On the Implications of Model Misspecifications and Mixed Indicator Types for Finding Standardized Model Parameters

Model estimations given mixed data types proceed in two stages. Stage 1 obtains a covariance matrix of the continuous indicators and the variables underlying the categorical indicators. Stage 2 fits the researcher’s model to data with weighted, diagonally weighted, or unweighted least squares. For a misspecified model, three problems have been overlooked about defining and estimating population standardized parameters β. First, given the researcher’s model and population covariance matrix, only one fit function can yield unique population unstandardized parameters θ. Second, standardizing θ into β cannot remove the influence from data scales; β is not unique if data scales change. Third, the prevailing standard error method for θ̂ (known as the “robust SE”) may have large bias even for slightly misspecified models, and thus is not suitable for finding SE(β̂). Unless the researcher’s model is perfect, the three problems above will persist. We propose solutions and conduct simulations to verify the solutions are effective.