{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T14:08:43Z","timestamp":1790172523143,"version":"4.1.0"},"reference-count":0,"publisher":"Oxford University Press (OUP)","license":[{"start":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T00:00:00Z","timestamp":1790121600000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>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 class of two-step estimators are asymptotically at least as efficient as the complete-data estimator and are robust to misspecification of the working model. We propose a kernel-based method to construct a two-step estimator that achieves optimal efficiency within this class of estimators. 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.<\/jats:p>","DOI":"10.1093\/biomet\/asag057","type":"journal-article","created":{"date-parts":[[2026,9,22]],"date-time":"2026-09-22T11:42:29Z","timestamp":1790077349000},"source":"Crossref","is-referenced-by-count":0,"title":["An optimal two-step estimation approach for two-phase studies"],"prefix":"10.1093","author":[{"given":"Q","family":"Zhou","sequence":"first","affiliation":[{"name":"University of North Carolina at Charlotte Department of Mathematics and Statistics, , North Carolina 28223,","place":["U.S.A"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"K Y","family":"Wong","sequence":"additional","affiliation":[{"name":"The Hong Kong Polytechnic University, Hong KongHong Kong Polytechnic University Shenzhen Research Institute Department of Applied Mathematics, , Shenzhen,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2026,9,23]]},"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/biomet\/advance-article-pdf\/doi\/10.1093\/biomet\/asag057\/71347805\/asag057.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/advance-article-pdf\/doi\/10.1093\/biomet\/asag057\/71347805\/asag057.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T13:51:00Z","timestamp":1790171460000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/advance-article\/doi\/10.1093\/biomet\/asag057\/8832239"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,23]]},"references-count":0,"URL":"https:\/\/doi.org\/10.1093\/biomet\/asag057","relation":{},"ISSN":["0006-3444","1464-3510"],"issn-type":[{"value":"0006-3444","type":"print"},{"value":"1464-3510","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,23]]},"article-number":"asag057"}}