and that no sample equals zero. (Zeros and ties introduce several complications; see below.) The test is performed as follows:[7][8]Compute |X1|,\u2026,|Xn|{displaystyle |X_{1}|,dots ,|X_{n}|}.Sort |X1|,\u2026,|Xn|{displaystyle |X_{1}|,dots ,|X_{n}|}, and use this sorted list to assign ranks R1,\u2026,Rn{displaystyle R_{1},dots ,R_{n}}: The rank of the smallest observation is one, the rank of the next smallest is two, and so on.Let sgn{displaystyle operatorname {sgn} } denote the sign function: sgn\u2061(x)=1{displaystyle operatorname {sgn}(x)=1} if 0″\/> and sgn\u2061(x)=\u22121{displaystyle operatorname {sgn}(x)=-1} if x(Xi)Ri.{displaystyle T=sum _{i=1}^{N}operatorname {sgn}(X_{i})R_{i}.}Produce a p{displaystyle p}-value by comparing T{displaystyle T} to its distribution under the null hypothesis.The ranks are defined so that Ri{displaystyle R_{i}} is the number of j{displaystyle j} for which |Xj|\u2264|Xi|{displaystyle |X_{j}|leq |X_{i}|}.  Additionally, if \u03c3:{1,\u2026,n}\u2192{1,\u2026,n}{displaystyle sigma colon {1,dots ,n}to {1,dots ,n}} is such that |X\u03c3(1)|(n)|{displaystyle |X_{sigma (1)}|{displaystyle T^{-}} are defined by[9]T+=\u22111\u2264i\u2264n,\u00a0Xi>0Ri,T\u2212=\u22111\u2264i\u2264n,\u00a0Xi0}R_{i},\\T^{-}&=sum _{1leq ileq n, X_{i}Because T++T\u2212{displaystyle T^{+}+T^{-}} equals the sum of all the ranks, which is 1+2+\u22ef+n=n(n+1)\/2{displaystyle 1+2+dots +n=n(n+1)\/2}, these three statistics are related by:[10]T+=n(n+1)2\u2212T\u2212=n(n+1)4+T2,T\u2212=n(n+1)2\u2212T+=n(n+1)4\u2212T2,T=T+\u2212T\u2212=2T+\u2212n(n+1)2=n(n+1)2\u22122T\u2212.{displaystyle {begin{aligned}T^{+}&={frac {n(n+1)}{2}}-T^{-}={frac {n(n+1)}{4}}+{frac {T}{2}},\\T^{-}&={frac {n(n+1)}{2}}-T^{+}={frac {n(n+1)}{4}}-{frac {T}{2}},\\T&=T^{+}-T^{-}=2T^{+}-{frac {n(n+1)}{2}}={frac {n(n+1)}{2}}-2T^{-}.end{aligned}}}Because T{displaystyle T}, T+{displaystyle T^{+}}, and T\u2212{displaystyle T^{-}} carry the same information, any of them may be used as the test statistic.The positive-rank sum and negative-rank sum have alternative interpretations that are useful for the theory behind the test. Define the Walsh average Wij{displaystyle W_{ij}} to be 12(Xi+Xj){displaystyle {tfrac {1}{2}}(X_{i}+X_{j})}. Then:[11]T+=#{Wij>0:1\u2264i\u2264j\u2264n},T\u2212=#{Wiji\u2264j\u2264n}.{displaystyle {begin{aligned}T^{+}=#{W_{ij}>0colon 1leq ileq jleq n},\\T^{-}=#{W_{ij}Null and alternative hypotheses[edit]One-sample test[edit]The one-sample Wilcoxon signed-rank test can be used to test whether data comes from a symmetric population with a specified median.[12] If the population median is known, then it can be used to test whether data is symmetric about its center.[13]To explain the null and alternative hypotheses formally, assume that the data consists of independent and identically distributed samples from a distribution F{displaystyle F}. If X1{displaystyle X_{1}} and X2{displaystyle X_{2}} are IID F{displaystyle F}-distributed random variables, define F(2){displaystyle F^{(2)}} to be the cumulative distribution function of 12(X1+X2){displaystyle {tfrac {1}{2}}(X_{1}+X_{2})}. 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