Multiple imputation for measurement error correction based on a calibration sample

In much of applied statistics variables of interest are measured with error. In particular, regression with covariates that are subject to measurement error requires adjustment to avoid biased estimates and invalid inference. We consider two aspects of this problem. Detection Limits (DL) arise in epidemiological or other empirical studies that involve measurements of an analyte. Measurements below the DL are often reported as missing, since they are subject to unacceptable measurement error. This approach ignores the fact that values above DL are also subject to error. We describe a Bayesian measurement error model for data subject to detection limits, which allows for heteroscedastic measurement error throughout the range of the variable. Application of our model to calibration data for fat-soluble vitamins suggests that the prediction uncertainty is actually higher for true values above the DL than values below the DL, suggesting that the classical approach to this problem is flawed from a statistical perspective.Second, we consider the estimation of regression parameters when the covariate is measured with error. Most of the previous work assumes the measurement error has a constant variance, an assumption that does not hold in many situations. For the constant variance case, we develop a multiple imputation method based on the non-differential measurement error assumption and show that this method compares favorably with regression calibration. We propose extensions of regression calibration arid multiple imputation for heteroscedastic measurement error, and compare their performance via a simulation study. The multiple imputation method is shown to provide better inferences in this setting.We also consider the situation where interest concerns regression of outcomes on a variable subject to measurement error arid other covariates, and information about measurement error is provided in the form of summary statistics from a bivariate calibration sample. This data scenario is often encountered in epidemiological applications. We propose and evaluate a simple multiple imputation method for this problem that yields valid inferences for the regression model parameters, using multiple imputation combining rules proposed by Reiter (2008). The method is based on normal assumptions, and hence robustness to lack of normality is also assessed.