Abstract
Over the last two decades, a large number of estimators have been proposed to assess brain connectivity from electroencephalography (EEG) and magnetoencephalography (MEG) data. The statistical theory underlying these estimators, however, is relatively underdeveloped. In particular, the theoretical relationships between different estimators were unknown until very recently. In a recent preprint, Nolte et al. derived formulas for such relationships under the assumption that the data are Gaussian. One of the results was that the phase-lag index and the lagged coherence concentrate around identical values for large sample sizes, i.e. have identical asymptotic limits. A proof of this statement, however, was only sketched, the model assumptions were not checked in experimental data, and sampling properties of the estimators were not considered. We derive the probability density of the relative-phase in the Gaussian model and use it to provide an alternative proof of the asymptotic equality of the phase-lag index and the lagged coherence. The proof is based on power series expansions of the Fourier coefficients of the phase-lag index and the lagged coherence, which are demonstrated to be hypergeometric functions. We also assess the sampling properties of the phase-lag index and the lagged coherence through numerical simulations from the Gaussian model. These demonstrate that throughout the entire parameter space of the model and for all sample sizes, the standard error of the phase-lag index is higher than that of the lagged coherence and thus establish that the lagged coherence is a uniformly better estimator than the phase-lag index. We use experimental EEG and MEG data to verify to what extent the Gaussian assumption is appropriate. Deviations from normality were observed precisely at frequencies for which EEG/MEG power spectra had local maxima, i.e. at oscillatory resonances. Depending on the data-set, the resonances were located in the delta, alpha, and beta frequency bands and correspond to the respective brain rhythms. Based on these observations, we propose to model EEG/MEG data with exponential power densities, which include the Gaussian and Laplace densities as special cases. Lastly, we demonstrate that the asymptotic equality of the phase-lag index and the lagged coherence, as well as the large relative standard errors of the phase lag index, also hold in experimental EEG/MEG data. This establishes that the lagged coherence is not only a better estimator for Gaussian data, but for experimental EEG/MEG data as well.