Statistica Sinica 19 (2009), 1077-1093

SPATIAL LINEAR MIXED MODELS WITH

COVARIATE MEASUREMENT ERRORS

Yi Li$^{1, 3}$, Haicheng Tang$^2$ and Xihong Lin$^3$

$^1$Dana Farber Cancer Institute, $^2$American Express

and $^3$Harvard School of Public Health

Abstract: Spatial data with covariate measurement errors have been commonly observed in public health studies. Existing work mainly concentrates on parameter estimation using Gibbs sampling, and no work has been conducted to understand and quantify the theoretical impact of ignoring measurement error on spatial data analysis in the form of the asymptotic biases in regression coefficients and variance components when measurement error is ignored. Plausible implementations, from frequentist perspectives, of maximum likelihood estimation in spatial covariate measurement error models are also elusive. In this paper, we propose a new class of linear mixed models for spatial data in the presence of covariate measurement errors. We show that the naive estimators of the regression coefficients are attenuated while the naive estimators of the variance components are inflated, if measurement error is ignored. We further develop a structural modeling approach to obtaining the maximum likelihood estimator by accounting for the measurement error. We study the large sample properties of the proposed maximum likelihood estimator, and propose an EM algorithm to draw inference. All the asymptotic properties are shown under the increasing-domain asymptotic framework. We illustrate the method by analyzing the Scottish lip cancer data, and evaluate its performance through a simulation study, all of which elucidate the importance of adjusting for covariate measurement errors.

Key words and phrases: Asymptotic bias, consistency and asymptotic normality, EM algorithmMeasurement error, increasing domain asymptotics, spatial data, structural modeling, variance components.