Reduced Reparameterized Unified Model Applied to Learning Spatial Rotation Skills

  • Susu Zhang
  • Jeff DouglasEmail author
  • Shiyu Wang
  • Steven Andrew Culpepper
Part of the Methodology of Educational Measurement and Assessment book series (MEMA)


There has been a growing interest in measuring students in a learning context. Cognitive diagnosis models (CDMs) are traditionally used to measure students’ skill mastery at a static time point, but recently, they have been combined with longitudinal models to track students’ changes in skill acquisition over time. In this chapter, we propose a longitudinal learning model with CDMs. We consider different kinds of measurement models, including the reduced-reparameterized unified model (r-RUM) and the noisy input, deterministic-“and”-gate (NIDA) model. We also consider the incorporation of theories on skill hierarchies. Different models are fitted to a data set collected from a computer-based spatial rotation learning program (Wang S, Yang Y, Culpepper SA, Douglas JA, J Educ Behav Stat, 2016. and we evaluate and compare these models using several goodness-of-fit indices.


  1. Chen, Y., Culpepper, S., Wang, S., & Douglas, J. (2017). A hidden Markov model for learning trajectories with application to spatial rotation skills. Applied Psychological Measurement.
  2. Corbett, A. T., & Anderson, J. R. (1994). Knowledge tracing: Modeling the acquisition of procedural knowledge. User Modeling and User-Adapted Interaction, 4(4), 253–278.CrossRefGoogle Scholar
  3. Culpepper, S. A., & Hudson, A. (2017). An improved strategy for Bayesian estimation of the reduced reparameterized unified model. Applied Psychological Measurement.
  4. DiBello, L. V., Stout, W. F., & Roussos, L. A. (1995). Unified cognitive/psychometric diagnostic assessment likelihood-based classification techniques. In P. D. Nichols, S. F. Chipman, & R. L. Brennan (Eds.), Cognitively diagnostic assessment (pp. 361–389). New York: Routledge.Google Scholar
  5. González-Brenes, J., Huang, Y., & Brusilovsky, P. (2014). General features in knowledge tracing to model multiple subskills, temporal item response theory, and expert knowledge. In The 7th International Conference on Educational Data Mining (pp. 84–91).Google Scholar
  6. González-Brenes, J. P., & Mostow, J. (2013). What and when do students learn? Fully data-driven joint estimation of cognitive and student models. In Proceedings of the 6th International Conference on Educational Data Mining (pp. 236–240).Google Scholar
  7. Haertel, E. H. (1989). Using restricted latent class models to map the skill structure of achievement items. Journal of Educational Measurement, 26(4), 301–321.CrossRefGoogle Scholar
  8. Hartz, S. (2002). A Bayesian framework for the unified model for assessing cognitive abilities: Blending theory with practicality. (Unpublished doctoral dissertation). University of Illinois at Urbana-Champaign.Google Scholar
  9. Junker, B. W., & Sijtsma, K. (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory. Applied Psychological Measurement, 25(3), 258–272.CrossRefGoogle Scholar
  10. Kaya, Y., & Leite, W. L. (2016). Assessing change in latent skills across time with longitudinal cognitive diagnosis modeling an evaluation of model performance. Educational and Psychological Measurement.
  11. Leighton, J., & Gierl, M. (Eds.). (2007). Cognitive diagnostic assessment for education: Theory and applications. Cambridge University Press.Google Scholar
  12. Leighton, J. P., Gierl, M. J., & Hunka, S. M. (2004). The attribute hierarchy method for cognitive assessment: A variation on Tatsuoka’s rule-space approach. Journal of Educational Measurement, 41(3), 205–237.CrossRefGoogle Scholar
  13. Li, F., Cohen, A., Bottge, B., & Templin, J. (2015). A latent transition analysis model for assessing change in cognitive skills. Educational and Psychological Measurement, 76(2), 181–204.CrossRefGoogle Scholar
  14. Macready, G. B., & Dayton, C. M. (1977). The use of probabilistic models in the assessment of mastery. Journal of Educational and Behavioral Statistics, 2(2), 99–120.CrossRefGoogle Scholar
  15. Maris, E. (1999). Estimating multiple classification latent class models. Psychometrika, 64(2), 187–212.CrossRefGoogle Scholar
  16. Pardos, Z. A., & Heffernan, N. T. (2010). Modeling individualization in a Bayesian networks implementation of knowledge tracing. In International Conference on User Modeling, Adaptation, and Personalization (pp. 255–266).Google Scholar
  17. Sinharay, S., Johnson, M. S., & Stern, H. S. (2006). Posterior predictive assessment of item response theory models. Applied Psychological Measurement, 30(4), 298–321.CrossRefGoogle Scholar
  18. Spiegelhalter, D. J., Best, N. G., Carlin, B. P., & van der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(4), 583–639.CrossRefGoogle Scholar
  19. Studer, C. (2012). Incorporating learning over time into the cognitive assessment framework (Unpublished doctoral dissertation). Carnegie Mellon University.Google Scholar
  20. Templin, J. L., & Henson, R. A. (2006). Measurement of psychological disorders using cognitive diagnosis models. Psychological Methods, 11(3), 287.CrossRefGoogle Scholar
  21. Wang, S., Yang, Y., Culpepper, S. A., & Douglas, J. A. (2016). Tracking skill acquisition with cognitive diagnosis models: A higher-order, hidden Markov model with covariates. Journal of Educational and Behavioral Statistics.
  22. Xu, Y., & Mostow, J. (2012). Comparison of methods to trace multiple subskills: Is LR-DBN best? In Proceedings of the 5th International Conference on Educational Data Mining (pp. 41–48).Google Scholar
  23. Yoon, S. Y. (2011). Psychometric properties of the revised Purdue Spatial Visualization Tests: Visualization of Rotations (The Revised PSVT-R). (Unpublished doctoral dissertation). Purdue University.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Susu Zhang
    • 1
  • Jeff Douglas
    • 2
    Email author
  • Shiyu Wang
    • 3
  • Steven Andrew Culpepper
    • 2
  1. 1.Department of StatisticsColumbia UniversityNew YorkUSA
  2. 2.Department of StatisticsUniversity of Illinois at Urbana-ChampaignChampaignUSA
  3. 3.Department of Educational PsychologyUniversity of GeorgiaAthensUSA

Personalised recommendations