{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T10:17:59Z","timestamp":1771064279601,"version":"3.50.1"},"reference-count":23,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2024,4,11]],"date-time":"2024-04-11T00:00:00Z","timestamp":1712793600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,12,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The probability of benefit can be a valuable and meaningful measure of treatment effect. Particularly for an ordinal outcome, it can have an intuitive interpretation. Unfortunately, this measure, and variations of it, are not identifiable even in randomized trials with perfect compliance. There is, for this reason, a long literature on nonparametric bounds for unidentifiable measures of benefit. These have primarily focused on perfect randomized trial settings and one or two specific estimands. We expand these bounds to observational settings with unmeasured confounders and imperfect randomized trials for all three estimands considered in the literature: the probability of benefit, the probability of no harm and the relative treatment effect.<\/jats:p>","DOI":"10.1093\/biomet\/asae020","type":"journal-article","created":{"date-parts":[[2024,4,11]],"date-time":"2024-04-11T21:13:30Z","timestamp":1712870010000},"page":"1429-1436","source":"Crossref","is-referenced-by-count":6,"title":["Sharp symbolic nonparametric bounds for measures of benefit in observational and imperfect randomized studies with ordinal outcomes"],"prefix":"10.1093","volume":"111","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0504-8404","authenticated-orcid":false,"given":"Erin E","family":"Gabriel","sequence":"first","affiliation":[{"name":"Section of Biostatistics, Department of Public Health, University of Copenhagen , \u00d8ster Farimagsgade 5 , K\u00f8benhavn DK-1353,","place":["Denmark"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1279-8676","authenticated-orcid":false,"given":"Michael C","family":"Sachs","sequence":"additional","affiliation":[{"name":"Section of Biostatistics, Department of Public Health, University of Copenhagen , \u00d8ster Farimagsgade 5 , K\u00f8benhavn DK-1353,","place":["Denmark"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andreas Kryger","family":"Jensen","sequence":"additional","affiliation":[{"name":"Section of Biostatistics, Department of Public Health, University of Copenhagen , \u00d8ster Farimagsgade 5 , K\u00f8benhavn DK-1353,","place":["Denmark"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2024,4,11]]},"reference":[{"key":"2024112207520442700_asae020-B1","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1111\/1468-0262.00114","article-title":"Inconsistency of the bootstrap when a parameter is on the boundary of the parameter space","volume":"68","author":"Andrews","year":"2000","journal-title":"Econometrica"},{"key":"2024112207520442700_asae020-B2","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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