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A Copula Model for Residual Dependency in DINA Model

  • Zhihui FuEmail author
  • Ya-Hui Su
  • Jian Tao
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 265)

Abstract

Cognitive diagnosis models (CDMs) have been received the increasing attention by educational and psychological assessment. In practice, most CDMs are not robust to violations of local item independence. Many approaches have been proposed to deal with the local item dependence (LID), such as conditioning on other responses and additional random effects (Hansen In Hierarchical item response models for cognitive diagnosis. University of California, LA, 2013); however, these have some drawbacks, such as non-reproducibility of marginal probabilities and interpretation problem. (Braeken et al. In Psychometrika 72(3): 393–411 2007) introduced a new class of marginal models that makes use of copula functions to capture the residual dependence in item response models. In this paper, we applied the copula methodology to model the item dependencies in DINA model. It is shown that the proposed copula model could overcome some of the dependency problems in CDMs, and the estimated model parameters recovered well through simulations. Furthermore, we have extended the R package CDM to fit the proposed copula DINA model.

Keywords

Cognitive diagnosis models Copula model Local item dependence 

References

  1. Bradow, E. T., Wainer, H., & Wang, X, A Bayesian random effects model for testlets. Psychometrika 64, 153-168(1999)Google Scholar
  2. Braeken, J., Tuerlinckx, F., & De Boeck, P. (2007). Copula Functions for Residual Dependency. Psychometrika, 72(3), 393–411.MathSciNetCrossRefGoogle Scholar
  3. Chen, Y., Culpepper, S. A., Chen, Y., & Douglas, J. (2018). Bayesian Estimation of the DINA Q matrix. Psychometrika, 83, 89–108.MathSciNetCrossRefGoogle Scholar
  4. 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.MathSciNetCrossRefGoogle Scholar
  5. Chen, J., & de la Torre, J., & Zhang, Z.,. (2013). Relative and absolute fit evaluation in cognitive diagnosis modeling. Journal of Educational Measurement, 50, 123–140.Google Scholar
  6. Culpepper, S. A., & Chen, Y. (2018). Development and Application of an Exploratory Reduced Reparameterized Unified Model. Journal of Educational and Behavioral Statistics,.  https://doi.org/10.3102/1076998618791306.CrossRefGoogle Scholar
  7. de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76, 179–199.MathSciNetCrossRefGoogle Scholar
  8. Dempster, A.P., Laird, N.M. and Rubin, D.B., Maximum likelihood wiih incomplete data via the EM algorithm. Journal of the Royal Statistical Society 39, 1-38. Series B (1977)Google Scholar
  9. Hansen, M., Hierarchical item response models for cognitive diagnosis. Unpublished doctoral dissertation, University of California, LA (2013)Google Scholar
  10. 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.MathSciNetCrossRefGoogle Scholar
  11. Joe, H., & Xu, J. (1996). The Estimation Method of Inference Functions for Margins for Multivariate Models, Technical Report 166. Department of Statistics: University of British Columbia.Google Scholar
  12. 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.MathSciNetCrossRefGoogle Scholar
  13. Maydeu-Olivares, A., & Coffman, D. L. (2006). Random intercept item factor analysis. Psychological Methods, 11, 344–362.CrossRefGoogle Scholar
  14. Nelsen RB, An Introduction to Copulas(Springer-Verlag, New York, 2006)Google Scholar
  15. Sklar, A. W. (1959). Fonctions de répartition àn dimension et leurs marges. Publications de lInstitut de Statistique de lUniversitéde Paris, 8, 229–231.zbMATHGoogle Scholar
  16. von Davier, M. (2008). A general diagnostic model applied to language testing data. British Journal of Mathematical and Statistical Psychology, 61, 287–307.MathSciNetCrossRefGoogle Scholar
  17. Xu, G., & Shang, Z. (2018). Identifying latent structures in restricted latent class models. Journal of the American Statistical Association, 113, 1284–1295.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Statistics, School of Mathematics and System ScienceShenyang Normal UniversityShenyangPeople’s Republic of China
  2. 2.Department of PsychologyNational Chung Cheng UniversityChiayi CountyTaiwan
  3. 3.Key Laboratory of Applied Statistics of MOE, School of Mathematics and StatisticsNortheast Normal UniversityChangchunPeople’s Republic of China

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