[R] Wilcoxon signed rank test and its requirements

[Joris Meys]

I do agree that one should not trust solely on sources like wikipedia
and graphpad, although they contain a lot of valuable information.

This said, it is not too difficult to illustrate why, in the case of
the one-sample signed rank test, the differences should be not to far
away from symmetrical. It just needs some reflection on how the
statistic is calculated. If you have an asymmetrical distribution, you
have a lot of small differences with a negative sign and a lot of
large differences with a positive sign if you test against the median
or mean. Hence the sum of ranks for one side will be higher than for
the other, leading eventually to a significant result.

An extreme example :

> set.seed(100)
> y <- rnorm(100,1,2)^2
> wilcox.test(y,mu=median(y))

        Wilcoxon signed rank test with continuity correction

data:  y
V = 3240.5, p-value = 0.01396
alternative hypothesis: true location is not equal to 1.829867

> wilcox.test(y,mu=mean(y))

        Wilcoxon signed rank test with continuity correction

data:  y
V = 1763, p-value = 0.008837
alternative hypothesis: true location is not equal to 5.137409

Which brings us to the question what location is actually tested in
the wilcoxon test. For the measure of location to be the mean (or
median), one has to assume that the distribution of the differences is
rather symmetrical, which implies your data has to be distributed
somewhat symmetrical. The test is robust against violations of this
-implicit- assumption, but in more extreme cases skewness does matter.

Cheers
Joris

On Thu, Jun 24, 2010 at 7:40 PM, David Winsemius <dwinsemius at comcast.net> wrote:
>
>
> You are being misled. Simply finding a statement on a statistics software
> website, even one as reputable as Graphpad (???), does not mean that it is
> necessarily true. My understanding (confirmed reviewing "Nonparametric
> statistical methods for complete and censored data" by M. M. Desu, Damaraju
> Raghavarao, is that the Wilcoxon signed-rank test does not require that the
> underlying distributions be symmetric. The above quotation is highly
> inaccurate.
>
> --
> David.
>
>>

-- 
Joris Meys
Statistical consultant

Ghent University
Faculty of Bioscience Engineering
Department of Applied mathematics, biometrics and process control

tel : +32 9 264 59 87
Joris.Meys at Ugent.be
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