Multiple membership multiple classification (MMMC) models

In the social and other sciences many data are collected with a known but complex underlying structure. Over the past two decades there has been an increase in the use of multilevel modelling techniques that account for nested data structures. Often however the underlying data structures are more complex and cannot be fitted into a nested structure. First, there are cross-classified models where the classifications in the data are not nested. Secondly, we consider multiple membership models where an observation does not belong simply to one member of a classification. These two extensions when combined allow us to fit models to a large array of underlying structures. Existing frequentist modelling approaches to fitting such data have some important computational limitations. In this paper we consider ways of overcoming such limitations using Bayesian methods, since Bayesian model fitting is easily accomplished using Monte Carlo Markov chain (MCMC) techniques. In examples where we have been able to make direct comparisons, Bayesian methods in conjunction with suitable 'diffuse' prior distributions lead to similar inferences to existing frequentist techniques. In this paper we illustrate our techniques with examples in the fields of education, veterinary epidemiology, demography, and public health illustrating the diversity of models that fit into our framework.