Generalized latent variable models with non-linear effects

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