[R-sig-eco] Test for circular data

[Dixon, Philip M [STAT]]

Thiago,

I can see why you're not having much luck in the usual circular statistics literature, which focuses on responses that are circular quantities.  In your case, the explanatory variable (time) is circular. 

I would answer your second question by fitting a periodic regression, i.e. using sin(2 pi* Time / 24) and cos(2 pi*Time/24) as the two X variables.  Choose the appropriate distribution for your response and include other fixed or random effects as appropriate.   The amplitude and phase of the apparent oscillation can be computed from the regression coefficients.  That model allows one peak per 24 hours.  You can add more components or change the frequency as appropriate for the biology and your hypothesis about the within-day pattern.  The period model also provides a simple test for 'aggregation' (i.e., test both periodic coefficients = 0).  There are many other possible tests of aggregation, depending on the form of the presumed 'no aggregation' distribution of your response.

That model is the temporal equivalent of a simple linear regression model: simple, able to detect various forms of departure from a constant mean, but definitely not an exact description of the pattern over time.  The model can be made more flexible by using a spline function that is appropriately smooth across the 24:00 hr to the 0:30 period.  Look in the smoothing spline and/or GAM literature if you want to explore this.

Philip