Abstract:

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 ofmultilevel 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
ofa classification. These two extensions when combined allow us to fit models to a large array ofunderlying
structures. Existing frequentist modelling approaches to fitting such data have some important computational
limitations. In this paper we consider ways ofovercoming 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 ofeducation, veterinary epidemiology, demography, and public health
illustrating the diversity ofmodels that fit into our framework