[R] Wilcoxon signed rank test and its requirements

[S Ellison]

There is a potentially useful remark from Peter Dalfgaard at
http://www.mail-archive.com/r-help@stat.math.ethz.ch/msg86359.html :

Summarising:
"[The Wilcoxon paired rank sign test assumes symmetry]  
...of differences, and under the null hypothesis. This is usually
rather uncontroversial. "


My rider to this: It's uncontroversial because differences between
random samples from the same asymmetric distribution would form a
symmetric distribution of differences, and the null for the wilcoxon is
essentially that the distributions are the same. Symmetry of differences
at the null follows.
BUT the corollary is that location might not be the only thing that can
cause a wilcoxon test to show a significant difference. 

set.seed(1023)

x<-rlnorm(50)
z<-rlnorm(50, sdlog=3)
z<-z-mean(z)+mean(x)

mean(x)
mean(z)
#Same mean..

wilcox.test(x,z)
#Strongly significant test result.

#Not a perfect example, as the test relates to true means, not data set
means. 
#But very different skew and scale will make for a very significant
test result as well as very different means



On Thu, Jun 24, 2010 at 4:16 AM, Atte Tenkanen <attenka at utu.fi> wrote:
> PS.
>
> Mayby I can somehow try to transform data and check it, for example,
using the skewness-function of timeDate-package?
>
>> Thanks. What I have had to ask is that
>>
>> how do you test that the data is symmetric enough?
>> If it is not, is it ok to use some data transformation?
>>
>> when it is said:
>>
>> "The Wilcoxon signed rank test does not assume that the data are
>> sampled from a Gaussian distribution. However it does assume that
the
>> data are distributed symmetrically around the median. If the
>> distribution is asymmetrical, the P value will not tell you much
about
>> whether the median is different than the hypothetical value."
>>
>> > On Wed, Jun 23, 2010 at 10:27 PM, Atte Tenkanen <attenka at utu.fi>
wrote:
>> > > Hi all,
>> > >
>> > > I have a distribution, and take a sample of it. Then I compare
>> that
>> > sample with the mean of the population like here in "Wilcoxon
signed
>>
>> > rank test with continuity correction":
>> > >
>> > >> wilcox.test(Sample,mu=mean(All), alt="two.sided")
>> > >
>> > >        Wilcoxon signed rank test with continuity correction
>> > >
>> > > data:  AlphaNoteOnsetDists
>> > > V = 63855, p-value = 0.0002093
>> > > alternative hypothesis: true location is not equal to 0.4115136
>> > >
>> > >> wilcox.test(Sample,mu=mean(All), alt = "greater")
>> > >
>> > >        Wilcoxon signed rank test with continuity correction
>> > >
>> > > data:  AlphaNoteOnsetDists
>> > > V = 63855, p-value = 0.0001047
>> > > alternative hypothesis: true location is greater than 0.4115136
>> > >
>> > > What assumptions are needed for the population?
>> >
>> > wikipedia says:
>> > "The Wilcoxon signed-rank test is a _non-parametric_ statistical
>> > hypothesis test for... "
>> > it also talks about the assumptions.
>> >
>> > > What can we say according these results?
>> > > p-value for the "less" is 0.999.
>> >
>> > That the p-value for less and greater seem to sum up to one, and
that
>> > the p-value of greater is half of that for two-sided. You
shouldn't
>> > ask what we can say. You should ask yourself "What was the
question
>> > and is this test giving me an answer on that question?"
>> >
>> > Cheers
>> > Joris
>> >
>> > --
>> > 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 
>> > -------------------------------
>> > Disclaimer : http://helpdesk.ugent.be/e-maildisclaimer.php 
>



-- 
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
-------------------------------
Disclaimer : http://helpdesk.ugent.be/e-maildisclaimer.php 

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