Limited utility of small-variance priors to detect local misspecification in Bayesian structural equation models

In a highly influential paper on current practice in Bayesian structural equation modeling (BSEM), Muthén and Asparouhov (Psychol Methods 17:313–335, 2012) proposed using small-variance priors to constrain non-target parameters to be close to (rather than exactly) zero, with the “side product” (p. 313) that the posterior distributions of such nontarget parameters could be used analogous to modification indices. This chapter presents 2 simulation studies of their utility, in the context of (a) constraining cross-loadings to be nearly zero and (b) constraining factor loadings and intercepts to be equivalent across groups or occasions. The first study reinforced earlier findings that small-variance priors can prevent detecting important misspecifications (i.e., global-fit indices indicate better fit as priors become less restrictive). In contrast, these local indicators have greater power to detect invalid constraints when priors are less restrictive. Study 2 revealed similar patterns in the context of detecting invalid equality constraints and showed limited utility of small-variance priors over modification indices under maximum-likelihood estimation. Our advice is to evaluate global fit in BSEM without small-variance priors, and only when hypothesized models are rejected, utilize small-variance priors to search for clues about possible respecification. We recommend exploring other tools for local-fit evaluation in BSEM, which might detect misspecifications without introducing additional complications of small-variance priors (e.g., propagation of bias).