Maximum likelihood estimation for nonlinear structural equation models with normal latent variables

Abstract

Structural equation analysis has been widely used in behavioral and social sciences. In practice, for continuous observed variables, linear structural equation models have been used nearly exclusively. The use of models that are nonlinear in latent variables has been limited to simple situations with a linear measurement model and with a single polynomial structural relationship, usually containing only one quadratic or cross-product term. This paper introduces a general structural equation model with a nonlinear measurement model and a simultaneous system of nonlinear and non-polynomial structural relationships. For such a model, a parameterization useful for identification and interpretation is presented. For model fitting and parameter inferences, the maximum likelihood approach is considered. A method for obtaining parameter estimates and their asymptotic covariance matrix estimate is developed based on a new version of the Monte Carlo EM algorithm. The performance of the algorithm is examined using simulation studies.