2.7. Parameter Estimations—Parametric Empirical Bayes (PEB), Bayesian Model Reduction (BMR) and Bayesian Model Averaging (BMA)

After the model specification, we took the first level individual (within-subjects) estimated fully connected DCMs to a second level (between-subjects) analysis using parametric empirical Bayes (PEB) [45]. PEB is a hierarchical approach to model how individual subjects’ connections relate to the group level by using the individual DCMs as priors (first level) to constrain the variables in the Bayesian linear regression model (second level) [46]. This method allows individual variability in connection strengths to then influence the group (second level) analysis. After group level connection strengths (parameters) have been estimated, hypotheses are tested by comparing evidence for different variations of these parameters in a process known as Bayesian model comparison.

In order to assess the differences in connectivity strength between patients and healthy controls we first built a PEB model including the average connectivity in the healthy group (as baseline), the intercept of differences between healthy controls (0) and patients (1), and the mean centered age as non-interest regressor (to account for the effect of differences in age between the groups, p = 0.002).

We subsequently ran a second PEB model including only PDOC patients to assess the canonical connectivity in this group, also using mean centered age as covariate. This was used to extract connectivity parameters to conduct correlations with clinical variables (see CRS-R below).

For both PEBs, after fitting each model we used Bayesian model comparison to prune away parameters that are not contributing to the model evidence. This was achieved by performing a search over nested models where parameters are removed from the full model hierarchically. We then used Bayesian Model Reduction (BMR) to prune connections from the full model until there are no more improvements in model-evidence. This approach uses the group parameters as empirical priors to re-estimate the individual parameters, and in doing so, limits the effects of outliers [47]. Parameters from the best models following BMR were then taken, weighted by their model evidence, and combined using Bayesian model averaging (BMA). To define statistical significance, we applied a threshold of a posterior probability > 0.95 (strong evidence) for free energy for each connection (i.e., comparing the evidence for all models in which the particular connection is on with those where the connection is switched off). For more information on BMR and BMA, see Friston et al., 2016 and Zeidman et al., 2019 [45,46].