Multilevel item response models: An approach to errors in variables regression

In this article we show how certain analytic problems that arise when one attempts to use latent variables as outcomes in regression analyses can be addressed by taking a multilevel perspective on item response modeling. Under a multilevel, or hierarchical, perspective we cast the item response model as a within-student model and the student population distribution as a between-student model. Taking this perspective leads naturally to an extension of the student population model to include a range of student level variables, and it invites the possibility of further extending the models to additional levels so that multilevel models can be applied with latent outcome variables. In the two-level case, he model that we employ is formally equivalent to the plausible value procedures that are used as part of the National Assessment of Educational Progress (NAEP), but we present the method for a different class of measurement models, and we use a simultaneous estimation method rather than two-step estimation. In our application of the models to the appropriate treatment of measurement error in the dependent variable of a between-student regression, we also illustrate the adequacy of some approximate procedures that are used in NAEP.