  In the first stage of a two-stage study, the researcher uses a statistical
model to impute the unobserved exposures. In the second stage, imputed
exposures serve as covariates in epidemiological models. Imputation error in
the first stage operate as measurement errors in the second stage, and thus
bias exposure effect estimates. This study aims to improve the estimation of
exposure effects by sharing information between the first and second stage. At
the heart of our estimator is the observation that not all second-stage
observations are equally important to impute. We thus borrow ideas from the
optimal-experimental-design theory, to identify individuals of higher
importance. We then improve the imputation of these individuals using ideas
from the machine-learning literature of domain-adaptation. Our simulations
confirm that the exposure effect estimates are more accurate than the current
best practice. An empirical demonstration yields smaller estimates of PM effect
on hyperglycemia risk, with tighter confidence bands. Sharing information
between environmental scientist and epidemiologist improves health effect
estimates. Our estimator is a principled approach for harnessing this
information exchange, and may be applied to any two stage study.
