  Regression calibration as developed by Rosner, Spiegelman and Willet is used
to correct the bias in effect estimates due to measurement error in continuous
exposures. The method involves two models: a measurement error model (MEM)
relating the mismeasured exposure to the true exposure and an outcome model
relating the mismeasured exposure to outcome. However, no comprehensive
guidance exists for determining which covariates should be included in each
model. In this paper, we investigate the selection of the minimal and most
efficient covariate adjustment sets under a causal inference framework. We show
that in order to correct for the measurement error, researchers must adjust
for, in both MEM and outcome model, any common causes (1) of true exposure and
the outcome and (2) of measurement error and the outcome. When such variable(s)
are only available in the main study, researchers should still adjust for them
in the outcome model to reduce bias, provided that these covariates are at most
weakly associated with measurement error. We also show that adjusting for so
called prognostic variables that are independent of true exposure and
measurement error in outcome model, may increase efficiency, while adjusting
for any covariates that are associated only with true exposure generally
results in efficiency loss in realistic settings. We apply the proposed
covariate selection approach to the Health Professional Follow-up Study dataset
to study the effect of fiber intake on cardiovascular disease. Finally, we
extend the originally proposed estimators to a non-parametric setting where
effect modification by covariates is allowed.
