# When are circular statistics needed (reference)?

I have a set of data of measurements for the lateral angle of the internal acoustic meatus (it's an angle between the canal of our inner ear and another bone structure in our skull). The measurements were taken by three different evaluators at three different points in time. Values vary from 16° to 115° (the anatomy of our skull would not let the angle vary much beyond these values - the literature actually only reports a range from 30° to 70°). The final goal is to see if there are any sex differences in lateral angle.

Previous studies use linear statistics, but my co-authors are adamant that I should use circular statistics (Watson-Williams tests and similar).

Question 1 Are circular statistics always necessary when dealing with angular data? I don't have full circularity, and in itself, direction is not the main question here, but a difference in measured angle. This response from this same forum makes me think that linear statistics are sufficient, but I'd like a fuller explanation as well as a supporting reference (to justify my statistical decision). I saw a few references here, but they seem to dive straight into the statistics.

Question 2 If circular statistics are required, what is the circular equivalent of the intraclass correlation coefficient (ICC) to test for inter and intra-observer reliability? This is the one that seems the most reasonable to my data as my outcome variable is continuous.

Question 2.1 Would it be appropriate to project my data as suggested here, by converting the angles to (sin(x),cos(x)), and thus transforming it to "linear"? But if I do this, I'm not sure how to operate it or interpret it since I will have two outcome variables rather than one? Would it look like this?

• Some quick thoughts: (1) pubmed.ncbi.nlm.nih.gov/18378102 did an ANOVA on their dataset, so at least there is precedent. Do you have normality on the degree measures? (2.1) Since your data are between 0 and 180 degrees, cosine alone would be sufficient I would think, however, it would tend to squeeze the outliers closer to the middle. Or you could subtract the mean so the values center at 0 degrees and range between +/-90 degrees and use sine. – Ron Jensen Apr 9 at 20:19
• Thank you @RonJensen for such a quick response. (1) Yes, there is plenty of precedent, but all papers I read just went ahead and used linear statistics without dwelling on the angular nature of their data. I wanted to understand the justification to do (or not do) this, My data is skewed right with outliers for higher values, although I have well over 2000 data points. I've been using robust statistics using trimmed means and bootstrap to deal with that (from WRS2 package in R) but of course it will depend on the circular bit. 2.1 That would solve the lack of normality, I think? Thank you – answer42 Apr 9 at 20:38