Illustrating the Value of Prior Predictive Checking for Bayesian Structural Equation Modeling

Abstract

A unique feature of Bayesian estimation is the inclusion of prior knowledge through prior distributions. These prior distributions can benefit or impair many components of the ensuing analysis. Priors are especially important to assess in the context of structural equation models (SEMs), which often carry data and modeling complexities where priors can be particularly influential. In this article, we illustrate a statistical approach to assess the impact of our prior specifications: the prior predictive checking procedure. We introduce a comprehensive prior predictive checking workflow that organizes the procedure into clear steps, and we relate this workflow to the SEM framework. Through three examples, we demonstrate how the workflow can aid researchers in more fully understanding different aspects of their prior specification within SEM. Code and additional resources are provided in the online Supplemental Materials to facilitate future application of the prior predictive checking procedure within the SEM framework.

Keywords:

• Bayes
• prior predictive checking
• structural equation modeling

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 We use the word “optimally” in quotations because we want to emphasize that priors, by nature, are inherently subjective. In this sense, there is no single “right” or “wrong” prior. The methods outlined in the current paper should be placed in the context of priors representing a continuum of appropriateness (as opposed to a “right” versus “wrong” dichotomy).

2 There are also other ways in which a prior that is too informative can be problematic, for example when the prior distribution only supports parameter values that are not supported by the information in the (to be collected) data. Once data are included in the analysis, such priors can cause issues with convergence and overly wide posterior distributions. The presence of this kind of overly informative prior can be assessed through a prior sensitivity analysis after data are collected.

3 It is also possible to use the Gibbs sampler through the “rjags” package. However, several features of “blavaan,” such as generating prior predictive samples, are not available with this sampler.

4 Although the default prior settings appear reasonable for the linear and quadratic slopes, we could have justified a more nuanced approach to the prior setting. Specifically, it is common that a quadratic slope contains less variance as compared to the linear slope. To reflect this point, we could have opted to include this artifact in our prior-building phase by specifying a smaller standard deviation hyperparameter for the quadratic slope.

5 In addition, Veen et al. (2020b) also specified a shape parameter and used a skew-normal distribution for their priors. To align this example with the previous two, we decided to focus on just the mean and standard deviation parameters.