[BioC] Minimal group sizes in permutation tests

Claus-Dieter Mayer claus at bioss.ac.uk
Tue Nov 13 15:04:43 CET 2007

```Hi Benjamin,

Benjamin Otto wrote:
> Hey all,
>
> 1. For statistical tests there are usually minimal group sizes recommended
> for appropriate working. For a chi-square test as example the lower level
> was 10 obersvations in each field of the table, if I remember correctly.
> What about permutation tests? Is there some kind of minimal recommendation
> for group sizes? I can't find any hint on that.
>
The group sizes determine how many different possible permutations they
are, eg. with 3 samples each in 2 groups you only have 20 permutations.
If you would use a 1-sided permutation-test in that situation the
smallest possible p-value thus would be 5%, i.e. you have no chance to
ever find a significant result at a 5% level (for a 2-sided test you
wouldn't even be able to get below 10%). In my opinion the number of
different permutations should be at least in the hundreds, so for a
2-group comparison I wouldn't use a permutation test for anything less
than 5 per group (in which case you have 252 permutations). For other
designs you would have to calculate the number of possible permutations
to see whether it makes sense.
Apart from that I see little other constraints in using a permutation
test as long as you are sure that under the nullhypothesis you are
testing the variables are "i.i.d" (= independently indentically distributed)
> 2. As far as I understand the permutation p-value is given by the quantile
> describing the position of the native p-value in the permutation p-value
> distribution. So for 100 permutations and 5 values smaller than the native
> one the new p-value would be 0.05. What happens when the original p-value is
> the absolut minimum? Is such a thing like p-value equals zero defined?
>
The classical definition of the p-value calculates the probability of
observing an outcome as extrem or even more extrem as the observed one.
So if the observed value is the smalles of the 100 permutations your
p-value would 1%.
> 3. Given a design of 3x3 samples (20 permutations), will the test return
> reasonable values? Doesn't look like it to me.
>
for 2x3 samples it would be 20 permutations, but for 3x3 the number will
be bigger

Claus
> Best regards,
>
> Benjamin Otto
>
>
> ======================================
> Benjamin Otto
> University Hospital Hamburg-Eppendorf
> Institute For Clinical Chemistry
> Martinistr. 52
> D-20246 Hamburg
>
> Tel.: +49 40 42803 1908
> Fax.: +49 40 42803 4971
> ======================================
>
>
>
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--
***********************************************************************************
Dr Claus-D. Mayer                    | http://www.bioss.ac.uk
Biomathematics & Statistics Scotland | email: claus at bioss.ac.uk
Rowett Research Institute            | Telephone: +44 (0) 1224 716652
Aberdeen AB21 9SB, Scotland, UK.     | Fax: +44 (0) 1224 715349

```