## Nonparametric item response theory axioms and properties under nonlinearity and their exemplification with knowledge space theory

- This paper investigates the dichotomous Mokken nonparametric item response theory (IRT) axioms and properties under incomparabilities among latent trait values and items. Generalized equivalents of the unidimensional nonparametric IRT axioms and properties are formulated for nonlinear (quasi-ordered) person and indicator spaces. It is shown that monotone likelihood ratio (MLR) for the total score variable and nonlinear latent trait implies stochastic ordering (SO) of the total score variable, but may fail to imply SO of the nonlinear latent trait. The reason for this and conditions under which the implication holds are specified, based on a new, simpler proof of the fact that in the unidimensional case MLR implies SO. The approach is applied in knowledge space theory (KST), a combinatorial test theory. This leads to a (tentative) Mokken-type nonparametric axiomatization in the currently parametric theory of knowledge spaces. The nonparametric axiomatization is compared with theThis paper investigates the dichotomous Mokken nonparametric item response theory (IRT) axioms and properties under incomparabilities among latent trait values and items. Generalized equivalents of the unidimensional nonparametric IRT axioms and properties are formulated for nonlinear (quasi-ordered) person and indicator spaces. It is shown that monotone likelihood ratio (MLR) for the total score variable and nonlinear latent trait implies stochastic ordering (SO) of the total score variable, but may fail to imply SO of the nonlinear latent trait. The reason for this and conditions under which the implication holds are specified, based on a new, simpler proof of the fact that in the unidimensional case MLR implies SO. The approach is applied in knowledge space theory (KST), a combinatorial test theory. This leads to a (tentative) Mokken-type nonparametric axiomatization in the currently parametric theory of knowledge spaces. The nonparametric axiomatization is compared with the assumptions of the parametric basic local independence model which is fundamental in KST. It is concluded that this paper may provide a first step toward a basis for a possible fusion of the two split directions of psychological test theories IRT and KST.…

Author: | Ali ÜnlüGND |
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URN: | urn:nbn:de:bvb:384-opus4-4562 |

Frontdoor URL | https://opus.bibliothek.uni-augsburg.de/opus4/563 |

Series (Serial Number): | Preprints des Instituts für Mathematik der Universität Augsburg (2007-32) |

Type: | Preprint |

Language: | English |

Publishing Institution: | Universität Augsburg |

Release Date: | 2007/07/24 |

Tag: | Nichtparametrische Item-Response-Theorie; Monotone-Likelihood-Verhältnis; Parametrische und Nichtparametrische Wissensraumtheorienonparametric item response theory; monotone likelihood ratio; parametric and nonparametric knowledge space theory |

GND-Keyword: | Probabilistische Testtheorie; Nichtparametrisches Verfahren; Likelihood-Quotient; Wissensrepräsentation |

Institutes: | Mathematisch-Naturwissenschaftlich-Technische Fakultät |

Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik | |

Dewey Decimal Classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |