Advertisement

Psychometrika

, Volume 55, Issue 4, pp 577–601 | Cite as

On the sampling theory roundations of item response theory models

  • Paul W. Holland
Article

Abstract

Item response theory (IT) models are now in common use for the analysis of dichotomous item responses. This paper examines the sampling theory foundations for statistical inference in these models. The discussion includes: some history on the “stochastic subject” versus the random sampling interpretations of the probability in IRT models; the relationship between three versions of maximum likelihood estimation for IRT models; estimating θ versus estimating θ-predictors; IRT models and loglinear models; the identifiability of IRT models; and the role of robustness and Bayesian statistics from the sampling theory perspective.

Key words

stochastic subjects marginal maximum likelihood (MML) conditional maximum likelihood (CML) unconditional maximum likelihood (UML) joint maximum likelihood (JML) probability simplex loglinear models robustness 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andersen, E. B. (1970). Asymptotic properties of conditional maximum likelihood estimators.Journal of the Royal Statistical Society, Series B, 32, 283–301.Google Scholar
  2. Andersen, E. B. (1980). Discrete statistical models with social science applications. Amsterdam: North Holland.Google Scholar
  3. Birch, M. W. (1964). A new proof of the Pearson-Fisher theorem.Annals of Mathematical Statistics, 35, 718–824.Google Scholar
  4. Birnbaum, Z. W. (1967).Statistical theory for logistic mental test models with a prior distribution of ability (ETS Research Bulletin RB-67-12). Princeton, NJ: Educational Testing Service.Google Scholar
  5. Bock, R. D. (1967, March). Fitting a response model for n dichotomous items. Paper read at the Psychometric Society Meeting, Madison, WI.Google Scholar
  6. Bock, R. D., & Aitken, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm.Psychometrika, 46, 443–459.Google Scholar
  7. Bock, R. D., & Lieberman, M. (1970). Fitting a response model for n dichotomously scored items.Psychometrika, 35, 179–197.Google Scholar
  8. Bush, R. R., & Mosteller, F. (1955).Stochastic models for learning. New York: Wiley.Google Scholar
  9. Cressie, N., & Holland, P. W. (1983). Characterizing the manifest probabilities of latent trait models.Psychometrika, 48, 129–141.Google Scholar
  10. de Leeuw, J., & Verhelst, N. (1986). Maximum likelihood estimation in generalized Rasch models.Journal of Educational Statistics, 11, 183–196.Google Scholar
  11. Follman, D. A. (1988). Consistent estimation in the Rasch model based on nonparametric margins.Psychometrika, 53, 553–562.Google Scholar
  12. Guttman, L. (1941). The quantification of a class of attributes: A theory and method of scale construction. In P. Horst, et al. (Ed.),The prediction of personal adjustment (pp. 319–348). New York: Social Science Research Council.Google Scholar
  13. Guttman, L. (1950). The basis for scalogram analysis. In S. A. Stoufer, et al. (Ed.),Studies in social psychology in World War II, Vol. 4, measurement and prediction (pp. 60–90). Princeton, NJ: Princeton University Press.Google Scholar
  14. Haberman, S. J. (1977). Maximum likelihood estimates in exponential response models.Annals of Statistics, 5, 815–841.Google Scholar
  15. Holland, P. W. (1981). when are item response models consistent with observed data?Psychometrika, 46, 79–92.Google Scholar
  16. Holland, P. W. (1990). The Dutch Identity: A new tool for the study of item response models.Psychometrika, 55, 5–18.Google Scholar
  17. Holland, P. W., & Rosenbaum, P. R. (1986). Conditional association and unidimensionality in monotone latent variable models.Annals of Statistics, 14, 1523–1543.Google Scholar
  18. Junker, B. W. (1988). Statistical aspects of a new latent trait model. Unpublished doctoral dissertation, University of Illinois at Urbana-Champaign, Department of Statistics.Google Scholar
  19. Junker, B. W. (1989).conditional association, essential independence and local independence, Unpublished manuscript, University of Illinois at Urbana-Champaign, Department of Statistics.Google Scholar
  20. Junker, B. W. (in press). Essential independence and likelihood-based ability estimation for polytomous items.Psychometrika.Google Scholar
  21. Lawley, D. N. (1943). On problems connected with item selection and test construction.Proceedings of the Royal Statistical Society of Edinburgh, 61, 273–287.Google Scholar
  22. Lazarsfeld, P. F. (1950). The logical and mathematical foundations of latent structure analysis. In S. A. Stoufer, et al. (Ed.),Studies in social psychology in Wold War II, Vol. 4, measurement and prediction (pp. 362–412). Princeton, NJ: Princeton University Press.Google Scholar
  23. Lazarsfeld, P. F. (1959). Latent structure analysis. In S. Koch (Ed.),Psychology: A study of a science, Volume 3 (pp. 476–543). New York: McGraw Hill.Google Scholar
  24. Leonard, T. (1975). Bayesian estimation methods for two-way contingency tables.Journal of the royal Statistical Society, Series B, 37, 23–37.Google Scholar
  25. Levine, M. V. (1989).Ability distribution, pattern probabilities and quasidensities (Final Report.) Champaign, IL: University of Illinois, Model Based Measurement Laboratory.Google Scholar
  26. Lewis, C. (1985). Developments in nonparametric ability estimation. In D. J. Weiss (Ed.),Proceedings of the 1982 IRT/CAT conference (pp. 105–122). Minneapolis, MN: University of Minnesota.Google Scholar
  27. Lewis, C. (1990).A discrete, ordinal IRT model. Paper presented at the Annual Meeting of the American Educational Research Association, Boston, MA.Google Scholar
  28. Lindsay, B., Clogg, C. C., & Grego, J. (in press). Semi-parametric estimation in the Rasch model and related exponential response models, including a simple latent class model for item analysis.Journal of the American Statistical Association.Google Scholar
  29. Lord, F. M. (1952). A theory of test scores.Psychometrika Monograph No. 7, 17 (4, Pt. 2).Google Scholar
  30. Lord, F. M. (1967).An analysis of the Verbal Scholastic Aptitude Test using Brinbaum's three-parametric logistic model (ETS Research Bulletin RB-67-34). Princeton, NJ: Education Testing Service.Google Scholar
  31. Lord, F. M. (1974). Estimation of latent ability and item parameters when they are omitted responses.Psychometrika, 39, 247–264.Google Scholar
  32. Lord, F. M., & Novick, M. R. (1968).Statistical theories of mental test scores. Reading, MA: Addison-Wesley.Google Scholar
  33. Mislevy, R., & Stocking, M. (1989).A consumer's guide to LOGIST and BILOG.Applied Psychological Measurement, 13, 57–75.Google Scholar
  34. Oakes, D. (1988). Semi-parametric models. In S. Kotz & N. L. Johnson (Eds.),Encyclopedia of statistical science, Volume 8 (pp. 367–369). New York: Wiley.Google Scholar
  35. Rasch, G. (1960).Probabilistic medoels for some intelligence and attainment tests. Copenhagen: Nielson and Lydiche. (for Danmarks Paedagogiske Institut).Google Scholar
  36. Rosenbaum, P. R. (1984). Testing the conditional independence and monotonicity assumptions of item response theory.Psychometrika, 49, 425–436.Google Scholar
  37. Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores.Psychometrika Monograph No. 17, 33, (4, Pt. 2).Google Scholar
  38. Samejima, F. (1972). A general model for free response data.Psychometrika Monograph No. 18, 34, (4, Pt. 2).Google Scholar
  39. Samejima, F. (1983). Some methods and approaches of estimating the operating characteristics of discrete item responses. In H. Wainer & S. Messick (Eds.),Principals (sic) of modern psychological measurement (pp. 154–182). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  40. Stout, W. (1987). A nonparametric approach for assessing latent trait unidimensionality.Psychometrika, 52, 589–617.Google Scholar
  41. Stout, W. (1990). A new item response theory modeling approach with applications to unidimensionality assesment and ability estimation.Psychometrika, 55, 293–325.Google Scholar
  42. Thissen, D. (1982). Marginal maximum liklihood estimation for the one-parameter logistic model.Psychometrika, 47, 175–186.Google Scholar
  43. Tjur, T. (1982). A connection between Rasch's item analysis model and a multiplicative Poisson model.Scandinavian Journal of Statistics, 9, 23–30.Google Scholar
  44. Tsao, R. (1967). A second order exponental model for multidimensional dichotomous contingency tables with applications in medical diagnosis. Unpublished doctoral disseration, Harvard University, Department of Statistics.Google Scholar
  45. Tucker, L. R. (1964). Maximum validity of a test with equivlent items.Psychometrika, 11, 1–14.Google Scholar
  46. Wainer, H., et al. (1990).Computerized adaptive testing: A primer. Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  47. Wright, B. D. (1977). Solving meassurement problems with the Rasch model.Journal of Educational Measurement, 14, 97–116.Google Scholar
  48. Wright, B. D., & Douglas, G. A. (1977). Best procedures for sample-free item analysis.Applied Psychological Measurement, 1, 281–295.Google Scholar
  49. Wright, B. D., & Stone, M. H. (1979).Best test design. Chicago: Mesa Press.Google Scholar

Copyright information

© The Psychometric Society 1990

Authors and Affiliations

  • Paul W. Holland
    • 1
  1. 1.Educational Testing ServicePrinceton

Personalised recommendations