Conditions of completeness of the Q-matrix of tests for cognitive diagnosis

Hans Friedrich Köhn, Chia Yi Chiu

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


The associations between the items of a test based on the cognitive diagnosis framework and the skills required to solve them are documented in the Q-matrix. If the items have skill profiles that allow for the identification of all possible proficiency classes among examinees, then the Q-matrix of the test is said to be complete. An incomplete Q-matrix causes examinees to be assigned to proficiency classes to which they do not belong. Thus, completeness of the Q-matrix is an integral requirement of any cognitively diagnostic test. However, completeness of the Q-matrix is often difficult to establish, especially, for tests with a large number of items involving multiple skills. As an additional complication, completeness is not an intrinsic property of the Q-matrix, but can only be assessed in reference to a specific diagnostic classification model (DCM) supposed to underlie the data—that is, the Q-matrix of a given test can be complete for one model but incomplete for another. For different types of DCMs, conditions of Q-completeness are studied. Rules are derived to determine the completeness of a given Q-matrix.

Original languageEnglish (US)
Title of host publicationQuantitative Psychology Research - The 80th Annual Meeting of the Psychometric Society
EditorsWen-Chung Wang, Jeffrey A. Douglas, Marie Wiberg, L. Andries van der Ark, Daniel M. Bolt
Number of pages10
ISBN (Print)9783319387574
StatePublished - 2016
Event80th annual meeting of the Psychometric Society, 2015 - Beijing, China
Duration: Jul 12 2015Jul 16 2015

Publication series

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


Other80th annual meeting of the Psychometric Society, 2015


  • Cognitive diagnosis
  • Diagnostic classification models
  • Identifiability
  • Q-Matrix completeness

ASJC Scopus subject areas

  • General Mathematics


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