The Wilcoxon signed-rank test is a non-parametric rank test for statistical hypothesis testing 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. The one-sample version serves a purpose similar to that of the one-sample Student's t-test. 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 is a good alternative to the t-test when the normal distribution of the differences between paired individuals cannot be assumed. Instead, it assumes a weaker hypothesis that the distribution of this difference is symmetric around a central value and it aims to test whether this center value differs significantly from zero. The Wilcoxon test is a more powerful alternative to the sign test because it considers the magnitude of the differences, but it requires this moderately strong assumption of symmetry.
History The test is named after Frank Wilcoxon (1892–1965) who, in a single paper, proposed both it and the rank-sum test for two independent samples. The test was popularized by Sidney Siegel (1956) in his influential textbook on non-parametric statistics. Siegel used the symbol T for the test statistic, and consequently, the test is sometimes referred to as the Wilcoxon T-test.
Test procedure 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 a sample ( X 1 , Y 1 ) , … , ( X n , Y n ) {\displaystyle (X_{1},Y_{1}),\dots ,(X_{n},Y_{n})} . Each data point in the 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 ( X i , Y i ) {\displaystyle (X_{i},Y_{i})} by its difference X i − Y i {\displaystyle X_{i}-Y_{i}} . 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. The data for a one-sample test is a sample in which each observation is a real number: X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} . Assume for simplicity that the observations in the sample have distinct absolute values and that no observation equals zero. (Zeros and ties introduce several Complications; see below.) The test is performed as follows:
Compute | X 1 | , … , | X n | . {\displaystyle |X_{1}|,\dots ,|X_{n}|.}
Sort | X 1 | , … , | X n | {\displaystyle |X_{1}|,\dots ,|X_{n}|} , and use this sorted list to assign ranks R 1 , … , R n {\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 ( x ) = 1 {\displaystyle \operatorname {sgn}(x)=1} if x > 0 {\displaystyle x>0} and sgn ( x ) = − 1 {\displaystyle \operatorname {sgn}(x)=-1} if x < 0 {\displaystyle x<0} . The test statistic is the signed-rank sum T {\displaystyle T} : T = ∑ i = 1 N sgn ( X i ) R i . {\displaystyle T=\sum _{i=1}^{N}\operatorname {sgn}(X_{i})R_{i}.}
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