Implications of alternative parameterizations in structural equation models for longitudinal categorical variables

When analyzing scaling conditions in latent variable Structural Equation Models (SEMs) with continuous observed variables, analysts scaling a latent variable typically set the factor loading of one indicator to one and either set its intercept to zero or the mean of its latent variable to zero. When binary and ordinal observed variables are part of SEMs, the identification and scaling choices are more varied. Longitudinal data further complicate this. In SEM software, such as lavaan and Mplus, fixing the underlying variables’ variances or the error variances to one are two primary scaling conventions. As demonstrated in this paper, choosing between these constraints can significantly impact longitudinal analysis, affecting model fit, degrees of freedom, and assumptions about the dynamic process and error structure. We explore alternative parameterizations and conditions of model equivalence with categorical repeated measures. Using data from the National Longitudinal Survey of Youth 1997, we empirically explore how different parameterizations lead to varying conclusions in longitudinal categorical analysis. More specifically, we provide insights into the specifications of the autoregressive latent trajectory model and its special cases - the linear growth curve and first-order autoregressive models - for categorical repeated measures. These findings have broader implications for a wide range of longitudinal models.