Equivalence of regression calibration methods in main study/external validation study designs

Measurement error occurs when one or more regression model covariates are measured with error, and is a common problem in occupational health and other fields. If the effects of measurement error are not corrected, estimates of the regression coefficients and their variances are biased. We consider the situation when an external validation study provides information about the measurement error process. Two related approaches for correcting measurement error, both called regression calibration, have been proposed. Although the estimators for the corrected regression coefficients can be shown to be the same under the two approaches, the asymptotic variances are derived using two different methods and have not been compared. In this paper, we show that these two methods give algebraically identical estimators for the corrected regression coefficients and their asymptotic variances, in the general case of multiple covariates measured with error, for a generalized linear model of exponential family form.