{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,10]],"date-time":"2026-10-10T05:50:55Z","timestamp":1791611455755,"version":"4.3.4"},"reference-count":44,"publisher":"Wiley","license":[{"start":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T00:00:00Z","timestamp":1791504000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/"},{"start":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T00:00:00Z","timestamp":1791504000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["bpspsychub.onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Brit J Math &amp;amp; Statis"],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Amortized variational inference (AVI) has become a popular, computationally efficient alternative to marginal maximum likelihood estimation (MML) for estimating high\u2010dimensional item response theory models on large datasets. However, currently the AVI framework lacks methods to assess model fit and determine parameter uncertainty. Therefore, it would be desirable to have the well established, classical goodness\u2010of\u2010fit based modelling assessment tools from item response theory available to assess models fit with AVI. However, AVI optimizes a lower\u2010bound to the marginal likelihood, and the asymptotic properties that are required to use these tools are not guaranteed. In this paper, we use AVI estimates in a likelihood\u2010based Fisher scoring procedure, ensuring asymptotic guarantees associated with MML. Large\u2010sample theory for one\u2010step estimators shows that when the initial estimator is sufficiently accurate (within an  neighbourhood of the true parameter), a single Fisher scoring update is sufficient to obtain a consistent, and asymptotically efficient estimator. We investigate empirically whether one scoring step suffices to restore the validity of classical goodness\u2010of\u2010fit measures and standard error estimation. Using simulation studies and an application to large\u2010scale Narcissism Personality Inventory data, we show that the proposed AVI\u2010initialized one\u2010step Fisher scoring approach yields fit statistics and uncertainty estimates that are on par with MML while retaining the computational advantage of AVI.<\/jats:p>","DOI":"10.1111\/bmsp.70078","type":"journal-article","created":{"date-parts":[[2026,10,10]],"date-time":"2026-10-10T05:24:08Z","timestamp":1791609848000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Model fit and parameter uncertainty in amortized variational inference based item response theory"],"prefix":"10.1111","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-7554-2262","authenticated-orcid":false,"given":"K.","family":"Veldkamp","sequence":"first","affiliation":[{"name":"Department of Psychology University of Amsterdam  The Netherlands"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7458-1272","authenticated-orcid":false,"given":"R. P. P. P.","family":"Grasman","sequence":"additional","affiliation":[{"name":"Department of Psychology University of Amsterdam  The Netherlands"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8900-2106","authenticated-orcid":false,"given":"R.","family":"Debelak","sequence":"additional","affiliation":[{"name":"Department of Developmental and Educational Research Methods University of Zurich  Zurich Switzerland"},{"name":"MLO Lab, EPFL  Lausanne Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7168-3238","authenticated-orcid":false,"given":"D.","family":"Molenaar","sequence":"additional","affiliation":[{"name":"Department of Psychology University of Amsterdam  The Netherlands"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,10,9]]},"reference":[{"issue":"1","key":"e_1_2_12_2_1","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1177\/1073191110382845","article-title":"What does the narcissistic personality inventory really measure?","volume":"18","author":"Ackerman R. A.","year":"2011","journal-title":"Assessment"},{"key":"e_1_2_12_3_1","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1214\/aoms\/1177706538","article-title":"Maximum\u2010likelihood estimation of parameters subject to restraints","volume":"29","author":"Aitchison J.","year":"1958","journal-title":"The Annals of Mathematical Statistics"},{"issue":"2","key":"e_1_2_12_4_1","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1037\/0033-2909.107.2.238","article-title":"Comparative fit indexes in structural models","volume":"107","author":"Bentler P. M.","year":"1990","journal-title":"Psychological Bulletin"},{"issue":"518","key":"e_1_2_12_5_1","doi-asserted-by":"crossref","first-page":"859","DOI":"10.1080\/01621459.2017.1285773","article-title":"Variational inference: A review for statisticians","volume":"112","author":"Blei D. M.","year":"2017","journal-title":"Journal of the American Statistical Association"},{"issue":"2","key":"e_1_2_12_6_1","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/BF02291262","article-title":"Fitting a response model for n dichotomously scored items","volume":"35","author":"Bock D.","year":"1970","journal-title":"Psychometrika"},{"issue":"4","key":"e_1_2_12_7_1","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1007\/BF02293801","article-title":"Marginal maximum likelihood estimation of item parameters: Application of an em algorithm","volume":"46","author":"Bock R. D.","year":"1981","journal-title":"Psychometrika"},{"key":"e_1_2_12_8_1","doi-asserted-by":"crossref","DOI":"10.1002\/9781118619179","volume-title":"Structural equations with latent variables","author":"Bollen K. A.","year":"1989"},{"issue":"2","key":"e_1_2_12_9_1","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1177\/0049124192021002005","article-title":"Alternative ways of assessing model fit","volume":"21","author":"Browne M. W.","year":"1992","journal-title":"Sociological Methods & Research"},{"key":"e_1_2_12_10_1","unstructured":"Burda Y. Grosse R. &Salakhutdinov R.(2015).Importance weighted autoencoders.arXiv preprint arXiv:1509.00519."},{"issue":"3","key":"e_1_2_12_11_1","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1007\/s11121-021-01253-4","article-title":"Incremental model fit assessment in the case of categorical data: Tucker\u2013lewis index for item response theory modeling","volume":"24","author":"Cai L.","year":"2023","journal-title":"Prevention Science"},{"key":"e_1_2_12_12_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v048.i06","article-title":"mirt: A multidimensional item response theory package for the r environment","volume":"48","author":"Chalmers R. P.","year":"2012","journal-title":"Journal of Statistical Software"},{"issue":"3","key":"e_1_2_12_13_1","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1111\/bmsp.12127","article-title":"Numerical approximation of the observed information matrix with oakes' identity","volume":"71","author":"Chalmers R. P.","year":"2018","journal-title":"British Journal of Mathematical and Statistical Psychology"},{"key":"e_1_2_12_14_1","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1111\/bmsp.12219","article-title":"Gaussian variational estimation for multidimensional item response theory","volume":"74","author":"Cho A. E.","year":"2021","journal-title":"British Journal of Mathematical and Statistical Psychology"},{"issue":"4","key":"e_1_2_12_15_1","doi-asserted-by":"crossref","first-page":"417","DOI":"10.2307\/3001616","article-title":"Some methods for strengthening the common \u03c72 tests","volume":"10","author":"Cochran W. G.","year":"1954","journal-title":"Biometrics"},{"issue":"6","key":"e_1_2_12_16_1","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1080\/00223890802388590","article-title":"The factor structure of the narcissistic personality inventory","volume":"90","author":"Corry N.","year":"2008","journal-title":"Journal of Personality Assessment"},{"key":"e_1_2_12_17_1","first-page":"1","volume-title":"2019 international joint conference on neural networks","author":"Curi M.","year":"2019"},{"issue":"3","key":"e_1_2_12_18_1","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1207\/s15327752jpa4803_11","article-title":"Factor analysis and construct validity of the narcissistic personality inventory","volume":"48","author":"Emmons R. A.","year":"1984","journal-title":"Journal of Personality Assessment"},{"key":"e_1_2_12_19_1","first-page":"221","volume-title":"Proceedings of the fifth berkeley symposium on mathematical statistics and probability","author":"Huber P. J.","year":"1967"},{"key":"e_1_2_12_20_1","doi-asserted-by":"crossref","DOI":"10.4324\/9781315812274","volume-title":"Handbook of psychological testing","author":"Kline P.","year":"2013"},{"key":"e_1_2_12_21_1","volume-title":"Test equating: Methods and practices","author":"Kolen M. J.","year":"2013"},{"issue":"4","key":"e_1_2_12_22_1","doi-asserted-by":"crossref","first-page":"857","DOI":"10.1016\/S0191-8869(03)00158-2","article-title":"The narcissistic personality inventory: Factor structure in a non\u2010clinical sample","volume":"36","author":"Kubarych T. S.","year":"2004","journal-title":"Personality and Individual Differences"},{"key":"e_1_2_12_23_1","doi-asserted-by":"crossref","DOI":"10.3389\/fpsyg.2022.935419","article-title":"Estimating three\u2010and four\u2010parameter mirt models with importance\u2010weighted sampling enhanced variational auto\u2010encoder","volume":"13","author":"Liu T.","year":"2022","journal-title":"Frontiers in Psychology"},{"key":"e_1_2_12_24_1","doi-asserted-by":"crossref","unstructured":"Luo N. &Ji F.(2025).Generative adversarial networks for high\u2010dimensional item factor analysis: A deep adversarial learning algorithm.arXiv preprint arXiv:2502.10650.","DOI":"10.1017\/psy.2025.10059"},{"key":"e_1_2_12_25_1","first-page":"1","article-title":"A note on improving variational estimation for multidimensional item response theory","volume":"89","author":"Ma C.","year":"2023","journal-title":"Psychometrika"},{"issue":"3","key":"e_1_2_12_26_1","doi-asserted-by":"crossref","first-page":"490","DOI":"10.1037\/0033-2909.111.3.490","article-title":"Model modifications in covariance structure analysis: The problem of capitalization on chance","volume":"111","author":"MacCallum R. C.","year":"1992","journal-title":"Psychological Bulletin"},{"issue":"4","key":"e_1_2_12_27_1","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1007\/s11336-005-1295-9","article-title":"Limited information goodness\u2010of\u2010fit testing in multidimensional contingency tables","volume":"71","author":"Maydeu\u2010Olivares A.","year":"2006","journal-title":"Psychometrika"},{"issue":"4","key":"e_1_2_12_28_1","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1080\/00273171.2014.911075","article-title":"Assessing approximate fit in categorical data analysis","volume":"49","author":"Maydeu\u2010Olivares A.","year":"2014","journal-title":"Multivariate Behavioral Research"},{"issue":"1","key":"e_1_2_12_29_1","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1080\/00273171.2022.2163477","article-title":"Generalizability of dynamic fit index, equivalence testing, and hu & bentler cutoffs for evaluating fit in factor analysis","volume":"58","author":"McNeish D.","year":"2023","journal-title":"Multivariate Behavioral Research"},{"issue":"1","key":"e_1_2_12_30_1","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1037\/met0000425","article-title":"Dynamic fit index cutoffs for confirmatory factor analysis models","volume":"28","author":"McNeish D.","year":"2023","journal-title":"Psychological Methods"},{"key":"e_1_2_12_31_1","unstructured":"Muth\u00e9n L. K. &Muth\u00e9n B. O.(2017).Mplus user's guide (8th ed.) [Computer software manual]. Los Angeles CA. Version 8."},{"key":"e_1_2_12_32_1","unstructured":"Open\u2010Source Psychometrics Project. (2012).Open psychology data: Raw data from online personality tests.https:\/\/openpsychometrics.org\/_rawdata\/"},{"key":"e_1_2_12_33_1","volume-title":"Sparse factor autoencoders for item response theory","author":"Paaben B.","year":"2022"},{"key":"e_1_2_12_34_1","first-page":"8024","volume-title":"Advances in neural information processing systems 32","author":"Paszke A.","year":"2019"},{"issue":"5","key":"e_1_2_12_35_1","doi-asserted-by":"crossref","first-page":"890","DOI":"10.1037\/0022-3514.54.5.890","article-title":"A principal\u2010components analysis of the narcissistic personality inventory and further evidence of its construct validity","volume":"54","author":"Raskin R.","year":"1988","journal-title":"Journal of Personality and Social Psychology"},{"issue":"2","key":"e_1_2_12_36_1","doi-asserted-by":"crossref","first-page":"590","DOI":"10.2466\/pr0.1979.45.2.590","article-title":"A narcissistic personality inventory","volume":"45","author":"Raskin R. N.","year":"1979","journal-title":"Psychological Reports"},{"key":"e_1_2_12_37_1","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-89976-3","volume-title":"Multidimensional item response theory models","author":"Reckase M. D.","year":"2009"},{"issue":"1","key":"e_1_2_12_38_1","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1146\/annurev.clinpsy.032408.153553","article-title":"Item response theory and clinical measurement","volume":"5","author":"Reise S. P.","year":"2009","journal-title":"Annual Review of Clinical Psychology"},{"issue":"12","key":"e_1_2_12_39_1","doi-asserted-by":"crossref","first-page":"1442","DOI":"10.1037\/pas0000597","article-title":"Advances in applications of item response theory to clinical assessment","volume":"31","author":"Thomas M. L.","year":"2019","journal-title":"Psychological Assessment"},{"issue":"1","key":"e_1_2_12_40_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11336-021-09748-3","article-title":"A deep learning algorithm for high\u2010dimensional exploratory item factor analysis","volume":"86","author":"Urban C. J.","year":"2021","journal-title":"Psychometrika"},{"key":"e_1_2_12_41_1","doi-asserted-by":"crossref","DOI":"10.1201\/9781315374512","volume-title":"Handbook of item response theory","author":"Van der Linden W. J.","year":"2016"},{"key":"e_1_2_12_42_1","volume-title":"Asymptotic statistics","author":"Van der Vaart A. W.","year":"2000"},{"issue":"1","key":"e_1_2_12_43_1","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1111\/bmsp.12363","article-title":"Handling missing data in variational autoencoder based item response theory","volume":"78","author":"Veldkamp K.","year":"2025","journal-title":"British Journal of Mathematical and Statistical Psychology"},{"key":"e_1_2_12_44_1","doi-asserted-by":"crossref","DOI":"10.4324\/9781410605931","volume-title":"Computerized adaptive testing: A primer","author":"Wainer H.","year":"2000"},{"key":"e_1_2_12_45_1","first-page":"337","volume-title":"Advances in computerized scoring of complex item formats","author":"Zenisky A. L.","year":"2014"}],"container-title":["British Journal of Mathematical and Statistical Psychology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/bpspsychub.onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/bmsp.70078","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/bpspsychub.onlinelibrary.wiley.com\/doi\/full-xml\/10.1111\/bmsp.70078","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/bpspsychub.onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/bmsp.70078","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,10]],"date-time":"2026-10-10T05:24:14Z","timestamp":1791609854000},"score":1,"resource":{"primary":{"URL":"https:\/\/bpspsychub.onlinelibrary.wiley.com\/doi\/10.1111\/bmsp.70078"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10,9]]},"references-count":44,"alternative-id":["10.1111\/bmsp.70078"],"URL":"https:\/\/doi.org\/10.1111\/bmsp.70078","archive":["Portico"],"relation":{},"ISSN":["0007-1102","2044-8317"],"issn-type":[{"value":"0007-1102","type":"print"},{"value":"2044-8317","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,10,9]]},"assertion":[{"value":"2026-01-19","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-09-16","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-10-09","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"bmsp.70078"}}