Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-14T21:12:14.478Z Has data issue: false hasContentIssue false
  • English
  • Français

Confidence Distribution for the Ability Parameter of the Rasch Model

Published online by Cambridge University Press:  01 January 2025

Piero VeroneseOpen the ORCID record for Piero Veronese[Opens in a new window] and
Eugenio Melilli
Piero Veronese*
Affiliation:
Bocconi University
Eugenio Melilli
Affiliation:
Bocconi University
*
Correspondence should be made to Piero Veronese, Department of Decision Sciences, Bocconi University, via Roentgen 1, 20136 Milano, Italy. Email: piero.veronese@unibocconi.it
Get accessRights & Permissions [Opens in a new window]

Abstract

In this paper, we consider the Rasch model and suggest novel point estimators and confidence intervals for the ability parameter. They are based on a proposed confidence distribution (CD) whose construction has required to overcome some difficulties essentially due to the discrete nature of the model. When the number of items is large, the computations due to the combinatorics involved become heavy, and thus, we provide first- and second-order approximations of the CD. Simulation studies show the good behavior of our estimators and intervals when compared with those obtained through other standard frequentist and weakly informative Bayesian procedures. Finally, using the expansion of the expected length of the suggested interval, we are able to identify reasonable values of the sample size which lead to a desired length of the interval.

Keywords

asymptotic expansionconfidence intervalcoverageextreme scorefiducial distributionitem response theorynatural exponential familyobjective Bayesian inference

Information

Type
Theory and Methods
Information
Psychometrika , Volume 86 , Issue 1 , March 2021, pp. 131 - 166
Copyright
Copyright © 2021 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.)

Article purchase

Temporarily unavailable

References

Andrich, D.(2010).Sufficiency and conditional estimation of person parameters in the polytomous Rasch model.Psychometrika,75,292–308.CrossRefGoogle Scholar
Baker, F. B., &Kim, S. H. (2004).Item Response Theory: Parameter Estimation Techniques.New York:Dekker.CrossRefGoogle Scholar
Berger, J. O.(2006).The case for objective Bayesian analysis.Bayesian Analysis,1,385–402.CrossRefGoogle Scholar
Bond, T. G., &Fox, C. M.(2015).Applying the Rasch model: Fundamental measurement in the human sciences.(3)New York:Routledge.CrossRefGoogle Scholar
Brown, L. D. (1986. Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory. Lecture Notes—Monograph Series, vol. 9. Hayward, CA: Institute of Mathematical Statistics.Google Scholar
Brown, L. D.,Cai, T. T., &DasGupta, A.(2001).Interval estimation for a binomial proportion.Statistical Science,16,101–117.CrossRefGoogle Scholar
Christensen, K. B.,Kreiner, S., &Mesbah, M.(2013).Rasch Models in Health.Hoboken:Wiley.Google Scholar
Cohen, J.,Chan, T.,Jiang, T., &Seburn, M.(2008).Consistent estimation of Rasch item parameters and their standard errors under complex sample designs.Applied Psychological Measurement,32,289–310.CrossRefGoogle Scholar
Deheuvels, P.,Puri, M. L., &Ralescu, S. S.(1989).Asymptotic expansions for sums of nonidentically distributed Bernoulli random variables.Journal of Multivariate Analysis,28,282–303.CrossRefGoogle Scholar
Doebler, A.,Doebler, P., &Holling, H.(2013).Optimal and most exact confidence intervals for person parameters in item response theory models.Psychometrika,78,98–115.CrossRefGoogle ScholarPubMed
Finch, H., &Edwards, J. M.(2016).Rasch model parameter estimation in the presence of a nonnormal latent trait using a nonparametric bayesian approach.Educational and Psychological Measurement,76,662–684.CrossRefGoogle ScholarPubMed
Fischer, G. H.,Molenaar, I. W.(2012).Rasch Models: Foundations, Recent Developments, and Applications.New York:Springer.Google Scholar
Fisher, R. A.(1930).Inverse probability.Mathematical Proceedings of the Cambridge Philosophical Society,26,528–535.CrossRefGoogle Scholar
Fox, J.-P.(2010).Bayesian Item Response Modeling: Theory and Applications.Berlin:Springer.CrossRefGoogle Scholar
Fraser, D. A. S.(2011).Is Bayes posterior just quick and dirty confidence?.Statistical Science,26,299–316.CrossRefGoogle Scholar
Gelman, A.,Carlin, J. B.,Stern, H. S.,Dunson, D. B.,Vehtari, A., &Rubin, D. B.(2013).Bayesian Data Analysis.Boco Raton:CRC Press.CrossRefGoogle Scholar
Hall, P.(2013).The Bootstrap and Edgeworth Expansion.New York:Springer.Google Scholar
Hambleton, R. K.,Swaminathan, H., &Rogers, H. J.(1991).Fundamentals of Item Response Theory.Newbury Park:Sage Publications.Google Scholar
Hannig, J.(2009).On generalized fiducial inference.Statistica Sinica,19,491–544.Google Scholar
Hannig, J.,Iyer, H. K.,Lai, R. C. S., &Lee, T. C. M.(2016).Generalized fiducial inference: A review and new results.Journal of the American Statistical Association,44,476–483.Google Scholar
Hoijtink, H., & Boomsma, A. (1995. On person parameter estimation in the dichotomous Rasch model. In Rasch Models (pp. 53–68). Springer.CrossRefGoogle Scholar
Jannarone, R. J.,Kai, F. Y., &Laughlin, J. E.(1990).Easy Bayes estimation for Rasch-type models.Psychometrika,55,449–460.CrossRefGoogle Scholar
Johnson, R. A.(1967).An asymptotic expansion for posterior distributions.The Annals of Mathematical Statistics,38,1899–1906.CrossRefGoogle Scholar
Klauer, K. C.(1991).Exact and best confidence intervals for the ability parameter of the Rasch model.Psychometrika,56,535–547.CrossRefGoogle Scholar
Kolassa, J. E.(2006).Series Approximation Methods in Statistics.New York:Springer.Google Scholar
Liu, Y., &Hannig, J.(2016).Generalized fiducial inference for binary logistic item response models.Psychometrika,81,290–324.CrossRefGoogle ScholarPubMed
Lord, F. M.(1975).Evaluation with Artificial Data of a Procedure for Estimating Ability and Item Characteristic Curve Parameters.Princeton:ETS Research Bulletin Series.Google Scholar
Lord, F. M.(1980).Application of Item Response Theory to Practical Testing Problems.Hillsdale:Lawrence Erlbaum Ass.Google Scholar
Lord, F. M.(1983).Unbiased estimators of ability parameters, of their variance, and of their parallel-forms reliability.Psychometrika,48,233–245.CrossRefGoogle Scholar
Mair, P., & Hatzinger, R. (2007. Extended Rasch modeling: The eRm package for the application of IRT models in R.CrossRefGoogle Scholar
Mair, P., &Strasser, H.(2018).Large-scale estimation in Rasch models: asymptotic results, approximations of the variance-covariance matrix of item parameters, and divide-and-conquer estimation.Behaviormetrika,45,189–209.CrossRefGoogle Scholar
Ogasawara, H.(2012).Asymptotic expansions for the ability estimator in item response theory.Computational Statistics,27,661–683.CrossRefGoogle Scholar
Rasch, G.(1960).Probabilistic Models for Some Intelligence and Attainment Tests.Copenhagen:The Danish Institute of Educational Research.Google Scholar
Schweder, T., &Hjort, N. L.(2002).Confidence and likelihood.Scandinavian Journal of Statistics,29,309–332.CrossRefGoogle Scholar
Schweder, T., &Hjort, N. L.(2016).Confidence, Likelihood and Probability.London:Cambridge University Press.CrossRefGoogle Scholar
Serfling, R. J.(1980).Approximation Theorems of Mathematical Statistics.New York:Wiley.CrossRefGoogle Scholar
Severini, T. A.(2000).Likelihood Methods in Statistics.Oxford:Oxford University Press.CrossRefGoogle Scholar
Shen, J.,Liu, R. Y., &Xie, M.(2018).Prediction with confidence—A general framework for predictive inference.Journal of Statistical Planning and Inference,195,126–140.CrossRefGoogle Scholar
Sheng, Y.(2012).An empirical investigation of Bayesian hierarchical modeling with unidimensional IRT models.Behaviormetrika,40,19–40.CrossRefGoogle Scholar
Singh, K.,Xie, M., &Strawderman, M.(2005).Combining information through confidence distributions.Annals of Statistics,33,159–183.CrossRefGoogle Scholar
Singh, K.,Xie, M., &Strawderman, W. E.(2007).Confidence distribution (CD)—Distribution estimator of a parameter.Complex Datasets and Inverse Problems: Tomography, Networks and Beyond,Hayward:Institute of Mathematical Statistics.132–150.CrossRefGoogle Scholar
Swaminathan, H., &Gifford, J. A.(1982).Bayesian estimation in the Rasch model.Journal of Educational Statistics,7,175–191.CrossRefGoogle Scholar
Swaminathan, H., &Gifford, J. A.(1985).Bayesian estimation in the two-parameter logistic model.Psychometrika,50,349–364.CrossRefGoogle Scholar
Thissen, D., &Wainer, H.(1982).Some standard errors in item response theory.Psychometrika,47,397–412.CrossRefGoogle Scholar
Veronese, P., &Melilli, E.(2015).Fiducial and confidence distributions for real exponential families.Scandinavian Journal of Statistics,42,471–484.CrossRefGoogle Scholar
Veronese, P., &Melilli, E.(2018).Fiducial, confidence and objective Bayesian posterior distributions for a multidimensional parameter.Journal of Statistical Planning and Inference,195,153–173.CrossRefGoogle Scholar
Warm, T. A.(1989).Weighted likelihood estimation of ability in item response theory.Psychometrika,54,427–450.CrossRefGoogle Scholar
Wright, B.(1998).Estimating measures for extreme scores.Rasch Measurement Transactions,12,632–633.Google Scholar
Wright, B. D., &Stone, M. H.(1979).Best Test Design.Chicago:Mesa press.Google Scholar
Xie, M., &Singh, K.(2013).Confidence distribution, the frequentist distribution estimator of a parameter: A review.International Statistical Review,81,3–39.CrossRefGoogle Scholar