# Why are ties so difficult in nonparametric statistics?

My nonparametric text, Practical Nonparametric Statistics, often gives clean formulas for expectations, variances, test statistics, and the like, but includes the caveat that this only works if we ignore ties. When calculating the Mann-Whitney U Statistic, it is encouraged that you throw out tied pairs when comparing which is bigger.

I get that ties don't really tell us much about which population is bigger (if that's what we're interested in) since neither group is bigger than the other, but it doesn't seem like that would matter when developing asymptotic distributions.

Why then is it such a quandary dealing with ties in some nonparametric procedures? Is there a way of extracting any useful information from ties, rather than simply throwing them away?

EDIT: In regards to @whuber's comment, I checked my sources again, and some procedures use an average of ranks instead of dropping the tied values completely. While this seems more sensible in reference to retaining information, it also seems to me that it lacks rigor. The spirit of the question still stands, however.

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Are you saying that Practical Nonparametric Statistics tells you to "throw out" data when they are tied? Could you perhaps be mis-interpreting its advice? Could you quote it exactly? –  whuber May 31 '11 at 16:59
Yes, it is possible that I am misinterpreting the advice. From the same author: jstor.org/stable/2284536 "Wilcoxon suggested dropping the zeros from the data initially, and performing the test on the reduced set of data. If there are no nonzero ties this procedure results in a conditionally (given the number of zeros) distribution free test, and enables the existing exact tables of critical values to be used. For this reason, most books on nonparametric statistics incorporate Wilcoxon's method into their description of the test" –  Christopher Aden May 31 '11 at 17:43
Granted, this is in reference to the Wilcoxon Signed Rank test, but I have heard similar advice used in other NP procedures. In regards to the Mann-Whitney example, I went back and checked in the book, and you are correct that I am mistaken. With Mann-Whitney, the book recommends averaging the ranks of the tied values, ie: if ranks 6 and 7 are tied, giving each one a value of 6.5. –  Christopher Aden May 31 '11 at 18:29
Thank you. There are rigorous ways to account for tied groups. They are important when working with censored (but continuous) data, because frequently the censored values constitute a large tied group. For the Kruskal-Wallis and Wilcoxon Rank Sum tests, see chapter 18 of R. O. Gilbert, *Statistical Methods for Environmental Pollution Monitoring." Formulas involving tied data can get complicated, but in some cases (like the K-W test) all you need to do is compute an ANOVA table for the ranks. –  whuber May 31 '11 at 19:17