Measurement and Uncertainty Preserving Parametric Modeling for Continuous Latent Variables With Discrete Indicators and External Variables

Research in education and behavioral sciences often involves the use of latent variable models that are related to indicators, as well as related to covariates or outcomes. Such models are subject to interpretational confounding, which occurs when fitting the model with covariates or outcomes alters the results for the measurement model. This has received attention in models for continuous observable variables but to date has not been examined in the context of discrete variables. This work demonstrates that interpretational confounding can occur in models for discrete variables, and develops a multistage Bayesian estimation approach to deal with this problem. The key features of this approach are that it is (a) measurement preserving, in that it precludes the possibility of interpretational confounding, and (b) uncertainty preserving, in that the uncertainty from the earlier stage of estimating the measurement model is propagated to the second stage of estimating the relations between the latent variable(s) and any covariates or outcomes. Previous work on these methods had only considered models for continuous observed variables, and software was limited to models with a single latent variable and either covariates or outcomes. This work extends the approach and software to a more general class of solutions, including discrete variables, illustrating the procedures with analyses of real data. Functions for conducting the analyses in widely available software are provided.