Bayesian graphical models for regression on multiple data sets with different variables

Routinely collected administrative data sets, such as national registers, aim to collect information on a limited number of variables for the whole population. In contrast, survey and cohort studies contain more detailed data from a sample of the population. This paper describes Bayesian graphical models for fitting a common regression model to a combination of data sets with different sets of covariates. The methods are applied to a study of low birth weight and air pollution in England and Wales using a combination of register, survey, and small-area aggregate data. We discuss issues such as multiple imputation of confounding variables missing in one data set, survey selection bias, and appropriate propagation of information between model components. From the register data, there appears to be an association between low birth weight and environmental exposure to NO(2), but after adjusting for confounding by ethnicity and maternal smoking by combining the register and survey data under our models, we find there is no significant association. However, NO(2) was associated with a small but significant reduction in birth weight, modeled as a continuous variable.