Glen Barnett <[EMAIL PROTECTED]> wrote in message a930vq$25b$[EMAIL PROTECTED]">news:a930vq$25b$[EMAIL PROTECTED]... > > JC <[EMAIL PROTECTED]> wrote in message > [EMAIL PROTECTED]">news:[EMAIL PROTECTED]... > > Does anyone have a suggestion on what procedure to use in order to > > report effect size from a Mann-Whitney?
Here's an example of calculating a point estimate and an interval estimate for the location difference with the Mann-Whitney test. For this calculation, the hypothesis is as follows: Let X, Y be random variables from the two populations under consideration. Let the two populations have the same distribution, apart (perhaps) from location - that is, F(X) = F(Y - theta). This is the usual assumption for the Mann-Whitney - you don't normally need this much if you're just doing the test, but for the estimate and interval I think it's necessary. The two-tailed Mann-Whitney test is then: H0: theta = 0 vs H1: theta <> 0 Consider subtracting a value, t, from the Y sample, y*(i) = y(i) - t. Take as your point estimate of the location difference (theta) that value of t which makes the Mann-Whitney statistic take its expected value under H0. When (as often happens) there is a range of values for which this is true, take the centre of that range of t as your estimate of theta. Now for a 100(1-alpha)% interval estimate: Work out your upper and lower level alpha critical values. Find values t_u and t_l that make the Mann- Whitney equal to those critical values (note that due to discreteness of the Mann-Whitney, you will be restricted to a discrete set of possible confidence levels). Where there's a range of values that satisfy the conditions, if I'm thinking straight (and I may not be right now, as I have an illness) you take the outermost extremes. In each case (point estimate and upper and lower limits) there are simple procedures for finding the values that you need from the sample values, but if you only need to do it once, something like binary search should be quick enough if you have a computer to do the calculation of the Mann-Whitney statistic each time (as long as you take advantage of the fact that the statistic only changes when t is such that the ranks change - there's no point in trying anything smaller). In the case of the point estimate, if I recall correctly (and again, I may not be recalling correctly), the estimate of theta turns out to be the median of pairwise differences, y(i)-x(j). For the interval estimate it's more complicated, but it can be found in a number of texts on nonparametrics if you need to do it a lot. If it's not in Conover's book, it might be in Neave and Worthington. Failing that, try Lehmann's book. [Note that we're simply finding the Hodges-Lehmann estimator and CI for the Mann-Whitney two sample test - the same basic ideas work for just about any test.] Glen . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
