The problem of measurement error arises whenever data are measured with greater variability than the true quantities of interest, View the MathML sourceX. It can be attributed to sources like device error, assay error, and biological fluctuations. Summary statistics and regression models based on mis-measured data, View the MathML sourceW, may have biased parameter estimators, reduced power and insufficient confidence interval coverage (Fuller, 1987, Armstrong, 2003 and Carroll et al., 2006).
Many strategies to adjust for measurement error have been proposed. Correction for measurement error in covariates, View the MathML sourceX, in linear and generalized linear models is commonly achieved by regression calibration (RC), which substitutes an estimate of the conditional mean View the MathML sourceE(X|W) for the unknown View the MathML sourceX (Carroll and Stefanski, 1990 and Gleser, 1990). When estimation of the density of View the MathML sourceX is of interest, this is not particularly useful, as View the MathML sourceE(X|W) is over-corrected in terms of having reduced spread (Eddington, 1940 and Tukey, 1974). However, the linear regression based on this substitution estimates the underlying parameters of interest. RC is also implemented in non-linear models because of its simplicity, but is typically most effective for general linear models when the measurement error is not large (Rosner et al., 1989 and Carroll et al., 2006). Alternatives for non-linear models are thoroughly reviewed by Carroll et al. (2006) and include maximum likelihood, conditional score, SIMEX and Bayesian methods. Density estimation is addressed in a separate literature, where deconvolution methods are a prominent approach for scalar variables (Carroll and Hall, 1988 and Stefanski and Carroll, 1990). Recent papers add to the theoretical basis for deconvolution estimators and offer an extension to heteroscedastic measurement error (Carroll and Hall, 2004, Delaigle, 2008, Delaigle et al., 2008 and Delaigle and Meister, 2008).
Despite the variety of available methodology, measurement error correction is rarely implemented (Jurek et al., 2004), and when it is, RC remains popular. Convenience may be a priority for a method to achieve widespread use. The preceding methods target estimation of parameters in a specific regression context; a different method must be implemented for every type of regression model in which View the MathML sourceX is used, and density estimation is treated separately. An alternative approach is to focus on re-creating the true View the MathML sourceX from the observed View the MathML sourceW, at least approximately, as the primary quantity of interest or as a means to improving parameter estimation (Louis, 1984, Bay, 1997, Shen and Louis, 1998 and Freedman et al., 2004). Most recently, Thomas et al. (2011) introduce a Moment Adjusted Imputation (MAI) method that aims to replace scalar, mis-measured data, WW, with estimators, View the MathML sourceX̂, that have asymptotically the same joint distribution with a response, YY, and potentially error-free covariates View the MathML sourceZ, as does the latent variable, XX, up to some number of moments. Originally developed in an unpublished dissertation, (Bay, 1997), MAI extends the idea of moment reconstruction (MR), which focuses on the first two moments of the joint distribution (Freedman et al., 2004 and Freedman et al., 2008). Thomas et al. (2011) investigate the performance of MAI in logistic regression and demonstrate superior results to RC and MR when the distribution of XX is non-normal. Moreover, the View the MathML sourceX̂ can be used for density estimation, with a guarantee of matching the latent variable mean, variance and potentially higher moments.
Thomas et al. (2011) discuss an important application where measurement error is likely present in multiple covariates and a scalar adjustment method would not be adequate. In this example, Gheorghiade et al. (2006) studied blood pressure at admission in patients hospitalized with acute heart failure using data from the Organized Program to Initiate Lifesaving Treatment in Hospitalized Patients with Heart Failure (OPTIMIZE-HF) registry. Logistic regression was used to describe the relationship in-hospital mortality and both systolic and diastolic blood pressure. These variables, when measured at the same time, are likely to have correlated measurement error due to common biological fluctuations and measurement facilities. The original analysis by Gheorghiade et al. (2006) regards both variables as error free. Thomas et al. (2011) revise the analysis to account for measurement error in systolic blood pressure, which is of primary interest, but not diastolic blood pressure. Adjustment of both variables could be important. When multiple covariates are measured with error, one could apply a univariate adjustment separately. However, this would not account for correlation between the latent variable measurement errors.
Here, we introduce the extension of MAI to multivariate mis-measured data and provide a computationally convenient approach to implementation. The result is quite similar to univariate MAI, but unique challenges are addressed. In Section 2, we define a set of moments that are feasible for matching with multivariate measurement error and introduce the natural extension of the MAI algorithm, used to obtain adjusted data with appropriate moments. In Section 3, a numerically convenient method for the implementation of multivariate MAI is proposed. In Section 4, the alternative implementations of multivariate MAI are evaluated and MAI is compared to other imputation methods via simulation in applications to density estimation and logistic regression. In Section 5, we revise the OPTIMIZE-HF analysis to account for measurement error and obtain estimates that describe the features of “true” diastolic and systolic blood pressure.
