Abstract
Diagnostic classification models in educational measurement describe ability in a knowledge domain as a composite of specific binary skills called “cognitive attributes,” each of which an examinee may or may not have mastered. Attribute Hierarchy Models (AHMs) account for the possibility that attributes are dependent by imposing a hierarchical structure such that mastery of one or more attributes is a prerequisite of mastering one or more other attributes. Thus, the number of meaningfully defined attribute combinations is reduced, so that constructing a complete Q-matrix may be challenging. (The Q-matrix of a cognitively diagnostic test documents which attributes are required for solving which item; the Q-matrix is said to be complete if it guarantees the identifiability of all realizable proficiency classes among examinees.) For structured Q-matrices (i.e., the item attribute profiles are restricted to reflect the hierarchy postulated to underlie the attributes), the conditions of completeness have been established. However, sometimes, a structured Q-matrix cannot be assembled because the items of the test in question have attribute profiles that do not conform to the prerequisite structure imposed by the postulated attribute hierarchy. A Q-matrix composed of such items is called “unstructured.” In this article, the completeness conditions of unstructured Q-matrices for the DINA model are presented. Specifically, there exists an entire range of Q-matrices that are all complete for DINA-AHMs. Thus, the theoretical results presented here can be combined with extant insights about Q-completeness for models without attribute hierarchies into a unified framework on the completeness of Q-matrices for the DINA model.
Similar content being viewed by others
References
Cai, Y., Tu, D., & Ding, S. (2018). Theorems and methods of a complete Q matrix with attribute hierarchies under restricted Q-matrix design. Frontiers in Psychology, 9, Article 1413.
Chen, Y., Culpepper, S.A., Chen, Y., & Douglas, J. (2018). Bayesian estimation of DINA Q matrix. Psychometrika, 83, 89–108.
Chen, Y., Liu, J., Xu, G., & Ying, Z. (2015). Statistical analysis of Q-matrix based diagnostic classification models. Journal of the American Statistical Association, 110, 850–866.
Chiu, C.-Y. (2013). Statistical refinement of the Q-matrix in cognitive diagnosis. Applied Psychological Measurement, 37, 598–618.
Chiu, C.-Y., Douglas, J., & Li, X. (2009). Cluster analysis for cognitive diagnosis: Theory and applications. Psychometrika, 74, 633–665.
Chiu, C.-Y., & Köhn, H.-F. (2015). Consistency of cluster analysis for cognitive diagnosis: The DINO model and the DINA model revisited. Applied Psychological Measurement, 39, 465–479.
Chiu, C.-Y., & Köhn, H.-F. (2016). The Reduced RUM as a logit model:, Parameterization and constraints. Psychometrika, 81, 350–370.
Chung, M. (2014). Estimating the Q-matrix for Cognitive Diagnosis Models in a Bayesian Framework. Doctoral dissertation. Columbia University.
Chung, M. (2019). A Gibbs sampling algorithm that estimates the Q-matrix for the DINA model. Journal of Mathematical Psychology, 93, 1–8.
DeCarlo, L.T. (2012). Recognizing uncertainty in the Q-matrix via a Bayesian extension of the DINA model. Applied Psychological Measurement, 36, 447–468.
de la Torre, J. (2008). An empirically based method of Q-matrix validation for the DINA model: Development and applications. Journal of Educational Measurement, 45, 343–362.
de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76, 179–199.
de la Torre, J., & Chiu, C.-Y. (2016). A general method of empirical Q-matrix validation. Psychometrika, 81, 253–73.
de la Torre, J., & Douglas, J.A. (2004). Higher-order latent trait models for cognitive diagnosis. Psychometrika, 69, 333–353.
de la Torre, J., Hong, Y., & Deng, W. (2010). Factors affecting the item parameter estimation and classification accuracy of the DINA model. Journal of Educational Measurement, 47, 227–249.
DiBello, L.V., Roussos, L.A., & Stout, W.F. (2007). Review of cognitively diagnostic assessment and a summary of psychometric models. In Rao, C.R., & Sinharay, S. (Eds.) Handbook of Statistics: Vol.26. Psychometrics (pp. 979–1030). Amsterdam: Elsevier.
Fu, J., & Li, Y. (2007). An integrative review of cognitively diagnostic psychometric models. Chicago: Paper presented at the Annual Meeting of the National Council on Measurement in Education.
Gierl, M. (2007). Making diagnostic inferences about cognitive attributes using the rule-space model and attribute hierarchy method. Journal of Educational Measurement, 44, 325–340.
Gross, J., & George, A.C. (2014). On prerequisite relations between attributes in noncompensatory diagnostic classification. Methodology, 10(3), 100–107.
Gu, Y., & Xu, G. (2020). Partial identifiability of restricted latent class models. Annals of Statistics, 48(4), 2082–2107.
Gu, Y., & Xu, G. (2019a). The sufficient and necessary condition for the identifiability and estimability of the DINA model. Psychometrika, 84, 468–483.
Gu, Y., & Xu, G. (2019b). Identifiability of hierarchical latent attribute models. https://arxiv.org/pdf/1906.07869.pdf.
Haberman, S.J., & von Davier, M. (2007). Some notes on models for cognitively based skill diagnosis. In Rao, C.R., & Sinharay, S. (Eds.) Handbook of Statistics: Vol.26. Psychometrics (pp. 1031–1038). Amsterdam: Elsevier.
Haertel, E.H. (1989). Using restricted latent class models to map the skill structure of achievement items. Journal of Educational Measurement, 26, 333–352.
Haertel, E.H., & Wiley, D.E. (1994). Representation of ability structure: Implications for testing. In Fredriksen, N., Mislevy, R., & Bejar, I. (Eds.) Testing theory for a new generation of tests (pp. 359–384). Hillsdale: Erlbaum.
Hartz, S.M. (2002). A Bayesian framework for the Unified Model for assessing cognitive abilities: Blending theory with practicality (Doctoral dissertation). Available from ProQuest Dissertations and Theses database. (UMI No.3044108).
Hartz, S.M., & Roussos, L.A. (2008). The Fusion Model for skill diagnosis: Blending theory with practicality. (Research report No. RR-08-71). Princeton: Educational Testing Service.
Henson, R.A., Templin, J.L., & Willse, J.T. (2009). Defining a family of cognitive diagnosis models using log-linear models with latent variables. Psychometrika, 74, 191–210.
Junker, B.W., & Sijtsma, K. (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory. Applied Psychological Measurement, 25, 258–272.
Köhn, H.-F., & Chiu, C.-Y. (2017). A procedure for assessing completeness of the Q-matrix of cognitive diagnosis models. Psychometrika, 82, 112–132.
Köhn, H.F., & Chiu, C.-Y. (2019). Attribute hierarchy models in cognitive diagnosis: identfiability of the latent attribute space and conditions for completeness of the Q-matrix. Journal of Classfication, 36, 541–565.
Leighton, J., & Gierl, M. (2007). Cognitive diagnostic assessment for education: Theory and applications. Cambridge: Cambridge University Press.
Leighton, J.P., Gierl, M.J., & Hunka, S. (2004). The attribute hierarchy model: An approach for integrating cognitive theory with assessment practice. Journal of Educational Measurement, 41, 205–236.
Liu, C.-W., Andersson, B., & Skrondal, A. (2020). A constrained Metropolis-Hastings Robbins-Monro algorithm for Q matrix estimation in DINA models. Psychometrika, 85, 322–357.
Liu, R., Huggins-Manley, A.C., & Bradshaw, L. (2016). The impact of q-matrix designs on diagnostic classification accuracy in the presence of attribute hierarchies. Educational and Psychological Measurement, 76, 220–240.
Liu, J., Xu, G., & Ying, Z. (2012). Data-driven learning of Q-matrix. Applied Psychological Measurement, 36, 548–564.
Liu, J., Xu, G., & Ying, Z. (2013). Theory of the self-learning Q-matrix. Bernoulli, 19, 1790–1817.
Macready, G.B., & Dayton, C.M. (1977). The use of probabilistic models in the assessment of mastery. Journal of Educational Statistics, 33, 379–416.
Maris, E. (1999). Estimating multiple classification latent class models. Psychometrika, 64, 187–212.
Nichols, P.D., Chipman, S.E., & Brennan, R.L. (1995). Cognitively diagnostic assessment. Mahwah: Erlbaum.
Rupp, A.A., & Templin, J.L. (2008). Unique characteristics of diagnostic classification models: A comprehensive review of the current state-of-the-art. Measurement, 6, 219–262.
Rupp, A.A., Templin, J.L., & Henson, R.A. (2010). Diagnostic Measurement. Theory, Methods, and Applications. New York: Guilford.
Sessoms, J., & Henson, R.A. (2018). Applications of Diagnostic Classification Models: A literature review and critical commentary. Measurement: Interdisciplinary Research and Perspectives, 16(1), 1–17.
Tatsuoka, K.K. (1985). A probabilistic model for diagnosing misconception in the pattern classification approach. Journal of Educational and Behavioral Statistics, 12, 55–73.
Tatsuoka, K.K. (1990). Toward an integration of item-response theory and cognitive error diagnosis. In Frederiksen, N., Glaser, R., Lesgold, A., & Shafto, M. (Eds.) Diagnostic monitoring of skill and knowledge acquisition (pp. 453–488). Erlbaum: Hillsdale.
Tatsuoka, K.K. (2009). Cognitive assessment. An introduction to the Rule Space Method. New York: Routledge/Taylor & Francis.
Templin, J., & Bradshaw, L. (2014). Hierarchical diagnostic classification models: A family of models for estimating and testing attribute hierarchies. Psychometrika, 79, 317–339.
von Davier, M. (2005). A general diagnostic model applied to language testing data (Research report No. RR-05-16). Princeton, NJ: Educational Testing Service.
von Davier, M. (2008). A general diagnostic model applied to language testing data. British Journal of Mathematical and Statistical Psychology, 61, 287–301.
Zhang, S. S., DeCarlo, L. T., & Ying, Z. (2013). Non-identifiability, equivalence classes, and attribute-specific classification in Q-matrix based cognitive diagnosis models. preprint, arXiv:1303.0426.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Köhn, H., Chiu, CY. A Unified Theory of the Completeness of Q-Matrices for the DINA Model. J Classif 38, 500–518 (2021). https://doi.org/10.1007/s00357-021-09384-7
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00357-021-09384-7