Circular/directional data analysis within a narrow arc

Question

I am studying the presence of an anatomical variation to the structure of a long bone in healthy and diseased individuals. I have two separate groups of individuals. The anatomical variation is torsion of long bone which sounds simple enough but not according to my reviewers. The big headache is that some individuals in both groups have a reversed direction of torsion. The majority are externally rotated; however; some are internally rotated.

1. How do I convert circular data to linear?
2. What do I use to test for Gaussian distribution? 3.Once I get past the distribution, what test do I use to compare? Is Mann-Whitney U test OK to use?
3. Side-to side variability - left to right. Each pair of limbs regardless of disease status are torsed in the same direction.

The range of both groups is fairly narrow but in two opposite directions. Healthy individuals range (-5.44 degrees internal to + 6.39 external) Diseased individuals (-26.55 internal to +25.15) However when plotted on a compass rose the two groups' ranges hardly overlap.

I used (+) to determine external and (-) to determine internal torsion I am not a professional statistician so please go easy on me.

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The original data set describing the geometry of the transverse axis (about 70 data pairs per specimen) was treated similarly as described for the longitudinal axis. Again, a normalised natural coordinate s was constructed such that 0 6 s 6 1, s = 0 at the medial border of the medial coronoid process and s = 1 at the lateral border of the medial coronoid process and the original Cartesian z–x reference frame (Fig. 2D) of each set of data was translated so that the medial border of the medial coronoid process was located at (z0, x0) = (0, 0). Sixth-order polynomial curve fits were performed (highest r2 = 0.9996, lowest r2 = 0.9933, mean r2 = 0.9981) and the x-coordinates were calculated for intervals of 0.025 (i.e. s = 0.025, 0.05, 0.075, 0.1, . . . , 1). The resulting series of 40 data set pairs characterised the geometry of the medial coronoid process at its transverse axis and was used for further calculations. The inclination angle b of the transverse axis of the medial coronoid process was calculated trigonometrically using the original coordinates of the first and last data set pair. Four joints (Table 1) were excluded from this analysis because the contour of the articular surface was not flat (like in all other joints) but had deep steps related to high degree subchondral bone erosion.

Answer 1

I am not a professional statistician either, but I've been dealing with directional data for a few decades.

I see no reason with this kind of data to over-emphasise their origin (pun intended) as angles. Two simple acid tests for deciding when you really need directional or circular statistics are that (a) in principle, essentially any direction is possible (b) you realise that the usual mean is not helpful, as when the mean of 1$^\circ$ and 359$^\circ$ is returned as 180$^\circ$.

That doesn't seem likely here. Indeed, in contrast to the usual situation for circular data in which the origin is arbitrary (e.g. South at 0$^\circ$ would be equally (un)satisfactory compared with North at 0$^\circ$), the origin at 0$^\circ$ seems entirely natural here.

I don't know why Mann-Whitney should spring to mind here. You give no information at all that could help us advise whether the Gaussian or normal is an acceptable reference distribution here.

Summary: When directions cover part of the circle, treating them as linear is both simpler and defensible. In your case, the origin doesn't seem arbitrary, so stick with it.

Note: Circular data being treated as linear does not make trigonometry irrelevant, although in this case, with small angles, the possible transformation sine of angle is essentially linear in angle over the ranges observed.