Application of the Stochastic EM Method to Latent Regression Models

The reporting methods used in large scale assessments such as the National Assessment of Educational Progress (NAEP) rely on a latent regression model. The first component of the model consists of a p-scale IRT measurement model that defines the response probabilities on a set of cognitive items in p scales depending on a p-dimensional latent trait variable θ = (θ1, . . . θp). In the second component, the conditional distribution of this latent trait variable θ is modeled by a multivariate, multiple regression on a set of predictor variables, which are usually based on student, school and teacher variables in assessments such as NAEP. In order to fit the latent regression model using the maximum (marginal) likelihood estimation technique, multivariate integrals have to be evaluated. In the computer program MGROUP used by ETS for fitting the latent regression model to data from NAEP and other sources, the integration is currently done either by numerical quadrature (for problems up to two dimensions) or by an approximation of the integral. CGROUP, the current operational version of the MGROUP program used in NAEP and other assessments since 1993, is based on Laplace approximation that may not provide fully satisfactory results, especially if the number of items per scale is small. This paper examines the application of stochastic expectation-maximization (EM) methods (where an integral is approximated by an average over a random sample) to NAEP-like settings. We present a comparison of CGROUP with a promising implementation of the stochastic EM algorithm that utilizes importance sampling. Simulation studies and real data analysis show that the stochastic EM method provides a viable alternative to CGROUP for fitting multivariate latent regression models.