Table of Contents
ISRN Applied Mathematics
Volume 2012 (2012), Article ID 269385, 14 pages
http://dx.doi.org/10.5402/2012/269385
Research Article

A Gibbs Sampler for the Multidimensional Item Response Model

Section on Statistics and Measurement, Department of EPSE, Southern Illinois University Carbondale, Wham 223, MailCode 4618, Carbondale, IL 62901-4618, USA

Received 2 March 2012; Accepted 26 March 2012

Academic Editors: S. He and X. Xue

Copyright © 2012 Yanyan Sheng and Todd C. Headrick. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. D. Bock and M. Aitkin, “Marginal maximum likelihood estimation of item parameters: application of an EM algorithm,” Psychometrika, vol. 46, no. 4, pp. 443–459, 1981. View at Publisher · View at Google Scholar
  2. R. J. Mislevy, “Estimation of latent group effects,” Journal of the American Statistical Association, vol. 80, no. 392, pp. 993–997, 1985. View at Publisher · View at Google Scholar
  3. R. J. Patz and B. W. Junker, “A straightforward approach to Markov chain Monte Carlo methods for item response models,” Journal of Educational and Behavioral Statistics, vol. 24, no. 2, pp. 146–178, 1999. View at Google Scholar · View at Scopus
  4. R. K. Tsutakawa and H. Y. Lin, “Bayesian estimation of item response curves,” Psychometrika, vol. 51, no. 2, pp. 251–267, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. J. Bafumi, A. Gelman, D. K. Park, and N. Kaplan, “Practical issues in implementing and understanding Bayesian ideal point estimation,” Political Analysis, vol. 13, no. 2, pp. 171–187, 2005. View at Publisher · View at Google Scholar · View at Scopus
  6. C. S. Martin, T. Chung, L. Kirisci, and J. W. Langenbucher, “Item response theory analysis of diagnostic criteria for alcohol and cannabis use disorders in adolescents: implications for DSM-V,” Journal of Abnormal Psychology, vol. 115, no. 4, pp. 807–814, 2006. View at Publisher · View at Google Scholar · View at Scopus
  7. U. Feske, L. Kirisci, R. E. Tarter, and P. A. Pilkonis, “An application of item response theory to the DSM-III-R criteria for borderline personality disorder,” Journal of Personality Disorders, vol. 21, no. 4, pp. 418–433, 2007. View at Publisher · View at Google Scholar · View at Scopus
  8. C. L. Beseler, L. A. Taylor, and R. F. Leeman, “An item-response theory analysis of DSM-IV Alcohol-Use disorder criteria and “binge” drinking in undergraduates,” Journal of Studies on Alcohol and Drugs, vol. 71, no. 3, pp. 418–423, 2010. View at Google Scholar · View at Scopus
  9. D. A. Gilder, I. R. Gizer, and C. L. Ehlers, “Item response theory analysis of binge drinking and its relationship to lifetime alcohol use disorder symptom severity in an american indian community sample,” Alcoholism, vol. 35, no. 5, pp. 984–995, 2011. View at Publisher · View at Google Scholar · View at Scopus
  10. A. T. Panter and B. B. Reeve, “Assessing tobacco beliefs among youth using item response theory models,” Drug and Alcohol Dependence, vol. 68, pp. S21–S39, 2002. View at Google Scholar · View at Scopus
  11. D. Courvoisier and J. F. Etter, “Using item response theory to study the convergent and discriminant validity of three questionnaires measuring cigarette dependence,” Psychology of Addictive Behaviors, vol. 22, no. 3, pp. 391–401, 2008. View at Publisher · View at Google Scholar · View at Scopus
  12. J. S. Rose and L. C. Dierker, “An item response theory analysis of nicotine dependence symptoms in recent onset adolescent smokers,” Drug and Alcohol Dependence, vol. 110, no. 1-2, pp. 70–79, 2010. View at Publisher · View at Google Scholar · View at Scopus
  13. S. E. Fienberg, M. S. Johnson, and B. W. Junker, “Classical multilevel and Bayesian approaches to population size estimation using multiple lists,” Journal of the Royal Statistical Society. Series A, vol. 162, no. 3, pp. 383–405, 1999. View at Google Scholar · View at Scopus
  14. M. Reiser, “An application of the item-response model to psychiatric epidemiology,” Sociological Methods and Research, vol. 18, pp. 66–103, 1989. View at Google Scholar
  15. M. Orlando, C. D. Sherbourne, and D. Thissen, “Summed-score linking using item response theory: application to depression measurement,” Psychological Assessment, vol. 12, no. 3, pp. 354–359, 2000. View at Publisher · View at Google Scholar · View at Scopus
  16. A. Tsutsumi, N. Iwata, N. Watanabe et al., “Application of item response theory to achieve cross-cultural comparability of occupational stress measurement,” International Journal of Methods in Psychiatric Research, vol. 18, no. 1, pp. 58–67, 2009. View at Publisher · View at Google Scholar · View at Scopus
  17. D. F. Andrade and H. R. Tavares, “Item response theory for longitudinal data: population parameter estimation,” Journal of Multivariate Analysis, vol. 95, no. 1, pp. 1–22, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. J. M. te Marvelde, C. A. W. Glas, G. Van Landeghem, and J. Van Damme, “Application of multidimensional item response theory models to longitudinal data,” Educational and Psychological Measurement, vol. 66, no. 1, pp. 5–34, 2006. View at Publisher · View at Google Scholar
  19. M. von Davier and S. Sinharay, “An importance sampling em algorithm for latent regression models,” Journal of Educational and Behavioral Statistics, vol. 32, no. 3, pp. 233–251, 2007. View at Publisher · View at Google Scholar · View at Scopus
  20. M. von Davier and S. Sinharay, “Stochastic approximation methods for latent regression item response models,” Journal of Educational and Behavioral Statistics, vol. 35, no. 2, pp. 174–193, 2010. View at Publisher · View at Google Scholar · View at Scopus
  21. R. Holman and C. A. W. Glas, “Modelling non-ignorable missing-data mechanisms with item response theory models,” The British Journal of Mathematical and Statistical Psychology, vol. 58, no. 1, pp. 1–17, 2005. View at Publisher · View at Google Scholar
  22. A. Birnbaum, “Statistical theory for logistic mental test models with a prior distribution of ability,” Journal of Mathematical Psychology, vol. 6, no. 2, pp. 258–276, 1969. View at Google Scholar · View at Scopus
  23. F. B. Baker and S.-H. Kim, Item response theory: Parameter Estimation Techniques, vol. 176, Marcel Dekker, New York, NY, USA, 2nd edition, 2004.
  24. I. W. Molenaar, “Estimation of item parameters,” in Rasch Models: Foundations, Recent Developments, and Applications, G. H. Fischer and I. W. Molenaar, Eds., pp. 39–51, Springer, New York, NY, USA, 1995. View at Google Scholar · View at Zentralblatt MATH
  25. A. F. M. Smith and G. O. Roberts, “Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods,” Journal of the Royal Statistical Society. Series B, vol. 55, no. 1, pp. 3–23, 1993. View at Google Scholar · View at Zentralblatt MATH
  26. L. Tierney, “Markov chains for exploring posterior distributions,” The Annals of Statistics, vol. 22, no. 4, pp. 1701–1762, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. J. H. Albert, “Bayesian estimation of normal ogive item response curves using gibbs sampling,” Journal of Educational Statistics, vol. 17, pp. 251–269, 1992. View at Google Scholar
  28. R. J. Patz and B. W. Junker, “Applications and extensions of MCMC in IRT: multiple item types, missing data, and rated responses,” Journal of Educational and Behavioral Statistics, vol. 24, no. 4, pp. 342–366, 1999. View at Google Scholar · View at Scopus
  29. S. K. Sahu, “Bayesian estimation and model choice in item response models,” Journal of Statistical Computation and Simulation, vol. 72, no. 3, pp. 217–232, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. A. A. Béguin and C. A. W. Glas, “MCMC estimation and some model-fit analysis of multidimensional IRT models,” Psychometrika, vol. 66, no. 4, pp. 541–561, 2001. View at Publisher · View at Google Scholar
  31. H. Lee, Markov Chain Monte Carlo Methods for Estimating Multidimensional Ability in Item Response Analysis, ProQuest LLC, Ann Arbor, Mich, USA, 1995.
  32. Y. Sheng and C. K. Wikle, “Comparing multiunidimensional and unidimensional item response theory models,” Educational and Psychological Measurement, vol. 67, no. 6, pp. 899–919, 2007. View at Publisher · View at Google Scholar
  33. Y. Sheng and C. K. Wikle, “Bayesian multidimensional IRT models with a hierarchical structure,” Educational and Psychological Measurement, vol. 68, no. 3, pp. 413–430, 2008. View at Publisher · View at Google Scholar · View at Scopus
  34. Y. Sheng and C. K. Wikle, “Bayesian IRT models in incorporating general and specific abilities,” Behaviormetrika, vol. 36, no. 1, pp. 27–48, 2009. View at Publisher · View at Google Scholar
  35. L. Yao, BMIRT: Bayesian Multivariate Item Response Theory [Computer Software], CTB/McGraw-Hill, Monterey, Calif, USA, 2003.
  36. L. Yao and K. A. Boughton, “A multidimensional item response modeling approach for improving subscale proficiency estimation and classification,” Applied Psychological Measurement, vol. 31, no. 2, pp. 83–105, 2007. View at Publisher · View at Google Scholar
  37. R. Levy, “The rise of markov chain monte carlo estimation for psychometric modeling,” Journal of Probability and Statistics, vol. 2009, Article ID 537139, 18 pages, 2009. View at Publisher · View at Google Scholar
  38. M. D. Reckase, Ed., Multidimensional Item Response Theory, Springer, New York, NY, USA, 2009.
  39. D. M. Bolt and V. F. Lall, “Estimation of compensatory and noncompensatory multidimensional item response models using Markov chain Monte Carlo,” Applied Psychological Measurement, vol. 27, no. 6, pp. 395–414, 2003. View at Publisher · View at Google Scholar
  40. S. Geman and D. Geman, “Stochastic relaxation, gibbs distributions, and the bayesian restoration of images,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 6, no. 6, pp. 721–741, 1984. View at Google Scholar · View at Scopus
  41. S. E. Embretson and S. P. Reise, Item Response Theory for Psychologists, Lawrence Erlbaum Associates, Hillside, NJ, USA, 2000.
  42. R. P. McDonald, Factor Analysis and Related Methods, Lawrence Erlbaum, Hillside, NJ, USA, 1985.
  43. R. K. Tsutakawa, “Estimation of two-parameter logistic item response curves,” Journal of Educational Statistics, vol. 9, pp. 263–276, 1985. View at Google Scholar
  44. Y. Sheng, “A MATLAB package for Markov chain Monte Carlo with a multi-unidimensional IRT model,” Journal of Statistical Software, vol. 28, no. 10, pp. 1–20, 2008. View at Google Scholar · View at Scopus
  45. Y. Sheng, “Bayesian estimation of MIRT models with general and specific latent traits in MATLAB,” Journal of Statistical Software, vol. 34, no. 3, pp. 1–27, 2010. View at Google Scholar · View at Scopus
  46. K. J. Holzinger and F. Swineford, “The Bi-factor method,” Psychometrika, vol. 2, no. 1, pp. 41–54, 1937. View at Publisher · View at Google Scholar · View at Scopus
  47. J. Barnard, R. McCulloch, and X.-L. Meng, “Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage,” Statistica Sinica, vol. 10, no. 4, pp. 1281–1311, 2000. View at Google Scholar · View at Zentralblatt MATH
  48. M. Pourahmadi, “Cholesky decompositions and estimation of a covariance matrix: orthogonality of variance-correlation parameters,” Biometrika, vol. 94, no. 4, pp. 1006–1013, 2007. View at Publisher · View at Google Scholar
  49. M. A. Tanner and W. H. Wong, “The calculation of posterior distributions by data augmentation,” Journal of the American Statistical Association, vol. 82, no. 398, pp. 528–550, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  50. A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2nd edition, 2004.
  51. W. R. Gilks, S. Richardson, and D. J. Spiedelhalter, Markov Chain Monte Carlo in Practice, Chapman and Hall, London, UK, 1996.
  52. A. Gelman and D. B. Rubin, “Inference from iterative simulation using multiple sequences,” Statistical Science, vol. 7, pp. 457–511, 1992. View at Google Scholar
  53. J. P. Fox, “Multilevel IRT modeling in practice with the package mlirt,” Journal of Statistical Software, vol. 20, no. 5, pp. 1–16, 2007. View at Google Scholar · View at Scopus
  54. S. Osterlind, “A national review of scholastic achievement in general education: how are we doing and why should we care?” vol. 25 of ASHE-ERIC Higher Education Report, George Washington University Graduate School of Education and Human Development, Washington, DC, USA, 1997. View at Google Scholar