Abstract: Estimation of the regression parameters and variance components in a longitudinal mixed model with measurement error in a time-varying covariate is considered. The positive bias in variance estimators caused by covariate measurement error in a normal linear mixed model has recently been identified and studied (Tosteson, Buonaccorsi and Demidenko (1997)). The methods suggested there for correction of the bias involve convenient adaptations of existing software for a particular model. In this paper, we study alternative methods of estimation which achieve higher efficiencies and extend readily to a more general class of models. Full and pseudo-maximum likelihood estimators under normality are considered as is a pseudo-moment approach relying on initial estimation of nuisance parameters. The latter lead to a ``regression calibration'' method for estimating the regression parameters, in which a substitution is made for the unknown covariates, followed by a correction for estimation of the variance parameters. It is shown that for some cases this yields the pseudo-maximum likelihood estimates and, in these cases, the resulting estimators are highly efficient relative to the full maximum likelihood estimators. We first consider a model with no additional data, where identifiability follows from assumptions about the longitudinal model for the unobserved true covariates, and then describe some extensions to cases where either replicate or validation data is available. We illustrate with an example investigating the relationship between dietary and serum beta-carotene.
Key words and phrases: Longitudinal model, maximum likelihood, measurement error, random effects.