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
Mail: richs...@earthlink.net
----- Original Message ----- From: "David Winsemius" <dwinsem...@comcast.net
>
To: "Murray Cooper" <myrm...@earthlink.net>
Cc: "Charlotta Rylander" <z...@nilu.no>; <r-help@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".
Thus if your two samples had say different variances,
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
questions, please e-mail your data and a more detailed
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
Mail: richs...@earthlink.net
----- Original Message ----- From: "Charlotta Rylander"
<z...@nilu.no>
To: <r-help@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]]
______________________________________________
R-help@r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
______________________________________________
R-help@r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
______________________________________________
R-help@r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
