Estimation in covariate-adjusted regression

Abstract

The method of covariate-adjusted regression was recently proposed for situations where both predictors and response in a regression model are not directly observed, but are observed after being contaminated by unknown functions of a common observable confounder in a multiplicative fashion. One example is data collected for a study on diabetes, where the variables of interest, systolic and diastolic blood pressures and glycosolated hemoglobin levels are known to be influenced by an observable confounder, body mass index. An estimation procedure based on equidistant binning (EB), currently available, gives consistent estimators for the regression coefficients adjusted for the confounder. In this paper, we propose two new estimation procedures based on nearest-neighbor binning (NB) and local polynomial modeling (LP). Even though, the three methods perform similarly in terms of their bias, it is shown through simulation studies that NB has smaller variance compared to EB, and LP yields substantially lower variance relative to the two binning methods for small to moderate sample sizes. The consistency and convergence rates of the proposed estimators of LP, with the smallest MSE, are also established. We illustrate the proposed method of LP with the above-mentioned diabetes data, where the goal is to uncover the regression relation between the response, glycosolated hemoglobin levels, and the predictors, systolic and diastolic blood pressures, adjusted for body mass index.