various portions of the model by using the Bayes rule. Conceptually, the approach imposes a distributional assumption on a set of parameters assumed to possess inherent variability. This allows one to effectively "borrow information" from the set as a whole, and then "pull back" the more-extreme and less-precise estimates of these parameters. This achieves a more-stable portrait of the pattern of variability as a whole. When there is information about errors in the exposure measures, measurement-error models are useful.

The empirical Bayes methodology has multiple uses in biomedical applications (Breslow, 1990) and may be of particular value in environmental epidemiology. For parametric models, the methods have been developed in some detail (Morris, 1983) and are known as "parametric empirical Bayes." Kass and Steffey (1989) refer to such a structure as a conditionally independent hierarchic model. The methods have been described for a variety of specific applications, including Poisson models (Albert, 1988; Gaver et al., 1990) where estimating mortality rates is of issue (Hui and Berger, 1983; Tsutakawa et al., 1985; Clayton and Kaldor, 1987; Desouza, 1991), particularly as regards geographic clustering or mapping of disease rates (Manton et al., 1989; Merril and Selvin, 1992); binomial-logistic models (Levin, 1986); and normal models (DuMouchel and Harris, 1983) of diseases that may be related to environmental factors.

Modeling Covariance Structures

The primary effect of an agent is not always to change the expected value of a health outcome. Rather, the response to a pollutant may be heterogeneous, and the effects of interest may include modifying the response to other exposures, or the effect may be indirect through modification of outcomes other than that of primary interest. These issues are discussed below in order of increasing complexity.

Heterogeneity of Response

Heterogeneity of response to environmental agents is well established. For example, chamber studies of ozone exposure of exercising young adults identified a sensitive subgroup that had the largest short-term reductions in lung function in response to ozone (McDonnell et al., 1985). These differential responses were reproducible in subsequent challenges of the individual subjects. The degree of sensitivity to ozone did not seem to be associated with preexisting respiratory conditions, and markers predicting enhanced sensitivity have not been identified to date (McDonnell et al., 1985).

These laboratory findings are mirrored by field epidemiology studies



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