tests of uniformity for circular data

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Pierre Mégevand
Pierre Mégevand on 25 Sep 2013
Dear all,
I have a question regarding tests of uniformity for circular data.
In essence, I am looking for a one-sample permutation test of the null hypothesis that there is no phase concentration at a given electrode and timepoint, that controls for the repetition of testing over electrodes and timepoints (family-wise type I error rate).
I am working with intracranial EEG data. If I want to assess which brain regions respond to an external stimulus, a possible approach is to extract the high-gamma power for each electrode at each trial following stimulus presentation, normalize it against a suitable baseline (e.g. before stimulus presentation), and then compute a one-sample permutation test based on the tmax statistic. The null hypothesis assessed by this test is that there is no change in high-gamma power following stimulus presentation. Such an approach provides strong control for the family-wise type I error rate (otherwise, the repetition of statistical testing at each electrode and timeframe would give rise to an unacceptably high risk of falsely rejecting the null hypothesis) (Groppe et al., Psychophysiology 2011).
Now, I am wondering whether something similar exists for phase concentration. In my experiment, there is a cue that signals imminent stimulus presentation (the cue is non-informative as to which condition the stimulus actually belongs to), and I am looking for signs that phase concentration (reflected by an increase in the intertrial coherence or phase-locking value) happened at some electrodes following the cue and before the actual stimulus was presented. If I have a strong prior hypothesis on the timing of the effect of the cue (as well as the frequency band that will undergo phase reset or concentration, and the electrode location), I can select a timepoint and electrode and use the Rayleigh test or the omnibus test for circular data (see e.g. Fisher, Statistical Analysis of Circular Data, 1993; a MATLAB toolbox implementing some of these tests has been developed: http://www.mathworks.com/matlabcentral/fileexchange/10676-circular-statistics-toolbox-directional-statistics). These tests assess the null hypothesis that a sample of circular data (such as phase angles of brain oscillations) come from a circular uniform distribution (the alternative hypothesis differs somewhat between these tests).
But, what if I don't have such a strong a priori hypothesis? Is there a way for me to compute the intertrial coherence at each timepoint and each electrode (similar to the high-gamma power), and then perform some sort of statistical test against a null hypothesis of uniformity, all the while correcting for multiple comparisons?
Specifically, bearing in mind the one-sample tmax statistic-based permutation test for changes in high-gamma power mentioned above: is there a way to design a permutation test for circular data that assesses the null hypothesis that the data do come from a circular uniform distribution?
Note that I am not looking to compare one condition against another (a permutation test based on a statistic such as Watson's Yr or U2 would probably work), but specifically to perform a 1-sample test.
Thank you for your thoughts, comments and advice!

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