# Testing the difference of (some) quantile-Q between groups?

For some Y variable, that is divided into 3 groups (X), I wish to compare the groups and for the hypothesis that the 90% quantile is the same between all three groups. What tests can I use?

One option I can think of is using quantile regression, are there other alternatives/approuches?

I imagine that if I had wanted to compare the median, I could have used the kruskal wallis test (though it is based on ranks, but if I remember correctly, it would give the same results when the residual distribution is symmetric)

Thanks.

## 2 Answers

You are right with word "median" on your mind, albeit Kruskal-Wallis is not the test for medians. What you need is median test. It tests (asymptotically by chi-square or exactly by permutations) whether several groups are same in regard to ratio of observations falling above/not above some value. By default, median of the combined sample is taken for that value (and hence is the name of the test, which is then the test for equality of population medians). But you could specify another value than median. Any quantile will do. The test then will compare groups in regard to the proportion of cases that fall not above the quantile.

• Thanks ttnphns, I forgat about the median test - you are right, I could use that. Regarding the kruskal wallis, as I've written - I know it is a test for the ranks. But if I remember correctly, there are some cases where its results are also valid for the median, is it not? – Tal Galili Sep 29 '11 at 15:30
• Mann-Whitney and its extension to several groups, Kruskal-Wallis, is the test of "location". "Location" (quotes are intentional because different statisticians differently define it) is, vaguely, a nonparametric counterpart of concept "mean" (rather than median): you may look in Wikipedia on Mann-Whitney - the key words there are "stochastically greater" and "Hodges-Lehmann" – ttnphns Sep 29 '11 at 15:42
• Interesting, I see how the Wikipedia page says that the test is for comparing medians... So should it say comparing the mean ranks? en.wikipedia.org/wiki/… – Tal Galili Sep 29 '11 at 16:01
• Not medians. Mann-Whitney can be significant when group medians are equal. Thus, it's not test of medians, generally. It's the test of "stochastical prevalence" or that Hodges-Lehmann difference estimate (HL) is 0. Difference in mean rank (DMR)? I think, it's almost correct. I once computed HL and DMR for many simulated pairs of samples and found they correlate almost linearly with r almost 1. – ttnphns Sep 29 '11 at 16:14
• Thanks ttnphns - so this clarifies for me why I had this in my head - but also that it is something to check more into... – Tal Galili Sep 29 '11 at 16:30

There's an approach for comparing all quantiles of two groups simultaneously:

Simultaneously compare all of the quantiles to get a global sense of where the distributions differ and by how much. For example, low scoring participants in group 1 might be very similar to low scoring participants in group 2, but for high scoring participants, the reverse might be true.

(taken from a script by Rand R. Wilcox)

The method was derived 1976 by Doksum and Sievers, and is implemented as the sband function in the WRS package for R. The method gives a comparison of all quantiles while controlling for overall $\alpha$ error.

However, you can only compare two groups at once. Maybe you can do pairwise comparisons by adjusting for $\alpha$ inflation.