Combining contingency tables with missing dimensions

We propose a methodology for estimating the cell probabilities in a multiway contingency table by combining partial information from a number of studies when not all of the variables are recorded in all studies. We jointly model the full set of categorical variables recorded in at least one of the studies, and we treat the variables that are not reported as missing dimensions of the study-specific contingency table. For example, we might be interested in combining several cohort studies in which the incidence in the exposed and nonexposed groups is not reported for all risk factors in all studies while the overall numbers of cases and cohort size is always available. To account for study-to-study variability, we adopt a Bayesian hierarchical model. At the first stage of the model, the observation stage, data are modeled by a multinomial distribution with fixed total number of observations. At the second stage, we use the logistic normal (LN) distribution to model variability in the study-specific cells' probabilities. Using this model and data augmentation techniques, we reconstruct the contingency table for each study regardless of which dimensions are missing, and we estimate population parameters of interest. Our hierarchical procedure borrows strength from all the studies and accounts for correlations among the cells' probabilities. The main difficulty in combining studies recording different variables is in maintaining a consistent interpretation of parameters across studies. The approach proposed here overcomes this difficulty and at the same time addresses the uncertainty arising from the missing dimensions. We apply our modeling strategy to analyze data on air pollution and mortality from 1987 to 1994 for six U.S. cities by combining six cross-classifications of low, medium, and high levels of mortality counts, particulate matter, ozone, and carbon monoxide with the complication that four of the six cities do not report all the air pollution variables. Our goals are to investigate the association between air pollution and mortality by reconstructing the tables with missing dimensions, to determine the most harmful pollutant combinations, and to make predictions about these key issues for a city other than the six sampled. We find that, for high levels of ozone and carbon monoxide, the number of cases with a high number of deaths increases as the levels of particulate matter, PM10, increases and that the most harmful combinations corresponds to high levels of PM10, confirming prior findings that levels of PM10 higher than the NAAQS standard are harmful.