Identifiability of the latent attribute space and conditions of Q-matrix completeness for attribute hierarchy models

Hans Friedrich Köhn, Chia Yi Chiu

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations


Educational researchers have argued that a realistic view of the role of attributes in cognitively diagnostic modeling should account for the possibility that attributes are not isolated entities, but interdependent in their effect on test performance. Different approaches have been discussed in the literature; among them the proposition to impose a hierarchical structure so that mastery of one or more attributes is a prerequisite of mastering one or more other attributes. A hierarchical organization of attributes constrains the latent attribute space such that several proficiency classes, as they exist if attributes are not hierarchically organized, are no longer defined, because the corresponding attribute combinations cannot occur with the given attribute hierarchy. Hence, the identification of the latent attribute space is often difficult—especially, if the number of attributes is large. As an additional complication, constructing a complete Q-matrix may not at all be straightforward if the attributes underlying the test items are supposed to have a hierarchical structure. In this article, the conditions of identifiability of the latent space if attributes are hierarchically organized and the conditions of completeness of the Q-matrix are studied.

Original languageEnglish (US)
Title of host publicationQuantitative Psychology - The 82nd Annual Meeting of the Psychometric Society, Zurich, Switzerland, 2017
EditorsJorge Gonzalez, Rianne Janssen, Marie Wiberg, Dylan Molenaar, Steven Culpepper
PublisherSpringer New York LLC
Number of pages13
ISBN (Print)9783319772486
StatePublished - 2018
Externally publishedYes
Event82nd Annual meeting of the Psychometric Society, 2017 - Zurich, Switzerland
Duration: Jul 17 2017Jul 21 2017

Publication series

NameSpringer Proceedings in Mathematics and Statistics
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017


Conference82nd Annual meeting of the Psychometric Society, 2017

Bibliographical note

Publisher Copyright:
© Springer International Publishing AG, part of Springer Nature 2018.


  • Attribute hierarchy
  • Cognitive diagnosis
  • Completeness
  • DINA model
  • Latent attribute space
  • Q-matrix


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