Nonlinear Effects in the Generalized Latent Variable Model

Until recently, latent variable models such as the factor analysis model for metric responses, the two-parameter logistic model for binary responses, the multinomial model for nominal responses considered only main effects of latent variables without allowing for interaction or polynomial latent variable effects. However, nonlinear relationships among the latent variables might be necessary in real applications. Methods for fitting models with nonlinear latent terms have been developed mainly under the structural equation modelling approach. In this paper, we consider a general latent variable model framework for mixed responses (metric and categorical) that allows inclusion of both nonlinear latent and covariate effects. The model parameters are estimated using full Maximum Likelihood based on a hybrid integration-maximization algorithm. Finally, a new method for obtaining factor scores based on Multiple Imputation is proposed here for the model with nonlinear terms.