# [R] Bootstrap or Wilcoxons' test?

David Winsemius dwinsemius at comcast.net
Sat Feb 14 08:09:09 CET 2009

```The Wilcoxon rank sum test is not "plain and simple a test equality of
distributions". If it were such, it would be able to test for
differences in variance when locations were similar. For that purpose
it would, in point of fact, be useless. Compare these simple
situations w.r.t. the WRS:

> x <- rnorm(100)  # mean=0, sd=1
> y <- rnorm(100, mean=0, sd=4)
> wilcox.test(x,y)

Wilcoxon rank sum test with continuity correction

data:  x and y
W = 4518, p-value = 0.2394
alternative hypothesis: true location shift is not equal to 0

> y <- rnorm(100, mean=.2, sd=0)
>
> wilcox.test(x,y)

Wilcoxon rank sum test with continuity correction

data:  x and y
W = 3900, p-value = 0.004079
alternative hypothesis: true location shift is not equal to 0

It is a test of the equality of location (and the median is a readily
understood non-parametric measure of location). The test is derived
under the *assumption* that the samples are drawn from the *same*
distribution differing only by a shift. If the distributions were not
of the same family, the test would be invalidated. The wilcox.test
help page is informative, saying "the null hypothesis is that the
distributions of xand y differ by a location shift of mu". The
pseudomedian is optionally estimated when conf.int is set to TRUE. I
also suggest looking at the formula for the statistic. It is available
with getAnywhere(wilcox.test.default).

If one wants a test for "equality of distribution", one could turn to
a more general test (with loss of power but with at least some
potential for detecting differences in dispersion) such as the
Kolmogorov-Smirnov or Kuiper tests. With x and y as above:

> ks.test(x,y)

Two-sample Kolmogorov-Smirnov test

data:  x and y
D = 0.61, p-value < 2.2e-16
alternative hypothesis: two-sided

Warning message:
In ks.test(x, y) : cannot compute correct p-values with ties

Returning to the OP's question, rather than worrying about normality
in samples, the greater threat to validity in regression methods is
unequal variances across groups or the range of continuous predictors.

--
David Winsemius

On Feb 13, 2009, at 11:12 PM, Murray Cooper wrote:

> First of all, sorry for my typing mistakes.
>
> Second, the WRS test is most certainly not a test for unequal medians.
> Although under specified models it would be. Just as under specified
> models it can be a test for other measures of location. Perhaps I
> did not
> word my explanation correctly, but I did not mean to imply that it
> would
> be a test of equality of variance. It is plain and simple a test for
> the equality
> of distributions. When the results of a properly applied parametric
> test do
> not agree with the WRS, it is usually do to a difference in the
> empirical
> density function of the two samples.
>
> Murray M Cooper, Ph.D.
> Richland Statistics
> 9800 N 24th St
> Richland, MI, USA 49083
>
> ----- Original Message ----- From: "David Winsemius" <dwinsemius at comcast.net
> >
> To: "Murray Cooper" <myrmail at earthlink.net>
> Cc: "Charlotta Rylander" <zcr at nilu.no>; <r-help at r-project.org>
> Sent: Friday, February 13, 2009 9:19 PM
> Subject: Re: [R] Bootstrap or Wilcoxons' test?
>
>
>> I must disagree with both this general characterization of the
>> Wilcoxon test and with the specific example offered. First, we
>> ought  to spell the author's correctly and then clarify that it is
>> the  Wilcoxon rank-sum test that is being considered. Next, the WRS
>> test is  a test for differences in the location parameter of
>> independent  samples conditional on the samples having been drawn
>> from the same  distribution. The WRS test would have no
>> discriminatory power for  samples drawn from the same distribution
>> having equal location  parameters but only different with respect
>> to unequal dispersion. Look  at the formula, for Pete's sake. It
>> summarizes differences in ranking,  so it is in fact designed NOT
>> to be sensitive to the spread of the  values in the sample. It
>> would have no power, for instance, to test  the variances of two
>> samples, both with a mean of 0, and one having a  variance of 1
>> with the other having a variance of 3.  One can think of  the WRS
>> as a test for unequal medians.
>>
>> --
>> David Winsemius, MD. MPH
>> Heritage Laboratories
>>
>>
>> On Feb 13, 2009, at 7:48 PM, Murray Cooper wrote:
>>
>>> Charlotta,
>>>
>>> I'm not sure what you mean when you say simple linear
>>> regression. From your description you have two groups
>>> of people, for which you recorded contaminant concentration.
>>> Thus, I would think you would do something like a t-test to
>>> compare the mean concentration level. Where does the
>>> regression part come in? What are you regressing?
>>>
>>> As for the Wilcoxnin test, it is often thought of as a
>>> nonparametric t-test equivalent. This is only true if the
>>> observations were drawn, from a population with the
>>> same probability distribution. The null hypothesis of
>>> the Wilcoxin test is actually "the observations were
>>> drawn, from the same probability distribution".
>>> there means could be the same, but since the variances
>>> are different, the Wilcoxin could give you a significant result.
>>>
>>> Don't know if this all makes sense, but if you have more
>>> description of what analysis you used and I'd be happy
>>> to try and help out.
>>>
>>> Murray M Cooper, Ph.D.
>>> Richland Statistics
>>> 9800 N 24th St
>>> Richland, MI, USA 49083
>>>
>>> ----- Original Message ----- From: "Charlotta Rylander"
>>> <zcr at nilu.no>
>>> To: <r-help at r-project.org>
>>> Sent: Friday, February 13, 2009 3:24 AM
>>> Subject: [R] Bootstrap or Wilcoxons' test?
>>>
>>>
>>>> Hi!
>>>>
>>>>
>>>>
>>>> I'm comparing the differences in contaminant concentration
>>>> between 2
>>>> different groups of people ( N=36, N=37). When using a simple
>>>> linear
>>>> regression model I found no differences between groups, but when
>>>> evaluating
>>>> the diagnostic plots of the residuals I found my independent
>>>> variable to
>>>> have deviations from normality (even after log transformation).
>>>> Therefore I
>>>> have used bootstrap on the regression parameters ( R= 1000 &
>>>> R=10000) and
>>>> this confirms my results , i.e., no differences between groups
>>>> ( and the
>>>> distribution is log-normal). However, when using wilcoxons' rank
>>>> sum test on
>>>> the same data set I find differences between groups.
>>>>
>>>>
>>>>
>>>> Should I trust the results from bootstrapping or from wilcoxons'
>>>> test?
>>>>
>>>>
>>>>
>>>> Thanks!
>>>>
>>>>
>>>>
>>>> Regards
>>>>
>>>>
>>>>
>>>> Lotta Rylander
>>>>
>>>>
>>>> [[alternative HTML version deleted]]
>>>>
>>>> ______________________________________________
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>>>>
>>>
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