A colleague wants to examine the directionality of amphibian dispersal to and from a pond. He set up a drift fence around the pond and had buckets at different positions to capture the amphibians. He measured the angle from the center of the pond to each bucket. There are 23 buckets arrayed around the pond; due to physical constraints, the angle between each bucket (and resulting arc length of the sector) is not perfectly regular.

He wants to use methods from circular statistics to examine directionality (or conversely, uniformity). The data can be treated as bearings to each bucket (with counts) or as sectors around each bucket (with counts). Unfortunately, I can't find a good discussion in the literature (or on CrossValidated) as to what sort of methods are preferred for this sort of design...as most methods are based on continuous measures of bearing (or time) around the unit circle. There seems to have been some work on circular methods for discrete variables, but they don't seem to apply to irregular data with relatively large intervals.

An ad hoc method seems to be the use of a chi-square goodness-of-fit test based on comparison of observed counts to expected counts (here based on the total sample size times the arc length)...but this seems sketchy due to the high number of zeros for some species (buckets where amphibians were not captured) and potential circular autocorrelation among neighboring buckets (and hence lack of independence).

I have seen some papers that have used Rayleigh's z test and Rao's spacing test for sector data, but I am not sure whether this is "kosher" or not (as I can't find a discussion about the pitfalls). I thought I might try examining the fit to a parametric distribution (such as von Mises) to see if I am in the ballpark, but I feel insecure about it.

I would be very grateful for any guidance as to which methods might be most applicable to this sort of design...or if somebody could point me in the direction of a paper or book chapter where this issue discussed. I have skimmed through Jammalamadaka and SenGupta...and the table of contents of more recent monographs, but I can't seem to find any obvious discussion about sector data...and I can't justify throwing down the money for a book without being sure if this content is covered.

  • $\begingroup$ Hi coreydevinanderson. Welcome to CV. I like your question. and I agree about the dearth of satisfactory info, although the ever generous @NickCox gave some real illumination. $\endgroup$
    – Alexis
    Commented Feb 3, 2017 at 0:45
  • $\begingroup$ To update...it seems that the main problem with testing with grouped data is that the critical values for many of the tests were calculated under the assumption that the data are continuous. For omnibus tests of uniformity, Pewsey et al. (2013) provide example R code to estimate P-values based on "parametric bootstrapping" (for Watson's U2, Watson, and Kuiper tests). Using this approach, I have found the test inferences to be, more or less, the same (regardless of which test you use). $\endgroup$ Commented Jul 5, 2017 at 16:22
  • $\begingroup$ I should also say that options for further testing become increasingly limited if your data are multimodal and not reflectively symmetric. For grouped data, with irregular class boundaries, and mutimodal patterns...about all you can do is a omnibus tests of uniformity. You might also be able to do some pairwise testing (e.g., via Watson U2) but this becomes intractable if you have lots of comparisons. Many of the traditional summary measures for circular data (such as mean resultant length) are also compromised for multimodal data that are grouped. $\endgroup$ Commented Jul 5, 2017 at 16:29
  • $\begingroup$ Lastly, I should note that using trigonometric predictors, as suggested by Nick Cox (below) is a reasonable approach if you your data set is relatively simple and your main goal is prediction.. In our case, where we are trying to analyze patterns of dispersion from a pond for multiple amphibian species over multiple years, such model fitting can become intractable (since multiple models have to be fit and many of the patterns require multiple harmonics)...and quite difficult to interpret (IMO). $\endgroup$ Commented Jul 5, 2017 at 16:36
  • $\begingroup$ coreydevinanderson that sounds the basis of a paper. :) $\endgroup$
    – Alexis
    Commented Jul 8, 2017 at 0:04

1 Answer 1


I would say that one issue you raise is not discreteness, but resolution of measurement. Consider that even e.g. directions to the nearest degree are themselves binned too, just as people's heights often come in cm or inches.

In other biological or Earth or environmental sciences, it's common that data arrive as compass points N, E, S, W or as 8 or as 16 points; or as hours of day (24 possible values). So, data as coarse as those mentioned are by no means exceptional. Analyses in Mardia's books show that the information loss from grouping is fairly slight.

I would just use the midpoint direction for each bucket.

Chi-square I take to be a poor method for directional data. You're throwing away all the information on circular order. The only possible advantages are that you don't need a dedicated program and that the method is probably already familiar. Rayleigh's and Kuiper's tests are much, much better.

But I don't see that these tests address the scientific problem. What is the main thinking here? As I understand it, the direction is a predictor, not a response, and the counts are a response. That seems to imply Poisson regression with sine and cosine terms for direction and offsets for different sector widths.

Other than implying offsets, the irregularity of spacing is not much more of a problem than irregularity of a predictor would be in plain or vanilla regression. (Indeed, the data are much better in terms of spacing than most datasets.)

Something not discussed commonly (and to me surprisingly so) in books on circular statistics is the use of direction as a predictor in otherwise mainstream regression-type models. As said, the most useful way forward is to use sine and cosine terms as predictors.

The book you mention specifically is the least valuable book on circular statistics I've ever encountered, being an idiosyncratic mix of material and chock-full of strange typos and inconsistencies.

See also

recent thread on directional statistics books

paper on trigonometric predictors

  • $\begingroup$ Thanks Nick! Unfortunately, the book I mentioned was the only one readily accessible via the USG library system. The previous analyst did try the Poisson approach you mentioned (with the offsets for sector widths), which I thought was pretty slick...albeit it seems as if he had trouble fitting the model, perhaps due to the small sample size and potentially high number of zeros (for some species). Very cool idea though. $\endgroup$ Commented Feb 2, 2017 at 19:37

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