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Rayleigh's Uniformity Test (RUT), when run on a circular distribution C, returns p ~= 0. Then, we can confidently assert that the data was not uniformly distributed around a circle but can we make any predictions on whether the distribution was unimodal or multimodal? In light of the fact that RUT has less power to detect multimodal distributions...

A practical case — Let's say, I am working with a hypothetical animal that shows trimodal bursts of activity centered around 12 PM, 3 PM, and 1 AM. Under my experimental conditions, I show by an angular histogram plot (visually) that for two such animals — out of hundred —, that the activity-distribution is relatively far dense only around the 3 PM mark. Now, can I simply run RUT for the rest of the animals and cite the extremely low p-value? Will it be a sufficient proof of experimental unimodality over the usual multimodality of the animal?

AvadaMouse
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    Why should you wish to ignore some of your data? I am not a biologist but it's no surprise at all that 2 / 100 animals might behave differently from the other 98. There's no statistical basis for ignoring awkward complications in your data. It's for you as a scientist to report them as details. – Nick Cox Nov 05 '23 at 14:54

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The RUT has uniformity as null hypothesis and unimodality as alternative. As all tests, it serves to distinguish the null hypothesis from the alternative, but it was not made to distinguish the underlying "model" (by which I mean the union of null hypothesis and alternative) from anything else. So it will not give information about unimodality vs. multimodality. Your angular histogram is informative about this but the test is not.

Nick Cox
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Christian Hennig
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  • Agreed. Multimodality for circular data, as for any kind of data, is best examined by linking graphical analysis (that doesn't have to be a histogram) with a substantive story. Whether it is traffic counts in terms of time of day or wind direction at a coastal station or anything else, is there process understanding that makes sense of two or more peaks? – Nick Cox Nov 05 '23 at 12:31
  • @NickCox I do. Under usual circumstances, the data won't be uniform but multimodal. Under my experimental condition, I suspect the data to be unimodal around 45°. Indeed a few plots reveal my expectations to be correct but how to go about from a statistical pov? Essentially, I need some test that distinguishes between multimodal and unimodal data. Proving just that the data is nonuniform won't cut because it always is. – AvadaMouse Nov 05 '23 at 13:16
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    @AvadaMouse If I had this problem I'd do some research on existing tests for unimodality and see whether they generalise easily to your situation. Maybe one can do something based on bootstrap, but I don't think there's a standard test for unimodality for circular data. Note however that any test will be invalid if you choose it based on what you have seen in the data already, i.e., you can't run a valid test if you only think you should test unimodality because the data look like they are unimodal (unless you run it on new data). – Christian Hennig Nov 05 '23 at 13:30
  • Being multimodal isn't a well-defined condition and hard to distinguish from uniformity in practice. What's the barrier here? A reviewer or supervisor who insists on seeing a significance test? Which significance test would that be? would be my answer to a reviewer. (I don't have a supervisor any more.) – Nick Cox Nov 05 '23 at 13:37
  • @NickCox There is quite some interest and literature on tests for clustering against homogeneity, which often translates into testing unimodality. In fact multimodality can be distinguished well from unimodality if the modes are strong and well separated. – Christian Hennig Nov 05 '23 at 13:45
  • @ChristianHennig Your last if ... is precisely my point too. But if no-one can identify a test that applies directly to the OP's question, they aren't better off, as their question is about where to draw the line. – Nick Cox Nov 05 '23 at 14:26