On Fri, 13 Feb 2009, David Winsemius wrote:
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.
One can, and it may be helpful to do so, as long as one knows it isn't actually
true. Unfortunately, some text books claim or strongly imply it is true.
To make the test consistent for differences in the median you have to know in
advance that the distributions differ only by a location shift, and then it is
also consistent for differences in mean (or in any other location parameter).
Also, the operating characteristics aren't particularly similar to a real test
for medians, which has pretty low efficiency at the Normal location-shift model
(2/pi, IIRC) and is much more sensitive to ties in the data.
And I could go on and on about non-transitivity, but I won't. Anyone who is
interested can Google for 'Efron dice'.
-thomas
Thomas Lumley Assoc. Professor, Biostatistics
tlum...@u.washington.edu University of Washington, Seattle
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