Dynamic Latent Class Structural Equation Modeling: A Hands-On Tutorial for Modeling Intensive Longitudinal Data

Abstract

In this tutorial, we provide a hands-on guideline on how to implement complex Dynamic Latent Class Structural Equation Models (DLCSEM) in the Bayesian software JAGS. We provide building blocks starting with simple Confirmatory Factor and Time Series analysis, and then extend these blocks to Multilevel Models and Dynamic Structural Equation Models (DSEM). Subsequently, we introduce Hidden Markov Switching Models (HMSM) and demonstrate their integration with DSEM to yield DLCSEM. Leading through the tutorial is an example from clinical psychology using data on a generalized anxiety treatment that includes scales on anxiety symptoms and the Working Alliance Inventory that measures alliance between therapists and patients. Within each block, we provide an overview, specific hypotheses we want to test, the resulting model and its implementation, as well as an interpretation of the results. The aim of this tutorial is to provide a step-by-step guide for applied researchers that enables them to use this flexible DLCSEM framework for their own analyses.

Figures (13)

• Figure 1: Path diagram of the confirmatory factor analysis (CFA) model at a single time point for $N$ individuals. The model includes four latent factors ($\eta$), each loading onto three observed indicators ($y$) with freely estimated factor loadings ($\lambda$). Residual variances are not depicted for simplicity.
• Figure 2: Path diagram for the AR(1) time series model for a single person and a single item $y$. (a) for $3$ explicit time points; (b) for $t\in\{1,...,N_t\}$ time points.
• Figure 3: Trajectories of individual scores across 15 measurement occasions for three different BAI items, illustrating heterogeneity across patients.
• Figure 4: Path diagram of the multilevel model for a single observed item $y$ using an AR(1) structure with random intercept ($\alpha_i$) and slope ($\beta_i$) across $t\in\{1,...,N_t\}$ time points and $i\in\{1,..., N\}$ individuals.
• Figure 5: Path diagram of the DSEM for the latent factor $\eta$ loading on three indicators $y$ using an AR(1) structure with random intercept ($\alpha_i$) and slope ($\beta_i$) across $t\in\{1,...,N_t\}$ time points and $i\in\{1,...,N\}$ individuals.