Bayesian Hypothesis Testing for Gaussian Graphical Models:Conditional Independence and Order Constraints

Gaussian graphical models (GGM; partial correlation networks) have become increasingly popular in the social and behavioral sciences for studying conditional (in)dependencies between variables. In this work, we introduce exploratory and confirmatory Bayesian tests for partial correlations. For the former, we first extend the customary GGM formulation that focuses on conditional dependence to also consider the null hypothesis of conditional independence for each partial correlation. Here a novel testing strategy is introduced that can provide evidence for a null, negative, or positive effect. We then introduce a test for hypotheses with order constraints on partial correlations. This allows for testing theoretical and clinical expectations in GGMs. The novel matrix$-F$ prior distribution is described that provides increased flexibility in specification compared to the Wishart prior. The methods are applied to PTSD symptoms. In several applications, we demonstrate how the exploratory and confirmatory approaches can work in tandem: hypotheses are formulated from an initial analysis and then tested in an independent dataset. The methodology is implemented in the R package BGGM.