Abstract
The recent surge of interests in cognitive assessment has led to the development of cognitive diagnosis models. Central to many such models is a specification of the Q-matrix, which relates items to latent attributes that have natural interpretations. In practice, the Q-matrix is usually constructed subjectively by the test designers. This could lead to misspecification, which could result in lack of fit of the underlying statistical model. To test possible misspecification of the Q-matrix, traditional goodness of fit tests, such as the Chi-square test and the likelihood ratio test, may not be applied straightforwardly due to the large number of possible response patterns. To address this problem, this paper proposes a new statistical method to test the goodness fit of the Q-matrix, by constructing test statistics that measure the consistency between a provisional Q-matrix and the observed data for a general family of cognitive diagnosis models. Limiting distributions of the test statistics are derived under the null hypothesis that can be used for obtaining the test p-values. Simulation studies as well as a real data example are presented to demonstrate the usefulness of the proposed method.
Similar content being viewed by others
References
Bartholomew, D. J., & Tzamourani, P. (1999). The goodness of fit of latent trait models in attitude measurement. Sociological Methods & Research, 27(4), 525–546.
Cai, L., Maydeu-Olivares, A., Coffman, D. L., & Thissen, D. (2006). Limited-information goodness-of-fit testing of item response theory models for sparse \(2^p\) tables. British Journal of Mathematical and Statistical Psychology, 59(1), 173–194.
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., Douglas, J., & Li, X. (2009). Cluster analysis for cognitive diagnosis: Theory and applications. Psychometrika, 74(4), 633–665.
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(2), 179–199.
de la Torre, J., & Chiu, C.-Y. (2016). A general method of empirical \(Q\)-matrix validation. Psychometrika, 81(2), 253–273.
de la Torre, J., & Douglas, J. (2004). Higher order latent trait models for cognitive diagnosis. Psychometrika, 69, 333–353.
DeCarlo, L. T. (2011). On the analysis of fraction subtraction data: The DINA model, classification, latent class sizes, and the Q-matrix. Applied Psychological Measurement, 35, 8–26.
DeCarlo, L. T. (2012). Recognizing uncertainty in the \(Q\)-matrix via a bayesian extension of the DINA model. Applied Psychological Measurement, 36(6), 447–468.
Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via EM algorithm. Journal of the Royal Statistical Society Series B-Methodological, 39(1), 1–38.
DiBello, L., Stout, W., & Roussos, L. (1995). Unified cognitive psychometric assessment likelihood-based classification techniques. In P. D. Nichols, S. F. Chipman, & R. L. Brennan (Eds.), Cognitively diagnostic assessment (pp. 361–390). Hillsdale, NJ: Erlbaum.
Gu, Y., & Xu, G. (2018). Partial identifiability of restricted latent class models. arXiv preprint arXiv:1803.04353.
Hartz, S. (2002). A Bayesian framework for the unified model for assessing cognitive abilities: Blending theory with practicality. Doctoral Dissertation, University of Illinois, Urbana-Champaign.
Henson, R., & Templin, J. (2005). Hierarchical log-linear modeling of the skill joint distribution. Technical report, External Diagnostic Research Group.
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(2), 191–210.
Junker, B., & Sijtsma, K. (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory. Applied Psychological Measurement, 25, 258–272.
Lehmann, E. L., & Romano, J. P. (2006). Testing statistical hypotheses. Berlin: Springer.
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, 205–237.
Liu, J., Xu, G., & Ying, Z. (2012). Data-driven learning of \(Q\)-matrix. Applied Psychological Measurement, 36(7), 548–564.
Liu, J., Xu, G., & Ying, Z. (2013). Theory of self-learning \(Q\)-matrix. Bernoulli, 19(5A), 1790–1817.
Maydeu-Olivares, A. (2001). Limited information estimation and testing of thurstonian models for paired comparison data under multiple judgment sampling. Psychometrika, 66(2), 209–227.
Maydeu-Olivares, A., & Joe, H. (2005). Limited-and full-information estimation and goodness-of-fit testing in \(2^n\) contingency tables: A unified framework. Journal of the American Statistical Association, 100(471), 1009–1020.
Roussos, L. A., Templin, J. L., & Henson, R. A. (2007). Skills diagnosis using IRT-based latent class models. Journal of Educational Measurement, 44, 293–311.
Rupp, A. (2002). Feature selection for choosing and assembling measurement models: A building-block-based organization. Psychometrika, 2, 311–360.
Rupp, A., & Templin, J. (2008a). Effects of \(q\)-matrix misspecification on parameter estimates and misclassification rates in the dina model. Educational and Psychological Measurement, 68, 78–98.
Rupp, A., & Templin, J. (2008b). Unique characteristics of diagnostic classification models: A comprehensive review of the current state-of-the-art. Measurement: Interdisciplinary Research and Perspective, 6, 219–262.
Rupp, A., Templin, J., & Henson, R. A. (2010). Diagnostic measurement: Theory, methods, and applications. New York City: Guilford Press.
Sen, B., Banerjee, M., Woodroofe, M., et al. (2010). Inconsistency of bootstrap: The Grenander estimator. The Annals of Statistics, 38(4), 1953–1977.
Sen, B., & Xu, G. (2015). Model based bootstrap methods for interval censored data. Computational Statistics & Data Analysis, 81, 121–129.
Stout, W. (2007). Skills diagnosis using IRT-based continuous latent trait models. Journal of Educational Measurement, 44, 313–324.
Tatsuoka, K. (1985). A probabilistic model for diagnosing misconceptions in the pattern classification approach. Journal of Educational Statistics, 12, 55–73.
Tatsuoka, K. (1990). Toward an integration of item-response theory and cognitive error diagnosis. In N. Frederiksen, R. Glaser, A. Lesgold, & M. Shafto (Eds.), Diagnostic monitoring of skill and knowledge acquisition, (pp. 453–488).
Tatsuoka, C. (2002). Data-analytic methods for latent partially ordered classification models. Applied Statistics (JRSS-C), 51, 337–350.
Tatsuoka, C. (2005). Corrigendum: Data analytic methods for latent partially ordered classification models. Journal of the Royal Statistical Society: Series C (Applied Statistics), 54(2), 465–467.
Tatsuoka, K. (2009). Cognitive assessment: An introduction to the rule space method. Boca Raton: CRC Press.
Templin, J. (2006). CDM: Cognitive diagnosis modeling with Mplus . Available from http://jtemplin.myweb.uga.edu/cdm/cdm.html.
Templin, J., He, X., Roussos, L., & Stout, W. (2003). The pseudo-item method: A simple technique for analysis of polytomous data with the fusion model. Technical report, External Diagnostic Research Group.
Templin, J., & Henson, R. (2006). Measurement of psychological disorders using cognitive diagnosis models. Psychological Methods, 11, 287–305.
Tollenaar, N., & Mooijaart, A. (2003). Type I errors and power of the parametric bootstrap goodness-of-fit test: Full and limited information. British Journal of Mathematical and Statistical Psychology, 56(2), 271–288.
Van der Vaart, A. W. (2000). Asymptotic statistics (Vol. 3). Cambridge: Cambridge university press.
von Davier, M. (2005). A general diagnosis model applied to language testing data. Research report, Educational Testing Service.
von Davier, M. (2008). A general diagnostic model applied to language testing data. British Journal of Mathematical and Statistical Psychology, 61, 287–307.
Xu, G. (2017). Identifiability of restricted latent class models with binary responses. The Annals of Statistics, 45(2), 675–707.
Xu, G., & Shang, Z. (2018). Identifying latent structures in restricted latent class models. Journal of the American Statistical Association. https://doi.org/10.1080/01621459.2017.1340889.
Zhang, S. S., DeCarlo, L. T., & Ying, Z. (2013). Non-identifiability, equivalence classes, and attribute-specific classification in Q-matrix based cognitive diagnosis models. ArXiv e-prints.
Acknowledgements
The authors thank the Editor, the Associate Editor, and four reviewers for many helpful and constructive comments. This work is partially supported by National Science Foundation (Grant No. SES-1659328, DMS-1712717, IIS-1633360, MMS-1826540), Institute of Education Sciences (Grant No. R305D160010), and Army Grant (Grant No. W911NF-15-1-0159).
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Gu, Y., Liu, J., Xu, G. et al. Hypothesis Testing of the Q-matrix. Psychometrika 83, 515–537 (2018). https://doi.org/10.1007/s11336-018-9629-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11336-018-9629-6