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

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.