Marginal Maximum Likelihood Estimation of a Latent Variable Model With Interaction

There has been considerable interest in nonlinear latent variable models specifying interaction between latent variables. Although it seems to be only slightly more complex than linear regression without the interaction, the model that includes a product of latent variables cannot be estimated by maximum likelihood assuming normality. Consequently, many approximate methods have been proposed. Recently, a maximum likelihood method of estimation based on the expectation–maximization algorithm has been suggested that is optimum if the distribution assumptions are true. In this article, the authors outline an alternative marginal maximum likelihood estimator using numerical quadrature. A key feature of the approach is that in the marginal distribution of the manifest variables the complicated integration can be reduced, often to a single dimension. This allows a direct approach to maximizing the log-likelihood and makes the method relatively straightforward to use.