Fig. 1

Type I error rates (a, c, e) and power for a von Mises distribution (b, d, f) of the traditional Rao and the simulation-based Rao test with different sample sizes. (a, b) First, we tested a continuous uniform distribution (type I error) and a von Mises distribution (power), p-values are evaluated by the R function rao.spacing.test in the package circular and by simulation, showing similar type I error probabilities and power. (c, d) Second, we tested simulated data rounded to the next degree (360 bins) and applied either the traditional or the simulation-based test. (e, f) Then we repeated the type I error and power estimation for data binned in 36 equal bins (rounded to the next 10°). The dashed line indicates the nominal 5% level for type I errors. The traditional Rao test shows inflated type I error rates when used on rounded data (c, e). The simulation-based test has low type I error rates and offers power similar to the traditional Rao test (d, f)