Doubly robust and locally efficient estimation with missing outcomes

Abstract

We consider parametric regression where the outcome is subject to miss- ingness. To achieve the semiparametric efficiency bound, most existing estimation methods require the correct modeling of certain second moments of the data, which can be very challenging in practice. We propose an estimation procedure based on the conditional empirical likelihood (CEL) method. Our method does not require us to model any second moments. We study the CEL-based inverse probability weighted (CEL-IPW) and augmented inverse probability weighted (CEL-AIPW) estimators in detail. Under some regularity conditions and the missing at ran- dom (MAR) mechanism, the CEL-IPW estimator is consistent if the missingness mechanism is correctly modeled, and the CEL-AIPW estimator is consistent if either the missingness mechanism or the conditional mean of the outcome is cor- rectly modeled. When both quantities are correctly modeled, the CEL-AIPW es- timator attains the semiparametric efficiency bound without modeling any second moments. The asymptotic distributions are derived. Numerical implementation through nested optimization routines using the Newton-Raphson algorithm is dis- cussed. We study the problem of parametric regression when the outcome is subject to missingness. The central interest is the estimation and inference of the re- gression coefficients. In practice there are many reasons that can lead to missing outcomes: budget or technique restrictions, subjects' failure to comply with the protocol, or simply the study design. Missing data usually bring big challenges to estimation and inference, as the application of statistical methods developed for data without missing values can lead to biased estimation and misleading conclusions. In addition to the outcome and covariates, we assume that some auxiliary variables are available for all subjects. Although the auxiliary variables are not of direct statistical interest, they may help to explain the missingness mecha- nism, and thus reduce the impact of missing data on estimation and inference.