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Psychometrika

, Volume 75, Issue 2, pp 280–291 | Cite as

The Impact of Fallible Item Parameter Estimates on Latent Trait Recovery

  • Ying ChengEmail author
  • Ke-Hai Yuan
Theory and Methods

Abstract

In this paper we propose an upward correction to the standard error (SE) estimation of \(\hat{\theta}_{\mathrm{ML}}\) , the maximum likelihood (ML) estimate of the latent trait in item response theory (IRT). More specifically, the upward correction is provided for the SE of \(\hat{\theta}_{\mathrm{ML}}\) when item parameter estimates obtained from an independent pretest sample are used in IRT scoring. When item parameter estimates are employed, the resulting latent trait estimate is called pseudo maximum likelihood (PML) estimate. Traditionally, the SE of \(\hat{\theta}_{\mathrm{ML}}\) is obtained on the basis of test information only, as if the item parameters are known. The upward correction takes into account the error that is carried over from the estimation of item parameters, in addition to the error in latent trait recovery itself. Our simulation study shows that both types of SE estimates are very good when θ is in the middle range of the latent trait distribution, but the upward-corrected SEs are more accurate than the traditional ones when θ takes more extreme values.

IRT scoring pseudo maximum likelihood (PML) standard error upward correction latent trait estimation 

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References

  1. Baker, F.B., & Kim, S.H. (2004). Item response theory: Parameter estimation techniques (2nd ed.). New York: Dekker. Google Scholar
  2. 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. 397–472). Reading: Addison-Wesley. Google Scholar
  3. Bock, R.D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443–459. CrossRefGoogle Scholar
  4. Cai, L. (2008). SEM of another flavour: Two new applications of the supplemented EM algorithm. British Journal of Mathematical and Statistical Psychology, 61, 309–329. CrossRefPubMedGoogle Scholar
  5. Cover, T.M., & Thomas, J.A. (1991). Elements of information theory. New York: Wiley. CrossRefGoogle Scholar
  6. Embretson, S.E., & Reise, S.P. (2000). Item response theory for psychologists. Mahwah: Erlbaum. Google Scholar
  7. Godambe, V.P., & Thompson, M.E. (1974). Estimating equations in the presence of a nuisance parameter. The Annals of Statistics, 2, 568–571. CrossRefGoogle Scholar
  8. Gong, G., & Samaniego, F. (1981). Pseudo-maximum likelihood estimation: Theory and applications. The Annals of Statistics, 9, 861–869. CrossRefGoogle Scholar
  9. Hambleton, R., & Swaminathan, H. (1985). Item response theory: Principles and applications. Boston: Kluwer-Nijhoff Publishing. Google Scholar
  10. Harwell, M.R., & Baker, F.B. (1991). The use of prior distributions in marginalized Bayesian item parameter estimation: A didactic. Applied Psychological Measurement, 15, 375–389. CrossRefGoogle Scholar
  11. Harwell, M.R., Baker, F.B., & Zwarts, M. (1988). Item parameter estimation via marginal maximum likelihood and an EM algorithm: A didactic. Journal of Educational Statistics, 13, 243–271. CrossRefGoogle Scholar
  12. Parke, W.R. (1986). Pseudo maximum likelihood estimation: The asymptotic distribution. The Annals of Statistics, 14, 355–357. CrossRefGoogle Scholar
  13. Patz, R.J., & Junker, B.W. (1999). A straightforward approach to Markov chain Monte Carlo methods for item response models. Journal of Educational and Behavioral Statistics, 24, 146–178. Google Scholar
  14. Tsutakawa, R.K., & Johnson, J.C. (1990). The effect of uncertainty of item parameter estimation on ability estimates. Psychometrika, 55, 371–390. CrossRefGoogle Scholar
  15. Yuan, K.-H., & Jennrich, R.I. (2000). Estimating equations with nuisance parameters: Theory and applications. Annals of the Institute of Statistical Mathematics, 52, 343–350. CrossRefGoogle Scholar
  16. Wingersky, M.S., Barton, M.A., & Lord, F.M. (1982). LOGIST user’s guide. Princeton: Educational Testing Service. Google Scholar
  17. Zimowski, M., Muraki, E., Mislevy, R.J., & Bock, R.D. (2003). BILOG-MG 3: Item analysis and test scoring with binary logistic models. Chicago: Scientific Software [Computer software]. Google Scholar

Copyright information

© The Psychometric Society 2010

Authors and Affiliations

  1. 1.University of Notre DameNotre DameUSA

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