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

["Nick Cox" <n.j.cox@durham.ac.uk>]

To underline one of Ronan's points: 

Scaling U to pr(X > Y) goes back over 50 years. There's a paper on it in
the Berkeley symposia by Z.W. Birnbaum 

<http://digitalassets.lib.berkeley.edu/math/ucb/text/math_s3_v1_article-
02.pdf> 

which gives yet earlier references. 

Nick 
n.j.cox@durham.ac.uk 

Ronan Conroy

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.


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