Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2025-01-05T20:27:52.146Z Has data issue: false hasContentIssue false
  • English
  • Français

Identifiability of Diagnostic Classification Models

Published online by Cambridge University Press:  01 January 2025

Gongjun Xu  and
Stephanie Zhang
Gongjun Xu*
Affiliation:
University of Minnesota
Stephanie Zhang
Affiliation:
Google Inc.
*
Correspondence should be made to Gongjun Xu, School of Statistics, University of Minnesota, 313 Ford Hall, 224 Church St SE, Minneapolis, MN 55455 USA. Email: xuxxx360@umn.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Diagnostic classification models (DCMs) are important statistical tools in cognitive diagnosis. In this paper, we consider the issue of their identifiability. In particular, we focus on one basic and popular model, the DINA model. We propose sufficient and necessary conditions under which the model parameters are identifiable from the data. The consequences, in terms of the consistency of parameter estimates, of fulfilling or failing to fulfill these conditions are illustrated via simulation. The results can be easily extended to the DINO model through the duality of the DINA and DINO models. Moreover, the proposed theoretical framework could be applied to study the identifiability issue of other DCMs.

Keywords

diagnostic classification modelsthe DINA modelmodel identifiabilityQ-matrix
Type
Original Paper
Information
Psychometrika , Volume 81 , Issue 3 , September 2016 , pp. 625 - 649
Copyright
Copyright © 2015 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bechger, T. M., Verstralen, H. H., & Verhelst, N. D., (2002). Equivalent linear logistic test models. Psychometrika, 67 1123–136CrossRefGoogle Scholar
Chen, Y., Liu, J., Xu, G., & Ying, Z. (2014). Statistical analysis of q-matrix based diagnostic classification models. Journal of the American Statistical Association, (to appear).Google Scholar
Chiu, C-Y (2013). Statistical refinement of the Q-matrix in cognitive diagnosis. Applied Psychological Measurement, 37 598–618CrossRefGoogle Scholar
Chiu, C-YDouglas, J. A., & Li, X. (2009). Cluster analysis for cognitive diagnosis: Theory and applications. Psychometrika, 74 633–665CrossRefGoogle Scholar
de la Torre, J. (2008). An empirically-based method of Q-matrix validation for the DINA model: Development and applications. Journal of Educational Measurement, 45 343–362CrossRefGoogle Scholar
de la Torre, J. (2009). DINA model and parameter estimation: A didactic. Journal of Educational and Behavioral Statistics, 34 115–130CrossRefGoogle Scholar
de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76 179–199CrossRefGoogle Scholar
de la Torre, J., & Douglas, J. A., (2004). Higher order latent trait models for cognitive diagnosis. Psychometrika, 69 333–35310.1007/BF02295640CrossRefGoogle Scholar
DeCarlo, L. T., (2011). On the analysis of fraction subtraction data: The DINA model, classification, class sizes, and the Q-matrix. Applied Psychological Measurement, 35 8–26CrossRefGoogle Scholar
DeCarlo, L. T., (2012). Recognizing uncertainty in the Q-matrix via a bayesian extension of the DINA model. Applied Psychological Measurement, 36 6447–468CrossRefGoogle Scholar
DiBello, L. V., Stout, W. F., Roussos, L. A.,Nichols, P. D., Chipman, S. F., & Brennan, R. L., (1995). Unified cognitive psychometric diagnostic assessment likelihood-based classification techniques. Cognitively diagnostic assessment, Hillsdale, NJErlbaum Associates 361–390Google Scholar
Gabrielsen, A. (1978). Consistency and identifiability. Journal of Econometrics, 8 2261–263CrossRefGoogle Scholar
Goodman, L. A., (1974). Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika, 61 215–231CrossRefGoogle Scholar
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, 74 191–210CrossRefGoogle Scholar
Junker, B. W., & Sijtsma, K. (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory. Applied Psychological Measurement, 25 258–272CrossRefGoogle Scholar
Koopmans, T. C.,Statistical inference in dynamic economic models, (1950). New York, USAWileyGoogle Scholar
Koopmans, T. C., & Reiersøl, O. (1950). The identification of structural characteristics. Annals of Mathematical Statistics, 21 165–181CrossRefGoogle Scholar
Leighton, J. P., Gierl, M. J., & Hunka, S. M., (2004). The attribute hierarchy model for cognitive assessment: A variation on Tatsuoka’s rule-space approach. Journal of Educational Measurement, 41 205–237CrossRefGoogle Scholar
Liu, J., Xu, G., & Ying, Z. (2012). Data-driven learning of Q-matrix. Applied Psychological Measurement, 36 548–564CrossRefGoogle ScholarPubMed
Liu, J., Xu, G., & Ying, Z. (2013). Theory of self-learning Q-matrix. Bernoulli, 19 5A1790–1817CrossRefGoogle ScholarPubMed
Maris, G., & Bechger, T. M., (2004). Equivalent MIRID models. Psychometrika, 69 4627–639CrossRefGoogle Scholar
Maris, G., & Bechger, T. M., (2009). Equivalent diagnostic classification models. Measurement, 7 41–46Google Scholar
McHugh, R. B., (1956). Efficient estimatio and local identification in latent class analysis. Psychometrika, 21 331–347CrossRefGoogle Scholar
Paulino, CDM, & de Bragançca Pereira, C. A., (1994). On identifiability of parametric statistical models. Journal of the Italian Statistical Society, 3 1125–151CrossRefGoogle Scholar
Rupp, A. A., & Templin, J. L., (2008). Effects of Q-matrix misspecification on parameter estimates and misclassification rates in the DINA model. Educational and Psychological Measurement, 68 78–98CrossRefGoogle Scholar
Rupp, A. A., Templin, J. L., & Henson, R. A., (2010). Diagnostic measurement: Theory, methods, and applications, New York, USAGuilford PressGoogle Scholar
San Martín, E., & Quintana, F. (2002). Consistency and identifiability revisited. Brazilian Journal of Probability and Statistics, 16 99–106Google Scholar
San Martín, E., Rolin, J-M, & Castro, L. M., (2013). Identification of the 1PL model with guessing parameter: Parametric and semi-parametric results. Psychometrika, 78 2341–379CrossRefGoogle ScholarPubMed
Tatsuoka, C. (2009). Diagnostic models as partially ordered sets. Measurement, 7 49–53Google Scholar
Tatsuoka, K. K., (1983). Rule space: An approch for dealing with misconceptions based on item response theory. Journal of Educational Measurement, 20 345–354CrossRefGoogle Scholar
Tatsuoka, K. K., (1985). A probabilistic model for diagnosing misconceptions in the pattern classification approach. Journal of Educational Statistics, 12 55–73CrossRefGoogle Scholar
Tatsuoka, K. K., Frederiksen, N., Glaser, R., Lesgold, A., & Shafto, M. (1990). Toward an integration of item-response theory and cognitive error diagnosis. Diagnostic monitoring of skill and knowledge acquisition, Hillsdale, NJErlbaum 453–488Google Scholar
Tatsuoka, K. K. (1991). Boolean algebra applied to determination of universal set of knowledge states (Research report No. RR-91-44-ONR). Princeton, NJ: Educational Testing Service.Google Scholar
Tatsuoka, K. K., (2009). Cognitive assessment: An introduction to the rule space method, New York, USA: RoutledgeCrossRefGoogle Scholar
Templin, J. L., & Henson, R. A., (2006). Measurement of psychological disorders using cognitive diagnosis models. Psychological Methods, 11 287–305CrossRefGoogle ScholarPubMed
U.S. Department of Education. (2009, November). Race to the top program executive summary. Washington, D.C. Retrieved May 9, 2014 from http://www2.ed.gov/programs/racetothetop/executive-summary.pdf.Google Scholar
von Davier, M. (2005). A general diagnostic model applied to language testing data (Research report), Princeton, NJ: Educational Testing ServiceGoogle Scholar
von Davier, M. (2008). A general diagnostic model applied to language testing data. British Journal of Mathematical and Statistical Psychology, 61 287–307CrossRefGoogle ScholarPubMed
von Davier, M., & Yamamoto, K. (2004). A class of models for cognitive diagnosis. In 4th spearman conference. Philadelphia, PA.Google Scholar
Zhang, S. S., DeCarlo, L. T., & Ying, Z. (2013, March). Non-identifiability, equivalence classes, and attribute-specific classification in Q-matrix based Cognitive Diagnosis Models. ArXiv e-prints.Google Scholar