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ALLSTAT  December 2008

ALLSTAT December 2008

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Subject:

Re: QUERY:Valid use of Mann-Whitney Test - RESPONSES

From:

kornbrot <[log in to unmask]>

Reply-To:

kornbrot <[log in to unmask]>

Date:

Sat, 20 Dec 2008 14:17:30 +0000

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text/plain

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text/plain (207 lines)

What was the question you were using the Mann-Whitney to try and answer?
Since you simulated normals, I assume you had unbounded metric data.
By metric I mean that  a difference beween N and N + x does NOT depend on N.
The Mann-Whitney will detect a shift in a measure of central tendency, the
median,  iff the distributions in the 2 groups have the SAME SHAPE (same var
and skew). 
If normal, but suspect different variance, can use Welch test
OR can use robust procedures that downweight extreme values [huber, wilcox],
these will tell one if the Œbulkš of central values have shifted
If you only want to know if 2 groups are different, use Kolmogorov-Smirnov
If you are convinced similar shapes, e.g. Incomes tend to be positively
skewed, so comparing 2 groups using M-W is reasonable
For metric data, am far from convinced that M-W is bets strategy. Why not
transform to normality and then use Welch if variance unequal? Or the robus
procedures?

If data really is only ordinal, the M-W only suitable if also unbounded.
In my view it is NEVER suitable for Likert data, e.g when people give
categories of agreement with statement of give categorized degree of
approval.
This is because if a group is more favourable than not, then dsitributionn
is likely to be negatively skewed, and vice verse. Hence, the Œsame shapeš
criteria is never met. For such data ordinal procedures that compare the
proportion of values below 1 or more criteria are much more informative.

So what is M_W good for?
My heretical view is NOTHING!  It was very useful in its time, but has been
superceded by new methods

Best

Diana


On 17/12/2008 09:28, "Henderson, Robin" <[log in to unmask]>
wrote:

> Dear Allstat
> 
> Sincere thanks to everyone who responded.  I deliberately simulated data from
> two normal populations with the same variance with n1 = 5 and n2 = 10.  The
> ratio of the larger IQR to the smaller IQR exceeded 2 in around one in three
> cases.  Thus, in a scenario where a Mann-Whitney test would be appropriate
> (albeit inferior to a t-test), the guideline would "advise" against its use.
> This led me doubt its validity and post the query.  Responses are given below.
> 
> Stephen Senn
> I suspect that the guideline is not very good. However, it is
> probably to do with heteroscedasticity and you have programmed
> homoscedasticty into your simulation.
> 
> It is well-known that the two-sample  t-test is not robust if the
> population variances are different unless the samples sizes are the
> same. If the smaller sample has the larger variance there is a
> problem. I suspect that what is being done here is to give some
> guidance to deal with possible differences in variances but I also
> suspect that it is a pretty useless rule of thumb.
> 
> It might be worth looking at Gerald van Belle's
> http://www.vanbelle.org/ book to see if there is a mention .
> 
> Nick Longford
> I have, and ignored it, for the good of the science.
> 
> Paul Wilson
> I have never heard of this restriction on the Mann-Whitney.
> Please let me know what the general opinion of your respondents is. I
> would hazard a guess that whoever penned the "guideline" was trying to
> be clever and "transfer" the guideline that a t-test should not be used
> if the standard deviation of one sample is more than twice that of the
> other, but I could be wrong!
> 
> Allan White
> From a theoretical perspective, I can see why this recommendation was made.
> The Mann-Whitney test is a member of the class of permutation tests. This
> class of tests has the property that, under the null hypothesis, all the
> rearrangements of the data performed by the test must be equally likely.
> This condition is met if, and only if, the data from the different groups
> is drawn from the same error distribution. (I mean by this that the
> distributions of scores within each group/condition should not differ in
> any respect, except that of location). In the scenario where the spread
> differs obviously (and substantially) between the groups, this condition is
> clearly violated and the test degenerates into one that merely tests for
> significant differences (OF ANY SORT) between the groups/conditions. Thus,
> under these circumstances, the test ceases to be a simple test for differences
> in location (medians).
> 
> Roger Newson
> I have not seen this guideline as such. However, I have long been aware
> that the Mann-Whitney U-statistic, and the associated confidence
> interval for the Hodges-Lehmann median difference, are robust to
> non-Normality and non-robust to unequal variability. I have developed a
> package (somersd) in the Stata statistical language to calculate
> confidence intervals for rank statistics that are robust to unequal
> variability. The theory is written up in Newson (2002), Newson (2006a)
> and Newson (2006b), and also in some manuals distributed with the
> package. All of these can be downloaded from my website (see my
> signature below). If you have Stata, then you can download the package
> by typing in Stata
> 
> ssc describe somersd
> ssc install somersd, replace
> 
> I have done some simulations, and submitted the results for publication
> in Computational Statistics and Data Analysis, on the performance, under
> a wide range of scenarios, of various confidence intervals for median
> differences (my package, the Lehmann formula, and the equal-variance and
> unequal-variance t-tests. The message of these simulations is that the
> method implemented in the somersd package is robust to non-Normality and
> to unequal variability, at the price of being non-robust to tiny sample
> numbers, under which conditions the confidence intervals may extend from
> minus infinity to plus infinity. This is because, under those
> conditions, if we are not allowed to assume Normality and/or equal
> variability, then the median difference really could be anywhere.
> 
> Chris Lloyd
> Top of my head - MW is appropriate for a SHIFT model. For testing, the
> null hypothesis is identical distributions i.e. shift zero. I did some
> simulations years ago that convinced me that it was not robust to scale
> differences - the size of the test can be pretty badly compromised,
> especially when one population is contaminated with skew (holding the
> medians equal).
> 
> But the twice-scale rule. Never heard of it.
> 
> Lisa Yelland
> I have not come across this guideline. However I have read a paper* which
> looks
> at the performance of the Mann-Whitney test in a range of scenarios using
> simulations and shows that it performs poorly when the variances in the two
> groups are unequal.
> 
> I am curious about the simulation you did. Firstly, if the data are normal
> then
> the t-test would be preferable to the Mann-Whitney test. Secondly, the
> guideline
> you mentioned relates to unequal inter-quartile ranges and yet you simulated
> data with equal variances (perhaps this was a typo?). I think it would be more
> relevant to simulate non-normal data with a range of differences in the
> inter-quartile range between groups to judge whether the guideline is
> appropriate (depending on how much time you want to spend investigating
> this!).
> 
> * Skovlund E & Fenstad G (2001). Should we always choose a nonparametric test
> when comparing two apparently nonnormal distributions?. Journal of Clinical
> Epidemiology; 54:86-92.
> 
> Martin Bland
> This is nonsense, as the Mann Whitney test can be used for ordinal
> data.  The notion of interquartile range involves subtraction, so can
> apply only to interval data.  However, there is a condition that if you
> want to use the test as testing the null hypothesis that the medians are
> the same, the two distributions must differ only in location.  This
> clearly applies to interval data only as ordinal data do not have a
> shape.  Under these circumstances the test also tests the null
> hypothesis that the means are equal.  As the standard deviations must be
> identical, you might at well do a t test and get a confidence interval.
> 
> Dorothy Middleton
> In theory, the bootstrap is the only technique that should be used to
> compare the means of two populations that have quite different variances
> (that is, the Behrens-Fisher problem).  Student's t, a permutation test
> using the original observations, and the permutation using ranks
> (Mann-Whitney, Wilcoxon) are all likely to yield inexact significance
> levels.  Still simulations have shown that permutation tests are almost
> exact even when the variance of one population is twice that of the
> other.  See http://statisticsonline.info/application.htm.
> 
> 
> Best Wishes
> 
> Robin
> 
> G Robin Henderson
> Audit Coordinator
> Scottish National Stroke Audit
> Royal Infirmary of Edinburgh
> 0131 242 6934
> 
> 
> 
> *****************************************************************
> The information contained in this message may be confidential or
> legally privileged and is intended for the addressee only. If you
> have received this message in error or there are any problems
> please notify the originator immediately. The unauthorised use,
> disclosure, copying or alteration of this message is
> strictly forbidden.
> *****************************************************************

Professor Diana Kornbrot
 School of Psychology
 University of Hertfordshire
 College Lane, Hatfield, Hertfordshire AL10 9AB, UK

 email:  [log in to unmask]
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