Incorporating item response theory into structural equation modes and modeling the nonlinear relationship among latent variables

In structural equation models (SEM), linearity between indicators and latent variables and among latent variables is often assumed. This assumption is often too restrictive. In this study, we incorporated item response theory into SEM so that the relationship between item responses and latent variables can be better described and the relationship among latent variables can be nonlinear. Parameters of the new class of models can be estimated using Bayesian methods. In a series of simulations, we demonstrated how the Markov Chain Monte Carlo estimation procedure and the freeware WinBUGS can be applied to estimate parameters. The simulation results showed that all parameters can be recovered very well, suggesting great flexibility of the new class of models.