# Re: [R] Wilcoxon signed rank test and its requirements

From: Joris Meys <jorismeys_at_gmail.com>
Date: Fri, 25 Jun 2010 00:09:52 +0200

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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