Metric Transformations and the Filtered Monotonic Polynomial Item Response Model

• Published in: Psychometrika 2019-03, Vol.84 (1), p.105-123
• Main Author: Feuerstahler, Leah M.
• Format: Article
• Language: English
• Subjects: Adolescent; Assessment; Behavioral Science and Psychology; Humanities; Humans; Item response theory; Law; Male; Models, Statistical; Nonlinear Dynamics; Nonparametric statistics; Personality Tests; Polynomials; Psychology; Psychometrics; Psychometrics - methods; Quantitative psychology; Self Concept; Statistical Theory and Methods; Statistics for Social Sciences; Statistics, Nonparametric; Testing and Evaluation; True Scores
• ISSN: 0033-3123; 1860-0980
• DOI: 10.1007/s11336-018-9642-9

The θ 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 & Browne in J Educ Behav Stat 40:5–34, 2015 ), can be used to specify item response models on metrics other than the θ 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 θ 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.