Two-phase sampling is commonly adopted for reducing cost and improving estimation efficiency. In many two-phase studies, the outcome and some cheap covariates are observed for a large sample in Phase I, and expensive covariates are obtained for a selected subset of the sample in Phase II. As a result, the analysis of the association between the outcome and covariates faces a missing data problem. Complete-case analysis, which relies solely on the Phase II sample, is generally inefficient. In this paper, we study a two-step estimation approach, which first obtains an estimator using the complete data, and then updates it using an asymptotically mean-zero estimator obtained from a working model between the outcome and cheap covariates using the full data. This two-step estimator is asymptotically at least as efficient as the complete-data estimator and is robust to misspecification of the working model. We propose a kernel-based method to construct a two-step estimator that achieves optimal efficiency. Additionally, we develop a simple joint update approach based on multiple working models to approximate the optimal estimator when a fully nonparametric kernel approach is infeasible. We illustrate the proposed methods with various outcome models. We demonstrate their advantages over existing approaches through simulation studies and provide an application to a major cancer genomics study.

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