Estimation and Comparison of Endogenous Ordered Category Multilevel Models

Data often take the form of ordered categories. For instance, in education, test results are often reported as grades. Where a hierarchical structure exists for the data, multilevel modelling of such ordered categorisations can be carried out using macros in MLwiN. The ordered categorisation can be seen as the realisation of an unknown underlying latent variable. A link function is used to relate the two and this determines the scale of the latent variable. This causes a difficulty because whatever model is fitted to the data, the latent variable is rescaled to have the same variance, meaning that developments in the parameter estimates for different models cannot be followed. A heuristic way of overcoming this difficulty has been used by Fielding (2003), using Conditional Mean Scoring (CMS), for models where regressors are not related to the random part of the multilevel model (the exogenous case). In this paper the endogenous case is examined. The use of an instrumental variable approach to overcoming the estimation problems associated with endogenous variables together with the CMS method is shown to be successful in producing a method that allows successive models to be compared. Simulated data and a practical application are used.