The equivalence of linear Gaussian connectivity techniques

The theoretical basis of linear Gaussian connectivity methods for the analysis of fMRI data is examined in this article, resulting in a clarification of methodological dependencies between techniques. In particular, Granger causality connectivity procedures, which describe instantaneous and directed influence between sets of voxel timeseries, are shown to be remappings of correlation-based metrics. Furthermore, the statistical inference tests applied to pairwise Granger causality measures are theoretically shown to be equivalent to inference tests applied to correlation-based metrics. These results are demonstrated empirically using receiver operating characteristic curves derived from vector autoregressive models of various lags, sample size, and noise covariance values. The equivalence of linear Granger causality and correlation-based methods, in both metric and test statistic, renders linear Granger causality a restatement of traditional data-driven methodologies in the context of brain connectivity studies. Furthermore, the equivalence highlights the centrality of partial correlation and partial variance in linear connectivity analyses and bridges the gap between functional and effective connectivity techniques. Consequently, rather than a distinction rooted in methodological difference, the dichotomy between functional and effective connectivity methods is ultimately a function of model configuration realized in choices such as the selection of nodes, the choice to model instantaneous and/or directed influence, and the choice to employ many bivariate models or a single multivariate model. While these theoretical results may be unsurprising to the reader with advanced statistical knowledge, they highlight the importance of a clear understanding of the theoretical basis of connectivity analysis methods for human brain mapping researchers.