{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T16:36:56Z","timestamp":1790181416683,"version":"4.1.0"},"reference-count":60,"publisher":"Oxford University Press (OUP)","issue":"5","license":[{"start":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T00:00:00Z","timestamp":1750204800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100008982","name":"NSF","doi-asserted-by":"publisher","award":["2242776"],"award-info":[{"award-number":["2242776"]}],"id":[{"id":"10.13039\/501100008982","id-type":"DOI","asserted-by":"publisher"}]},{"name":"NIH-NIEHS","award":["1R01ES031651"],"award-info":[{"award-number":["1R01ES031651"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,11,11]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Principal stratification is essential for revealing causal mechanisms involving post-treatment intermediate variables, in real-world applications like surrogate marker evaluation. Principal stratification analysis with continuous intermediate variables is increasingly common but challenging due to the infinite principal strata and the nonidentifiability and nonregularity of principal causal effects (PCEs). Inspired by recent research, we resolve these challenges by first using a flexible copula-based principal score model to identify PCE under weak principal ignorability. We then target the local functional substitute of PCE, which is statistically regular and can accurately approximate PCE with vanishing bandwidth. We simplify the full efficient influence function of the local functional substitute by considering its oracle-scenario alternative. This leads to a computationally efficient and straightforward estimator for the local functional substitute and PCE with vanishing bandwidth. We prove the double robustness of our proposed estimator, and derive its asymptotic normality for inferential purposes. With a vanishing bandwidth, our method attains minimax optimality for the nonparametric estimation of the PCE. With a fixed bandwidth, it achieves semiparametric efficiency in estimating its local functional substitute. We demonstrate the strong performance of our proposed estimator through simulations and apply it to surrogate analysis of short-term CD4 count in ACTG 175.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkaf034","type":"journal-article","created":{"date-parts":[[2025,5,31]],"date-time":"2025-05-31T07:41:04Z","timestamp":1748677264000},"page":"1655-1677","source":"Crossref","is-referenced-by-count":3,"title":["Semiparametric localized principal stratification analysis with continuous strata"],"prefix":"10.1093","volume":"87","author":[{"given":"Yichi","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Statistics, Indiana University Bloomington , Bloomington, IN 47405 ,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7703-707X","authenticated-orcid":false,"given":"Shu","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Statistics, North Carolina State University , Raleigh, NC 27695 ,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2025,6,18]]},"reference":[{"issue":"2","key":"2025111101214689000_qkaf034-B1","doi-asserted-by":"publisher","first-page":"477","DOI":"10.1111\/biom.v77.2","article-title":"Evaluation of longitudinal surrogate markers","volume":"77","author":"Agniel","year":"2021","journal-title":"Biometrics"},{"key":"2025111101214689000_qkaf034-B2","author":"Athey"},{"issue":"4","key":"2025111101214689000_qkaf034-B3","doi-asserted-by":"publisher","first-page":"605","DOI":"10.1093\/biostatistics\/kxx010","article-title":"Bayesian inference for causal mechanisms with application to a randomized study for postoperative pain control","volume":"18","author":"Baccini","year":"2017","journal-title":"Biostatistics"},{"issue":"494","key":"2025111101214689000_qkaf034-B4","doi-asserted-by":"publisher","first-page":"469","DOI":"10.1198\/jasa.2011.ap09094","article-title":"Modeling partial compliance through copulas in a principal stratification framework","volume":"106","author":"Bartolucci","year":"2011","journal-title":"Journal of the American Statistical Association"},{"key":"2025111101214689000_qkaf034-B5","volume-title":"Efficient and adaptive inference in semiparametric models","author":"Bickel","year":"1993"},{"key":"2025111101214689000_qkaf034-B6","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4614-4818-1","volume-title":"Essential statistical inference","author":"Boos","year":"2013"},{"key":"2025111101214689000_qkaf034-B7","doi-asserted-by":"crossref","DOI":"10.1007\/b138566","volume-title":"The evaluation of surrogate endpoints","author":"Burzykowski","year":"2005"},{"issue":"5","key":"2025111101214689000_qkaf034-B8","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1257\/aer.p20171038","article-title":"Double\/debiased\/neyman machine learning of treatment effects","volume":"107","author":"Chernozhukov","year":"2017","journal-title":"American Economic Review"},{"issue":"1","key":"2025111101214689000_qkaf034-B9","doi-asserted-by":"publisher","first-page":"C1","DOI":"10.1111\/ectj.12097","article-title":"Double\/debiased machine learning for treatment and structural parameters: Double\/debiased machine learning","volume":"21","author":"Chernozhukov","year":"2018","journal-title":"The Econometrics Journal"},{"issue":"3","key":"2025111101214689000_qkaf034-B10","doi-asserted-by":"publisher","first-page":"576","DOI":"10.1093\/ectj\/utac002","article-title":"Debiased machine learning of global and local parameters using regularized Riesz representers","volume":"25","author":"Chernozhukov","year":"2022","journal-title":"The Econometrics Journal"},{"issue":"1","key":"2025111101214689000_qkaf034-B11","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1093\/biomet\/asac033","article-title":"A simple and general debiased machine learning theorem with finite-sample guarantees","volume":"110","author":"Chernozhukov","year":"2023","journal-title":"Biometrika"},{"issue":"3","key":"2025111101214689000_qkaf034-B12","doi-asserted-by":"publisher","first-page":"589","DOI":"10.1093\/biomet\/68.3.589","article-title":"Nonparametric estimates of standard error: The jackknife, the bootstrap and other methods","volume":"68","author":"Efron","year":"1981","journal-title":"Biometrika"},{"issue":"413","key":"2025111101214689000_qkaf034-B13","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1080\/01621459.1991.10474996","article-title":"Compliance as an explanatory variable in clinical trials","volume":"86","author":"Efron","year":"1991","journal-title":"Journal of the American Statistical Association"},{"issue":"4","key":"2025111101214689000_qkaf034-B14","doi-asserted-by":"publisher","first-page":"1161","DOI":"10.1111\/biom.2006.62.issue-4","article-title":"Augmented designs to assess immune response in vaccine trials","volume":"62","author":"Follmann","year":"2006","journal-title":"Biometrics"},{"issue":"452","key":"2025111101214689000_qkaf034-B15","doi-asserted-by":"publisher","first-page":"1101","DOI":"10.1080\/01621459.2000.10474306","article-title":"On the effect of treatment among would-be treatment compliers: An analysis of the multiple risk factor intervention trial","volume":"95","author":"Follmann","year":"2000","journal-title":"Journal of the American Statistical Association"},{"issue":"4","key":"2025111101214689000_qkaf034-B16","doi-asserted-by":"publisher","first-page":"979","DOI":"10.1093\/biomet\/asy053","article-title":"Principal ignorability in mediation analysis: Through and beyond sequential ignorability","volume":"105","author":"Forastiere","year":"2018","journal-title":"Biometrika"},{"issue":"1","key":"2025111101214689000_qkaf034-B17","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1111\/biom.2002.58.issue-1","article-title":"Principal stratification in causal inference","volume":"58","author":"Frangakis","year":"2002","journal-title":"Biometrics"},{"issue":"498","key":"2025111101214689000_qkaf034-B18","doi-asserted-by":"publisher","first-page":"450","DOI":"10.1080\/01621459.2011.643719","article-title":"Evaluating the effect of training on wages in the presence of noncompliance, nonemployment, and missing outcome data","volume":"107","author":"Frumento","year":"2012","journal-title":"Journal of the American Statistical Association"},{"issue":"4","key":"2025111101214689000_qkaf034-B19","doi-asserted-by":"publisher","first-page":"1146","DOI":"10.1111\/biom.2008.64.issue-4","article-title":"Evaluating candidate principal surrogate endpoints","volume":"64","author":"Gilbert","year":"2008","journal-title":"Biometrics"},{"issue":"2","key":"2025111101214689000_qkaf034-B20","doi-asserted-by":"publisher","first-page":"315","DOI":"10.2307\/2998560","article-title":"On the role of the propensity score in efficient semiparametric estimation of average treatment effects","volume":"66","author":"Hahn","year":"1998","journal-title":"Econometrica: Journal of the Econometric Society"},{"issue":"15","key":"2025111101214689000_qkaf034-B21","doi-asserted-by":"publisher","first-page":"1081","DOI":"10.1056\/NEJM199610103351501","article-title":"A trial comparing nucleoside monotherapy with combination therapy in HIV-infected adults with CD4 cell counts from 200 to 500 per cubic millimeter","volume":"335","author":"Hammer","year":"1996","journal-title":"New England Journal of Medicine"},{"issue":"26","key":"2025111101214689000_qkaf034-B22","doi-asserted-by":"publisher","first-page":"5290","DOI":"10.1002\/sim.v41.26","article-title":"Identifying surrogate markers in real-world comparative effectiveness research","volume":"41","author":"Han","year":"2022","journal-title":"Statistics in Medicine"},{"issue":"4","key":"2025111101214689000_qkaf034-B23","doi-asserted-by":"publisher","first-page":"601","DOI":"10.1002\/pam.v21:4","article-title":"Differential effects of high-quality child care","volume":"21","author":"Hill","year":"2002","journal-title":"Journal of Policy Analysis and Management: The Journal of the Association for Public Policy Analysis and Management"},{"issue":"14","key":"2025111101214689000_qkaf034-B24","doi-asserted-by":"publisher","first-page":"1823","DOI":"10.1097\/00002030-199814000-00014","article-title":"Cd4 cell count as a surrogate endpoint in HIV clinical trials: A meta-analysis of studies of the aids clinical trials group","volume":"12","author":"Hughes","year":"1998","journal-title":"AIDS"},{"issue":"1","key":"2025111101214689000_qkaf034-B25","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1214\/10-STS321","article-title":"Identification, inference and sensitivity analysis for causal mediation effects","volume":"25","author":"Imai","year":"2010","journal-title":"Statistical Science"},{"issue":"1","key":"2025111101214689000_qkaf034-B26","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1162\/003465304323023651","article-title":"Nonparametric estimation of average treatment effects under exogeneity: A review","volume":"86","author":"Imbens","year":"2004","journal-title":"Review of Economics and Statistics"},{"key":"2025111101214689000_qkaf034-B27","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-12465-5","volume-title":"Copula theory and its applications","author":"Jaworski","year":"2010"},{"issue":"4","key":"2025111101214689000_qkaf034-B28","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1214\/20-STS810","article-title":"Identification of causal effects within principal strata using auxiliary variables","volume":"36","author":"Jiang","year":"2021","journal-title":"Statistical Science"},{"issue":"4","key":"2025111101214689000_qkaf034-B29","doi-asserted-by":"publisher","first-page":"1423","DOI":"10.1111\/rssb.12538","article-title":"Multiply robust estimation of causal effects under principal ignorability","volume":"84","author":"Jiang","year":"2022","journal-title":"Journal of the Royal Statistical Society: Series B, Statistical Methodology"},{"issue":"481","key":"2025111101214689000_qkaf034-B30","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1198\/016214507000000347","article-title":"Principal stratification for causal inference with extended partial compliance","volume":"103","author":"Jin","year":"2008","journal-title":"Journal of the American Statistical Association"},{"issue":"23","key":"2025111101214689000_qkaf034-B31","doi-asserted-by":"publisher","first-page":"2857","DOI":"10.1002\/sim.v28:23","article-title":"On the use of propensity scores in principal causal effect estimation","volume":"28","author":"Jo","year":"2009","journal-title":"Statistics in Medicine"},{"issue":"2","key":"2025111101214689000_qkaf034-B32","doi-asserted-by":"publisher","first-page":"480","DOI":"10.1093\/jrsssb\/qkae099","article-title":"On the role of surrogates in the efficient estimation of treatment effects with limited outcome data","volume":"87","author":"Kallus","year":"2024","journal-title":"Journal of the Royal Statistical Society: Series B, Statistical Methodology"},{"key":"2025111101214689000_qkaf034-B33","volume-title":"Statistical causal inferences and their applications in public health research","author":"Kennedy","year":"2016"},{"issue":"3","key":"2025111101214689000_qkaf034-B34","doi-asserted-by":"publisher","first-page":"1927","DOI":"10.1214\/19-AOAS1260","article-title":"Bayesian methods for multiple mediators: Relating principal stratification and causal mediation in the analysis of power plant emission controls","volume":"13","author":"Kim","year":"2019","journal-title":"The Annals of Applied Statistics"},{"issue":"4","key":"2025111101214689000_qkaf034-B35","doi-asserted-by":"publisher","first-page":"1677","DOI":"10.1111\/rssa.12547","article-title":"Health effects of power plant emissions through ambient air quality","volume":"183","author":"Kim","year":"2020","journal-title":"Journal of the Royal Statistical Society: Series A (Statistics in Society)"},{"issue":"19","key":"2025111101214689000_qkaf034-B36","doi-asserted-by":"publisher","first-page":"3837","DOI":"10.1002\/sim.v41.19","article-title":"Using principal stratification in analysis of clinical trials","volume":"41","author":"Lipkovich","year":"2022","journal-title":"Statistics in Medicine"},{"key":"2025111101214689000_qkaf034-B37","author":"Lu"},{"key":"2025111101214689000_qkaf034-B38","volume-title":"Handbook of matching and weighting adjustments for causal inference","author":"Mattei","year":"2023"},{"issue":"5","key":"2025111101214689000_qkaf034-B39","doi-asserted-by":"publisher","first-page":"729","DOI":"10.1111\/j.1467-9868.2011.00780.x","article-title":"Augmented designs to assess principal strata direct effects","volume":"73","author":"Mattei","year":"2011","journal-title":"Journal of the Royal Statistical Society: Series B, Statistical Methodology"},{"key":"2025111101214689000_qkaf034-B40","doi-asserted-by":"crossref","DOI":"10.1201\/9780203753736","volume-title":"Generalized linear models","author":"McCullagh","year":"2019"},{"issue":"1","key":"2025111101214689000_qkaf034-B41","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1515\/1557-4679.1380","article-title":"A refreshing account of principal stratification","volume":"8","author":"Mealli","year":"2012","journal-title":"The International Journal of Biostatistics"},{"key":"2025111101214689000_qkaf034-B42","volume-title":"An introduction to copulas","author":"Nelsen","year":"2007"},{"issue":"10","key":"2025111101214689000_qkaf034-B43","doi-asserted-by":"publisher","first-page":"1637","DOI":"10.1002\/sim.v35.10","article-title":"Robust estimation of the proportion of treatment effect explained by surrogate marker information","volume":"35","author":"Parast","year":"2016","journal-title":"Statistics in Medicine"},{"key":"2025111101214689000_qkaf034-B44","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1214\/193940307000000527","article-title":"Higher order influence functions and minimax estimation of nonlinear functionals","volume":"66","author":"Robins","year":"2008","journal-title":"Probability and Statistics: Essays in Honor of David A. Freedman"},{"issue":"2","key":"2025111101214689000_qkaf034-B45","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1097\/00001648-199203000-00013","article-title":"Identifiability and exchangeability for direct and indirect effects","volume":"3","author":"Robins","year":"1992","journal-title":"Epidemiology"},{"issue":"1","key":"2025111101214689000_qkaf034-B46","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1093\/biomet\/70.1.41","article-title":"The central role of the propensity score in observational studies for causal effects","volume":"70","author":"Rosenbaum","year":"1983","journal-title":"Biometrika"},{"issue":"5","key":"2025111101214689000_qkaf034-B47","doi-asserted-by":"publisher","first-page":"688","DOI":"10.1037\/h0037350","article-title":"Estimating causal effects of treatments in randomized and nonrandomized studies","volume":"66","author":"Rubin","year":"1974","journal-title":"Journal of Educational Psychology"},{"issue":"371","key":"2025111101214689000_qkaf034-B48","doi-asserted-by":"publisher","first-page":"591","DOI":"10.2307\/2287653","article-title":"Randomization analysis of experimental data: The fisher randomization test comment","volume":"75","author":"Rubin","year":"1980","journal-title":"Journal of the American Statistical Association"},{"issue":"2","key":"2025111101214689000_qkaf034-B49","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1111\/sjos.2004.31.issue-2","article-title":"Direct and indirect causal effects via potential outcomes","volume":"31","author":"Rubin","year":"2004","journal-title":"Scandinavian Journal of Statistics"},{"key":"2025111101214689000_qkaf034-B50","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1214\/088342306000000114","article-title":"Causal inference through potential outcomes and principal stratification: Application to studies with \u201ccensoring\u201d due to death","volume":"21","author":"Rubin","year":"2006","journal-title":"Statistical Science"},{"issue":"496","key":"2025111101214689000_qkaf034-B51","doi-asserted-by":"publisher","first-page":"1331","DOI":"10.1198\/jasa.2011.ap10425","article-title":"A Bayesian semiparametric approach to intermediate variables in causal inference","volume":"106","author":"Schwartz","year":"2011","journal-title":"Journal of the American Statistical Association"},{"key":"2025111101214689000_qkaf034-B52","first-page":"229","article-title":"Fonctions de repartition an dimensions et leurs marges","volume":"8","author":"Sklar","year":"1959","journal-title":"Publications de L\u2019Institut de Statistique de L\u2019Universit\u00e9 de Paris"},{"issue":"3","key":"2025111101214689000_qkaf034-B53","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1007\/s11121-014-0513-z","article-title":"Assessing the generalizability of randomized trial results to target populations","volume":"16","author":"Stuart","year":"2015","journal-title":"Prevention Science"},{"key":"2025111101214689000_qkaf034-B54","volume-title":"Introduction to nonparametric estimation","author":"Tsybakov","year":"2008"},{"issue":"1","key":"2025111101214689000_qkaf034-B55","doi-asserted-by":"publisher","first-page":"25","DOI":"10.2202\/1544-6115.1309","article-title":"Super learner","volume":"6","author":"Van der Laan","year":"2007","journal-title":"Statistical Applications in Genetics and Molecular Biology"},{"issue":"1","key":"2025111101214689000_qkaf034-B56","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2202\/1557-4679.1329","article-title":"Principal stratification\u2013uses and limitations","volume":"7","author":"VanderWeele","year":"2011","journal-title":"The International Journal of Biostatistics"},{"issue":"3","key":"2025111101214689000_qkaf034-B57","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1111\/biom.v69.3","article-title":"Surrogate measures and consistent surrogates","volume":"69","author":"VanderWeele","year":"2013","journal-title":"Biometrics"},{"issue":"1","key":"2025111101214689000_qkaf034-B58","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1093\/biomet\/asz065","article-title":"Model-free approach to quantifying the proportion of treatment effect explained by a surrogate marker","volume":"107","author":"Wang","year":"2020","journal-title":"Biometrika"},{"issue":"4","key":"2025111101214689000_qkaf034-B59","doi-asserted-by":"publisher","first-page":"803","DOI":"10.1111\/j.0006-341X.2002.00803.x","article-title":"A measure of the proportion of treatment effect explained by a surrogate marker","volume":"58","author":"Wang","year":"2002","journal-title":"Biometrics"},{"issue":"3","key":"2025111101214689000_qkaf034-B60","doi-asserted-by":"publisher","first-page":"922","DOI":"10.1111\/biom.2012.68.issue-3","article-title":"A Bayesian approach to improved estimation of causal effect predictiveness for a principal surrogate endpoint","volume":"68","author":"Zigler","year":"2012","journal-title":"Biometrics"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/87\/5\/1655\/63515098\/qkaf034.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/87\/5\/1655\/63515098\/qkaf034.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,11]],"date-time":"2025-11-11T06:21:55Z","timestamp":1762842115000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/87\/5\/1655\/8166818"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,18]]},"references-count":60,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,6,18]]},"published-print":{"date-parts":[[2025,11,11]]}},"URL":"https:\/\/doi.org\/10.1093\/jrsssb\/qkaf034","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2025,11]]},"published":{"date-parts":[[2025,6,18]]}}}