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st: RE: two sample test under generalized Behrens-Fisher conditions

From   Nick Cox <>
To   "''" <>
Subject   st: RE: two sample test under generalized Behrens-Fisher conditions
Date   Tue, 14 Dec 2010 10:39:31 +0000

In this kind of territory, I would always 

1. Check out what is said in Rupert G. Miller, Beyond ANOVA. See on the CRC Press reissue 

Your library may hold a copy of the Wiley original. 

2. Be wary of the stark cookbooky alternative: data if normal, ranks otherwise. What happened to the idea of transformations or link functions? How do you decide when the data are approximately normal any way? 

Here is an example of a different approach. In the auto data, -mpg- given -foreign- is neither normal nor heteroscedastic. But these are secondary issues. Consider this set of results. In each -family(normal)- is implied. 

foreach v in "power 1" "power 0.5" "log" "power -0.5" "power -1" {
	qui glm mpg foreign, link(`v')
	mat b = e(b)
	mat V = e(V)
	di "`v'"    "{col 20}" %3.2f   b[1,1] / sqrt(V[1,1])

power 1            3.63
power 0.5          3.70
log                3.75
power -0.5         -3.78
power -1           -3.80

The change of sign of what -glm- calls the z statistic is an expected side-effect of changing to inverse transformations. More importantly, z changes only very slowly and the collective set of results points to the idea that 1/mpg is a more appropriate scale than mpg on which to test for differences. This of course matches basic science. 

Generalized linear models are nearly 40 years old as a family. When are they going to receive the recognition they deserve? 


Airey, David C

I was reading a little about what to do when you have both unequal variance and non-normality. Neither the equal variance t-test nor the Mann-Whitney U test are best when you want to interpret the difference in means or medians.

I had found the Stata command -fprank-, but it turns out this robust ranks test doesn't escape a symmetry assumption to interpret the location difference.

I found that some recommend using Welch's t-test on the ranked data (Zimmerman and Zumbo (1993) Rank transformations and the power of the Student's t test and the Welch t' test for non-normal populations with unequal variances. Canadian Journal of Experimental Psychology 47:3, 523-539).

This appears easy and satisfying solution to teach with: always use unequal variances t-test and use ranks if the data are also not normal.

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