Framing Structural Equation Models as Bayesian Non-Linear Multilevel Regression Models

This dissertation is a collection of three papers. The first is a conceptual paper, followed by two data analysis papers. All three papers examine the connection between structural equation models and regression models, and how one may better learn, research and apply structural equation models when structural equation models are thought of as regression models. Each paper contains unique contributions. In the first paper, I focus on conceptual issues related to estimating structural equation models (SEMs) as Bayesian multilevel regression models. I review prevailing views on the equivalence of the two model classes (SEM and multilevel regression), and show how a Bayesian approach allows for the unity of both model classes. Adopting a Bayesian approach introduces additional considerations for estimating SEMs which I review. Additionally, I lay out linear regression model specifications that are directly equivalent to commonplace SEMs. Finally, the paper ends with a discussion of open issues in SEMs that a Bayesian multilevel regression approach to SEMs more readily solves. The goal of the second paper is to frame structural equation models (SEMs) as Bayesian multilevel regression models using the example of a unidimensional confirmation factor model. Framing SEMs as Bayesian regression models provides an alternative approach to understanding SEMs that can improve model transparency and enhance innovation during modeling. For demonstration, I analyze six indicators of living standards data from 101 countries. I show how the unidimensional confirmatory factor analysis (CFA) with congeneric indicators is a nonlinear multilevel regression model. I fit this model using Bayesian estimation and conduct model diagnostics from the regression perspective. The model diagnostics identify misspecification that standard SEM misfit statistics are unable to detect and I extend the congeneric model to accommodate the unique features of the data under study. I also provide extensive guidance on prior specification, which is relevant for estimating complex regression models such as these. I end with discussion of the implications of a regression approach to modeling these data and data used in SEMs more broadly. In the third paper, I turn to bounded count indicators. Such indicators are common in the study of rare behaviours -- I develop structural equation models for such indicators. The models are developed as Bayesian non-linear multilevel regression models; I assume the indicators are beta-binomial variables and jointly model the location and ICC of the indicators, recommending latent variables for both parameters. Furthermore, I present an interval-censoring extension to the developed models to be used when binned or coarsened versions of the indicators are available in place of the true counts. I show that the models can recover population parameters using simulated data, and show how to perform a complete analysis using data from the Irish longitudinal study on ageing. Taken together, the papers form the beginnings of a syllabus on the dissertation topic. Ultimately, the papers demonstrate that to do better at structural equation modeling, one needs to better understand regression modeling. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]