Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-07T18:34:39.720Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimax D-Optimal Designs for Item Response Theory Models

Published online by Cambridge University Press:  01 January 2025

Martijn P. F. Berger ,
C. Y. Joy King  and
Weng Kee Wong
Martijn P. F. Berger*
Affiliation:
Department of Methodology and Statistics, University of Maastricht
C. Y. Joy King
Affiliation:
Department of Biostatistics, UCLA
Weng Kee Wong
Affiliation:
Department of Biostatistics, UCLA
*
Requests for reprints should be sent to Martijn R F. Berger, Department of Methodology and Statistics, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands. E-mail: martijn.berger@stat.unimaas.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

Various different item response theory (IRT) models can be used in educational and psychological measurement to analyze test data. One of the major drawbacks of these models is that efficient parameter estimation can only be achieved with very large data sets. Therefore, it is often worthwhile to search for designs of the test data that in some way will optimize the parameter estimates. The results from the statistical theory on optimal design can be applied for efficient estimation of the parameters.

A major problem in finding an optimal design for IRT models is that the designs are only optimal for a given set of parameters, that is, they are locally optimal. Locally optimal designs can be constructed with a sequential design procedure. In this paper minimax designs are proposed for IRT models to overcome the problem of local optimality. Minimax designs are compared to sequentially constructed designs for the two parameter logistic model and the results show that minimax design can be nearly as efficient as sequentially constructed designs.

Keywords

optimal designIRT modelsminimaxsequential designs
Type
Original Paper
Information
Psychometrika , Volume 65 , Issue 3 , September 2000 , pp. 377 - 390
Copyright
Copyright © 2000 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adema, J. J. (1990). Models and algorithms for the construction of achievement tests. Unpublished doctoral dissertation, University of Twente.Google Scholar
Andrich, D. (1978). A rating scale formulation for ordered response categories. Psychometrika, 43, 561–573CrossRefGoogle Scholar
Atkinson, A. C., Donev, A. N. (1996). Optimum experimental designs. Oxford: Clarendon PressGoogle Scholar
Bartholomew, D. J. (1987). Latent variable models and factor analysis. London: Oxford University PressGoogle Scholar
Berger, M. P. F. (1992). Sequential sampling designs for the two-parameters item response theory model. Psychometrika, 57, 521–538CrossRefGoogle Scholar
Berger, M. P. F. (1994). D-optimal sequential sampling designs for item response theory models. Journal of Educational Statistics, 19, 43–56CrossRefGoogle Scholar
Berger, M. P. F. (1994). A general approach to algorithmic design of fixed-form tests, adaptive tests and testlets. Applied Psychological Measurement, 18, 141–153CrossRefGoogle Scholar
Berger, M. P. F., Mathijssen, E. (1997). Optimal test designs for polytomously scored items. British Journal of Mathematical and Statistical Psychology, 50, 127–141CrossRefGoogle Scholar
Berger, M. P. F., Veerkamp, W. J. J. (1996). A review of selection methods for optimal test design. In Wilson, M., Engelhard, G. (Eds.), Objective measurement: theory into practice, Volume III (pp. 437–455). Norwoord, NJ: Ablex PublishingGoogle Scholar
Boekkooi-Timminga, E. (1989). Models for computerized test construction, Unpublished doctoral dissertation, University of Twente.Google Scholar
Bock, R.D. (1972). Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika, 37, 29–51CrossRefGoogle Scholar
Chaloner, K., Larntz, K. (1989). Optimal Bayesian design applied to logistic regression experiments. Journal of Statistical Planning and Inference, 21, 191–208CrossRefGoogle Scholar
Fedorov, V.V. (1980). Convex design theory. Mathematische Operations forschung und Statistics, Serie Statistics, 11, 403–413Google Scholar
Ford, I., Kitsos, C. P., Titterington, D. M. (1989). Recent advances in nonlinear experimental design. Technometrics, 31, 49–60CrossRefGoogle Scholar
Heinen, A. G. J. J. (1996). Latent class and discrete latent variable models. London: Sage PublicationsGoogle Scholar
Jones, D. H., Jin, Z. (1994). Optimal sequential designs for on-line item estimation. Psychometrika, 59, 59–75CrossRefGoogle Scholar
Kiefer, J. (1974). General equivalence theory for optimum designs (Approximate theory). The Annals of Statistics, 2, 849–879CrossRefGoogle Scholar
Kiefer, J., Wolfowitz, (1960). The equivalence of two extremum problems. Canadian Journal of Mathematics, 12, 363–366CrossRefGoogle Scholar
King, C. Y. Joy (1996)Minimax optimal designs. Unpublished doctoral disseration, UCLA Department of Biostatistics.Google Scholar
Lord, F. M. (1980). Applications of item response theory to practical testing problems. Hillsdale, NJ: Lawrence ErlbaumGoogle Scholar
Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174CrossRefGoogle Scholar
Parthasarathy, K. R. (1967). Probability measures on metric spaces. New York: Academic PressCrossRefGoogle Scholar
Pazman, (1986). Foundations of optimum experimental design. Dordrecht/Boston/Lancaster: D. Reidel PublishingGoogle Scholar
Pshenichnyi, B. N. (1971). Necessary conditions for an extremum. New York: Marcel DekkerGoogle Scholar
Sitter, R. R. (1992). Robust designs for binary data. Biometrics, 48, 1145–1155CrossRefGoogle Scholar
Stefanski, L. A., Carroll, R. J. (1985). Covariate measurement error in logistic regression. Annals of Statistics, 13, 1335–1351CrossRefGoogle Scholar
Silvey, S. D. (1980). Optimal design. London: Chapman & HallCrossRefGoogle Scholar
Stocking, M. L. (1990). Specifying optimum examinees for item parameter estimation in item response theory. Psychometrika, 55, 461–475CrossRefGoogle Scholar
Theunissen, T.J.J.M. (1985). Binary programming and test design. Psychometrika, 50, 411–420CrossRefGoogle Scholar
Thissen, D., Steinberg, L. (1986). A taxonomy of item response models. Psychometrika, 51, 567–577CrossRefGoogle Scholar
Thissen, D., Wainer, H. (1982). Some standard errors in item response theory. Psychometrika, 47, 397–412CrossRefGoogle Scholar
Van der Linden, W. J., Boekkooi-Timminga, E. (1989). A maximin model for test design with practical constraints. Psychometrika, 53, 237–247CrossRefGoogle Scholar
Wingersky, M. S., Lord, F. M. (1984). An investigation of methods for reducing sampling error in certain IRT procedures. Applied Psychological Measurement, 8, 347–364CrossRefGoogle Scholar
Wong, W. K. (1992). A unified approach to the construction of minimax designs. Biometrika, 79, 611–619CrossRefGoogle Scholar
Wong, W. K., Cook, R. D. (1993). Heteroscedastic G-optimal designs. Journal of the royal Statistical Society, Series B, 55, 871–880CrossRefGoogle Scholar
Wynn, H. P. (1970). The sequential generation of D-optimum experimental designs. Annals of Mathematical Statistics, 41, 1655–1664CrossRefGoogle Scholar
Wu, C.F.J. (1985). Efficient sequential designs with binary data. Journal of the American Statistical Association, 80, 974–984CrossRefGoogle Scholar
Zhu, W., Ahn, H., Wong, W. K. (1998). Multiple objective optimal designs for the logit model. Communications in Statistics (Theory and Methods), 27, 1581–1592CrossRefGoogle Scholar