{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T21:50:02Z","timestamp":1790805002337,"version":"4.1.0"},"reference-count":38,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"unspecified","delay-in-days":2119,"URL":"https:\/\/www.cambridge.org\/core\/terms"},{"start":{"date-parts":[[2019,3,15]],"date-time":"2019-03-15T00:00:00Z","timestamp":1552608000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Psychometrika"],"published-print":{"date-parts":[[2019,3,15]]},"abstract":"<jats:p>The <jats:inline-formula><jats:alternatives><mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\"><mml:mi>\u03b8<\/mml:mi><\/mml:math><jats:tex-math>\\documentclass[12pt]{minimal}\n\\usepackage{amsmath}\n\\usepackage{wasysym}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsbsy}\n\\usepackage{mathrsfs}\n\\usepackage{upgreek}\n\\setlength{\\oddsidemargin}{-69pt}\n\\begin{document}$$\\theta $$\\end{document}<\/jats:tex-math><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S003331230000675X_IEq1.png\"\/><\/jats:alternatives><\/jats:inline-formula> metric in item response theory is often not the most useful metric for score reporting or interpretation. In this paper, I demonstrate that the filtered monotonic polynomial (FMP) item response model, a recently proposed nonparametric item response model (Liang &amp; Browne in J Educ Behav Stat 40:5\u201334, 2015), can be used to specify item response models on metrics other than the <jats:inline-formula><jats:alternatives><mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\"><mml:mi>\u03b8<\/mml:mi><\/mml:math><jats:tex-math>\\documentclass[12pt]{minimal}\n\\usepackage{amsmath}\n\\usepackage{wasysym}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsbsy}\n\\usepackage{mathrsfs}\n\\usepackage{upgreek}\n\\setlength{\\oddsidemargin}{-69pt}\n\\begin{document}$$\\theta $$\\end{document}<\/jats:tex-math><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S003331230000675X_IEq2.png\"\/><\/jats:alternatives><\/jats:inline-formula> metric. Specifically, I demonstrate that any item response function (IRF) defined within the FMP framework can be re-expressed as another FMP IRF by taking monotonic transformations of the latent trait. I derive the item parameter transformations that correspond to both linear and nonlinear transformations of the latent trait metric. These item parameter transformations can be used to define an item response model based on any monotonic transformation of the <jats:inline-formula><jats:alternatives><mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\"><mml:mi>\u03b8<\/mml:mi><\/mml:math><jats:tex-math>\\documentclass[12pt]{minimal}\n\\usepackage{amsmath}\n\\usepackage{wasysym}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsbsy}\n\\usepackage{mathrsfs}\n\\usepackage{upgreek}\n\\setlength{\\oddsidemargin}{-69pt}\n\\begin{document}$$\\theta $$\\end{document}<\/jats:tex-math><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S003331230000675X_IEq3.png\"\/><\/jats:alternatives><\/jats:inline-formula> metric, so long as the metric transformation is approximated by a monotonic polynomial. I demonstrate this result by defining an item response model directly on the approximate true score metric and discuss the implications of metric transformations for applied testing situations.<\/jats:p>","DOI":"10.1007\/s11336-018-9642-9","type":"journal-article","created":{"date-parts":[[2018,11,9]],"date-time":"2018-11-09T20:01:06Z","timestamp":1541793666000},"page":"105-123","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Metric Transformations and the Filtered Monotonic Polynomial Item Response Model"],"prefix":"10.1017","volume":"84","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7001-8519","authenticated-orcid":false,"given":"Leah M.","family":"Feuerstahler","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2025,1,1]]},"reference":[{"key":"S003331230000675X_CR12","doi-asserted-by":"publisher","DOI":"10.1111\/j.1745-3984.1988.tb00295.x"},{"key":"S003331230000675X_CR3","volume-title":"MMPI-A (Minnesota Multiphasic Personality Inventory-Adolescent): Manual for administration, scoring, and interpretation","author":"Butcher","year":"1992"},{"key":"S003331230000675X_CR38","first-page":"205","article-title":"Statistical and psychometric issues in the measurement of educational achievement trends: Examples from the National Assessment of Educational Progress","volume":"17","author":"Zwick","year":"1992","journal-title":"Journal of Educational Statistics"},{"key":"S003331230000675X_CR1","first-page":"267","volume-title":"Second international symposium on information theory","author":"Akaike","year":"1973"},{"key":"S003331230000675X_CR26","doi-asserted-by":"publisher","DOI":"10.1007\/BF02294480"},{"key":"S003331230000675X_CR30","doi-asserted-by":"publisher","DOI":"10.1126\/science.103.2684.677"},{"key":"S003331230000675X_CR25","doi-asserted-by":"publisher","DOI":"10.3102\/1076998616680841"},{"key":"S003331230000675X_CR7","doi-asserted-by":"publisher","DOI":"10.1007\/s11336-014-9428-7"},{"key":"S003331230000675X_CR32","doi-asserted-by":"publisher","DOI":"10.1177\/014662168300700208"},{"key":"S003331230000675X_CR20","doi-asserted-by":"publisher","DOI":"10.1515\/9783110813203"},{"key":"S003331230000675X_CR11","first-page":"233","article-title":"Fitting monotonic polynomials to data","volume":"9","author":"Hawkins","year":"1994","journal-title":"Computational Statistics"},{"key":"S003331230000675X_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/BF02291567"},{"key":"S003331230000675X_CR15","volume-title":"A semi-parametric approach to estimating item response functions (Unpublished doctoral dissertation)","author":"Liang","year":"2007"},{"key":"S003331230000675X_CR35","doi-asserted-by":"publisher","DOI":"10.1007\/s11336-015-9469-6"},{"key":"S003331230000675X_CR28","doi-asserted-by":"publisher","DOI":"10.1037\/1082-989X.8.2.164"},{"key":"S003331230000675X_CR10","doi-asserted-by":"publisher","DOI":"10.4992\/psycholres1954.22.144"},{"key":"S003331230000675X_CR17","doi-asserted-by":"publisher","DOI":"10.1007\/BF02291708"},{"key":"S003331230000675X_CR5","volume-title":"A method of distribution and density estimation (Unpublished dissertation)","author":"Elphinstone","year":"1985"},{"key":"S003331230000675X_CR29","doi-asserted-by":"publisher","DOI":"10.1111\/j.1745-3984.1997.tb00521.x"},{"key":"S003331230000675X_CR22","doi-asserted-by":"publisher","DOI":"10.1080\/00949655.2016.1139582"},{"key":"S003331230000675X_CR16","doi-asserted-by":"publisher","DOI":"10.3102\/1076998614556816"},{"key":"S003331230000675X_CR37","doi-asserted-by":"publisher","DOI":"10.1111\/j.1745-3984.2001.tb01127.x"},{"key":"S003331230000675X_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/s00180-012-0390-5"},{"key":"S003331230000675X_CR2","doi-asserted-by":"publisher","DOI":"10.1111\/j.1745-3984.1997.tb00515.x"},{"key":"S003331230000675X_CR4","doi-asserted-by":"publisher","DOI":"10.1080\/03610928308828450"},{"key":"S003331230000675X_CR33","doi-asserted-by":"publisher","DOI":"10.1007\/BF02294081"},{"key":"S003331230000675X_CR9","doi-asserted-by":"crossref","unstructured":"Feuerstahler, L. 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