On 2 Jan 2003 13:42:22 -0800, [EMAIL PROTECTED] (Donald Burrill) wrote: > I suspect that something like an assumption of symmetry may be needed, > on the following argument: > > Assumptions generally deal with the situation if the null hypothesis be > true: in this case, if the two subgroup medians are equal. If the > subgroups were not symmetrical and had the same median, how badly could > the test go wrong?
I'm confused. I see several notes about symmetry - are these attempting to preserve or defend or qualify the Rank-transformation as providing a test on the *median*? That is not an explicit property of MW .... The MW test is a test on the average rank, as we all agree. - I have long been satisfied by regarding the tests as tests for "stochastic superiority". What happens with the medians can be inconsistent, if you push the examples. The example given below might as well be symmetric around the *mean*. With 1st and 3rd quartiles belonging to one sample, etc., the two samples do seem to fit a shift-model that can be tested, with fairly good power, after the rank transformation. However, Don says, "symmetry around the median"; is that a conventional qualifier that has escaped my awareness? It seems obvious to me that when comparing symmetric distributions with vastly different variances, where the variance has to reflect the effect of a Scaling difference, which is above and beyond the Location difference, then simple means testing, of ranks or otherwise, is not going to capture the whole difference. - back to the original questioner - It seems to me that symmetry of the samples has little to do with practical applications of the tests. > > The worst case I can conjure up on the spur of the moment is this: > suppose the two groups to be equal in size, and that we imagine the > total set of data to be divided into quarters at the overall quartiles; > and suppose Group A consists of the 1st and 3rd quarters while Group B > comprises the 2nd and 4th quarters. A and B will have the same median > (or if not, we can arrange that by swapping at most two observations > between the groups, in what I suppose to be an obvious way), so the null > hypothesis is true in this case. > Now for such a case I rather suspect that the sum of the ranks on which > the test is based might be interestingly far from the value expected > under an assumption of symmetry about the median for each group. > (Haven't tried to demonstrate that with fictitious data, but leave it as > an exercise for an interested reader.) > > What I cannot tell (on the spur of the moment) is whether this situation > can reasonably be included in the "100*alpha %" of cases for which one > expects to reject the null hypothesis falsely; but surely if one had > reason to expect some such distribution on a systematic basis that > reason would be sufficient to invalidate one's calculation of any > p-value based on the standard assumption(s). > > Comments, anyone? -- Don. [snip, preceding posts] -- Rich Ulrich, [EMAIL PROTECTED] http://www.pitt.edu/~wpilib/index.html . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
