What Margaret seems to be looking for is something that illustrates the
difference between a median pairwise difference (which is zero under the
null hypothesis tested by the Mann-Whitney U-test) and a difference
between medians (which is equal to the median pairwise difference for
some pairs of distributions but not others). A counterexample might be
comparing 2 exponential (or shifted exponential) subpopulations. See
Newson (2008), downloadable from my website, for the mathematical details.
For more about the detailed maths of computing confidence intervals for
median differences, differences between medians and other parameters
behind "nonparametric" statistics, Margaret might like to check out
Newson (2017), also downloadable from my website.
These documents (and a few more) are part of my repository of
miscellaneous unrefereed mathematical jottings at
http://rogernewsonresources.org.uk/papers.htm#miscellaneous_documents
I hope this helps.
Best wishes
Roger
References
Newson RB. Hodges–Lehmann median differences between exponential
subpopulations. Uploaded 12 October, 2008. Download from
http://rogernewsonresources.org.uk/miscdocs/mdexp1.pdf
Newson RB. Asymptotic distributions of two–sample rank statistics for
continuous outcomes. Uploaded 27 April, 2017. Download from
http://rogernewsonresources.org.uk/miscdocs/asy2con1.pdf
-------- Forwarded Message --------
Subject: Re: Mann-Whitney U-test illustrations for special case only,
please
Date: Tue, 18 Sep 2018 15:40:09 +0000
From: Robert Newcombe <[log in to unmask]>
Reply-To: Robert Newcombe <[log in to unmask]>
To: [log in to unmask]
It's best to think of this in ROC terms. The ROC curve has to pass through (0, 0) and (1, 1) anyway. Identical medians implies it also passes through (0.5, 0.5). But, although constrained by these 3 points, one or both halves of the curve must lie substantially above the main diagonal. It is quite possible to construcy sets of numbers that satisfy this constraint. But I think it would be difficult to find a convincing real example in which such a departure from H0 would be plausible.
Robert Newcombe
Cardiff
-----Original Message-----
From: A UK-based worldwide e-mail broadcast system mailing list [mailto:[log in to unmask]] On Behalf Of MACDOUGALL Margaret
Sent: 29 August 2018 21:34
To: [log in to unmask]
Subject: Mann-Whitney U-test illustrations for special case only, please
Hello
I am in search of good illustrations for teaching purposes where for two groups, the medians are identical but when the two groups are compared using the Mann-Whitney U-test, there is a significant difference between these groups.
Thanks in advance
Best wishes
Margaret
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Dr Margaret MacDougall
Medical Statistician and Researcher in Education (Senior Lecturer) Centre for Population Health Sciences The Usher Institute of Population Health Sciences and Informatics University of Edinburgh Medical Shool Teviot Place Edinburgh EH8 9AG
Tel: +44(0)131 650 3211
Fax: +44(0)131 650 6909
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