A note on monotonicity of item response functions for ordered polytomous item response theory models

Hyeon Ah Kang, Ya Hui Su, Hua Hua Chang

Research output: Contribution to journalArticlepeer-review


A monotone relationship between a true score (τ) and a latent trait level (θ) has been a key assumption for many psychometric applications. The monotonicity property in dichotomous response models is evident as a result of a transformation via a test characteristic curve. Monotonicity in polytomous models, in contrast, is not immediately obvious because item response functions are determined by a set of response category curves, which are conceivably non-monotonic in θ. The purpose of the present note is to demonstrate strict monotonicity in ordered polytomous item response models. Five models that are widely used in operational assessments are considered for proof: the generalized partial credit model (Muraki, 1992, Applied Psychological Measurement, 16, 159), the nominal model (Bock, 1972, Psychometrika, 37, 29), the partial credit model (Masters, 1982, Psychometrika, 47, 147), the rating scale model (Andrich, 1978, Psychometrika, 43, 561), and the graded response model (Samejima, 1972, A general model for free-response data (Psychometric Monograph no. 18). Psychometric Society, Richmond). The study asserts that the item response functions in these models strictly increase in θ and thus there exists strict monotonicity between τ and θ under certain specified conditions. This conclusion validates the practice of customarily using τ in place of θ in applied settings and provides theoretical grounds for one-to-one transformations between the two scales.

Original languageEnglish (US)
Pages (from-to)523-535
Number of pages13
JournalBritish Journal of Mathematical and Statistical Psychology
Issue number3
StatePublished - Nov 2018


  • item response theory
  • monotonicity
  • polytomous response models
  • true score

ASJC Scopus subject areas

  • Statistics and Probability
  • Arts and Humanities (miscellaneous)
  • General Psychology


Dive into the research topics of 'A note on monotonicity of item response functions for ordered polytomous item response theory models'. Together they form a unique fingerprint.

Cite this