Psychometrika

, Volume 50, Issue 3, pp 349–364 | Cite as

Bayesian estimation in the two-parameter logistic model

  • Hariharan Swaminathan
  • Janice A. Gifford
Article

Abstract

A Bayesian procedure is developed for the estimation of parameters in the two-parameter logistic item response model. Joint modal estimates of the parameters are obtained and procedures for the specification of prior information are described. Through simulation studies it is shown that Bayesian estimates of the parameters are superior to maximum likelihood estimates in the sense that they are (a) more meaningful since they do not drift out of range, and (b) more accurate in that they result in smaller mean squared differences between estimates and true values.

Key words

Bayesian estimates item response model two-parameter logistic model modal estimates maximum likelihood estimates 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andersen, E. B. (1972). The numerical solution of a set of conditional estimation equations.The Journal of the Royal Statistical Society (Series B),34, 42–54.Google Scholar
  2. Birnbaum, A. (1969). Statistical theory for logistic mental test models with a prior distribution of ability.Journal of Mathematical Psychology, 6, 258–276.Google Scholar
  3. Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: An application of the EM algorithm.Psychometrika, 46, 443–460.Google Scholar
  4. Bock, R. D., & Lieberman, M. (1970). Fitting a response model forn dichotomously scored items.Psychometrika, 35, 179–197.Google Scholar
  5. Haberman, S. (1975). Maximum likelihood estimates in exponential response models.The Annals of Statistics, 5, 815–841.Google Scholar
  6. Hambleton, R. K., & Rovinelli, R. (1973). A FORTRAN IV program for generating examinees response data from logistic test models.Behavioral Science, 18, 74.Google Scholar
  7. Johnson, N. L., & Welch, B. L. (1939). On the calculation of the cumulants of the x distribution.Biometrika, 31, 216–218.Google Scholar
  8. Kendall, M. G., & Stuart, A. (1973).The advanced theory of statistics: Vol. I. New York: Hafner.Google Scholar
  9. Kiefer, J., & Wolfowitz, J. (1956). Consistency of the maximum likelihood estimates in the presence of infinitely many incidental parameters.Annals of Mathematical Statistics, 27, 887–890.Google Scholar
  10. Lindley, D. V. (1971). The estimation of many parameters. In V. P. Godambe & D. A. Sprott (Eds.),Foundations of Statistical Inference. Toronto: Holt, Rinehart, & Winston, pp. 435–455.Google Scholar
  11. Lindley, D. V., & Smith, A. F. (1972). Bayesian estimates for the linear model.Journal of the Royal Statistical Society (Series B),34, 1–41.Google Scholar
  12. Lord, F. M. (1968). An analysis of the Verbal Scholastic Aptitude Test using Birnbaum's three-parameter logistic model.Educational and Psychological Measurement, 28, 989–1020.Google Scholar
  13. Lord, F. M. (1980).Applications of item response theory to practical testing problems. New Jersey: Lawrence Erlbaum.Google Scholar
  14. Lord, F. M. & Novick, M. R. (1968).Statistical theories of mental test scores. Reading, MA: Addison-Wesley.Google Scholar
  15. Loyd, B. H., & Hoover, H. D. (1980). Vertical equating using the Rasch model.Journal of Educational Measurement, 17, 179–193.Google Scholar
  16. Neyman, J., & Scott, E. L. (1948). Consistent estimates based on partially consistent observations.Econometrica, 16, 1–32.Google Scholar
  17. Novick, M. R., & Jackson, P. (1974).Statistical methods for educational and psychological research. New York: McGraw-Hill.Google Scholar
  18. Novick, M. R., Lewis, C., & Jackson, P. H. (1973). The estimation of proportions inm groups.Psychometrika, 38, 19–46.Google Scholar
  19. Owen, R. (1975). A Bayesian sequential procedure for quantal response in the context of adaptive mental testing.Journal of the American Statistical Association, 70, 351–356.Google Scholar
  20. Slinde, J. A. & Linn, R. L. (1979). A note on vertical equating via the Rasch model for groups of quite different ability and tests of quite different difficulty.Journal of Educational Measurement, 16, 159–165.Google Scholar
  21. Swaminathan, H., & Gifford, J. A. (1982). Bayesian estimation in the Rasch model.Journal of Educational Statistics, 7, 175–191.Google Scholar
  22. Swaminathan, H., & Gifford, J. A. (1983). Estimation of parameters in the three-parameter latent trait model. In D. J. Weiss (Ed.),New horizons in testing. New York: Academic Press.Google Scholar
  23. Wood, R. L., Wingersky, M. S., & Lord, F. M. (1976). A computer program for estimating examinee ability and item characteristic curve parameters (Research Memorandum 76-6). Princeton, NJ: Educational Testing Service. (Revised 1978)Google Scholar
  24. Wright, B. D. (1977). Solving measurement problems with the Rasch model.Journal of Educational Measurement, 14, 97–116.Google Scholar
  25. Zellner, A. (1971).An introduction to Bayesian inference in econometrics, New York: John Wiley & Sons.Google Scholar

Copyright information

© The Psychometric Society 1985

Authors and Affiliations

  • Hariharan Swaminathan
    • 2
  • Janice A. Gifford
    • 1
  1. 1.Mount Holyoke CollegeUSA
  2. 2.School of EducationUniversity of MassachusettsAmherst

Personalised recommendations