Quasilikelihood estimation in measurement error models with correlated replicates

We consider quasilikelihood models when some of the predictors are measured with error. In many cases, the true but fallible predictor is impossible to measure, and the best one can do is to obtain replicates of the fallible predictor. We consider the case that the replicates are not independent. If one assumes that replicates are independent and they are not, one typically underestimates the extent of the measurement error, leading to an inconsistent errors in variables correction. We devise techniques for estimating the measurement error covariance matrix. In addition, we discuss how one might perform a quasilikelihood analysis by computing the mean and variance functions of the observed data, both using approximations and also exactly through a Monte Carlo method. The methods are illustrated on a data set involving systolic blood pressure and urinary sodium chloride, where the measurement errors appear to be approximately normally distributed but highly correlated, and the distribution of the true predictor is reasonably modeled as a mixture of normals.