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?

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    $\begingroup$ 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. $\endgroup$
    – Ron Jensen
    Apr 9 at 20:19
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    $\begingroup$ 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 $\endgroup$
    – answer42
    Apr 9 at 20:38

Circular statistics are useful to describe random processes with a rotational symmetry or a modular arithmetic such that multiple additions can result in an identity. E.g. for a clock, if you add twelve times an hour then you return at the same point.

This is not relevant for 'the lateral angle of the internal acoustic meatus' (you might want to simplify that term. I had to look it up). I can not imagine that there is a random process that causes this angle to make a full turn. The angle will be around some mean value and due to random processes it might be slightly tilted in one way or the other, but not a full circle.

What you might want to consider is use an appropriate transformation of the raw data. This decision depends mostly on knowledge of the field (how/why does this angle vary, and what is a suitable variable to describe it), and not so much on something standard in statistics (e.g. like Fisher transformation or the arcsine transformation to compute Cohen's H).

  • $\begingroup$ Note for values on the range stated cosine will stretch higher values further apart and is thus unlikely to help. Also, as cosine decreases with angle, negating it might be a better idea -- which doesn't solve the first problem. $\endgroup$
    – Nick Cox
    Apr 10 at 17:14
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    $\begingroup$ Thank you for your input Sextus Empiricus. Tried to explain the lateral angle without a lot of anatomical jargon. No, the angle will never have a full turn. That would not be pleasant for its bearer. From the literature, it seems the angle follows a bimodal distribution (due to sex differences as mentioned) but that's it. Would that require a transformation? @NickCox, would Ron Jensen's suggestion to center at 0 and use sine be more appropriate? But again, is it necessary? Thank you! $\endgroup$
    – answer42
    Apr 10 at 20:42
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    $\begingroup$ @answer42 transformations are done to make computations easier or to make expressions more intuïtieve/interpretable. I do not see what a transformation would achieve. But that said, in order to see what a transformation might achieve, one would need to see and understand the raw data (and I don't). In any case, if you do not have an idea why you are doing a transformation and are just doing the transformation just for the sake of doing a transformation then you are probably not doing the right transformation... $\endgroup$ Apr 10 at 21:48
  • $\begingroup$ ... An example of a useful transformation (and based on the background/mechanics of the problem) would be a log transform when working with sound/decibels. An example of a bad transformation would be trying to get a bimodal distribution look linear because you believe that the output should be normal distributed, which is a false believe (Gung's answer here shows an example of a multimodal distribution being completely ok). $\endgroup$ Apr 10 at 21:53
  • $\begingroup$ Occasionally transformations are useful for directions over most or all of the circle. In this case I was pointing out that cosine will make matters worse. I don't see a strong reason for centring or for any transformation. $\endgroup$
    – Nick Cox
    Apr 11 at 0:22

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