Automatic search for fMRI connectivity mapping: An alternative to Granger causality testing using formal equivalences among SEM path modeling, VAR, and unified SEM

Abstract

Modeling the relationships among brain regions of interest (ROIs) carries unique potential to explicate how the brain orchestrates information processing. However, hurdles arise when using functional MRI data. Variation in ROI activity contains sequential dependencies and shared influences on synchronized activation. Consequently, both lagged and contemporaneous relationships must be considered for unbiased statistical parameter estimation. Identifying these relationships using a data-driven approach could guide theory-building regarding integrated processing. The present paper demonstrates how the unified SEM attends to both lagged and contemporaneous influences on ROI activity. Additionally, this paper offers an approach akin to Granger causality testing, Lagrange multiplier testing, for statistically identifying directional influence among ROIs and employs this approach using an automatic search procedure to arrive at the optimal model. Rationale for this equivalence is offered by explicating the formal relationships among path modeling, vector autoregression, and unified SEM. When applied to simulated data, biases in estimates which do not consider both lagged and contemporaneous paths become apparent. Finally, the use of unified SEM with the automatic search procedure is applied to an empirical data example.

Introduction

Connectivity mapping provides insight into how the brain orchestrates information processing. “Effective connectivity” maps in particular glean valuable information by identifying the influence that one region of interest's (ROI) activity may have on another (Friston and Stephan, 2007). In this manner, effective connectivity mapping attempts to establish causal directions of regional activity. The majority of current statistical techniques for assessing effective connectivity with functional MRI data identify either contemporaneous or lagged effects, which is problematic since both must be considered simultaneously for unbiased estimation.

Each of the various statistical methods used to model effective connectivity begin by obtaining representative time series from anatomically or statistically identified ROIs (see Goncalves and Hall, 2003). Path diagrams fit by means of structural equation modeling (SEM) appear to be the most straightforward application and more common approach. Here, covariance patterns of contemporaneous blood-oxygen-level dependant (BOLD) time series illustrate brain functioning via directed pathways (McIntosh and Gonzalez-Lima, 1994). Outside of the general linear model approach, dynamic causal modeling (DCM) uses deterministic differential equations to assess how regions relate and estimate external modulation of connections (Friston, 2007). DCM attempts to include neuronal-hemodynamic activity in the model, making the model perhaps the most comprehensive to date (Sarty, 2007).

A third approach, vector (or “multivariate”) autoregression (VAR), estimates the influence that data from ROIs at previous time points have on a given ROI's BOLD activity (Penny and Harrison, 2007). The use of VAR represents an important development in connectivity mapping for two reasons. One, BOLD activity contains sequential dependencies (Harrison et al., 2007) and VAR takes into account these autocorrelations (Shumway and Stoffer, 2006). Two, to be able to make causal inferences between ROIs, at minimum, temporal ordering must be established, i.e., a cause cannot occur later than its effect (Roebroeck et al., 2005). VARs identified by means of Granger causality offer an improvement upon DCM by not requiring a priori selection of directional associations among ROIs. Granger causality necessitates that including past information from one ROI offers a statistically unique contribution in explaining variance in a second ROI which is better than using solely the second ROI to predict itself (Goebel et al., 2003).

In what follows we review a fourth approach, the unified SEM approach of Kim et al. (2007) to model contemporaneous and sequential relationships among ROIs, and present several extensions. In particular, we introduce a new automatic search procedure to identify optimal unified SEM models based on Lagrange multiplier testing. This automatic search procedure constitutes a powerful alternative to Granger causality testing in VAR modeling (Goebel et al., 2003). Additionally, the formal relationships among SEM path modeling, VAR, and unified SEM are explained and illustrated with applications to simulated and empirical data.