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Such work often proceeds in stages, where the results of an analysis from an earlier stage are fed into the analysis at a later stage. However, common practices in single\u2010stage and multistage estimation have weaknesses, including when viewed from a Bayesian perspective. This work extends recent developments in 2\u2010Stage Bayesian approaches in structural equation modelling (SEM) to advance a general multistage approach where models are viewed as comprised of fragments that can be assembled in a modular way. The proposed approach is more in line with Bayesian principles and offers advantages over existing approaches. The approach is illustrated by applications in several different modelling scenarios, including: SEM for a wider class of situations than existing approaches have considered; and calibration and scoring situations encountered in operational assessment using item response theory (IRT). R functions for executing these analyses by interfacing with M\n                    <jats:italic>plus<\/jats:italic>\n                    are provided and documented, as is code for running the examples.\n                  <\/jats:p>","DOI":"10.1111\/bmsp.70054","type":"journal-article","created":{"date-parts":[[2026,5,23]],"date-time":"2026-05-23T06:53:20Z","timestamp":1779519200000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["<scp>Modular<\/scp>\n                    item response and structural equation modelling via measurement and uncertainty preserving parametric modelling"],"prefix":"10.1111","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7737-9176","authenticated-orcid":false,"given":"Roy","family":"Levy","sequence":"first","affiliation":[{"name":"Arizona State University  Tempe Arizona USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,5,22]]},"reference":[{"key":"e_1_2_12_2_1","unstructured":"AERA APA & NCME. 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