Covariate measurement error in generalized linear models for longitudinal data: a latent Markov approach

One common approach to handle covariate measurement error in Generalized Linear Models (GLM) is classical error modeling. In the past 20 years, classical error modeling has been brought to (Non-Parametric) Maximum Likelihood (NPML) es- timation, by means of finite mixture modeling: the supposedly continuous true score is modeled as a discrete (multinomial) static latent variable, and is handled as a part of the model. Nonetheless, the true score is not allowed to vary over time: if the true score has own underlying dynamics, these are either unaccounted for or mistaken for measurement error, or possibly both. The aim of the present paper is to formulate a joint model for the outcome variable, the covariate observed with error (measure- ment model), and the true score model that accounts for the underlying dynamics in the true score. The true score and its dynamics are modeled non-parametrically as a first-order latent (hidden) Markov chain. Estimation is done extending the NPML approach, in a full maximum likelihood environment with a well-know modification of the EM algorithm (forward-backward algorithm). From an applied researcher per- spective, our methodology can safely handle both the case where the latent underly- ing characteristic is stable over time, as well as providing a suitable framework even when changes across measurement occasions are substantial. Within a GLM frame- work, it is demonstrated, by means of extensive simulation studies, that this is cru- cial to get correct estimates of the regression coefficients, as well as good coverages. In the real-data application, the effect of heart rate on the occurrence of cardiovas- cular diseases in a sample of Chinese elderly patients is measured. Modeling the true (unobserved) heart rate and its dynamics - which, in elderly patients, are likely to be non negligible - will be showed to allow a correct assessment of risk factors of cardio- vascular diseases occurrence.