Circular data in biology: advice for effectively implementing statistical procedures

Abstract

Circular data are common in biological studies. The most fundamental question that can be asked of a sample of circular data is whether it suggests that the underlying population is uniformly distributed around the circle, or whether it is concentrated around at least one preferred direction (e.g. a migratory goal or activity phase). We compared the statistical power of five commonly used tests (the Rayleigh test, the V-test, Watson's test, Kuiper's test and Rao's spacing test) across a range of different unimodal scenarios. The V-test showed higher power for symmetrical distributions, Rao's spacing performed worst for all explored unimodal distributions tested and the remaining three tests showed very similar performance. However, the V-test only applies if the hypothesis is restricted to one (pre-specified) direction of interest. In all other unimodal cases, we recommend using the Rayleigh test. Much less explored is the multimodal case with data concentrated around several directions. We performed power simulations for a variety of multimodal situations, testing the performance of the widely used Rayleigh, Rao's, Watson, and Kuiper's tests as well as the more recent Bogdan and Hermans-Rasson tests. Our analyses of alternative statistical methods show that the commonly used tests lack statistical power in many of multimodal cases. Transformation of the raw data (e.g. doubling the angles) can overcome some of the issues, but only in the case of perfect f-fold symmetry. However, the Hermans-Rasson method, which is not yet implemented in any software package, outcompetes the alternative tests (often by substantial margins) in most of the multimodal situations explored. We recommend the wider uptake of the powerful but hitherto neglected Hermans-Rasson method. In summary, we provide guidance for biologists helping them to make decisions when testing circular data for single or multiple departures from uniformity.

Keywords: Animal navigation; Biostatistics; Chronobiology; Circadian; Emlen funnel; Magnetoreception.

Fig. 1

The estimated statistical type 1 error, i.e. proportion of tests generating p < 0.05. Estimates are based on 100,000 uniform samples for each of a range of sample sizes: 10, 15, 20, 25, 30, 40, 80 and 100. We compare the Rayleigh test, the three omnibus tests (Kuiper’s, Watson’s and Rao’s spacing tests), the V-test, the Bogdan, and Hermans-Rasson tests

Fig. 2

Examples for the distributions used in this analysis. The von Mises distribution, wrapped Cauchy and skewed normal distribution, over a range of concentration/dispersion parameters (values given in the plots, n = 100 for each)

Fig. 3

The estimated statistical power to reject the null hypothesis of uniformity based on a sample from a von Mises distribution. Estimates are based on 10,000 samples for each of four sample sizes: a 15, b 25, c 40 and d 100. We compare the Rayleigh test, the three omnibus tests (Kuiper’s, Watson’s and Rao’s spacing tests) and three different situations for the V-test (where the test mean value and mean value of the underlying distribution either exactly coincide, differ by 20° or differ by 45°). We obtain estimates for a range of different values of the parameter κ that defines the concentration of values for a von Mises distribution

Fig. 4

The estimated statistical power to reject the null hypothesis of uniformity based on a sample with two modes each from a von Mises distribution. Estimates are based on 10,000 samples with a sample size of 25. We compare the Rayleigh (2× = modification for two symmetrical modes), Kuiper’s, Watson’s, Rao’s spacing, Bogdan and Hermans-Rasson tests. We obtain estimates for a range of different values of the parameter κ that defines the concentration of values for a von Mises distribution. a Power estimates for two symmetric modes 180° apart. b Power estimates for two asymmetric modes 90° apart

Fig. 5

The estimated statistical power to reject the null hypothesis of uniformity based on a sample with two modes each from a von Mises distribution. Estimates are based on 10,000 samples with a sample size of 25. We compare the Rayleigh, Kuiper’s, Watson’s, Rao’s spacing, Bogdan and Hermans-Rasson tests. We obtain estimates for a different values of the distance (Δ) between the two modes, b different values of the concentration κ of one of the modes while holding the concentration of the other mode constant at 3.0, and c different weight ratios of the modes (equal weight = a delta proportion of 0.5)