Locally ancillary quasi-score models for errors-in-covariates

We use the notion of locally ancillary estimating functions to develop a quasi-score method for fitting regression models containing measurement error in the covariates. Suppose that interest is on the model E(Y|u, w) for response Y, the observed data are (y, x, w), and X is a mismeasured surrogate for u. We take a functional modeling approach, treating the u as a fixed nuisance parameter. Beginning with quasi-scores for the regression parameter and the unknown u, we derive a bias-corrected quasi-score for the regression parameter that is second-order locally ancillary for the nuisance u. Our method for this requires only the correct specification of the mean and variance functions for Y and X in terms of u, w, and the regression parameter. When an estimator for u is plugged into the corrected quasi-score, local approximations show that the bias is small. We present simulations verifying this result and an example from child psychiatry, both using log-linear regression models.