Modular item response and structural equation modelling via measurement and uncertainty preserving parametric modelling

Latent variable item response and structural equation models are widely used to model constructs and acknowledge measurement error in research settings and operational assessments. Such work often proceeds in stages, where the results of an analysis from an earlier stage are fed into the analysis at a later stage. However, common practices in single-stage and multistage estimation have weaknesses, including when viewed from a Bayesian perspective. This work extends recent developments in 2-Stage Bayesian approaches in structural equation modelling (SEM) to advance a general multistage approach where models are viewed as comprised of fragments that can be assembled in a modular way. The proposed approach is more in line with Bayesian principles and offers advantages over existing approaches. The approach is illustrated by applications in several different modelling scenarios, including: SEM for a wider class of situations than existing approaches have considered; and calibration and scoring situations encountered in operational assessment using item response theory (IRT). R functions for executing these analyses by interfacing with Mplus are provided and documented, as is code for running the examples.