Advertisement

Psychometrika

, Volume 77, Issue 4, pp 710–723 | Cite as

Optimal Designs for the Rasch Model

  • Ulrike Graßhoff
  • Heinz HollingEmail author
  • Rainer Schwabe
Article

Abstract

In this paper, optimal designs will be derived for estimating the ability parameters of the Rasch model when difficulty parameters are known. It is well established that a design is locally D-optimal if the ability and difficulty coincide. But locally optimal designs require that the ability parameters to be estimated are known. To attenuate this very restrictive assumption, prior knowledge on the ability parameter may be incorporated within a Bayesian approach. Several symmetric weight distributions, e.g., uniform, normal and logistic distributions, will be considered. Furthermore, maximin efficient designs are developed where the minimal efficiency is maximized over a specified range of ability parameters.

Key words

optimal design Bayesian design maximin efficient design item response theory Rasch model 

Notes

Acknowledgements

This research was supported by the Deutsche Forschungsgemeinschaft (DFG) under grant HO 1286/6-1. The authors would like to thank Norbert Gaffke for helpful discussions.

References

  1. Atkinson, A.C., Donev, A.N., & Tobias, R.D. (2007). Optimum experimental designs, with SAS. Oxford: Oxford University Press. Google Scholar
  2. Berger, M.P.F., King, C.Y.J., & Wong, W.K. (2000). Minimax D-optimal designs for item response theory models. Psychometrika, 65, 377–390. CrossRefGoogle Scholar
  3. Berger, M.P.F., & Wong, W.K. (2009). An introduction to optimal designs for social and biomedical research. Chichester: Wiley. CrossRefGoogle Scholar
  4. Braess, D., & Dette, H. (2007). On the number of support points of maximin and Bayesian optimal designs. Annals of Statistics, 35, 772–792. CrossRefGoogle Scholar
  5. Buyske, S. (2005). Optimal design in educational testing. In M.P.F. Berger & W.K. Wong (Eds.), Applied optimal designs (pp. 1–19). New York: Wiley. CrossRefGoogle Scholar
  6. Chaloner, K. (1993). A note on optimal Bayesian design for nonlinear problems. Journal of Statistical Planning and Inference, 37, 229–235. CrossRefGoogle Scholar
  7. Chaloner, K., & Larntz, K. (1989). Optimal Bayesian design applied to logistic regression experiments. Journal of Statistical Planning and Inference, 21, 191–208. CrossRefGoogle Scholar
  8. Dette, H. (1997). Designing experiments with respect to ‘standardized’ optimality criteria. Journal of the Royal Statistical Society. Series B, 59, 97–110. CrossRefGoogle Scholar
  9. Dette, H., & Neugebauer, H.-M. (1996). Bayesian optimal one point designs for one parameter nonlinear models. Journal of Statistical Planning and Inference, 52, 17–31. CrossRefGoogle Scholar
  10. Firth, D., & Hinde, J.P. (1997). On Bayesian D-optimum design criteria and the equivalence theorem in non-linear models. Journal of the Royal Statistical Society. Series B, 59, 793–797. CrossRefGoogle Scholar
  11. Graßhoff, U., & Schwabe, R. (2008). Optimal design for the Bradley–Terry paired comparison model. Statistical Methods and Applications, 17, 275–289. CrossRefGoogle Scholar
  12. Kiefer, J. (1974). General equivalence theory for optimum designs (approximate theory). Annals of Statistics, 2, 849–879. CrossRefGoogle Scholar
  13. van der Linden, W.J. (2005). Linear models for optimal test design. New York: Springer. Google Scholar
  14. van der Linden, W.J., & Pashley, P.J. (2010). Item selection and ability estimation adaptive testing. In W.J. van der Linden & C.A.W. Glas (Eds.), Elements of adaptive testing (pp. 3–30). New York: Springer. CrossRefGoogle Scholar

Copyright information

© The Psychometric Society 2012

Authors and Affiliations

  • Ulrike Graßhoff
    • 1
  • Heinz Holling
    • 2
    Email author
  • Rainer Schwabe
    • 1
  1. 1.Otto-von-Guericke University MagdeburgMagdeburgGermany
  2. 2.Institute of PsychologyUniversity of MünsterMünsterGermany

Personalised recommendations