I’ve been trying to find a sensible correlation coefficient to assess the reliability of the cylindrical axis on an optometric prescription. Glasses can have *spherical* correction, which has equal refractive power in all axes; but they can also have *cylindrical* correction, which has power in a specific axis to correct for astigmatism. So a prescription might include one or more cylindrical lenses, which need to be set at a specific angle. In our PERGENIC work, we corrected participants’ vision when necessary, but we wanted to be sure that we were doing it reliably. We asked 10% of participants to return for a second session, and when they came back we performed a second refraction without referring to the results of the first one—that way, we could compare the two sets of data to make sure they were consistent.

So far, so good. But you run in to a problem if you try to calculate, say, a Pearson product-moment coefficient.^{1} Here’s the problem: an angle is modular. For a cylinder angle, 0° is exactly the same as 180°. So if the first time I prescribe correction at 179°, and the second time I prescribe correction at 1°, there’s only a 2° difference. But it looks like a 178° difference. Which is not very good.

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