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A tutorial on group effective connectivity analysis, part 2: second level analysis with PEB

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

arXiv:1902.10604v1q-bio.QM

TL;DR

The paper addresses how to model commonalities and differences in effective connectivity across subjects after individual DCM estimation. It presents PEB as a hierarchical group-level framework that carries parameter estimates and uncertainty into Bayesian GLM analyses. PEB weights more precise subjects more strongly, can improve individual estimates through empirical priors, and is illustrated in an fMRI lateralization study.

  • Problem

    The challenge is quantifying commonalities and differences across subjects after each subject’s connectivity strengths have been inferred.

  • Method

    PEB hierarchically models DCM parameters at the group level using a Bayesian GLM over between-subject effects and random-effects variability.

  • Results

    More precise parameter estimates have greater influence on group-level results, while noisy and uncertain estimates are down-weighted.

  • Takeaways & Limitations

    Group-level parameters can serve as empirical priors that improve parameter estimates at the individual-subject level.

Abstract

from arXiv · show

This tutorial 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). This 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. We walk through this approach in detail, using data from a published fMRI experiment that characterised individual differences in hemispheric lateralization in a semantic processing task. The preliminary subject specific DCM analysis is covered in detail in a companion paper. This tutorial is accompanied by the example dataset and step-by-step instructions to reproduce the analyses.

1 Introduction

The tutorial introduces PEB as a hierarchical second-level framework for modelling commonalities and differences in effective connectivity across subjects after subject-specific DCM estimation. It explains how PEB preserves parameter uncertainty, tests between-subject hypotheses, and can refine individual estimates using group-level priors.

  • Subject-level modelling: Neuroimaging studies infer subject-specific neural responses and connectivity from measurements such as fMRI, EEG, and MEG.DCM models transformations from experimental stimulation through neural activity to observed data, estimating connection strengths probabilistically.
  • Tutorial scope: The tutorial illustrates group-level effective-connectivity modelling with a published fMRI experiment and addresses hierarchical analysis across imaging modalities.Its example considers individual differences in hemispheric lateralization and provides a worked setting for testing between-subject effects.
  • Hierarchical group analysis: PEB extends DCM with a hierarchical model in which subject-level parameters are collated and modelled at the group level using a General Linear Model.The model partitions individual differences into hypothesised group-level effects and unexplained random-effects variability.
  • Hierarchical group analysis: PEB carries both expected parameter values and their covariance to the group level, unlike analyses that retain only estimated connectivity peaks.This allows uncertainty in subject-level estimates to inform group-level modelling.
  • Benefits of PEB: More precise subject-level estimates receive greater influence in group-level results, while noisy and uncertain estimates are down-weighted.The hierarchy can also improve parameter estimates at the individual-subject level by using group-level parameters as empirical priors.
  • Relation to alternative group models: PEB assumes a shared basic architecture with subject differences expressed as connection-strength variation, whereas RFX BMS models variation in which architecture generated each subject’s data.PEB can represent between-subject effects using continuous behavioural or clinical covariates, such as the laterality index, and compare mixtures of first- and second-level effects.

2 Notation

The notation distinguishes vectors, matrices, variables, derivatives, and matrix elements, while implementation references connect the notation to SPM software.

  • Vectors use lowercase bold italics, matrices use uppercase bold italics, and other variables or function names use plain italics.
  • A dot denotes multiplication when standalone and a time derivative when placed above a variable.
  • The element in row m and column n of matrix A is written as A_mn.
  • Variables and dimensions are listed in Table 1, and MATLAB function names are provided to link methods with SPM implementation.

3 Experimental design

The experiment examined individual differences in language lateralisation using a 60-subject fMRI dataset and a factorial design separating within-subject stimulus and task effects from between-subject LI.

  • The dataset contained 60 subjects from a previously published fMRI experiment on language lateralisation.
  • Within subjects, the experiment used a balanced 2x2 design crossing Words versus Pictures with Semantic versus Perceptual matching.
  • LI was a between-subject measure of functional brain asymmetry, with positive values indicating left-hemisphere dominance and negative values indicating right-hemisphere dominance.LI was derived from the number of activated voxels in each hemisphere in an initial SPM analysis.
  • The central question was which neural circuitry underlies individual differences in the Laterality Index (LI).
  • Words and pictures were modelled in each subject’s first-level GLM and DCM, whereas LI, age, gender, and handedness entered the second-level PEB analysis.
  • The model space varied independently across stimulus type, dorsal versus ventral regions, and hemisphere, with a null model added as baseline.These three factors produced 3 × 3 × 3 + 1 = 28 candidate models.

4 First level analysis: DCM

First-level DCMs estimated stimulus-related changes in inhibitory self-connections across four frontal regions for each subject, while Bayesian inversion retained parameter uncertainty for group analysis.

  • One fMRI DCM was specified and inverted for each subject to estimate connectivity parameters from the observed neuroimaging timeseries.
  • The analysis focused on words- and pictures-related changes in inhibitory self-connections of left ventral, left dorsal, right ventral, and right dorsal frontal cortex.
  • Positive parameter values represented increased self-inhibition and reduced sensitivity to network inputs, whereas negative values represented disinhibition and increased sensitivity.
  • Variational Laplace balanced timeseries accuracy against complexity by maximising negative variational free energy, an approximation to log model evidence.
  • Each inverted DCM returned posterior parameter means and covariance, whose diagonal represented parameter uncertainty and off-diagonal elements represented covariance.
  • Parameter uncertainty arises because multiple settings can explain the data similarly and connectivity parameters may be collinear.

5 Second level analysis: Parametric Empirical Bayes (PEB)

PEB transferred subject-level DCM estimates and uncertainty into a hierarchical group GLM, testing commonalities and LI-related differences across predefined and automatically searched models. The analyses identified dorsal frontal modulation as the main shared task effect and right dorsal responses to words as the key LI-related difference.

  • 5 Second level analysis: Parametric Empirical Bayes (PEB): PEB models subject-level neural parameters with a second-level GLM whose design matrix encodes hypothesised between-subject sources of variation.
  • 5.2 PEB: Design matrix specification: The group design included commonalities, LI, handedness, gender, and age, while within-subject effects specified which connectivity parameters could vary.
  • 5 Second level analysis: Parametric Empirical Bayes (PEB): The hierarchical model partitions connectivity variability into designed group effects and zero-mean additive random effects, while allowing additional levels for nested groups.
  • 5 Second level analysis: Parametric Empirical Bayes (PEB): PEB carries subject-level parameter estimates and posterior covariance into group inference, with group parameters constraining estimates at lower levels.
  • 5.4 Inference: Comparing reduced PEB models: The best predefined model had commonalities from model 4 and LI differences from model 15, with 56% posterior probability.
  • 5.4 Inference: Comparing reduced PEB models: Model 4 indicated words and pictures modulated dorsal-region self-connections, whereas model 15 linked LI differences specifically to words modulating right dorsal frontal self-connection.
  • 5.5 Inference: family analysis: Family analysis assigned 86% probability to pictures-and-words modulation for commonalities and words-only modulation for LI differences.
  • 5.5 Inference: family analysis: A separate family analysis assigned 69% probability to dorsal regions explaining both commonalities and LI differences, with a 20% second-best model including dorsal and ventral regions for commonalities.

6 Prediction: Cross-validation

The analysis tested whether the laterality effect on right dorsal frontal region responses to words was large enough to predict individual subjects’ Laterality Index (LI). Leave-one-out cross-validation assessed predictive validity by fitting PEB models without each subject and predicting that subject’s covariates.

  • The prediction target was each subject’s LI, using modulation of rdF by words together with handedness, age, and gender.
  • The analysis focused on whether the neural laterality effect was sufficiently large to support prediction beyond the observed group-level effect.
  • Predictive validity was assessed with leave-one-out cross-validation, repeatedly fitting PEB to all but one subject.The left-out subject’s covariates were then predicted and prediction accuracy recorded.

Appendix 4: Leave-one out cross-validation with PEB.

Leave-one-out PEB predictions showed that DCM-derived neural parameters predicted subjects’ laterality scores above chance. However, substantial unexplained variability remained across subjects.

  • The out-of-sample Pearson correlation between actual LI and expected predicted LI was 0.34, p=0.004.
  • The DCM-estimated effect size was sufficiently large to predict left-out subjects’ LI with performance above chance.
  • Prediction retained substantial variability that the model did not explain.
  • Figure 8 compares predicted and actual mean-centred LI across subjects and shows the actual group effect as a dashed reference line.

7 Analysis summary

The analysis used PEB to characterise common and individual differences in effective connectivity during semantic processing. Common effects involved bilateral dorsal frontal responses to pictures and words, whereas LI-related differences were specific to right dorsal frontal responses to words.

  • The study examined whether subject differences in LI appeared in condition-specific connectivity changes for pictures or words across dorsal or ventral, left or right regions.
  • Across subjects, the frontal network was modulated by pictures and words in dorsal regions of both hemispheres.
  • Pictures inhibited ldF responses, whereas words inhibited rdF responses, forming a double dissociation between stimuli.
  • LI-related differences were expressed specifically in responses to words in the dorsal right hemisphere.
  • The LI-associated effect was positive: higher LI, indicating greater left lateralization, was associated with greater inhibition of rdF.
  • Cross-validation showed that this effect was sufficiently large to predict left-out subjects’ LI.

8 Discussion

The discussion presents DCM–PEB as a hierarchical framework for group connectivity analysis, extending from subject-level models to group-level hypothesis testing and deeper hierarchies. It also discusses model-search choices, predictive validation, possible applications, and limitations requiring further validation.

  • Hierarchical modelling: The framework models within-subject effects with DCM, then carries connectivity parameters to higher-level Bayesian GLMs for group analysis.PEB links individual subjects to the group level and partitions connectivity variability into designed effects and random effects.
  • Limitations: The suitability of particular priors in novel applications requires further validation, and the tutorial does not cover all framework features.
  • Extensions: Group-level posteriors can serve as empirical priors for first-level re-estimation, potentially helping recover subject parameters from local minima, especially in highly nonlinear DCMs.
  • Hierarchical modelling: PEB-of-PEBs extends the hierarchy by collating time-point parameters into subject-specific PEB models before modelling group-level commonalities and differences.
  • Model comparison: Group-level hypotheses can be tested by explicitly designing models or by automatically pruning parameters through Bayesian Model Reduction.
  • Model comparison: The two model-selection approaches produced similar results on the presented dataset, but this agreement is not guaranteed generally.
  • Model comparison: Automatic greedy searches are usually non-exhaustive because of the large model space, and can encourage post-hoc explanations for surviving parameters.
  • Extensions: The framework may also support unsupervised subject classification by searching over candidate group assignments, complementing finite-mixture hierarchical models.

9 Tables

The tables define the notation used for subject- and group-level PEB quantities and report Bayesian model averages of connectivity parameters across conditions and lateralization differences.

  • Notation: Table 1 defines symbols for subjects, DCM parameters, experimental conditions, group-level expectations, residual uncertainty, covariance, and variational free energy.It also distinguishes between-subject variability from residual uncertainty about group-level parameters.
  • Bayesian model averages: Table 2 reports Bayesian model-averaged expected values for estimated commonalities and lateralization-index differences.
  • Bayesian model averages: Table 3 reports Bayesian model-averaged average connectivity across experimental conditions for the DCM A-matrix.
  • Parameter units: Between-region connections are expressed in Hz, whereas self-connections are log scaling parameters relative to the default value -0.5Hz.

10 Appendix 1: PEB connectivity priors specification

PEB derives second-level priors from first-level priors while adjusting variance for the scaling of the group-level design matrix.

  • Prior construction: The second-level model is a Bayesian GLM whose parameters have priors derived from the first-level priors.
  • Prior construction: By default, second-level priors retain the first-level priors’ form, but their variance is adjusted according to the design matrix X.
  • Prior construction: The adjustment scales prior variance for parameters associated with a covariate by that covariate’s norm.The norm is defined using the sum-of-squares of the design-matrix columns and the number of subjects.
  • Prior construction: Setting the fixed parameter α to one specifies the tutorial’s chosen prior-scaling convention.

11 Appendix 2: PEB random effects specification

The tutorial specifies PEB random effects with connection-specific precision components, then uses Bayesian model reduction for model comparison, thresholding, and leave-one-out prediction.

  • Random-effects specification: PEB represents unexplained between-subject differences as zero-mean additive noise with precision matrix Π(2).
  • Random-effects specification: The tutorial selects one precision component per DCM connection, so between-subject variability is estimated separately for each connection.
  • Random-effects specification: Each connection-specific component has one non-zero entry, while its scaling parameter γ_j adjusts the corresponding precision component.The component value is set using the first-level parameter variance, with fixed β=16; exponentiation ensures positive precision.
  • Bayesian model comparison and reduction: Model evidence is approximated by negative variational free energy rather than computed exactly.
  • Bayesian model comparison and reduction: Bayesian model reduction computes reduced-model free energies analytically from a fitted full model by changing priors, including switching selected connections off.This makes it possible to evaluate large numbers of reduced models rapidly.
  • Bayesian model comparison and reduction: Parameter thresholding compares PEB models with each parameter switched on or off and uses full parameter covariance when assessing evidence.
  • Leave-one-out cross-validation: Leave-one-out PEB fits the group model without subject i and predicts the omitted design-matrix entry from that subject’s connectivity parameter.In the worked example, solving the prediction gives x_i=1, the correct group-membership assignment.
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