Wilcoxon Test and Mean Ratios
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