# Circular statistics for showing the directional mean lies outside a specified region of a duty cycle

I'm doing an analysis to examine if my data points are clustered along a polar axis (the cycle of a repeated playback sound followed by silence). I'm interested in:

1. Are the data points clustered along a specific region of the playback cycle. I'm using Rayleigh's test for this.

2. If they are clustered around a point on the axis, what is the probability that this point lies outside a specified region of the axis (essentially, is the data centered in a region of the playback cycle in which the playback sound is not playing)

For #2, I am considering doing bootstrapped estimates of the directional mean from my resampled data. Then using the distribution of the bootstrapped means to construct a 95% confidence interval, where the 95% of the points closest to the mean from my actual data are used to define the CI.

Essentially, if the CI does not overlap the playback period of my playback cycle, I can assume that there is a tendency to avoid the playback.

I feel that my approach may not be 100% correct and that there might already be better established methods to get what I'm looking for, but I've had a lot of trouble finding examples of circular statistics which are relevant to my question.

Any ideas? I'd also be super happy to see any references for studies which have applied circular statistics to similar questions!

### Edit: Ecological Relevance

So this question was in the context of playback experiments on singing birds. An area of ecology where there aren't really good statistical templates to follow in the literature (please let me know if I'm wrong here, and missed some good papers).

We had an $$x$$ minute track played on a loop, where the initial proportion of the cycle $$p$$ was some period of noise (0.5 for example), followed by silence for the remainder of the track.

I was interested if the individuals in the experiment would actively avoid singing in the noisey section of the track. So $$p$$ is the proportion of the track that we expect the bird to avoid singing in our alternative hypothesis, $$H_1$$.

• Could you add some (maybe ecological?) context to this question? That might help getting some more interest ... we find it easier/more interesting to think about a problem with some story to it! Jul 25, 2020 at 16:17
• Very old question, but I just added the solution I finally went with. Hopefully its useful to someone! Jul 31, 2020 at 15:03
• Nice, but it would be better if you posted your answer as a formal answer to the question (that is, in the answer box, so that it can get upvoted and then the question stands as resolved. Jul 31, 2020 at 15:31

### Thoughts on my proposed method (original question)

In my proposed approach, computing the mean singing direction via bootstrapped CI's had an issue. In this formualtion, I did not include the expected observations that would happen under a null hypothesis (the bird is singing randomly with no avoidance behaviour). Hence, I could not work this into a null hypothsis rejection test.

Additionally, I now believe that using the bootstrapped values closest to the observed mean for the calculation of my CI's didn't really make much statistical sense.

As I had difficulty in finding a way to formulate a testable null hypothesis based around the mean singing direction, I oped to instead focus on the expected vs observed singing overlap with the avoidance region of the playback track.

### A better approach

This analysis is now done and published (paper here, section 2.4.2 and figure 4)

Focusing on proportion of singing overlap, as opposed to mean singing direction: Each song has a start and end time, so songs are comprised of their start time, and duration:

1. I did a non-parametric bootstrap on the singing data (sampling with replacement, to make 10,000 new datasets.

2. In each bootstrap, I calculate the proportion of singing time which overlaps with the region-to-avoid (regions of the playback with noise in this case). This gives me: a) a bootstrapped mean singing overlap with the avoidance region, and 2) confidence intervals for this mean.

3. I then set a null hypothesis, where if the bird was singing in a random uniform manner (with respect to the playback track cycles), the amount of song overlap would be equal to the proportion of the track that is occupied by the region-to-avoid. So the $$H_0$$ for song overlap is $$p$$

4. If the bootstrapped confidence intervals for the bootstrapped song overlap were lower than the null hypothesis ($$< p$$), then I considered this as a rejection of the $$H_0$$, and support for the $$H_1$$ that the individual is actively trying to reduce song overlap with the region-to-avoid.

For more context, below I've pasted a figure from the paper, so you can see what our data looked like.