{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,22]],"date-time":"2026-10-22T20:47:51Z","timestamp":1792702071246,"version":"4.5.0"},"reference-count":41,"publisher":"Wiley","issue":"3-4","license":[{"start":{"date-parts":[[2025,1,24]],"date-time":"2025-01-24T00:00:00Z","timestamp":1737676800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Statistics in Medicine"],"published-print":{"date-parts":[[2025,2,10]]},"abstract":"<jats:title>ABSTRACT<\/jats:title><jats:p>The recent U.S. Food and Drug Administration guidance on complex innovative trial designs acknowledges the use of Bayesian strategies to incorporate historical information based on clinical expertise and data similarity. Also, data from multiple previous studies with similar settings often qualify for historical borrowing. Although several classes of informative priors can semi\u2010automatically leverage historical information based on data compatibility, it is common that some exogenous factors, such as the year of patient enrollment, can also influence the relevance of each historical study to the current trial. Consequently, a natural a priori ordering among historical trials often arises, a constraint that many current informative priors fail to accommodate. Motivated by a pediatric lupus clinical study and an oncology trial, we introduce a variant of the power prior, named the <jats:italic>ordered normalized power prior<\/jats:italic>, which ensures a targeted order restriction on the power parameters and maintains data\u2010adaptive borrowing. We further explore and compare two distinct normalization strategies and outline computational details with efficient sampling algorithms. The clinical datasets mentioned are analyzed, and extensive simulations are conducted for comparison. An efficient implementation is provided in our updated package <jats:styled-content>NPP<\/jats:styled-content> available on the Comprehensive R Archive Network.<\/jats:p>","DOI":"10.1002\/sim.10302","type":"journal-article","created":{"date-parts":[[2025,1,25]],"date-time":"2025-01-25T06:35:12Z","timestamp":1737786912000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Bayesian Borrowing With Multiple Heterogeneous Historical Studies Using Order Restricted Normalized Power Prior"],"prefix":"10.1002","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3715-8931","authenticated-orcid":false,"given":"Zifei","family":"Han","sequence":"first","affiliation":[{"name":"School of Statistics University of International Business and Economics  Beijing China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qiang","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Statistics University of International Business and Economics  Beijing China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ram","family":"Tiwari","sequence":"additional","affiliation":[{"name":"Global Stat Solutions  Reston Virginia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-2079-7647","authenticated-orcid":false,"given":"Tianyu","family":"Bai","sequence":"additional","affiliation":[{"name":"U.S. Food and Drug Administration  Silver Spring Maryland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2025,1,24]]},"reference":[{"key":"e_1_2_10_2_1","unstructured":"U.S. Food and Drug Administration \u201cGuidance for Industry: Interacting With the FDA on Complex Innovative Trial Designs for Drugs and Biological Products \u201d 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