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

 From "Feiveson, Alan H. (JSC-SK311)" To "statalist@hsphsun2.harvard.edu" Subject RE: st: RE: RE: median equality test for non normal variables Date Tue, 25 May 2010 11:04:47 -0500

```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?)

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