 2005 Royal Statistical Society 0035–9254/05/54021 Appl. Statist. (2005) 54, Part 1, pp. 21–30 Assessing interaction effects in linear measurement error models Li-Shan Huang and Hongkun Wang University of Rochester, USA and Christopher Cox National Institutes of Health, Bethesda, and University of Rochester Medical Center, USA [Received June 2002. Revised November 2003] Summary. In a linear model, the effect of a continuous explanatory variable may vary across groups defined by a categorical variable, and the variable itself may be subject to measure- ment error. This suggests a linear measurement error model with slope-by-factor interactions. The variables that are defined by such interactions are neither continuous nor discrete, and hence it is not immediately clear how to fit linear measurement error models when interactions are present. This paper gives a corollary of a theorem of Fuller for the situation of correcting measurement errors in a linear model with slope-by-factor interactions. In particular, the error- corrected estimate of the coefficients and its asymptotic variance matrix are given in a more easily assessable form. Simulation results confirm the asymptotic normality of the coefficients in finite sample cases. We apply the results to data from the Seychelles Child Development Study at age 66 months, assessing the effects of exposure to mercury through consumption of fish on child development for females and males for both prenatal and post-natal exposure. Keywords: Asymptotic normality; Interaction; Regression calibration; Simulation extrapolation 1. Introduction Linear measurement error models (MEMs) with continuous predictor variables measured with error are well developed in Fuller (1987). Situations in which categorical variables are measured with error are called ‘misclassifications’. Carroll et al. (1995) mentioned that, when all the vari- ables are discrete, correcting errors in a misclassified contingency table can be handled by the method of maximum likelihood. In environmental health studies, a subject’s true level of exposure is rarely known accurately and often a biomarker or an indirect measure of exposure is taken, which may deviate from the true value that actually produces the toxic effect. Moreover, the effect of exposure may differ by gender or some other categorical covariates. This leads us to consider correcting mea- surement errors in linear models with slope-by-factor interactions, i.e. we have a model with separate slopes that are defined by the levels of a categorical variable for a continuous predictor measured with error. The interaction terms are neither continuous nor discrete, as they were set equal to the predictor variable for the level specified (with measurement error) and to 0 otherwise (without measurement error). Fuller (1987) gave an example (example 3.1.3) on how to correct measurement error for interaction terms and pointed out that Address for correspondence: Li-Shan Huang, Department of Biostatistics and Computational Biology, Uni- versity of Rochester, 601 Elmwood Avenue, Box 630, Rochester, NY 14642, USA. E-mail: Lhuang@bst.rochester.edu Downloaded from https://academic.oup.com/jrsssc/article/54/1/21/7113004 by guest on 13 April 2023