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

From: David Winsemius <dwinsemius_at_comcast.net>
Date: Thu, 24 Jun 2010 18:51:43 -0400

On Jun 24, 2010, at 6:42 PM, Joris Meys wrote:

> On Fri, Jun 25, 2010 at 12:17 AM, David Winsemius
> <dwinsemius_at_comcast.net> wrote:
>>
>> On Jun 24, 2010, at 6:09 PM, Joris Meys wrote:
>>
>>> 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,
>>
>> That is a key point. I was assuming that you were using the paired
>> sample
>> version of the WSRT and I may have been misleading the OP. For the
>> one-sample situation, the assumption of symmetry is needed but for
>> the
>> paired sampling version of the test, the location shift becomes the
>> tested
>> hypothesis, and no assumptions about the form of the hypothesis are
>> made
>> except that they be the same.
>
> I believe you mean the form of the distributions. The assumption that
> the distributions of both samples are the same (or similar, it is a
> robust test) implies that the differences x_i - y_i are more or less
> symmetrically distributed. Key point here that we're not talking about
> the distribution of the populations/samples (as done in the OP) but
> about the distribution of the difference. I may not have been clear
> enough on that one.

What I meant about different hypotheses was that in the single sample case the H0 was mean (or median) = mu_pop and in the paired two sample the H0 was mean(distr_A_i - distr_B_1) =0. And yes, I did miss the OP's point. My apologies.

-- 
David.

>
> Cheers
> Joris
>
>> Any consideration of median or mean (which
>> will be the same in the case of symmetric distributions) gets lost
>> in the
>> paired test case.
>>
>> --
>> David.
>>
>>
>>> 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
>>> -------------------------------
>>> 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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