{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,17]],"date-time":"2026-10-17T01:42:17Z","timestamp":1792201337910,"version":"4.4.0"},"reference-count":47,"publisher":"Wiley","issue":"23-24","license":[{"start":{"date-parts":[[2026,10,16]],"date-time":"2026-10-16T00:00:00Z","timestamp":1792108800000},"content-version":"vor","delay-in-days":15,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"funder":[{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["R37\u2010AI131771"],"award-info":[{"award-number":["R37\u2010AI131771"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["P30\u2010AI110527"],"award-info":[{"award-number":["P30\u2010AI110527"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Statistics in Medicine"],"published-print":{"date-parts":[[2026,10]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>Large observational datasets, including those derived from electronic health records, are a valuable resource for medical research but are often affected by missingness, measurement error, and misclassification. Two\u2010phase sampling with generalized raking (GR) estimation is an efficient and robust approach to statistical inference in such settings. In this approach, variables that are unavailable or measured with error in a large Phase 1 cohort are obtained with higher\u2010quality measurements in a Phase 2 subsample. Previous research has studied optimal Phase 2 sampling designs for inverse\u2010probability\u2010weighted (IPW) estimators in nonadaptive, multiparameter settings, and for GR estimators in single\u2010parameter settings. In this work, we extend these results by deriving optimal adaptive, multiwave sampling designs for IPW and GR estimators when multiple parameters are of interest. We propose several practical allocation strategies, examine the design choices that arise when implementing them, and evaluate their performance through extensive simulations and an example using data from people living with HIV attending the Vanderbilt Comprehensive Care Clinic (VCCC). Our results show that independently optimizing allocation for each parameter improves efficiency over traditional case\u2010control sampling. We also derive an integer\u2010valued, A\u2010optimal allocation method that typically outperforms independent optimization. Notably, we find that optimal designs for GR can differ substantially from those for IPW, and that this distinction can meaningfully affect estimator efficiency in the multiple\u2010parameter setting. These findings offer practical guidance for future two\u2010phase studies involving incomplete or error\u2010prone data.<\/jats:p>","DOI":"10.1002\/sim.70786","type":"journal-article","created":{"date-parts":[[2026,10,17]],"date-time":"2026-10-17T01:10:23Z","timestamp":1792199423000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Optimal Two\u2010Phase Sampling Designs for Generalized Raking Estimators With Multiple Parameters of Interest"],"prefix":"10.1002","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-5736-3574","authenticated-orcid":false,"given":"Jasper B.","family":"Yang","sequence":"first","affiliation":[{"name":"Department of Biostatistics University of Washington  Seattle Washington USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3758-5992","authenticated-orcid":false,"given":"Bryan E.","family":"Shepherd","sequence":"additional","affiliation":[{"name":"Department of Biostatistics Vanderbilt University  Nashville Tennessee USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4255-5437","authenticated-orcid":false,"given":"Thomas","family":"Lumley","sequence":"additional","affiliation":[{"name":"Department of Statistics University of Auckland  Auckland New Zealand"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1883-8410","authenticated-orcid":false,"given":"Pamela A.","family":"Shaw","sequence":"additional","affiliation":[{"name":"Department of Biostatistics University of Washington  Seattle Washington USA"},{"name":"Biostatistics Division Kaiser Permanente Washington Health Research Institute  Seattle Washington USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,10,16]]},"reference":[{"issue":"1","key":"e_1_2_11_2_1","first-page":"610","article-title":"Unlocking the Potential of Electronic Health Records for Health Research","volume":"5","author":"Lee S.","year":"2020","journal-title":"International Journal of Population Data Science"},{"issue":"1","key":"e_1_2_11_3_1","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1136\/amiajnl-2011-000681","article-title":"Methods and Dimensions of Electronic Health Record Data Quality Assessment: Enabling Reuse for Clinical Research","volume":"20","author":"Weiskopf N. G.","year":"2013","journal-title":"Journal of the American Medical Informatics Association"},{"issue":"6","key":"e_1_2_11_4_1","doi-asserted-by":"crossref","DOI":"10.1001\/jamanetworkopen.2020.5867","article-title":"Frequency and Types of Patient\u2010Reported Errors in Electronic Health Record Ambulatory Care Notes","volume":"3","author":"Bell S. K.","year":"2020","journal-title":"JAMA Network Open"},{"issue":"15","key":"e_1_2_11_5_1","doi-asserted-by":"crossref","first-page":"1580","DOI":"10.1001\/jama.2012.489","article-title":"Use of Administrative Data to Estimate the Incidence of Statin\u2010Related Rhabdomyolysis","volume":"307","author":"Floyd J. S.","year":"2012","journal-title":"Journal of the American Medical Association"},{"issue":"5","key":"e_1_2_11_6_1","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1093\/aje\/kwy292","article-title":"Challenges and Opportunities for Using Big Health Care Data to Advance Medical Science and Public Health","volume":"188","author":"Shortreed S. M.","year":"2019","journal-title":"American Journal of Epidemiology"},{"key":"e_1_2_11_7_1","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1080\/01621459.1938.10503378","article-title":"Contribution to the Theory of Sampling Human Populations","volume":"33","author":"Neyman J.","year":"1938","journal-title":"Journal of the American Statistical Association"},{"issue":"2","key":"e_1_2_11_8_1","doi-asserted-by":"crossref","first-page":"1045","DOI":"10.1214\/20-AOAS1343","article-title":"Accounting for Dependent Errors in Predictors and Time\u2010To\u2010Event Outcomes Using Electronic Health Records, Validation Samples, and Multiple Imputation","volume":"14","author":"Giganti M. J.","year":"2020","journal-title":"Annals of Applied Statistics"},{"issue":"3","key":"e_1_2_11_9_1","doi-asserted-by":"crossref","first-page":"2649","DOI":"10.1111\/biom.13713","article-title":"Multiwave Validation Sampling for Error\u2010Prone Electronic Health Records","volume":"79","author":"Shepherd B. E.","year":"2023","journal-title":"Biometrics"},{"key":"e_1_2_11_10_1","unstructured":"T.Lumley \u201cRobustness of Semiparametric Efficiency in Nearly\u2010True Models for Two\u2010Phase Samples \u201dPre\u2010Print arXiv:1707.05924 (2017) https:\/\/arxiv.org\/abs\/1707.05924."},{"issue":"520","key":"e_1_2_11_11_1","doi-asserted-by":"crossref","first-page":"1468","DOI":"10.1080\/01621459.2017.1295864","article-title":"Efficient Semiparametric Inference Under Two\u2010Phase Sampling, With Applications to Genetic Association Studies","volume":"112","author":"Tao R.","year":"2017","journal-title":"Journal of the American Statistical Association"},{"issue":"30","key":"e_1_2_11_12_1","doi-asserted-by":"crossref","first-page":"6777","DOI":"10.1002\/sim.9210","article-title":"Combining Multiple Imputation With Raking of Weights: An Efficient and Robust Approach in the Setting of Nearly True Models","volume":"40","author":"Han K.","year":"2021","journal-title":"Statistics in Medicine"},{"key":"e_1_2_11_13_1","unstructured":"B. D.Williamson C.Krakauer E.Johnson et al. \u201cAssessing Treatment Effects in Observational Data With Missing Confounders: A Comparative Study of Practical Doubly\u2010Robust and Traditional Missing Data Methods \u201dPreprint arXiv:2412.15012 (2024) https:\/\/arxiv.org\/abs\/2412.15012."},{"key":"e_1_2_11_14_1","volume-title":"Sampling Techniques","author":"Cochran W. G.","year":"1977"},{"key":"e_1_2_11_15_1","unstructured":"J. B.Yang T.Lumley B. E.Shepherd andP. A.Shaw \u201cImproving Optimal Subsampling Through Stratification \u201dPreprint arXiv:2512.20837 (2025) https:\/\/arxiv.org\/abs\/2512.20837."},{"issue":"4","key":"e_1_2_11_16_1","doi-asserted-by":"crossref","first-page":"558","DOI":"10.2307\/2342192","article-title":"On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection","volume":"97","author":"Neyman J.","year":"1934","journal-title":"Journal of the Royal Statistical Society"},{"key":"e_1_2_11_17_1","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.spl.2017.04.026","article-title":"Exact Optimal Sample Allocation: More Efficient Than Neyman","volume":"129","author":"Wright T.","year":"2017","journal-title":"Statistics & Probability Letters"},{"key":"e_1_2_11_18_1","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1007\/s12561-009-9001-6","article-title":"Improved Horvitz\u2013Thompson Estimation of Model Parameters From Two\u2010Phase Stratified Samples: Applications in Epidemiology","volume":"1","author":"Breslow N. E.","year":"2009","journal-title":"Statistics in Biosciences"},{"issue":"8","key":"e_1_2_11_19_1","doi-asserted-by":"crossref","first-page":"1482","DOI":"10.1002\/sim.9300","article-title":"Optimal Sampling for Design\u2010Based Estimators of Regression Models","volume":"41","author":"Chen T.","year":"2022","journal-title":"Statistics in Medicine"},{"issue":"21","key":"e_1_2_11_20_1","doi-asserted-by":"crossref","first-page":"2899","DOI":"10.1002\/sim.6523","article-title":"Adaptive Sampling in Two\u2010Phase Designs: A Biomarker Study for Progression in Arthritis","volume":"34","author":"McIsaac M. A.","year":"2015","journal-title":"Statistics in Medicine"},{"issue":"3","key":"e_1_2_11_21_1","doi-asserted-by":"crossref","first-page":"857","DOI":"10.1177\/0962280220978500","article-title":"Two\u2010Phase Analysis and Study Design for Survival Models With Error\u2010Prone Exposures","volume":"30","author":"Han K.","year":"2021","journal-title":"Statistical Methods in Medical Research"},{"issue":"18","key":"e_1_2_11_22_1","doi-asserted-by":"crossref","first-page":"4090","DOI":"10.1002\/sim.9016","article-title":"Optimal Allocation in Stratified Cluster\u2010Based Outcome\u2010Dependent Sampling Designs","volume":"40","author":"Sauer S.","year":"2021","journal-title":"Statistics in Medicine"},{"issue":"12","key":"e_1_2_11_23_1","doi-asserted-by":"crossref","first-page":"2400","DOI":"10.1177\/09622802221122423","article-title":"Optimal Sampling Allocation for Outcome\u2010Dependent Designs in Cluster\u2010Correlated Data Settings","volume":"31","author":"Rivera\u2010Rodriguez C.","year":"2022","journal-title":"Statistical Methods in Medical Research"},{"issue":"522","key":"e_1_2_11_24_1","doi-asserted-by":"crossref","first-page":"829","DOI":"10.1080\/01621459.2017.1292914","article-title":"Optimal Subsampling for Large Sample Logistic Regression","volume":"113","author":"Wang H. Y.","year":"2018","journal-title":"Journal of the American Statistical Association"},{"issue":"537","key":"e_1_2_11_25_1","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1080\/01621459.2020.1773832","article-title":"Optimal Distributed Subsampling for Maximum Quasi\u2010Likelihood Estimators With Massive Data","volume":"117","author":"Yu J.","year":"2022","journal-title":"Journal of the American Statistical Association"},{"key":"e_1_2_11_26_1","unstructured":"R.Wang Q.Wang andW.Miao \u201cA Maximin Optimal Approach for Sampling Designs in Two\u2010Phase Studies \u201d(2025) https:\/\/arxiv.org\/abs\/2312.10596."},{"key":"e_1_2_11_27_1","unstructured":"T.ZrnicandE. J.Cand\u00e8s \u201cActive Statistical Inference \u201darXiv Preprint arXiv:2403.03208 (2024)."},{"key":"e_1_2_11_28_1","first-page":"68727","article-title":"Robust Sampling for Active Statistical Inference","volume":"38","author":"Li P.","year":"2026","journal-title":"Advances in Neural Information Processing Systems"},{"key":"e_1_2_11_29_1","unstructured":"D. M.KlugerandS.Bates \u201cM\u2010Estimation Under Two\u2010Phase Multiwave Sampling With Applications to Prediction\u2010Powered Inference \u201darXiv Preprint arXiv:2602.16933 (2026)."},{"issue":"423","key":"e_1_2_11_30_1","doi-asserted-by":"crossref","first-page":"1013","DOI":"10.1080\/01621459.1993.10476369","article-title":"Generalized Raking Procedures in Survey Sampling","volume":"88","author":"Deville J. C.","year":"1993","journal-title":"Journal of the American Statistical Association"},{"key":"e_1_2_11_31_1","unstructured":"T.Lumley \u201cSurvey: Analysis of Complex Survey Samples \u201dR package version 4.4 (2024) https:\/\/CRAN.R\u2010project.org\/package=survey."},{"issue":"427","key":"e_1_2_11_32_1","doi-asserted-by":"crossref","first-page":"846","DOI":"10.1080\/01621459.1994.10476818","article-title":"Estimation of Regression Coefficients When Some Regressors Are Not Always Observed","volume":"89","author":"Robins J. M.","year":"1994","journal-title":"Journal of the American Statistical Association"},{"issue":"2","key":"e_1_2_11_33_1","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1111\/j.1751-5823.2011.00138.x","article-title":"Connections Between Survey Calibration Estimators and Semiparametric Models for Incomplete Data","volume":"79","author":"Lumley T.","year":"2011","journal-title":"International Statistical Review"},{"issue":"467","key":"e_1_2_11_34_1","doi-asserted-by":"crossref","first-page":"832","DOI":"10.1198\/016214504000000584","article-title":"Improving the Efficiency of Relative\u2010Risk Estimation in Case\u2010Cohort Studies","volume":"99","author":"Kulich M.","year":"2004","journal-title":"Journal of the American Statistical Association"},{"issue":"11","key":"e_1_2_11_35_1","doi-asserted-by":"crossref","first-page":"1398","DOI":"10.1093\/aje\/kwp055","article-title":"Using the Whole Cohort in the Analysis of Case\u2010Cohort Data","volume":"169","author":"Breslow N. E.","year":"2009","journal-title":"American Journal of Epidemiology"},{"issue":"3","key":"e_1_2_11_36_1","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1002\/sim.8793","article-title":"Raking and Regression Calibration: Methods to Address Bias From Correlated Covariate and Time\u2010To\u2010Event Error","volume":"40","author":"Oh E. J.","year":"2021","journal-title":"Statistics in Medicine"},{"key":"e_1_2_11_37_1","volume-title":"Model Assisted Survey Sampling","author":"S\u00e4rndal C. E.","year":"2003"},{"issue":"30","key":"e_1_2_11_38_1","doi-asserted-by":"crossref","first-page":"4912","DOI":"10.1002\/sim.8760","article-title":"Optimal Multiwave Sampling for Regression Modeling in Two\u2010 Phase Designs","volume":"39","author":"Chen T.","year":"2020","journal-title":"Statistics in Medicine"},{"issue":"4","key":"e_1_2_11_39_1","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1080\/00031305.2012.733679","article-title":"The Equivalence of Neyman Optimum Allocation for Sampling and Equal Proportions for Apportioning the US House of Representatives","volume":"66","author":"Wright T.","year":"2012","journal-title":"American Statistician"},{"issue":"4","key":"e_1_2_11_40_1","doi-asserted-by":"crossref","first-page":"1368","DOI":"10.1111\/rssa.12689","article-title":"Two\u2010Phase Sampling Designs for Data Validation in Settings With Covariate Measurement Error and Continuous Outcome","volume":"184","author":"Amorim G.","year":"2021","journal-title":"Journal of the Royal Statistical Society: Series A (Statistics in Society)"},{"issue":"2","key":"e_1_2_11_41_1","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1214\/aoms\/1177706252","article-title":"Optimum Designs in Regression Problems","volume":"30","author":"Kiefer J.","year":"1959","journal-title":"Annals of Mathematical Statistics"},{"issue":"1","key":"e_1_2_11_42_1","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/0022-247X(82)90152-4","article-title":"A\u2010Optimality for Regression Designs","volume":"87","author":"Chan N. N.","year":"1982","journal-title":"Journal of Mathematical Analysis and Applications"},{"key":"e_1_2_11_43_1","first-page":"1","article-title":"Optimum Allocation for Adaptive Multi\u2010Wave Sampling in R: The R Package Optimall","volume":"114","author":"Yang J. B.","year":"2025","journal-title":"Journal of Statistical Software"},{"issue":"5","key":"e_1_2_11_44_1","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1089\/apc.2009.0283","article-title":"Postpartum Discontinuation of Antiretroviral Therapy and Risk of Maternal AIDS\u2010Defining Events, Non\u2010AIDS\u2010Defining Events, and Mortality Among a Cohort of HIV\u20101\u2010Infected Women in the United States","volume":"24","author":"Melekhin V. V.","year":"2010","journal-title":"AIDS Patient Care and STDs"},{"issue":"408","key":"e_1_2_11_45_1","doi-asserted-by":"crossref","first-page":"1074","DOI":"10.1080\/01621459.1989.10478874","article-title":"The Robust Inference for the Cox Proportional Hazards Model","volume":"84","author":"Lin D. Y.","year":"1989","journal-title":"Journal of the American Statistical Association"},{"issue":"2","key":"e_1_2_11_46_1","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1214\/aoms\/1177731454","article-title":"On the Efficient Design of Statistical Investigations","volume":"14","author":"Wald A.","year":"1943","journal-title":"Annals of Mathematical Statistics"},{"key":"e_1_2_11_47_1","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719109","volume-title":"Optimal Design of Experiments","author":"Pukelsheim F.","year":"2006"},{"issue":"514","key":"e_1_2_11_48_1","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1080\/01621459.2015.1136632","article-title":"I\u2010Optimal Design of Mixture Experiments","volume":"111","author":"Goos P.","year":"2016","journal-title":"Journal of the American Statistical Association"}],"container-title":["Statistics in Medicine"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/sim.70786","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1002\/sim.70786","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/sim.70786","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,17]],"date-time":"2026-10-17T01:10:29Z","timestamp":1792199429000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/sim.70786"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10]]},"references-count":47,"journal-issue":{"issue":"23-24","published-print":{"date-parts":[[2026,10]]}},"alternative-id":["10.1002\/sim.70786"],"URL":"https:\/\/doi.org\/10.1002\/sim.70786","archive":["Portico"],"relation":{},"ISSN":["0277-6715","1097-0258"],"issn-type":[{"value":"0277-6715","type":"print"},{"value":"1097-0258","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,10]]},"assertion":[{"value":"2026-03-20","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-10-08","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-10-16","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e70786"}}