A mixed-effects regression model for three-level ordinal response data

Three-level data occur frequently in behaviour and medical sciences. For example, in a multi-centre trial, subjects within a given site are randomly assigned to treatments and then studied over time. In this example, the repeated observations (level-1) are nested within subjects (level-2) who are nested within sites (level-3). Similarly, in twin studies, repeated measurements (level-1) are taken on each twin (level-2) within each twin pair (level-3). A three-level mixed-effects regression model is described here. Random effects at the second and third level are included in the model. Additionally, both proportional odds and non-proportional odds models are developed. The latter allows the effects of explanatory variables to vary across the cumulative logits of the model. A maximum marginal likelihood (MML) solution is described and Gauss-Hermite numerical quadrature is used to integrate over the distribution of random effects. The random effects are normally distributed in this instance. Features of this model are illustrated using data from a school-based smoking prevention trial and an Alzheimer's disease clinical trial.