F_circ statistic for steady-state evoked potentials; a generalized version of T c i r c 2 statistic

Steady-state evoked potentials (ssEP) provide objective tools for studying brain function in different experimental conditions. Frequency components of brain response to repetitive stimuli have been analyzed using $T_{circ}^2$ statistic; however, $T_{circ}^2$ statistic is limited to comparisons between two means. Here, we present a generalized version of $T_{circ}^2$ statistic which enables us to compare multiple means of Fourier estimates corresponding to multiple conditions within participant(s) or multiple groups of participants.

Frequency components of brain response are extracted from ssEP data using Fourier transform. Discrete Fourier measurements at frequency of interest are represented on the complex plane for statistical analyses. We present a new statistic called F_circ statistic to compare three or more clusters of Fourier estimates whether they have equal or unequal variances or/and numbers of samples. F_circ statistic derives from Welch's test but for multiple comparisons.

We demonstrate the validity of F_circ statistic using simulated and empirical clusters of Fourier estimates with equal and unequal variances and numbers of samples. Type-I error remains 0.05 for all the conditions. Furthermore, we illustrate that the probability of achieving a significant difference among multiple means when the true means are unequal depends on the total length of ssEP data but is independent of the duration chosen for performing Fourier transform on a fixed length of ssEP data.

$F_{circ}$ statistic is useful for multiple intra- and inter-participant and group comparisons of brain response at any frequency component extracted from ssEP data whether the group means have equal or unequal variances.