Hierarchical models for combining ecological and case-control data.

The ecological study design suffers from a broad range of biases that result from the loss of information regarding the joint distribution of individual-level outcomes, exposures, and confounders. The consequent nonidentifiability of individual-level models cannot be overcome without additional information; we combine ecological data with a sample of individual-level case-control data. The focus of this article is hierarchical models to account for between-group heterogeneity. Estimation and inference pose serious computational challenges. We present a Bayesian implementation based on a data augmentation scheme where the unobserved data are treated as auxiliary variables. The methods are illustrated with a dataset of county-specific infant mortality data from the state of North Carolina.