We propose and illustrate an approach for combining information from several regression studies, each considering only a subset of the variables of interest. Our approach uses a combination of Bayesian hierarchical modeling and data augmentation. Hierarchical models are a flexible tool for modeling study-to-study as well as within-study variability. Data augmentation methods address fully the uncertainty resulting from missing data and provide venues for combining information in a way that preserves the meaning of the regression coefficients across studies. We discuss in detail a normal-normal model, we suggest a simple and efficient numerical implementation based on a block Gibbs sampler, and we provide explicit full conditional distributions for an arbitrary pattern of variables missing by study.

We discuss an application of our model to investigating the level of chlorophyll-a in water quality management. Chlorophyll-a is one of the most important indicators of lake water quality. Scientists have developed a number and variety of forecasting models relating chlorophyll-a to nutrients such as phosphorus and nitrogen. These models often have to rely on sparse information from multiple sources—in this case lakes. We study the relationship among chlorophyll-a and phosphorus in 12 northern temperate lakes by using data from the literature. An important covariate is nitrogen, which is reported only in some of the studies.

Key Words
Hierarchical models; Missing covariates; Water quality.

Francesca Dominici is Visiting Assistant Professor, Department of Biostatistics, Johns Hopkins University. Giovanni Parmigiani is Assistant Professor, Institute of Statistics and Decision Sciences and Center for Health Policy Research and Education; Robert Wolpert is Associate Professor, Institute of Statistics and Decision Sciences; and Kenneth Reckhow is Associate Professor, Nicholas School of the Environment and Institute of Statistics and Decision Sciences; Duke University, Durham, NC 27708-0251. This work was completed while Francesca Dominici was a visiting scholar at the Institute of Statistics and Decision Sciences.