Wilcoxon–Mann–Whitney Test

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A Wilcoxon–Mann–Whitney Test is a non-parametric hypothesis test for comparing medians of two samples from the same population based on a U statistic.



References

2017a

  • (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/ranksum.htm

  • The t-test is the standard test for testing that the difference between population means for two non-paired samples are equal. If the populations are non-normal, particularly for small samples, then the t-test may not be valid. The rank sum 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 rank sum test is also commonly called the Mann-Whitney rank sum test or simply the Mann-Whitney test. Note that even though this test is commonly called the Mann-Whitney test, it was in fact developed by Wilcoxon.
To form the rank sum test, rank the combined samples. Then compute the sum of the ranks for sample one, [math]\displaystyle{ T_1 }[/math], and the sum of the ranks for sample two, [math]\displaystyle{ T_2 }[/math]. If the sample sizes are equal, the rank sum test statistic is the minimum of [math]\displaystyle{ T_1 }[/math] and [math]\displaystyle{ T_2 }[/math]. If the sample sizes are unequal, then find [math]\displaystyle{ T_1 }[/math] equal the sum of the ranks for the smaller sample. Then compute [math]\displaystyle{ T_2 = n_1(n_1 + n_2 + 1) - T1 }[/math]. [math]\displaystyle{ T }[/math] is the minimum of [math]\displaystyle{ T_1 }[/math] and [math]\displaystyle{ T_2 }[/math]. Sufficiently small values of [math]\displaystyle{ T }[/math] cause rejection of the null hypothesis that the sample means are equal.
Significance levels have been tabulated for small values of [math]\displaystyle{ n_1 }[/math] and [math]\displaystyle{ n_2 }[/math]. For sufficiently large [math]\displaystyle{ n_1 }[/math] and [math]\displaystyle{ n_2 }[/math], the following normal approximation is used:
[math]\displaystyle{ Z=\frac{|\mu−T|−0.5}{\sigma} }[/math]
where
[math]\displaystyle{ \mu=n1(n_1+n_2+1)/2 }[/math]
[math]\displaystyle{ \sigma = \sqrt{n_2\mu/6} }[/math]

2017b

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