  Public health researchers often estimate health effects of exposures (e.g.,
pollution, diet, 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 based on covariates
and/or spatio-temporal 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 error 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- 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 (1) corrects for finite-sample bias and (2) correctly
estimates standard errors. We demonstrate the utility of our methodology in
simulations and an example from air pollution epidemiology.
