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.
- AKA: MWW Rank-Sum Test, Rank-Sum Test, Mann-Whitney U Test, U Test, Wilcoxon Rank-Sum Test.
- …
- Counter-Example(s):
- an ANCOVA Test.
- a Wald Test.
- a Wilcoxon Signed-Rank Test.
- See: Statistical Hypothesis Test, Continuous Distribution Comparison Test, Analysis of variance, Bartlett's Test, F-test of Equality of Variances, Brown–Forsythe Test, Normal Distribution, Statistics, Nonparametric Statistics, Statistical Hypothesis Test, Null Hypothesis, t-Test.
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
- (CM,2017) ⇒ Retrieved 2017-01-08 from http://changingminds.org/explanations/research/analysis/mann-whitney.htm
- The Mann-Whitney test compares the medians of two groups of ordinal, non-parametric data to determine if they are statistically different.
The Mann-Whitney test statistic is called U, which is calculated as follows:
Put all the scores together (keeping note of their group identities) into rank order.
Calculate U as the sum of the numbers of scores from the experimental group that are less than each of the control group scores (or the other way round, whichever gives the smaller value of U.
Discussion: The Mann-Whitney test can be thought of as the non-parametric equivalent of the independent-measures t-test.
- The Mann-Whitney test compares the medians of two groups of ordinal, non-parametric data to determine if they are statistically different.
2016
- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Mann–Whitney_U_test Retrieved:2016-9-14.
- In statistics, the Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric test of the null hypothesis that two samples come from the same population against an alternative hypothesis, especially that a particular population tends to have larger values than the other.
Unlike the t-test it does not require the assumption of normal distributions. It is nearly as efficient as the t-test on normal distributions.
- In statistics, the Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric test of the null hypothesis that two samples come from the same population against an alternative hypothesis, especially that a particular population tends to have larger values than the other.
2005
- (Vickers, 2005) ⇒ Andrew J Vickers. (2005). “Parametric versus Non-parametric Statistics in the Analysis of Randomized Trials with Non-normally Distributed Data." BioMed Central Ltd. doi:10.1186/1471-2288-5-35
- QUOTE: It has generally been argued that parametric statistics should not be applied to data with non-normal distributions. Empirical research has demonstrated that Mann-Whitney generally has greater power than the t-test unless data are sampled from the normal. In the case of randomized trials, we are typically interested in how an endpoint, such as blood pressure or pain, changes following treatment. ... Change between skewed baseline and post-treatment data tended towards a normal distribution. ANCOVA was generally superior to Mann-Whitney in most situations, especially where log-transformed data were entered into the model. The estimate of the treatment effect from ANCOVA was not importantly biased.