Extended mixture factor analysis model with covariates for mixed binary and continuous responses

Finite mixture factor analysis provides a parsimonious model to explore latent group structures of high-dimensional data. In this modeling framework, we can explore latent structures for continuous responses. However, dichotomous items are often used to define latent domains in practice. This paper proposes an extended finite mixture factor analysis model with covariates to model mixed continuous and binary responses. We use a Monte Carlo expectation-maximization (MCEM) algorithm to estimate the model. In the E step, closed-form solutions are not available for the conditional expectation of complete data log likelihood, so it is approximated by sample means, which are in turn generated by the Gibbs sampler from the joint conditional distribution of latent variables. To monitor the convergence of the MCEM algorithm, we use bridge sampling to calculate the log likelihood ratio of two successive iterations. We adopt a diagnostic plot of the log likelihood ratio against iterations for monitoring the convergence of the MCEM algorithm. We compare different models based on BIC, in which we approximate the observed data log likelihood by using a Monte Carlo method. We investigate the computational properties of the MCEM algorithm by simulation studies. We use a real data example to illustrate the practical usefulness of the model. Finally, we discuss limitations and possible extensions.