Re: st: RE: RE: median equality test for non normal variables

[Roger Newson <r.newson@imperial.ac.uk>]

Yes, the Wilcoxon ranksum test calculates its P-value using a 
permutation method. Therefore, if we calculate a confidence interval for 
the Hodges-Lehmann median difference by inverting the ranksum test, then 
we are assuming that the 2 sub-population distributions differ only in 
location. This is in contrast to the -cendif- confidence interval for 
the Hodges-Lehmann median difference, which inverts a confidence 
interval for Somers' D, and which therefore contrasts to the usual 
Lehmann confidence interval formula as the unequal-variance t-test 
contrasts to the equal-variance t-test.

For a comparison of the performance of formulas for estimating the 
Hodges-Lehmann median difference, you might like to look at my 
presentation on this subject at the 2007 UK Stata User Meeting, 
downloadable from SSC at
http://ideas.repec.org/p/boc/usug07/01.html

Best wishes

Roger


Roger B Newson BSc MSc DPhil
Lecturer in Medical Statistics
Respiratory Epidemiology and Public Health Group
National Heart and Lung Institute
Imperial College London
Royal Brompton Campus
Room 33, Emmanuel Kaye Building
1B Manresa Road
London SW3 6LR
UNITED KINGDOM
Tel: +44 (0)20 7352 8121 ext 3381
Fax: +44 (0)20 7351 8322
Email: r.newson@imperial.ac.uk
Web page: http://www.imperial.ac.uk/nhli/r.newson/
Departmental Web page:
http://www1.imperial.ac.uk/medicine/about/divisions/nhli/respiration/popgenetics/reph/

Opinions expressed are those of the author, not of the institution.

On 25/05/2010 17:04, Feiveson, Alan H. (JSC-SK311) wrote:
> Isn't it true that the Wilcoxon rank sum test is designed only for possibilities of one distribution being a translation of the other? So the null would be identical distributions; the alternatives would be that the distributions differ only by a translation.
>
> So if distributions have different shapes but the same medians one might naively assume the "null" is true, but as this example shows, such a condition will likely be rejected by -ranksum-.
>
> Here's another example with continuous data:
>
> One distribution is gamma(1,1), while the other is a reflection of the first plus a translation so that both have the same median.
>
>
> drop _all
>   set obs 100
>   gen y=rgamma(1,1)
>   summ y,det
>   local med = r(p50)
>   set obs 200
>   gen group = 1 in 1/100
>   replace group=2 in 101/200
>   gen negy = -y[_n-100] if group==2
>   replace y = 2*`med'+negy if group==2
>   noi sum y if group==1,det
>   noi sum y if group==2,det
>   noi ranksum y,by(group)
>   noi qreg y group
>
>
> Note -ranksum- rejects its null (that the two distributions are identical, not that the medians are the same), whereas -qreg- accepts its null of equal medians.
>
> Al Feiveson
>
>
>
> -----Original Message-----
> From: owner-statalist@hsphsun2.harvard.edu [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Ronan Conroy
> Sent: Tuesday, May 25, 2010 5:06 AM
> To: statalist@hsphsun2.harvard.edu
> Subject: Re: st: RE: RE: median equality test for non normal variables
>
> <..>
>
> There is an interesting question concerning the difference between
> what people think they are doing when applying a 'nonparametric' test
> and what is actually happening.
>
> Consider the following data:
>
> input var group
> 1 0
> 2 0
> 3 0
> 4 0
> 4 0
> 4 0
> 4 0
> 4 1
> 4 1
> 4 1
> 4 1
> 5 1
> 6 1
> 7 1
> end
>
> Note that the median coincides with the highest value in group zero
> and the lowest value in group 1.
>
> What we get now depends critically on what we ask for:
>
> Test for equality of medians using -qreg- : P=1.000 (the medians are
> the same)
> Wilcoxon rank sum test : Prob>  |z| =   0.0196
> Median test (which does not test for equality of medians, NB) :
> Pearson chi2(1) =   3.8182   Pr = 0.051
> Median test, continuity corrected : Pearson chi2(1) =   1.6970   Pr =
> 0.193
> Ordered logit regression with group as a predictor : P =  0.997
> 'Harrell's C' (as calculated by -somersd-) : .76, P<  0.001
>
>
> I have put quotes around Harrell's C, as this quantity is simply a
> rescaling of Mann Whitney's U, dividing it by its maximum possible
> value, and was first proposed by Richard Herrnstein in 1976
> (Herrnstein, R. J., Loveland, D. H.,&  Cable, C. (1976). Natural
> concepts in pigeons. Journal of Experimental Psychology: Animal
> Behavior Processes, 2, 285-302), who termed it rho. Fans of
> terminological chaos will also recognise the entity as the area under
> the ROC curve. Harrell's C is identical with rho only when the data
> are uncensored (James A. Koziol, Zhenyu Jia.T he Concordance Index C
> and the Mann-Whitney Parameter Pr(X>Y) with Randomly Censored Data
> Biometrical Journal 2009:51(3);467 - 474.)
>
> I fancy that there is an amusing paper on this, clarifying the
> hypotheses being tested in each case, if anyone has time to write one...
>
> I am looking again at the t-test, which, after a couple of Kolmogorov-
> Smirnovs, is beginning to look more and more attractive.
>
>
> Ronan Conroy
> =================================
>
> rconroy@rcsi.ie
> Royal College of Surgeons in Ireland
> Epidemiology Department,
> Beaux Lane House, Dublin 2, Ireland
> +353 (0)1 402 2431
> +353 (0)87 799 97 95
> +353 (0)1 402 2764 (Fax - remember them?)
> http://rcsi.academia.edu/RonanConroy
>
> P    Before printing, think about the environment
>
>
>
>
>
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