Measurement error models with interactions

An important use of measurement error models is to correct regression models for bias due to covariate measurement error. Most measurement error models assume that the observed error-prone covariate ($W$) is a linear function of the unobserved true covariate ($X$) plus other covariates ($Z$) in the regression model. In this paper, we consider models for $W$ that include interactions between $X$ and $Z$. We derive the conditional distribution of $X$ given $W$ and $Z$ and use it to extend the method of regression calibration to this class of measurement error models. We apply the model to dietary data and test whether self-reported dietary intake includes an interaction between true intake and body mass index. We also perform simulations to compare the model to simpler approximate calibration models.