{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,2]],"date-time":"2026-09-02T15:48:41Z","timestamp":1788364121983,"version":"build-2803163510"},"reference-count":23,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T00:00:00Z","timestamp":1676592000000},"content-version":"vor","delay-in-days":2924,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1206693"],"award-info":[{"award-number":["DMS-1206693"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000065","name":"National Institute of Neurological Disorders and Stroke","doi-asserted-by":"publisher","award":["R01-NS07367"],"award-info":[{"award-number":["R01-NS07367"]}],"id":[{"id":"10.13039\/100000065","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000065","name":"National Institute of Neurological Disorders and Stroke","doi-asserted-by":"publisher","award":["R01-NS073671"],"award-info":[{"award-number":["R01-NS073671"]}],"id":[{"id":"10.13039\/100000065","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000054","name":"National Cancer Institute","doi-asserted-by":"publisher","award":["R37-CA057030"],"award-info":[{"award-number":["R37-CA057030"]}],"id":[{"id":"10.13039\/100000054","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,1,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>We study the regression relationship between covariates in case\u2013control data: an area known as the secondary analysis of case\u2013control studies. The context is such that only the form of the regression mean is specified, so that we allow an arbitrary regression error distribution, which can depend on the covariates and thus can be heteroscedastic. Under mild regularity conditions we establish the theoretical identifiability of such models. Previous work in this context has either specified a fully parametric distribution for the regression errors, specified a homoscedastic distribution for the regression errors, has specified the rate of disease in the population (we refer to this as the true population) or has made a rare disease approximation. We construct a class of semiparametric estimation procedures that rely on none of these. The estimators differ from the usual semiparametric estimators in that they draw conclusions about the true population, while technically operating in a hypothetical superpopulation. We also construct estimators with a unique feature, in that they are robust against the misspecification of the regression error distribution in terms of variance structure, whereas all other non-parametric effects are estimated despite the biased samples. We establish the asymptotic properties of the estimators and illustrate their finite sample performance through simulation studies, as well as through an empirical example on the relationship between red meat consumption and hetero-cyclic amines. Our analysis verified the positive relationship between red meat consumption and two forms of hetro-cyclic amines, indicating that increased red meat consumption leads to increased levels of MeIQx and PhIP, both being risk factors for colorectal cancer. Computer software as well as data to illustrate the methodology are available from http:\/\/www.stat.tamu.edu\/~carroll\/matlab__programs\/software.php .<\/jats:p>","DOI":"10.1111\/rssb.12107","type":"journal-article","created":{"date-parts":[[2015,2,16]],"date-time":"2015-02-16T04:38:08Z","timestamp":1424061488000},"page":"127-151","source":"Crossref","is-referenced-by-count":14,"title":["Semiparametric Estimation in the Secondary Analysis of Case\u2013Control Studies"],"prefix":"10.1093","volume":"78","author":[{"given":"Yanyuan","family":"Ma","sequence":"first","affiliation":[{"name":"University of South Carolina , Columbia"},{"name":"Texas A&M University , College Station , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Raymond J.","family":"Carroll","sequence":"additional","affiliation":[{"name":"Texas A&M University , College Station , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2015,2,15]]},"reference":[{"key":"2023021801100229000_rssb12107-cit-0001","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1093\/carcin\/24.2.275","article-title":"Investigation of interaction between n-acetyltransferase 2 and heterocyclic amines as potential risk factors for colorectal cancer","volume":"24","author":"Barrett","year":"2003","journal-title":"Carcinogenesis"},{"key":"2023021801100229000_rssb12107-cit-0002","volume-title":"Efficient and Adaptive Estimation for Semiparametric Models","author":"Bickel","year":"1993"},{"key":"2023021801100229000_rssb12107-cit-0003","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1093\/biomet\/92.2.399","article-title":"Semiparametric maximum likelihood estimation in case-control studies of gene-environment interactions","volume":"92","author":"Chatterjee","year":"2005","journal-title":"Biometrika"},{"key":"2023021801100229000_rssb12107-cit-0004","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1198\/016214505000000835","article-title":"Comment on the paper Likelihood based inference on haplotype effects in genetic association studies by D. Y. Lin and D. Zeng","volume":"101","author":"Chatterjee","year":"2006","journal-title":"J. Am. Statist. Ass."},{"key":"2023021801100229000_rssb12107-cit-0005","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1093\/biostatistics\/kxm011","article-title":"Retrospective analysis of haplotype-based case-control studies under a flexible model for gene-environment association","volume":"9","author":"Chen","year":"2008","journal-title":"Biostatistics"},{"key":"2023021801100229000_rssb12107-cit-0006","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1198\/jasa.2009.0104","article-title":"Shrinkage estimators for robust and efficient inference in haplotype-based case-control studies","volume":"104","author":"Chen","year":"2009","journal-title":"J. Am. Statist. 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