Nonparametric and Semiparametric Models for Missing Covariates in Parametric Regression

Abstract

Robustness of covariate modeling for the missing-covariate problem in parametric regression is studied under the missing-at-random assumption. For a simple missing-covariate pattern, nonparametric covariate model is proposed and is shown to yield a consistent and semiparametrically efficient estimator for the regression parameter. Total robustness is achieved in this situation. For more general missingcovariate patterns, a novel semiparametric modeling approach is proposed for the covariates. In this approach, the covariate distribution is first decomposed into the product of a series of conditional distributions according to the overall missing-data patterns, and the conditional distributions are then represented in the general odds ratio form. The general odds ratios are modeled parametrically, and the other components of the covariate distribution are modeled nonparametrically. Maximum semiparametric likelihood is used to find the parameter estimates. The proposed method yields a consistent estimator for the regression parameter when the odds ratios are modeled correctly. In general, the semiparametric covariate modeling strategy increases the robustness against covariate model misspecification when compared with the parametric modeling strategy proposed by Lipsitz and Ibrahim. The new covariate modeling approach can also be incorporated into the doubly robust procedure of Robins et al. to increase protection against misspecification of the missing-data mechanism. In addition, the proposed modeling strategy avoids the usually intractable integrations involved in the maximization of the incomplete-data likelihood with parametric covariate models. The proposed method can be applied to many regression models to handle incomplete covariates.