{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T16:36:55Z","timestamp":1790181415927,"version":"4.1.0"},"reference-count":23,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2023,9,14]],"date-time":"2023-09-14T00:00:00Z","timestamp":1694649600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key R&D Program of China","doi-asserted-by":"publisher","award":["2022YFA1008100"],"award-info":[{"award-number":["2022YFA1008100"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key R&D Program of China","doi-asserted-by":"publisher","award":["2020YFE0204200"],"award-info":[{"award-number":["2020YFE0204200"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12101607"],"award-info":[{"award-number":["12101607"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071015"],"award-info":[{"award-number":["12071015"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["1232008"],"award-info":[{"award-number":["1232008"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Statistical Science Research","award":["2022LZ13"],"award-info":[{"award-number":["2022LZ13"]}]},{"name":"Hong Kong Innovation and Technology Commission with InnoHK Project CIMDA"},{"name":"Hong Kong Institute of Data Science","award":["9360163"],"award-info":[{"award-number":["9360163"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,5,13]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>As highlighted in Dawid (2000) and Pearl &amp; Mackenzie (2018), deducing the causes of given effects is a more challenging problem than evaluating the effects of causes in causal inference. Lu et\u00a0al. (2023) proposed an approach for deducing causes of a single effect variable based on posterior causal effects. In many applications, there are multiple effect variables, and they can be used simultaneously to more accurately deduce the causes. To retrospectively deduce causes from multiple effects, we propose multivariate posterior total, intervention and direct causal effects conditional on the observed evidence. We describe the assumptions of no confounding and monotonicity, under which we prove identifiability of the multivariate posterior causal effects and provide their identification equations. The proposed approach can be applied for causal attributions, medical diagnosis, blame and responsibility in various studies with multiple effect or outcome variables. Two examples are used to illustrate the proposed approach.<\/jats:p>","DOI":"10.1093\/biomet\/asad056","type":"journal-article","created":{"date-parts":[[2023,9,14]],"date-time":"2023-09-14T21:49:32Z","timestamp":1694728172000},"page":"573-589","source":"Crossref","is-referenced-by-count":8,"title":["Retrospective causal inference with multiple effect variables"],"prefix":"10.1093","volume":"111","author":[{"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, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3287-7363","authenticated-orcid":false,"given":"Zitong","family":"Lu","sequence":"additional","affiliation":[{"name":"Department of Systems Engineering, City University of Hong Kong , Tat Chee Avenue , Kowloon, Hong Kong SAR, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6554-2463","authenticated-orcid":false,"given":"Jinzhu","family":"Jia","sequence":"additional","affiliation":[{"name":"School of Public Health and Center for Statistical Science, Peking University , 38 Xueyuan Road , Beijing 100191, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Min","family":"Xie","sequence":"additional","affiliation":[{"name":"Department of Systems Engineering, City University of Hong Kong , Tat Chee Avenue , Kowloon, Hong Kong SAR, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1890-1192","authenticated-orcid":false,"given":"Zhi","family":"Geng","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Beijing Technology and Business University , Fangshan District, Beijing 102488, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2023,9,14]]},"reference":[{"key":"2024051307572220400_asad056-B1","doi-asserted-by":"crossref","first-page":"1611","DOI":"10.1111\/j.1539-6924.2005.00696.x","article-title":"Variance estimators for three \u201cprobabilities of causation\u201d","volume":"25","author":"Cai","year":"2005","journal-title":"Risk Anal"},{"key":"2024051307572220400_asad056-B2","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1080\/01621459.2000.10474210","article-title":"Causal inference without counterfactuals","volume":"95","author":"Dawid","year":"2000","journal-title":"J. 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