# Test for difference of distributions on a torus

I have two circular dependent variables and would like to test for a difference in the distributions (presumably circular means) between multiple treatment groups. There are a number of multivariate tests such as MANOVA that will work on $DV \in \mathbb{R}^2$ but I'm looking for a test of differences of means (or something) on $DV \in \mathbb{S}\times\mathbb{S}$. If it can generalize to higher orders spaces (e.g., $\mathbb{S}\times\mathbb{S}\times...\times\mathbb{S}$) that would be a plus.

The test employed should consider the multivariate interactions, so we can't use something like a Kuiper test from circular statistics on each DV by itself.

We've considered transforming the data into some linear metric to use in a test, such as a Wilcoxon rank-sum on within vs. across condition cosines. However, it would be greatly preferred if we can cite an article describing the test rather than describe it ourselves.

As an example dataset consider hatching fruit flies. We measure the time of hatching and the direction towards which they initially fly away. We then want to look for an effect of a drug applied to the eggs.

• More information about the data would be welcome: what type they have (which is all you have revealed so far) often is of less consequence than what they represent, how they are measured, what their distributions could be, and how the analysis is to be interpreted.
– whuber
Commented Dec 6, 2012 at 6:42
• There's really not much more to it. It's two directional measures that are orthogonal. Commented Dec 6, 2012 at 16:56
• Then there's not much we can go on. For instance, because $\mathbb{C}^2$ is the same as $\mathbb{R}^4$ as far as the numbers are concerned, you don't have "circular" variables at all and you aren't working on a torus. Given the information you have supplied, all we know is that you want to test for a difference in multivariate distributions and the only special aspect is that the number of variables is even. That's too broad to be answerable. Can't you be any more specific?
– whuber
Commented Dec 6, 2012 at 18:31
• I'm not sure what you mean by "as far as the numbers are concerned" but $\mathbb{C}^2$ is not homeomorphic to $\mathbb{R}^4$ although $\mathbb{C}^2$ is a subspace of $\mathbb{R}^4$. It's also true that $\mathbb{S}^2$ is a subspace of $\mathbb{R}^3$, but that doesn't mean they are the same. There is an entire field of circular statistics designed to handle univariate data of this (circular) type. I'm just looking for a multivariate expansion. The data are fairly complicated, but I'll try to come up with a concise explanation. Commented Dec 6, 2012 at 19:06
• For statistical applications, homeomorphism is not a relevant concept. This is not a topology site! And yes, readers here know about circular statistics. My point is that you haven't told us anything that distinguishes your situation from a $2n$ (real) variate problem, even though it appears there may be distinguishing factors. (E.g., your use of "torus" suggests your data are supported on a product of circles.)
– whuber
Commented Dec 6, 2012 at 19:16