[R] Wilcoxon Test and Mean Ratios

[Thomas Lumley]

On Thu, Sep 20, 2012 at 5:46 AM, Mohamed Radhouane Aniba
<aradwen at gmail.com> wrote:
> Hello All,
>
> I am writing to ask your opinion on how to interpret this case. I have two vectors "a" and "b" that I am trying to compare.
>
> The wilcoxon test is giving me a pvalue of 5.139217e-303 of a over b with the alternative "greater". Now if I make a summary on each of them I have the following
>
>> summary(a)
>      Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
> 0.0000000 0.0001411 0.0002381 0.0002671 0.0003623 0.0012910
>> summary(c)
>      Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
> 0.0000000 0.0000000 0.0000000 0.0004947 0.0002972 1.0000000
>
> The mean ratio is then around 0.5399031 which naively goes in opposite direction of the wilcoxon test ( I was expecting to find a ratio >> 1)
>

There's nothing conceptually strange about the Wilcoxon test showing a
difference in the opposite direction to the difference in means.  It's
probably easiest to think about this in terms of the Mann-Whitney
version of the same test, which is based on the proportion of pairs of
one observation from each group where the `a' observation is higher.
Your 'c' vector has a lot more zeros, so a randomly chosen observation
from 'c' is likely to be smaller than one from 'a', but the non-zero
observations seem to be larger, so the mean of 'c' is higher.

The Wilcoxon test probably isn't very useful in a setting like this,
since its results really make sense only under 'stochastic ordering',
where the shift is in the same direction across the whole
distribution.

  -thomas

-- 
Thomas Lumley
Professor of Biostatistics
University of Auckland