{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,3]],"date-time":"2026-08-03T12:50:27Z","timestamp":1785761427247,"version":"3.56.0"},"reference-count":50,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,7,6]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>Learning causal relationships between pairs of complex traits from observational studies is of great interest in many scientific fields. However, most existing methods assume the absence of unmeasured confounding and restrict causal relationships between two traits to be unidirectional, assumptions that may be violated in real-world systems. In this paper, we address the problem of bivariate causal discovery in the presence of unmeasured confounding and potential feedback loops, leveraging possibly invalid instrumental variables. We propose a novel doubly robust identification framework that guarantees identifiability of the causal direction as long as one of two complementary assumptions holds, without requiring knowledge of which assumption is correct. Moreover, the causal effects are point identified in the unidirectional case and identifiable up to two candidate solutions in the bidirectional case, unless additional information is available. Building on this framework, we develop a finite-sample procedure to detect the causal direction and conduct inference on causal effects. We prove that our method consistently recovers the true causal direction and yields valid confidence intervals for causal effects. Extensive simulations demonstrate its superior performance compared to existing methods. We finally apply our approach to analyse real datasets from the UK Biobank.<\/jats:p>","DOI":"10.1093\/biomet\/asag035","type":"journal-article","created":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T11:34:42Z","timestamp":1781004882000},"source":"Crossref","is-referenced-by-count":0,"title":["Doubly robust identification for bivariate causal discovery under unmeasured confounding"],"prefix":"10.1093","volume":"113","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2461-2522","authenticated-orcid":false,"given":"Wei","family":"Li","sequence":"first","affiliation":[{"name":"Center for Applied Statistics and School of Statistics, Renmin University of China , 59 Zhongguancun Street , Beijing 100872, \u00a0 weilistat@ruc.edu.cn","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9261-4864","authenticated-orcid":false,"given":"Rui","family":"Duan","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, Harvard University , 677 Huntington Avenue , Boston, Massachusetts 02138, U.S.A. rduan@hsph.harvard.edu"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sai","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Statistics and Data Science, Tsinghua University , Beijing 100084, \u00a0 saili@tsinghua.edu.cn","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2026,6,11]]},"reference":[{"key":"2026080308113234600_asag035-B1","doi-asserted-by":"crossref","first-page":"444","DOI":"10.1080\/01621459.1996.10476902","article-title":"Identification of causal effects using instrumental variables","volume":"91","author":"Angrist","year":"1996","journal-title":"J. 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