Incorporating measurement error in the estimation of autoregressive models for longitudinal data

Abstract

Abstract Rosner et al. ( Stat. Med. 4 (1985), 457–467) discussed the use of a first-order autoregressive model for longitudinal data with multiple time-dependent and time-independent covariates. The model assumes that both covariate and outcome variables are measured without error, a condition frequently not satisfied in practice. Here we extend their model to allow for measurement error on both covariate and outcome variables. We obtain maximum likelihood estimates of the model parameters given knowledge of the measurement error variance for both outcome and exposure variables. Our method differs from standard linear model measurement error methods which do not apply to autoregressive models. A simulation study comparing the maximum likelihood estimates (MLEs) with the ordinary least square (OLS) estimates that ignore measurement error shows that the OLS estimates are grossly biased in the presence of substantial measurement error, while the MLEs are essentially unbiased. Furthermore, whereas the coverage probability of the confidence intervals based on the OLS estimates tends to be significantly smaller than the nominal level, the MLEs provide adequate coverage. The relative advantage of MLE versus OLS increases as either the amount of measurement error or the number of visits increases. Finally, we demonstrate that the method corrects for a severe underestimate of the effect of airway responsiveness on pulmonary function decline when both are measured with error.