[BioC] Wilcoxon test [was loged data or not loged previous to use normalize.quantile]

Claus Mayer claus at bioss.ac.uk
Wed Apr 6 19:07:49 CEST 2005

Two comments on this discussion:

Gordon Smyth wrote:

> Over many years as a statistician, I've heard it said so many times 
> "the variances were not equal so I used a Wilcoxon two-sample test 
> instead of a t-test" or "I used a rank test which is assumption free". 
> Like Naomi, I find it frustating that this misunderstanding is so 
> common. The fact is that all tests make some assumptions, and 
> inequality of population variances under the null hypothesis breaks 
> the Wilcoxon test just as it does the pooled t-test. I don't know 
> which test breaks down more quickly -- I certainly haven't seen any 
> evidence that the Wilcoxon test is more robust than the t-test to 
> inequality of variances.

I guess the explanation for this misunderstanding is that there are (at 
least) two different kinds of variance heterogeneity. The first one is 
that caused by relationship between mean and variance, which can be 
often removed by an appropriate transformation (e.g. log-transformation 
or VSN in the case of microarray intensities). In this case the Wilcoxon 
test or any other rank test really is robust against this variance 
heterogeneity, because it is invariant under any transformation.

The second type of variance heterogeneity is the one Gordon talks about, 
i.e. differences in variances that can also occur under the 
nullhypothesis of equal means. This problem cannot be resolved by a 
transformation and thus there is no obvious reason why rank tests should 
be more robust/better in this case.

>> With respect to permutations tests...
>> I'm under the impression that you only need independence, not the 
>> assumption of
>> constant variance.
> No, independence is not enough, as you say yourself in the next sentence.

Pollard and van der Laan have investigated resampling tests under 
variance inequality. Although permutation methods do not give exact 
control of type 1 errors for unequal variances (or non independently 
identically distributed data in general), they control this error 
asymptotically for balanced sample sizes. A simulation study in their 
paper confirmed that, so as long as the sample sizes are balanced you 
might not be that bad off with using a permutation method. Otherwise 
bootstrapping (within group based on the residuals) should be preferred



 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

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