The authors respond:
We thank Liao et al1 for their interest in our article.2 Our main objective was to propose a simple multiple imputation method that corrects for covariate measurement error in regression analysis, when the calibration data provide information only about the true value (X) and a proxy (W) for X subject to error, and the main study has measurements on a set of outcomes Y, the proxy variable W, and other predictors Z. Our method addresses the case where Z and Y are not observed in the calibration sample, unlike the more commonly considered case of internal calibration, where values of Z and Y are recorded in the calibration sample.
Liao et al1 state that “Guo et al. applied two methods [classical calibration and regression calibration] to a setting where the methods are theoretically biased, and then showed that their method, designed for the case considered, was valid.”2 Our intent in the simulations was to compare the performance of our proposed method with three common alternatives, specifically ignoring the measurement error, classical calibration, and regression prediction. Although we acknowledge that asymptotic properties are important, we feel that showing the performance of methods in finite sample simulations has value.
Concerning the compared methods, we agree that the “classical calibration” method is theoretically biased, but this approach is widely applied in epidemiological and clinical studies,3,4 so it seems a reasonable comparator for our proposed method. Concerning the “regression prediction” method, we used that term rather than “regression calibration”5 because we chose to reserve that term for the case where covariate values Z are available in the calibration sample. When Z is not available in the calibration sample, imputations from regression prediction cannot condition on Z and hence can result in bias, as noted by Liao et al and seen in our simulations. Our multiple imputation method allows the conditioning on Z to be reflected in the imputations in a simple way and hence can provide valid inferences.
It is not quite accurate to say that we simulated under situations where multiple imputation is based on a correct model because (1) our method also requires a choice of prior distribution for X and the model parameters and (2) we did include some situations where the normality assumption is violated.
Concerning terminology, Liao et al state that the term “external calibration” applies to the case where Z is observed in the calibration sample but not Y. This terminology seems unfortunate to us because in many (perhaps even most) situations where the calibration is done by an external party (eg, the assay manufacturer), Z would not be available.
In summary, we believe that our multiple imputation method is a valuable tool that can be added to data analysts’ toolbox for measurement error correction.
Merck & Co., Inc.
Roderick J. Little
University of Michigan
School of Public Health
Ann Arbor, MI
1. Liao X, Spiegelman D, Carroll RJ. Regression calibration is valid when properly applied. Epidemiology. 2013;24:466–467
2. Guo Y, Little RJ, McConnell DS. On using summary statistics from an external calibration sample to correct for covariate measurement error. Epidemiology. 2012;23:165–174
3. Higgins KM, Davidian M, Chew G, Burge H. The effect of serial dilution error on calibration inference in immunoassay. Biometrics. 1998;54:19–32
4. Browne RW, Whitcomb BW. Procedures for determination of detection limits: application to high-performance liquid chromatography analysis of fat-soluble vitamins in human serum. Epidemiology. 2010;21(suppl 4):S4–S9
5. Rosner B, Spiegelman D, Willett WC. Correction of logistic regression relative risk estimates and confidence intervals for measurement error: the case of multiple covariates measured with error. Am J Epidemiol. 1990;132:734–745