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Re: two-sample nonparametric test on quantiles
Posted:
Feb 3, 2013 12:50 AM
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On Jan 31, 2:30 pm, Rich Ulrich <rich.ulr...@comcast.net> wrote:
>
> By the way -- for years I have been somewhat disparaging
> towards rank tests. Now it turns out that I underestimated
> how bad they could be.
I, too, am much less sanguine about the U-test than I used to be.
Here are some notes I made to myself a few years ago that explain
one of the reasons for my pessimism.
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Regardless of how we estimate its standard deviation, there are
potential problems for any test that takes U to be approximately
normal. Suppose we have a continuous dependent variable in two
populations, say A and B, and that the distribution in each
population is symmetric with the same mean, so that P(A > B) = 1/2.
If population A is even moderately more variable than population B,
and if nA is small and nB is not, then the distribution of U can
look extremely non-normal, especially in the tails. For instance,
consider two logistic populations with zero means and with sA ? 4sB,
where s is the usual logistic scale parameter. If nA = 3 and nB = 30
then the distribution of U will have spikes at 0,30,60,90 whose
frequencies are approximately proportional to 1,3,3,1, with the
87 other values of U between 0 and 90 having smaller frequencies.
(Think of a suspension bridge with towers whose heights are
binomially distributed, with each tower connected to the next by
very saggy too-long cables, but with no cables connecting the end
towers to the shores.)
The reason this happens can be seen by transforming the dependent
variable so that the transformed B-distribution (the one with the
larger sample size and smaller variance) is Uniform(0,1). Then the
transformed A-distribution will generally look something a Beta
variable with alpha = beta < 1. (If the original distributions are
logistic then the transformed pdfA will be r*(x*(1?x))^(r?1) /
(x^r + (1?x)^r)^2, where r = sB/sA.) As sB/sA gets smaller, the
transformed A-distribution becomes more like a simple Bernoulli
variable that takes on only the values 0 and 1, each with
probability 1/2. If nB is not small then the sample distribution
of the transformed B-variable will be approximately uniform, and
U will approximate nB times a Binomial(nA,1/2) variable.
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