Multilevel mixture item response theory models: an application in education testing

Skrondal and Rabe-Hesketh (2004) proposed a generalized latent variable modeling framework integrating 1) factor analytic and random coe cient models, 2) models with discrete and continuous unobserved variables, and 3) hierarchical models with unobserved variables at di erent levels. This framework is implemented in their GLLAMM software package. In this paper I describe a strongly related framework that is implemented in the syntax version of the Latent GOLD software (Vermunt and Magidson, 2007). The most important extension compared to the GLLAMM approach is that it allows de ning models with any combination of discrete and continuous latent variables at each level of the hierarchy. The approach is illustrated with a multilevel application in educational testing; that is, using a set of mathematics test items taken from pupils nested within schools. An item response theory model is constructed for the responses on the test items and the between-school di erences in pupils' abilities and item di culties is modeled using a discrete mixture distribution at the school level.