A guide to group effective connectivity analysis, part 2: Second level analysis with PEB

Hayriye Cagnan,Vladimir Litvak,Amirhossein Jafarian,Cathy J. Price,Karl J. Friston,Peter Zeidman,Mohamed L. Seghier

doi:10.1016/j.neuroimage.2019.06.032

Abstract

This paper provides a worked example of using Dynamic Causal Modelling (DCM) and Parametric Empirical Bayes (PEB) to characterise inter-subject variability in neural circuitry (effective connectivity). It steps through an analysis in detail and provides a tutorial style explanation of the underlying theory and assumptions (i.e, priors). The analysis procedure involves specifying a hierarchical model with two or more levels. At the first level, state space models (DCMs) are used to infer the effective connectivity that best explains a subject's neuroimaging timeseries (e.g. fMRI, MEG, EEG). Subject-specific connectivity parameters are then taken to the group level, where they are modelled using a General Linear Model (GLM) that partitions between-subject variability into designed effects and additive random effects. The ensuing (Bayesian) hierarchical model conveys both the estimated connection strengths and their uncertainty (i.e., posterior covariance) from the subject to the group level; enabling hypotheses to be tested about the commonalities and differences across subjects. This approach can also finesse parameter estimation at the subject level, by using the group-level parameters as empirical priors. The preliminary first level (subject specific) DCM for fMRI analysis is covered in a companion paper. Here, we detail group-level analysis procedures that are suitable for use with data from any neuroimaging modality. This paper is accompanied by an example dataset, together with step-by-step instructions demonstrating how to reproduce the analyses.

Highlights

The Parametric Empirical Bayes (PEB) approach differs fundamentally, in that is a hierarchical model with random effects on parameters rather than models
After introducing the example dataset, we describe the specification of a PEB model and demonstrate testing hypotheses using Bayesian Model Reduction (BMR) – a efficient form of Bayesian model selection
We have illustrated an empirical Bayesian procedure for conducting group connectivity analyses, using Dynamic Causal Modelling (DCM) and PEB. This begins with a first level analysis, modelling within-subject effects using neural models (DCMs)

Summary

Notation

Vectors are denoted by lower case letters in bold italics (a) and matrices by upper case letters in bold italics (A). Other variables and function names are written in plain italics (f). The dot symbol (⋅) on its own means multiplication and when positioned above a variable (e.g. ż) denotes the derivative of a variable with respect to time. An element in row m and column n of matrix A is denoted by Amn. An element in row m and column n of matrix A is denoted by Amn To help associate methods with their implementation in the SPM software (http://www.fil.ion.ucl.ac.uk/spm/software/), MATLAB function names are provided in bold text, such as (spm_dcm_fit.m)

Experimental design

First level analysis

Theory

PEB: Design matrix specification

PEB: Model estimation

Inference

Inference: family analysis

Factor 1

Factor 2

Factor 3

Interim summary

Prediction