Consequences of grouped data for testing for departure from circular uniformity

Abstract

Limits to the precision of circular data often cause grouping of data points into discrete categories, but the effects of grouping on tests for circular uniformity have been little explored. The Rayleigh test is often applied to grouped circular data, despite it being designed for continuous data and the statistical literature recommending a suite of alternative tests specifically designed for grouped data. Here, we investigated the performance of the Rayleigh test relative to four alternatives for testing the null hypothesis of uniformity in grouped circular data. We employed simulation, grouping data into a discrete number of same-sized categories and with samples drawn from a range of different distributions. We found that grouping had little effect on the type I error rate or the power of the Rayleigh test, and that the power of the Rayleigh test was very similar to that of the previously recommended alternative tests designed specifically for grouped circular data. It may thus be appropriate to apply the Rayleigh test to grouped data, provided the situation is one in which the test has substantial statistical power.

Keywords: Chi-squared test; Kolmogorov-Smirnov test; Ordered categories; Periodic data; Rayleigh test.

Fig. 1

Exploration of the performance of the Rayleigh test on grouped data (drawn from a continuous circular distribution then truncated into either 360, 12, 8 or 4 values representing equal-sized categories). Sample sizes of 5, 10, … 50 were considered. For each combination of the number of categories and sample size, estimates were based on 100,000 samples. a The type I error rate when the underlying distribution was uniform. b The statistical power when the underlying distribution was von Mises with concentration parameter κ = 0.8

Fig. 2

The statistical power of the four tests defined in the “Methods” section for detecting departures from uniformity in circular data grouped into four categories for sample sizes 5, 10, … 50 drawn from four different types of distribution described in full in the “Methods” section. Estimates were based on 1000 replicate samples

Fig. 3

The statistical power of the four tests defined in the “Methods” section for detecting departures from uniformity in circular data grouped into 12 categories for sample sizes 5, 10, … 50 drawn from four different types of distribution described in full in the “Methods” section. Estimates were based on 1000 replicate samples

Fig. 4

The statistical power of the standard Rayleigh test and the test with the correction factor for grouping recommended by Batschelet (1981) defined in the “Methods” section for detecting departures from uniformity in circular data grouped into four categories for sample sizes 5, 10, … 50 drawn from three different types of distribution described in full in the “Methods” section. Estimates were based on 1000 replicate samples