
Re: twosample nonparametric test on quantiles
Posted:
Feb 3, 2013 12:50 AM


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 Utest 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. ____________________________________________________________________
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 nonnormal, 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 toolong 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 Bdistribution (the one with the larger sample size and smaller variance) is Uniform(0,1). Then the transformed Adistribution 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 Adistribution 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 Bvariable will be approximately uniform, and U will approximate nB times a Binomial(nA,1/2) variable.

