Maximum likelihood analysis of a general latent variable model with hierarchically mixed data.

A general two-level latent variable model is developed to provide a comprehensive framework for model comparison of various submodels. Nonlinear relationships among the latent variables in the structural equations at both levels, as well as the effects of fixed covariates in the measurement and structural equations at both levels, can be analyzed within the framework. Moreover, the methodology can be applied to hierarchically mixed continuous, dichotomous, and polytomous data. A Monte Carlo EM algorithm is implemented to produce the maximum likelihood estimate. The E-step is completed by approximating the conditional expectations through observations that are simulated by Markov chain Monte Carlo methods, while the M-step is completed by conditional maximization. A procedure is proposed for computing the complicated observed-data log likelihood and the BIC for model comparison. The methods are illustrated by using a real data set.