{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T17:21:36Z","timestamp":1787505696405,"version":"build-2736575974"},"reference-count":25,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2022,7,25]],"date-time":"2022-07-25T00:00:00Z","timestamp":1658707200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100010269","name":"Wellcome Trust","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100010269","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Expanding Excellence in England"},{"DOI":"10.13039\/501100000737","name":"University of Exeter","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000737","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000883","name":"University of Bristol","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000883","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000265","name":"Medical Research Council","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000265","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,5,15]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>Many partial identification problems can be characterized by the optimal value of a function over a set where both the function and set need to be estimated by empirical data. Despite some progress for convex problems, statistical inference in this general setting remains to be developed. To address this, we derive an asymptotically valid confidence interval for the optimal value through an appropriate relaxation of the estimated set. We then apply this general result to the problem of selection bias in population-based cohort studies. We show that existing sensitivity analyses, which are often conservative and difficult to implement, can be formulated in our framework and made significantly more informative via auxiliary information on the population. We conduct a simulation study to evaluate the finite sample performance of our inference procedure, and conclude with a substantive motivating example on the causal effect of education on income in the highly selected UK Biobank cohort. We demonstrate that our method can produce informative bounds using plausible population-level auxiliary constraints. We implement this method in the $\\texttt{R}$ package $\\texttt{selectioninterval}$.<\/jats:p>","DOI":"10.1093\/biomet\/asac042","type":"journal-article","created":{"date-parts":[[2022,7,25]],"date-time":"2022-07-25T11:10:58Z","timestamp":1658747458000},"page":"485-498","source":"Crossref","is-referenced-by-count":2,"title":["Sample-constrained partial identification with application to selection bias"],"prefix":"10.1093","volume":"110","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7897-6180","authenticated-orcid":false,"given":"Matthew J","family":"Tudball","sequence":"first","affiliation":[{"name":"University of Bristol MRC Integrative Epidemiology Unit, , Oakfield Grove, Bristol, BS8 2BN, U.K"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rachael A","family":"Hughes","sequence":"additional","affiliation":[{"name":"University of Bristol MRC Integrative Epidemiology Unit, , Oakfield Grove, Bristol, BS8 2BN, U.K"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kate","family":"Tilling","sequence":"additional","affiliation":[{"name":"University of Bristol MRC Integrative Epidemiology Unit, , Oakfield Grove, Bristol, BS8 2BN, U.K"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jack","family":"Bowden","sequence":"additional","affiliation":[{"name":"University of Exeter College of Medicine and Health, , Heavitree Road, Exeter, EX1 2LU, U.K"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qingyuan","family":"Zhao","sequence":"additional","affiliation":[{"name":"University of Cambridge Department of Pure Mathematics and Mathematical Statistics, , Wilberforce Road, Cambridge CB3 0WB, U.K"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2022,7,25]]},"reference":[{"key":"2023051508565809800_B1","doi-asserted-by":"crossref","first-page":"609","DOI":"10.3982\/ECTA9370","article-title":"Inference based on conditional moment inequalities","volume":"81","author":"Andrews,","year":"2013","journal-title":"Econometrica"},{"key":"2023051508565809800_B2","doi-asserted-by":"crossref","first-page":"119","DOI":"10.3982\/ECTA7502","article-title":"Inference for parameters defined by moment inequalities using generalized moment selection","volume":"78","author":"Andrews,","year":"2010","journal-title":"Econometrica"},{"key":"2023051508565809800_B3","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1093\/biomet\/ass064","article-title":"Interval estimation of population means under unknown but bounded probabilities of sample selection","volume":"100","author":"Aronow,","year":"2013","journal-title":"Biometrika"},{"key":"2023051508565809800_B4","first-page":"2410","article-title":"Recovering from selection bias in causal and statistical inference","author":"Bareinboim,","year":"2014"},{"key":"2023051508565809800_B5","doi-asserted-by":"crossref","first-page":"1012","DOI":"10.1080\/01621459.1994.10476836","article-title":"$P$ values maximized over a confidence set for the nuisance parameter","volume":"89","author":"Berger,","year":"1994","journal-title":"J. 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