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st: RE: When is the t-test appropriate?


From   "Nick Cox" <n.j.cox@durham.ac.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: RE: When is the t-test appropriate?
Date   Thu, 29 Aug 2002 20:06:25 +0100

Jon Wainwright
 
> I would like to compare the means of two very small random 
> samples (n1=3,
> n2=7). Both samples were drawn from populations of unknown 
> distribution. Is
> Stata's ttest appropriate in such a situation, or does it 
> require the
> underlying populations to be normally distributed?  If ttest is not
> appropriate, can anyone suggest are more appropriate method 
> for testing the
> difference of the means?

Lots of answers are possible on different levels. Whatever 
one says in one sense is likely to be wrong in another, 
but here's my take: 

1. Supposedly, the t test takes account of your sample 
sizes. It is possible that your data are such that the answer 
is clearcut even with very small sizes, but perhaps 
more likely is that you will fail to get a significant 
result (at conventional levels). A common but not universal 
interpretation of that would be that it indicates not so much 
that there is no difference, but more that the sample sizes 
are not large enough to indicate it reliably. Some would 
say that this is the main (or even only) role of a significance
test. 

2. At the same time, there is a underlying assumption 
of normality, whatever the sample size. Nevertheless, 
there are many theoretical results, much practical
experience and a lot of folklore to suggest that 
moderate non-normality may not be a problem. (Naturally, 
what is "moderate"?) In addition, other assumptions 
such as equal variances and independence may be much 
more critical. For an outstanding discussion with much more
precision, see Rupert G. Miller's "Beyond ANOVA". 

3. Calculate confidence intervals and look at them
graphically. There are various possible warning signs, for 
example if one confidence limit is an unattainable 
value (e.g. physically, biologically, economically 
impossible) then you have an enigma wrapped inside 
a riddle within a mystery, or whatever it was that 
Winston Churchill said on a totally different topic. 

4. Bootstrap! 

5. As Ricardo Ovaldia suggested, you could try a 
different approach, such as Mann-Whitney. That, 
however, tests a different hypothesis and says 
nothing directly about means.

6. Show us the data. 

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

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