Skip to main content
SpringerLink
Log in

Bayesian conditional inference for Rasch models

  • Original Paper
  • Published:
AStA Advances in Statistical Analysis Aims and scope Submit manuscript

Abstract

This paper is concerned with Bayesian inference in psychometric modeling. It treats conditional likelihood functions obtained from discrete conditional probability distributions which are generalizations of the hypergeometric distribution. The influence of nuisance parameters is eliminated by conditioning on observed values of their sufficient statistics, and Bayesian considerations are only referred to parameters of interest. Since such a combination of techniques to deal with both types of parameters is less common in psychometrics, a wider scope in future research may be gained. The focus is on the evaluation of the empirical appropriateness of assumptions of the Rasch model, thereby pointing to an alternative to the frequentists’ approach which is dominating in this context. A number of examples are discussed. Some are very straightforward to apply. Others are computationally intensive and may be unpractical. The suggested procedure is illustrated using real data from a study on vocational education.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Albert, J.H., Chib, S.: Bayesian residual analysis for binary response regression models. Biometrika 82, 747–769 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  • Altham, P.M.E.: Exact Bayesian analysis of a 2 \(\times \) 2 contingency table, and Fisher’s “exact” significance test. J. R. Stat. Soc. Ser. B 31, 261–269 (1969)

    MathSciNet  Google Scholar 

  • Andersen, E.B.: A goodness of fit test for the Rasch model. Psychometrika 38, 123–140 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  • Bayarri, M.J., Berger, J.O.: P values for composite null models. J. Am. Stat. Assoc. 95, 1127–1142 (2000)

    MathSciNet  MATH  Google Scholar 

  • Box, G.E.P.: Sampling and Bayes’ inference in scientific modelling and robustness. J. R. Stat. Soc. Ser. A 143, 383–430 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  • Box, G.E.P., Tiao, G.C.: Bayesian Inference in Statistical Analysis. Addison-Wesley, Reading (1973)

    MATH  Google Scholar 

  • Chen, Y., Diaconis, P., Holmes, S.P., Liu, J.S.: Sequential Monte Carlo methods for statistical analysis of tables. J. Am. Stat. Assoc. 100, 109–120 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, Y., Small, D.: Exact tests for the Rasch model via sequential importance sampling. Psychometrika 70, 11–30 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Draxler, C.: Sample size determination for Rasch model tests. Psychometrika 75, 708–724 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Draxler, C.: A note on a discrete probability distribution derived from the Rasch model. Adv. Appl. Stat. Sci. 6, 665–673 (2011)

    MathSciNet  MATH  Google Scholar 

  • Draxler, C., Alexandrowicz, R.W.: Sample size determination within the scope of conditional maximum likelihood estimation with special focus on testing the Rasch model. Psychometrika 80, 897–919 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Draxler, C., Zessin, J.: The power function of conditional tests of the Rasch model. Adv. Stat. Anal. 99, 367–378 (2015)

    Article  MathSciNet  Google Scholar 

  • Dyer, D., Pierce, R.L.: On the choice of the prior distribution in hypergeometric sampling. Commun. Stat. Theory Methods 22, 2125–2146 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Fischer, G.H., Molenaar, I.W.: Rasch Models—Foundations, Recent Developments and Applications. Springer, New York (1995)

    MATH  Google Scholar 

  • Fox, J.-P.: Bayesian Item Response Modeling. Springer, New York (2010)

    Book  MATH  Google Scholar 

  • Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984)

    Article  MATH  Google Scholar 

  • Glas, C.A.W.: The derivation of some tests for the Rasch model from the multinomial distribution. Psychometrika 53, 525–546 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  • Glas, C.A.W., Verhelst, N.D.: Extensions of the partial credit model. Psychometrika 54, 635–659 (1989)

    Article  MATH  Google Scholar 

  • Glas, C.A.W., Verhelst, N.D.: Testing the Rasch model. In: Fischer, G.H., Molenaar, I.W. (eds.) Rasch Models—Foundations, Recent Developments and Applications, pp. 69–96. Springer, New York (1995a)

    Google Scholar 

  • Glas, C.A.W., Verhelst, N.D.: Tests of fit for polytomous Rasch models. In: Fischer, G.H., Molenaar, I.W. (eds.) Rasch Models—Foundations, Recent Developments and Applications, pp. 325–352. Springer, New York (1995b)

    Google Scholar 

  • Kelderman, H.: Loglinear Rasch model tests. Psychometrika 49, 223–245 (1984)

    Article  MATH  Google Scholar 

  • Kelderman, H.: Item bias detection using loglinear IRT. Psychometrika 54, 681–697 (1989)

    Article  MathSciNet  Google Scholar 

  • Levy, R., Mislevy, R.J.: Bayesian Psychometric Modeling. CRC Press, Boca Raton (2016)

    MATH  Google Scholar 

  • Longhai, L.: gibbs.met: Naive Gibbs Sampling with Metropolis Steps. https://cran.r-project.org/web/packages/gibbs.met/index.html (2012)

  • Lord, F.M., Novick, M.R.: Statistical Theories of Mental Test Scores. Addison-Wesley, Reading (1968)

    MATH  Google Scholar 

  • Mahmoud, H.M.: Polya Urn Models. Chapman & Hall/CRC Press, Boca Raton (2008)

    Book  MATH  Google Scholar 

  • Martin, A.D., Quinn, K.M., Park, J.H.: MCMCpack: Markov Chain Monte Carlo (MCMC) Package. https://cran.r-project.org/web/packages/mcmcpack/index.html (2016)

  • Martin-Löf, P.: Statistika Modeller. Institutet för Försäkringsmatematik och Mathematisk Statistisk vid Stockholms Universitet, Stockholm (1973)

    Google Scholar 

  • Miller, J.W., Harrison, M.T.: Exact sampling and counting for fixed-margin matrices. Ann. Stat. 41, 1569–1592 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Mislevy, R.J.: Randomization-based inference about latent variables from complex samples. Psychometrika 56, 177–196 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Mislevy, R.J., Beaton, A.E., Kaplan, B., Sheehan, K.M.: Estimating population characteristics from sparse matrix samples of item responses. J. Educ. Meas. 29, 133–161 (1992)

    Article  Google Scholar 

  • Molenaar, I.W.: Some improved diagnostics for failure of the Rasch model. Psychometrika 48, 49–72 (1983)

    Article  Google Scholar 

  • Plummer, M., Stukalov, A., Denwood, M.: rjags: Bayesian graphical models using MCMC. https://cran.r-project.org/web/packages/rjags/index.html (2016)

  • Patterson, B., Atmar, W.: Nested subsets and the structure of insular mammalian faunas and archipelagos. Biol. J. Linn. Soc. 28, 65–82 (1986)

    Article  Google Scholar 

  • Ponocny, I.: Nonparametric goodness-of-fit tests for the Rasch model. Psychometrika 66, 437–459 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Raiffa, H., Schlaifer, R.: Applied Statistical Decision Theory. Clinton Press, Boston (1961)

    MATH  Google Scholar 

  • Rasch, G.: Probabilistic Models for Some Intelligence and Attainment Tests. The Danish Institute of Education Research, Copenhagen (1960)

    Google Scholar 

  • Robert, C., Casella, G.: Monte Carlo Statistical Methods. Springer, New York (2004)

    Book  MATH  Google Scholar 

  • Rubin, D.B.: Bayesianly justifiable and relevant frequency calculations for the applied statistician. Ann. Stat. 12, 1151–1172 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  • Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6, 461–464 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  • Spiegelhalter, D.J., Best, N.G., Carlin, B.P., van der Linde, A.: Bayesian measures of model complexity and fit. J. R. Stat. Soc. Ser. B 64, 583–639 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • van den Wollenberg, A.L.: Two new test statistics for the Rasch model. Psychometrika 47, 123–140 (1982)

    Article  MATH  Google Scholar 

  • Verhelst, N.D.: An efficient MCMC algorithm to sample binary matrices with fixed marginals. Psychometrika 73, 705–728 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Verhelst, N.D., Glas, C.A.W.: One parameter logistic model. In: Fischer, G.H., Molenaar, I.W. (eds.) Rasch Models—Foundations, Recent Developments and Applications, pp. 215–238. Springer, New York (1995)

    Google Scholar 

  • Verhelst, N.D., Hatzinger, R., Mair, P.: The RaschSampler. J. Stat. Softw. 20, 1–14 (2007)

    Article  Google Scholar 

  • Weber, S., Draxler, C., Bley, S., Wiethe-Körprich, M., Weiß, C., Gürer, C.: Der Projektverbund CoBALIT: large scale-assessments in der kaufmännischen Berufsbildung – Intrapreneurship (CoBALIT). In: Beck, K., Landenberger, M., Oser, F. (eds.) Technologiebasierte Kompetenzmessung in der beruflichen Bildung – Resultate aus dem Forschungsprogramm ASCOT, pp. 75–92. Bertelsmann, Bielefeld (2016)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University for Health and Life Sciences, EWZ 1, 6060, Hall, Austria

    Clemens Draxler

Authors
  1. Clemens Draxler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Clemens Draxler.

Appendix

Appendix

See Figs. 3, 4, 5, 6, 7, 8, 9, 10 and 11.

Fig. 3
figure 3

Histogram of posterior sample with respect to conditional effect parameter of item 1

Full size image
Fig. 4
figure 4

Histogram of posterior sample with respect to conditional effect parameter of item 2

Full size image
Fig. 5
figure 5

Histogram of posterior sample with respect to conditional effect parameter of item 3

Full size image
Fig. 6
figure 6

Histogram of posterior sample with respect to conditional effect parameter of item 4

Full size image
Fig. 7
figure 7

Histogram of posterior sample with respect to conditional effect parameter of item 5

Full size image
Fig. 8
figure 8

Histogram of posterior sample with respect to conditional effect parameter of item 6

Full size image
Fig. 9
figure 9

Histogram of posterior sample with respect to conditional effect parameter of item 7

Full size image
Fig. 10
figure 10

Histogram of posterior sample with respect to conditional effect parameter of item 8

Full size image
Fig. 11
figure 11

Histogram of posterior sample with respect to conditional effect parameter of item 9

Full size image

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Draxler, C. Bayesian conditional inference for Rasch models. AStA Adv Stat Anal 102, 245–262 (2018). https://doi.org/10.1007/s10182-017-0303-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10182-017-0303-6

Keywords

Search

Navigation