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Wilcoxon signed rank test

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Non-parametric tests that extend the sign tests. The single-sample version (observations x1, x2,…), suitable for a symmetric distribution, tests the null hypothesis that the population median has a specified value (m0). The matched-pair (or paired-sample) version (observation pairs (x1, y1), (x2, y2),…) is concerned with the differences (x1y1), (x2y2),…. With the assumption that these differences are independent observations from a symmetric distribution, the null hypothesis is that this distribution has median zero.

To determine the value of the test statistic, z, the first step is to calculate the differences d1, d2, …, where dj=xjm0 (single sample) or dj=xjyj (matched pairs). After zero differences have been discarded, the remaining n are arranged in ascending order of |dj|. The magnitudes are replaced by the corresponding ranks, with tied ranks where necessary. The signs of d1, d2,…, are now attributed to the ranks, resulting in signed ranks. Let P be the sum of the positive signed ranks and let T be the smaller of P and ½n(n+1)-P. The test statistic, given by is an observation from the upper half of an approximate standard normal distribution. The ½ is a continuity correction.

As an example, suppose that a symmetric distribution is believed to have median 100. A random sample of eight observations is reported as consisting of the values 92.3, 57.6, 88.8, 110.5, 100.0, 181.0, 96.0, 105.7. The supposed median is subtracted from each observation to give −7.7, −42.4, −11.2, 10.5, 0, 81.0, −4.0, 5.7. The value 0 is discarded and the remainder are arranged in order of ascending absolute magnitude: −4.0, 5.7, −7.7, 10.5, −11.2, −42.4, 81.0. Retaining the signs while replacing the values by ranks gives −1, 2, −3, 4, −5, −6, 7, so that p = 13, ½n(n+1) − P=15 (since n=7) and T=13. The test statistic is

Comparing with the tables of upper-tail percentage points of the standard normal distribution (Appendix VI), we see that the null hypothesis that the median is 100 should not be rejected.

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