The Modified Structural Quasi Score Estimator for Poisson Regression Parameters with Covariate Measurement Error

Abstract

This article proposed the Modified Structural Quasi Score (MSQS) estimators for Poisson regression parameters when a covariate is subject to measurement error. We study the situation when the true covariate in the Poisson regression model is unobserved, and the surrogate for this covariate is related to the true covariate by the additive measurement error model. We assumed that true covariate as a random variable with unknown density function distribution and its observable values as surrogates, which also has Poisson distribution. We applied the Empirical Bayes Deconvolution (EBD) method for estimating the true covariate density with a finite discrete support set. To estimate Poisson regression parameters, we construct an MSQS estimating equation based on proper functions for the mean and variance of the Poisson distributed surrogate. The MSQS estimator for the Poisson regression parameter is the root of the quasi-score function based on the quasi-likelihood method. We did some simulation scenarios for assessing the MSQS estimator by assuming the true covariate comes from Gamma distribution as a conjugate before Poison distribution. We compute the standard error of the mean, standard deviation, and bias of the MSQS estimator for various sample sizes to examine the estimator's appropriateness. The simulation showed that a combination of the finite discrete support set of surrogates based on the range values and smaller-scale parameter of Gamma distribution yields smaller values of bias estimator and the estimated standard deviation.