Efficient estimation in a partially specified nonignorable propensity score model

Consider the regression setting where the response variable is subject to missing data and the covariates are fully observed. A nonignorable propensity score model, i.e., the probability that the response is observed conditional on all variables depends on the missing values themselves, is assumed throughout the paper. In such problems, model misspecification and model identifiability are two critical issues. A fully parametric approach can produce results that are sensitive to the model assumptions, while a fully nonparametric approach may not be sufficient for model identification. A new flexible semiparametric propensity score model is proposed where the relationship between the missingness indicator and the partially observed response is totally unspecified and estimated nonparametrically, while the relationship between the missingness indicator and the fully observed covariates is modeled parametrically. The proposed estimator is constructed via a semiparametric treatment and is proved to be semiparametrically efficient. Comprehensive simulation studies are conducted to examine the finite-sample performance of the estimators. While the naive parametric method leads to heavily biased estimator and poor coverage results, the proposed method produces estimator with negligible finite-sample biases and also correct inference results. The proposed method is further illustrated via an electronic health records (EHR) data application for the albumin level in the blood sample. The empirical analyses demonstrated that the proposed semiparametric propensity score model is more sensible than a purely parametric model. The proposed method could be very useful to uncover the unknown and possibly nonlinear dependence of the propensity score model to the albumin level, and is recommended for practical use.