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Public health researchers often estimate health effects of exposures (e.g., pollution, diet, and lifestyle) that cannot be directly measured for study subjects. A common strategy in environmental epidemiology is to use a first‐stage (exposure) model to estimate the exposure on the basis of covariates and/or spatiotemporal proximity and to use predictions from the exposure model as the covariate of interest in the second‐stage (health) model. This induces a complex form of measurement error. We propose an analytical framework and methodology that is robust to misspecification of the first‐stage model and provides valid inference for the second‐stage model parameter of interest.  We decompose the measurement error into components analogous to classical and Berkson errors and characterize properties of the estimator in the second‐stage model if the first‐stage model predictions are plugged in without correction. Specifically, we derive conditions for compatibility between the first‐stage and second‐stage models that guarantee consistency (and have direct and important real‐world design implications), and we derive an asymptotic estimate of finite‐sample bias when the compatibility conditions are satisfied. We propose a methodology that does the following: (i) corrects for finite‐sample bias; and (ii) correctly estimates standard errors. We demonstrate the utility of our methodology in simulations and an example from air pollution epidemiology. Copyright © 2013 John Wiley & Sons, Ltd.

Measurement error in two‐stage analyses, with application to air pollution epidemiology