[R] help comparing two median with R
Thomas Lumley
tlumley at u.washington.edu
Wed Apr 18 01:43:00 CEST 2007
On Tue, 17 Apr 2007, Frank E Harrell Jr wrote:
> The points that Thomas and Brian have made are certainly correct, if one is
> truly interested in testing for differences in medians or means. But the
> Wilcoxon test provides a valid test of x > y more generally. The test is
> consonant with the Hodges-Lehmann estimator: the median of all possible
> differences between an X and a Y.
>
Yes, but there is no ordering of distributions (taken one at a time) that
agrees with the Wilcoxon two-sample test, only orderings of pairs of
distributions.
The Wilcoxon test provides a test of x>y if it is known a priori that the
two distributions are stochastically ordered, but not under weaker
assumptions. Otherwise you can get x>y>z>x. This is in contrast to the
t-test, which orders distributions (by their mean) whether or not they are
stochastically ordered.
Now, it is not unreasonable to say that the problems are unlikely to occur
very often and aren't worth worrying too much about. It does imply that it
cannot possibly be true that there is any summary of a single distribution
that the Wilcoxon test tests for (and the same is true for other
two-sample rank tests, eg the logrank test).
I know Frank knows this, because I gave a talk on it at Vanderbilt, but
most people don't know it. (I thought for a long time that the Wilcoxon
rank-sum test was a test for the median pairwise mean, which is actually
the R-estimator corresponding to the *one*-sample Wilcoxon test).
-thomas
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