Measuring the complexity of generalized linear hierarchical models

Measuring a statistical model's complexity is important for model criticism and comparison. However, it is unclear how to do this for hierarchical models due to uncertainty about how to count the random effects. The authors develop a complexity measure for generalized linear hierarchical models based on linear model theory. They demonstrate the new measure for binomial and Poisson observables modeled using various hierarchical structures, including a longitudinal model and an areal-data model having both spatial clustering and pure heterogeneity random effects. They compare their new measure to a Bayesian index of model complexity, the effective number pD of parameters (Spiegelhalter, Best, Carlin & van der Linde 2002); the comparisons are made in the binomial and Poisson cases via simulation and two real data examples. The two measures are usually close, but differ markedly in some instances where pD is arguably inappropriate. Finally, the authors show how the new measure can be used to approach the difficult task of specifying prior distributions for variance components, and in the process cast further doubt on the commonly used vague inverse gamma prior.