Theoretical considerations on the detection of evoked responses by means of the Rayleigh test

One of the most powerful tools for the detection of an evoked response is the Rayleigh test which checks a set of phase angles for uniformity. Several important theoretical aspects of the Rayleigh test are illustrated in the present paper. First of all, probability density functions and distribution functions of the underlying test-statistic as well as receiver operating characteristics (ROC) are presented. Then it is exemplified how to derive a ROC for the case that two independent Rayleigh test-statistics (e.g. from two different channels or two different frequency bands or time windows) are available. Finally, it is demonstrated that the probability of a false-positive decision may increase by an order of magnitude if the Rayleigh test is not performed once, for a fixed number of epochs specified in advance, but is carried out repeatedly during an ongoing experiment until either one of the tests indicates the presence of an evoked response or the upper limit for the number of epochs is exceeded.