A Three-Parameter Speeded Item Response Model: Estimation and Application

  • Joyce Chang
  • Henghsiu Tsai
  • Ya-Hui SuEmail author
  • Edward M. H. Lin
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 167)


When given time constraints, it is possible that examinees leave the harder items till later and are not able to finish answering every item in time. In this paper, this situation was modeled by incorporating a speeded-effect term into a three-parameter logistic item response model. Due to the complexity of the likelihood structure, a Bayesian estimation procedure with Markov chain Monte Carlo method was presented. The methodology is applied to physics examination data of the Department Required Test for college entrance in Taiwan for illustration.


Item response model Markov chain Monte Carlo Test speededness 



The research was supported by Academia Sinica and the Ministry of Science and Technology of the Republic of China under grant number MOST 102-2118-M-001 -007 -MY2. The authors would like to thank the co-editor, Professor Wen-Chung Wang, and Dr. Yu-Wei Chang for their helpful comments and suggestions, and the College Entrance Examination Center (CEEC) for providing the data.


  1. Agresti, A. (2002). Categorical data analysis. Hoboken, NJ: Wiley.CrossRefzbMATHGoogle Scholar
  2. Angoff, W. H. (1989). Does guessing really help? Journal of Educational Measurement, 26, 323–336.CrossRefGoogle Scholar
  3. Bejar, I. I. (1985). Test speededness under number-right scoring: An analysis of the test of English as a foreign language (Research Rep. RR-85-11). Princeton: Educational Testing Service.Google Scholar
  4. Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee’s ability. In F. M. Lord, & M. R. Novick (Eds.), Statistical theories of mental test scores (pp. 397–479). Reading, MA: Addison-Wesley.Google Scholar
  5. Bolt, D. M., Cohen, A. S., & Wollack, J. A. (2002). Item parameter estimation under conditions of test speededness: Application of a mixture Rasch model with ordinal constraints. Journal of Educational Measurement, 39, 331–348.CrossRefGoogle Scholar
  6. Boughton, K. A., & Yamamoto, K. (2007). A HYBRID model for test speededness. In M. von Davier, & C. H. Carstensen (Eds.), Multivariate and mixture distribution Rasch models (pp. 147–156). New York: Springer.CrossRefGoogle Scholar
  7. Cao, J., & Stokes, S. L. (2008). Bayesian IRT guessing models for partial guessing behaviors. Psychometrika, 73, 209–230.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Chang, Y.-W., Tsai, R.-C., & Hsu, N.-J. (2014). A speeded item response model: Leave the harder till later. Psychometrika, 79, 255–274.MathSciNetCrossRefzbMATHGoogle Scholar
  9. de Ayala, R. J. (2009). The theory and practice of item response theory. New York: Guilford Press.Google Scholar
  10. Embretson, S. E., & Reise, S. P. (2000). Item response theory for psychologists. Mahwah: L. Erlbaum Associates.Google Scholar
  11. Fox, J.-P. (2010). Bayesian item response modeling-theory and applications. New York: Springer.CrossRefzbMATHGoogle Scholar
  12. Goegebeur, Y., De Boeck, P., Wollack, J. A., & Cohen, A. S. (2008). A speeded item response model with gradual process change. Psychometrika, 73, 65–87.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Huang, H.-Y., & Hung, S.-P. (2010). Implementation and application of Bayesian three-level IRT random intercept latent regression model. Chinese Journal of Psychology, 52, 309–326. (in Chinese)Google Scholar
  14. Lee, Y.-H., & Ying, Z. (2015). A mixture cure-rate model for responses and response times in time-limit tests. Psychometrika, 80, 748–775.MathSciNetCrossRefzbMATHGoogle Scholar
  15. Li, Y., Bolt, D. M., & Fu, J. (2006). A comparison of alternative models for testlets. Applied Psychological Measurement, 30, 3–21.MathSciNetCrossRefGoogle Scholar
  16. Lord, F. M. (1975). Formula scoring and number-right scoring. Journal of Educational Measurement, 12, 7–11.CrossRefGoogle Scholar
  17. Meyer, J. P. (2010). A mixture Rasch model with item response time components. Applied Psychological Measurement, 34, 521–538.CrossRefGoogle Scholar
  18. Mislevy, R. L. (1986). Bayes modal estimation in item response theory. Psychometrika, 51, 177–195.MathSciNetCrossRefzbMATHGoogle Scholar
  19. Rost, J. (1990). Rasch models in latent classes: an integration of two approaches to item analysis. Applied Psychological Measurement, 14, 271–282.CrossRefGoogle Scholar
  20. Sinharay, S., Johnson, M. S., & Stern, H. S. (2006). Posterior predictive assessment of item response theory models. Applied Psychological Measurement, 30, 298–321.MathSciNetCrossRefGoogle Scholar
  21. 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, 64, 583–616.MathSciNetCrossRefzbMATHGoogle Scholar
  22. Swaminathan, H., & Gifford, J. A. (1982). Bayesian estimation in the Rasch model. Journal of Educational Statistics, 7, 175–192.CrossRefGoogle Scholar
  23. Swaminathan, H., & Gifford, J. A. (1985). Bayesian estimation in the twoparameter logistic model. Psychometrika, 50, 349–364.CrossRefzbMATHGoogle Scholar
  24. Swaminathan, H., & Gifford, J. A. (1986). Bayesian estimation in the three-parameter logistic model. Psychometrika, 51, 589–601.MathSciNetCrossRefzbMATHGoogle Scholar
  25. van der Linden, W. J. (2007). A hierarchical framework for modeling speed and accuracy on test items. Psychometrika, 72, 287–308.MathSciNetCrossRefzbMATHGoogle Scholar
  26. Wang, W.-C. (2004). Rasch measurement theory and application in education and psychology. Journal of Education and Psychology, 27, 637–694 (in Chinese).Google Scholar
  27. Wang, C., & Xu, G. (2015). A mixture hierarchical model for response times and response accuracy. British Journal of Mathematical and Statistical Psychology, 68, 456–477.CrossRefGoogle Scholar
  28. Yamamoto, K. (1989). HYBRID model of IRT and latent class models (ETS Research Rep. No. RR-89-41). Princeton: Educational Testing Service.Google Scholar
  29. Yamamoto, K. (1995). Estimating the effects of test length and test time on parameter estimation using the HYBRID model (TOEFL Technical Rep. No. TR-10). Princeton: Educational Testing Service.Google Scholar
  30. Yamamoto, K., & Everson, H. (1997). Modeling the effects of test length and test time on parameter estimation using the hybrid model. In J. Rost (Ed.), Applications of latent trait and latent class models in the social sciences (pp. 89–98). Munster: Waxmann.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Joyce Chang
    • 1
  • Henghsiu Tsai
    • 2
  • Ya-Hui Su
    • 3
    Email author
  • Edward M. H. Lin
    • 4
  1. 1.Department of EconomicsThe University of Texas at AustinAustinUSA
  2. 2.Institute of Statistical Science, Academia SinicaTaipeiTaiwan
  3. 3.Department of PsychologyNational Chung Cheng UniversityChia-YiTaiwan
  4. 4.Institute of FinanceNational Chiao Tung UniversityHsinchuTaiwan

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