[R] Wilcoxon Test and Mean Ratios

[Henric (Nilsson) Winell]

On 2012-09-20 21:07, Thomas Lumley wrote:
> On Fri, Sep 21, 2012 at 6:43 AM, avinash barnwal
> <avinashbarnwal123 at gmail.com> wrote:
>> Hi,
>>
>> http://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test
>>
>> We can clearly see that null hypothesis is median different or not.
>> One way of proving non difference is P(X>Y) = P(X<Y) where X and Y are
>> ordered.
>
>
> Avinash.  No.
>
> Firstly, the Wikipedia link is for the WIlcoxon signed rank test,
> which is a different test and so is irrelevant. Even if the
> signed-rank test were the one being discussed, you are still
> incorrect. The signed rank test is on the median of differences, not
> the difference in medians.  These are not the same, and need not even
> be in the same direction.
>
> Secondly, it is easy to establish that the WIlcoxon rank sum test need
> not agree with the ordering in  medians, just by looking at examples,
> as Peter showed
>
> Thirdly,  there is a well-known demonstration originally due to Brad
> Efron, "Efron's non-transitive dice', which implies that the
> Mann-Whitney U test (which *is* equivalent to the Wilcoxon rank-sum
> test) need not agree with the ordering given by *any* one-sample
> summary statistic.
>
> In this case, assuming the sample sizes are not too small (which looks
> plausible given the p-value), the question is what summary the
> original poster want's to compare: the mean (in which case the t-test
> is the only option) or some other summary.

I'll just chime in here and point towards the Fay and Proschan (2010) 
paper discussing decision rules, and their assumptions, in the 
two-sample situation.  It's freely available at 
http://www.i-journals.org/ss/viewarticle.php?id=51


Henric



> It's not possible to work
> this out from the distribution of the data, so we need to ask the
> original poster.  With reasonably large sample sizes he can get a
> permutation test and bootstrap confidence interval for any summary
> statistic of interest, but for the mean these will just reduce to the
> t-test.
>
> Rank tests (apart from Mood's test for quantiles, which has different
> problems) can really behave very strangely in the absence of
> stochastic ordering, because without stochastic ordering there is no
> non-parametric way to define the direction of difference between two
> samples.  It's important to remember that all the beautiful theory for
> rank tests was developed under the (much stronger) a location shift
> model: the distribution can have any shape, but the shape is assumed
> to be identical in the two groups.  Or, as one of my colleagues puts
> it "you don't know whether the treatment raises or lowers the outcome,
> but you know it doesn't change anything else".
>
> Knowledgeable and sensible statisticians who like the Wilcoxon test
> (Frank Harrell comes to mind) like it because they believe stochastic
> ordering is a reasonable assumption in the problems they work in, not
> because they think you can do non-parametric testing in its absence.
>
>
>     -thomas
>