Single-trial phase values, however, cannot be entered into regression because the data are circular (e.g., radian phase values of −3.1 and 3.1 are closer to each other than are 0.1 and 1.0). Therefore, we used an alternative approach, based on the idea that under the null hypothesis of no relationship between, e.g., reaction time and phase values, reaction times across trials should be uniformly distributed across phase. The less uniform this distribution, the more evidence accumulates to reject the null hypothesis. Taking each reaction time–phase pair as a vector with the phase as the angle and reaction time as the length, the magnitude of the average vector can be taken as a modulation of reaction time by phase angle (under the null hypothesis of no relationship, the average vector length would be close to zero). Here, reaction time and luminance data were rank-transformed because this method cannot be used with negative-valued data. Two issues inherent in magnitude scaling and phase distribution require a non-parametric intervention prior to group-level statistical analyses. The first issue is that non-transformed magnitudes are difficult to
