Moment reconstruction and moment‐adjusted imputation when exposure is generated by a complex, nonlinear random effects modeling process

For the classical, homoscedastic measurement error model, moment reconstruction (Freedman et al., 2004, 2008) and moment‐adjusted imputation (Thomas et al., 2011) are appealing, computationally simple imputation‐like methods for general model fitting. Like classical regression calibration, the idea is to replace the unobserved variable subject to measurement error with a proxy that can be used in a variety of analyses. Moment reconstruction and moment‐adjusted imputation differ from regression calibration in that they attempt to match multiple features of the latent variable, and also to match some of the latent variable's relationships with the response and additional covariates. In this note, we consider a problem where true exposure is generated by a complex, nonlinear random effects modeling process, and develop analogues of moment reconstruction and moment‐adjusted imputation for this case. This general model includes classical measurement errors, Berkson measurement errors, mixtures of Berkson and classical errors and problems that are not measurement error problems, but also cases where the data‐generating process for true exposure is a complex, nonlinear random effects modeling process. The methods are illustrated using the National Institutes of Health–AARP Diet and Health Study where the latent variable is a dietary pattern score called the Healthy Eating Index‐2005. We also show how our general model includes methods used in radiation epidemiology as a special case. Simulations are used to illustrate the methods.