Likelihood based procedures for general nonlinear structural equation analysis

Valid Statistical inferences in nonlinear structural equation models are of great interest recently. This dissertation aims at fitting a nonlinear structural equation model consisting of two parts; a general nonlinear measurement model relating observed variables or indicators to unobserved concepts or latent variables, and a nonlinear simultaneous structural model describing relationships among the latent variables. For model identification, we assume an explicitly solved reduced structural model exists. This dissertation is composed of two papers.; The first paper deals with the case where the latent variables in the reduced structural model are normally distributed. We developed maximum likelihood estimation by a Monte Carlo EM algorithm. The asymptotic covariance matrix of the estimator is computed by the inverse of the empirical observed information matrix. Initial values of the parameters for general and special reduced structural models are presented. For a Monte Carlo EM algorithm, we developed a new procedure both to choose the Monte Carlo sample size for computing the expectation in the E-step, and to stop the algorithm. Simulation studies for structural equation models with a variety of structural models are presented to assess the performance of our stopping rule and the estimators.; The second paper develops distribution-free statistical procedures without specifying distribution forms of the latent variables. We use the normal-mixtures as a flexible distribution family. A pseudo maximum likelihood estimation procedure is introduced by first obtaining the measurement model parameters by factor analysis, then maximizing the pseudo likelihood, the likelihood evaluated at the measurement model parameters estimates, with respect to the structural equation model parameters. The asymptotic covariance matrix of the measurement parameters estimates is computed by non-parametric bootstrap, which is combined with the empirical information matrix of all parameters for the full likelihood to produce an estimate of the asymptotic covariance of the reduced model parameters estimates. Simulation studies are presented.