{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,15]],"date-time":"2026-09-15T15:50:09Z","timestamp":1789487409021,"version":"build-2803163510"},"reference-count":27,"publisher":"Wiley","issue":"5","license":[{"start":{"date-parts":[[2026,9,15]],"date-time":"2026-09-15T00:00:00Z","timestamp":1789430400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,9,15]],"date-time":"2026-09-15T00:00:00Z","timestamp":1789430400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["RGPIN\u20102022\u201003050"],"award-info":[{"award-number":["RGPIN\u20102022\u201003050"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Biometrical J"],"published-print":{"date-parts":[[2026,10]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>\n                    Accurately estimating the parameters of a continuous distribution from dichotomized or aggregated data is a common problem in biomedical and environmental research. Many studies report only the proportion of subjects exceeding a threshold, without releasing individual\u2010level measurements. To address this limitation, we develop a hierarchical binomial\u2010probit modeling framework to reconstruct the parameters of an underlying normal distribution from threshold\u2010based data. The framework considers two principal settings. In the\n                    <jats:italic>fixed\u2010mean model<\/jats:italic>\n                    , all studies are assumed to share a common mean and variance, and the parameters  and  are estimated using a maximum\u2010likelihood estimator (MLE) and a generalized linear model (GLM) approximation. In the\n                    <jats:italic>random\u2010mean model<\/jats:italic>\n                    , each study has its own mean  drawn from a population distribution  with a common within\u2010study variance ; parameters , , and  are estimated using MLE, a generalized linear mixed model (GLMM) approximation, and a fully Bayesian Markov chain Monte Carlo (MCMC) method. Extensive simulations varying the number of studies, sample size, and heterogeneity ratio  evaluate estimator bias, variance, mean squared error, and coverage probability. Results show that MLE performs efficiently under well\u2010identified conditions, whereas GLMM and Bayesian estimators are more robust with small samples or strong heterogeneity. The proposed framework provides a unified and practical approach for inferring latent distributions from aggregated or privacy\u2010restricted data, with applications in clinical trial design, biomarker analysis, environmental monitoring, and quality\u00a0control.\n                  <\/jats:p>","DOI":"10.1002\/bimj.70173","type":"journal-article","created":{"date-parts":[[2026,9,15]],"date-time":"2026-09-15T13:54:38Z","timestamp":1789480478000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Estimation of Underlying Normal Distribution Parameters From Dichotomized Data"],"prefix":"10.1002","volume":"68","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-1722-916X","authenticated-orcid":false,"given":"Zhaoze","family":"Liu","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics University of Victoria  British Columbia Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Longwen","family":"Shang","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics University of Victoria  British Columbia Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mary","family":"Lesperance","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics University of Victoria  British Columbia Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shuiqing","family":"Zhou","sequence":"additional","affiliation":[{"name":"Innovation Research Institute of Zhejiang University of Technology  Shengzhou China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4728-2343","authenticated-orcid":false,"given":"Xuekui","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics University of Victoria  British Columbia Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,9,15]]},"reference":[{"issue":"1","key":"e_1_2_12_2_1","doi-asserted-by":"crossref","first-page":"33","DOI":"10.21608\/cjmss.2023.241890.1022","article-title":"Bayesian Estimation and Prediction for Exponentiated Inverted Topp\u2010Leone Distribution","volume":"3","author":"Albadawy A.","year":"2024","journal-title":"Computational Journal of Mathematical and Statistical Sciences"},{"issue":"1","key":"e_1_2_12_3_1","doi-asserted-by":"crossref","first-page":"211","DOI":"10.3934\/math.2026009","article-title":"Statistical Inference for a One\u2010Parameter Lifetime Model Under Unified Progressive Hybrid Censoring With Binomial Random Removals","volume":"11","author":"Almetwally E. 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C.","year":"1995","journal-title":"Journal of Computational and Graphical Statistics"},{"key":"e_1_2_12_26_1","volume-title":"R: A Language and Environment for Statistical Computing","author":"R Core Team","year":"2025"},{"key":"e_1_2_12_27_1","unstructured":"Stan Development Team.2024. \u201cRStan: The R Interface to Stan.\u201d R Package Version 2.32.6.https:\/\/mc\u2010stan.org\/users\/interfaces\/rstan."},{"key":"e_1_2_12_28_1","doi-asserted-by":"crossref","DOI":"10.1002\/0470854200","volume-title":"Meta\u2010Analysis of Controlled Clinical Trials","author":"Whitehead A.","year":"2002"}],"container-title":["Biometrical Journal"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/bimj.70173","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1002\/bimj.70173","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/bimj.70173","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,15]],"date-time":"2026-09-15T13:54:46Z","timestamp":1789480486000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/bimj.70173"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,15]]},"references-count":27,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2026,10]]}},"alternative-id":["10.1002\/bimj.70173"],"URL":"https:\/\/doi.org\/10.1002\/bimj.70173","archive":["Portico"],"relation":{},"ISSN":["0323-3847","1521-4036"],"issn-type":[{"value":"0323-3847","type":"print"},{"value":"1521-4036","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,15]]},"assertion":[{"value":"2026-04-15","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-07-31","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-09-15","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e70173"}}