{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T16:37:05Z","timestamp":1790181425106,"version":"4.1.0"},"reference-count":19,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"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":[[2025,7,16]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>Although the existing causal inference literature focuses on the forward-looking perspective by estimating effects of causes, the backward-looking perspective can provide insights into causes of effects. In backward-looking causal inference, the probability of necessity measures the probability that a certain event is caused by the treatment, given the observed treatment and outcome. Most existing results focus on binary outcomes. Motivated by applications with ordinal outcomes, we propose a general definition of the probability of necessity. However, identifying the probability of necessity is challenging because it involves the joint distribution of the potential outcomes. We propose the novel assumption of a monotonic incremental treatment effect to identify the probability of necessity with ordinal outcomes. We also discuss the testable implications of this key identification assumption. When it fails, we derive explicit formulas of the sharp large-sample bounds on the probability of necessity.<\/jats:p>","DOI":"10.1093\/biomet\/asaf049","type":"journal-article","created":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T07:54:01Z","timestamp":1751615641000},"source":"Crossref","is-referenced-by-count":4,"title":["Identifying and bounding the probability of necessity for causes of effects with ordinal outcomes"],"prefix":"10.1093","volume":"112","author":[{"given":"Chao","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Beijing Technology and Business University , Fangshan District, Beijing 102488,","place":["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,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wei","family":"Li","sequence":"additional","affiliation":[{"name":"Center for Applied Statistics and School of Statistics, Renmin University of China , 59 Zhongguancun Street , Beijing 100872,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2704-2353","authenticated-orcid":false,"given":"Peng","family":"Ding","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of California, Berkeley , 425 Evans Hall , Berkeley, California 94720,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2025,7,9]]},"reference":[{"key":"2025082101572202100_asaf049-B1","doi-asserted-by":"crossref","first-page":"1611","DOI":"10.1111\/j.1539-6924.2005.00696.x","article-title":"Variance estimators for three \u2018probabilities of causation\u2019","volume":"25","author":"Cai","year":"2005","journal-title":"Risk Anal"},{"key":"2025082101572202100_asaf049-B3","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1177\/0049124113515188","article-title":"Fitting science into legal contexts: assessing effects of causes or causes of effects?","volume":"43","author":"Dawid","year":"2014","journal-title":"Sociol. 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