Imputation-based adjusted score equations in generalized linear models with nonignorable missing covariate values

We consider the estimation of unknown parameters in a generalized linear model when some covariates have nonignorable missing values. When an instrument, a covariate that helps identifying parameters under nonignorable missingness, is appropriately specified, a pseudo likelihood approach similar to that in Tang, Little and Raghunathan (2003) or Zhao and Shao (2015) can be applied. However, this approach does not work well when the instrument is a weak predictor of the response given other covariates. We show that the asymptotic variances of the pseudo likelihood estimators for the regression coefficients of covariates other than the instrument diverge to infinity as the regression coefficient of the instrument goes to 0. By an imputation-based adjustment for the score equations, we propose a new estimator for the regression coefficients of the covariates other than the instrument. This works well even if the instrument is a weak predictor. It is semiparametric since the propensity of missing covariate data is completely unspecified. To solve the adjusted score equation, we develop an iterative algorithm that can be applied by using standard softwares at each iteration. We establish some theoretical results on the convergence of the proposed iterative algorithm and asymptotic normality of the resulting estimators. A variance estimation formula is also derived. Some simulation results and a data example are presented for illustration.