Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Item Response Theory with Estimation of the Latent Population Distribution Using Spline-Based Densities

  • 569 Accesses

  • 66 Citations

Abstract

The purpose of this paper is to introduce a new method for fitting item response theory models with the latent population distribution estimated from the data using splines. A spline-based density estimation system provides a flexible alternative to existing procedures that use a normal distribution, or a different functional form, for the population distribution. A simulation study shows that the new procedure is feasible in practice, and that when the latent distribution is not well approximated as normal, two-parameter logistic (2PL) item parameter estimates and expected a posteriori scores (EAPs) can be improved over what they would be with the normal model. An example with real data compares the new method and the extant empirical histogram approach.

This is a preview of subscription content, log in to check access.

References

  1. Anscombe, F.J. (1956). On estimating binomial response relations. Biometrika, 43, 461–464.

  2. Bahadur, R.R., & Ranga Rao, R. (1960). On deviations of the sample mean. Annals of Mathematical Statistics, 31, 1015–1027.

  3. Chernoff, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Annals of Statistics, 23, 493–507.

  4. Cohen, J. (1960). A coefficient of agreement for nominal scales. Educational and Psychological Measurement, 20, 37–46.

  5. Cronbach, L.J., & Gleser, G.C. (1965). Psychological tests and personnel decisions. Chicago, IL: University of Illinois Press.

  6. Feng, X., Dorans, N.J., Patsula, L.N., & Kaplan, B. (2003). Improving the statistical aspects of e-rater superscript registered: Exploring alternative feature reduction and combination rules. Technical Report RR-03-15. Princeton, NJ: Educational Testing Service.

  7. Gilula, Z., & Haberman, S.J. (1995a). Dispersion of categorical variables and penalty functions: Derivation, estimation, and comparability. Journal of the American Statistical Association, 90, 1447–1452.

  8. Gilula, Z., & Haberman, S.J. (1995b). Prediction functions for categorical panel data. Annals of Statistics, 23, 1130–1142.

  9. Goodman, L.A., & Kruskal, W.H. (1954). Measures of association for cross-classifications. Journal of the American Statistical Association, 49, 732–764.

  10. Haberman, S.J. (1982a). Analysis of dispersion of multinomial responses. Journal of the American Statistical Association, 77, 568–580.

  11. Haberman, S.J. (1982b). Measures of association. In S. Kotz & N.L. Johnson, (Eds.), Encyclopedia of statistical sciences (Vol. 1, pp. 130–137.) New York: Wiley.

  12. Hambleton, R.K., Swaminathan, H., & Rogers, H.J. (1991). Fundamentals of item response theory. Newbury Park, CA: Sage.

  13. Lachenbruch, P.A. (1968). On expected probabilities of misclassification in discriminant analysis, necessary sample size, and a relation with the multiple correlation coefficient. Biometrics, 24, 823–834.

  14. Lord, F.M., & Novick, M.R. (1968). Statistical theories of mental test scores. Reading, MA: Addison-Wesley.

  15. Savage, L. (1971). Elicitation of personal probabilities and expectations. Journal of the American Statistical Association, 66, 783–801.

  16. Stephan, F.F. (1945). The expected value and variance of the reciprocal and other negative powers of a positive bernoullian variate. Annals of Mathematical Statistics, 16, 50–61.

Download references

Author information

Correspondence to Carol M. Woods.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Woods, C.M., Thissen, D. Item Response Theory with Estimation of the Latent Population Distribution Using Spline-Based Densities. Psychometrika 71, 281 (2006). https://doi.org/10.1007/s11336-004-1175-8

Download citation

Keywords

  • item response theory
  • marginal maximum likelihood
  • latent variable
  • population distribution
  • density estimation
  • splines