Advertisement

AStA Advances in Statistical Analysis

, Volume 102, Issue 2, pp 245–262 | Cite as

Bayesian conditional inference for Rasch models

  • Clemens Draxler
Original Paper
  • 94 Downloads

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.

Keywords

Bayesian inference Discrete conditional probability distribution Hypergeometric distribution Conditional likelihood function Rasch model 

References

  1. Albert, J.H., Chib, S.: Bayesian residual analysis for binary response regression models. Biometrika 82, 747–769 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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)MathSciNetGoogle Scholar
  3. Andersen, E.B.: A goodness of fit test for the Rasch model. Psychometrika 38, 123–140 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bayarri, M.J., Berger, J.O.: P values for composite null models. J. Am. Stat. Assoc. 95, 1127–1142 (2000)MathSciNetzbMATHGoogle Scholar
  5. Box, G.E.P.: Sampling and Bayes’ inference in scientific modelling and robustness. J. R. Stat. Soc. Ser. A 143, 383–430 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Box, G.E.P., Tiao, G.C.: Bayesian Inference in Statistical Analysis. Addison-Wesley, Reading (1973)zbMATHGoogle Scholar
  7. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Chen, Y., Small, D.: Exact tests for the Rasch model via sequential importance sampling. Psychometrika 70, 11–30 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Draxler, C.: Sample size determination for Rasch model tests. Psychometrika 75, 708–724 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Draxler, C.: A note on a discrete probability distribution derived from the Rasch model. Adv. Appl. Stat. Sci. 6, 665–673 (2011)MathSciNetzbMATHGoogle Scholar
  11. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Draxler, C., Zessin, J.: The power function of conditional tests of the Rasch model. Adv. Stat. Anal. 99, 367–378 (2015)MathSciNetCrossRefGoogle Scholar
  13. Dyer, D., Pierce, R.L.: On the choice of the prior distribution in hypergeometric sampling. Commun. Stat. Theory Methods 22, 2125–2146 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Fischer, G.H., Molenaar, I.W.: Rasch Models—Foundations, Recent Developments and Applications. Springer, New York (1995)zbMATHGoogle Scholar
  15. Fox, J.-P.: Bayesian Item Response Modeling. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  16. Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984)CrossRefzbMATHGoogle Scholar
  17. Glas, C.A.W.: The derivation of some tests for the Rasch model from the multinomial distribution. Psychometrika 53, 525–546 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Glas, C.A.W., Verhelst, N.D.: Extensions of the partial credit model. Psychometrika 54, 635–659 (1989)CrossRefzbMATHGoogle Scholar
  19. 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
  20. 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
  21. Kelderman, H.: Loglinear Rasch model tests. Psychometrika 49, 223–245 (1984)CrossRefzbMATHGoogle Scholar
  22. Kelderman, H.: Item bias detection using loglinear IRT. Psychometrika 54, 681–697 (1989)MathSciNetCrossRefGoogle Scholar
  23. Levy, R., Mislevy, R.J.: Bayesian Psychometric Modeling. CRC Press, Boca Raton (2016)zbMATHGoogle Scholar
  24. Longhai, L.: gibbs.met: Naive Gibbs Sampling with Metropolis Steps. https://cran.r-project.org/web/packages/gibbs.met/index.html (2012)
  25. Lord, F.M., Novick, M.R.: Statistical Theories of Mental Test Scores. Addison-Wesley, Reading (1968)zbMATHGoogle Scholar
  26. Mahmoud, H.M.: Polya Urn Models. Chapman & Hall/CRC Press, Boca Raton (2008)CrossRefzbMATHGoogle Scholar
  27. 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)
  28. Martin-Löf, P.: Statistika Modeller. Institutet för Försäkringsmatematik och Mathematisk Statistisk vid Stockholms Universitet, Stockholm (1973)Google Scholar
  29. Miller, J.W., Harrison, M.T.: Exact sampling and counting for fixed-margin matrices. Ann. Stat. 41, 1569–1592 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Mislevy, R.J.: Randomization-based inference about latent variables from complex samples. Psychometrika 56, 177–196 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 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)CrossRefGoogle Scholar
  32. Molenaar, I.W.: Some improved diagnostics for failure of the Rasch model. Psychometrika 48, 49–72 (1983)CrossRefGoogle Scholar
  33. Plummer, M., Stukalov, A., Denwood, M.: rjags: Bayesian graphical models using MCMC. https://cran.r-project.org/web/packages/rjags/index.html (2016)
  34. Patterson, B., Atmar, W.: Nested subsets and the structure of insular mammalian faunas and archipelagos. Biol. J. Linn. Soc. 28, 65–82 (1986)CrossRefGoogle Scholar
  35. Ponocny, I.: Nonparametric goodness-of-fit tests for the Rasch model. Psychometrika 66, 437–459 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  36. Raiffa, H., Schlaifer, R.: Applied Statistical Decision Theory. Clinton Press, Boston (1961)zbMATHGoogle Scholar
  37. Rasch, G.: Probabilistic Models for Some Intelligence and Attainment Tests. The Danish Institute of Education Research, Copenhagen (1960)Google Scholar
  38. Robert, C., Casella, G.: Monte Carlo Statistical Methods. Springer, New York (2004)CrossRefzbMATHGoogle Scholar
  39. Rubin, D.B.: Bayesianly justifiable and relevant frequency calculations for the applied statistician. Ann. Stat. 12, 1151–1172 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  40. Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6, 461–464 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  42. van den Wollenberg, A.L.: Two new test statistics for the Rasch model. Psychometrika 47, 123–140 (1982)CrossRefzbMATHGoogle Scholar
  43. Verhelst, N.D.: An efficient MCMC algorithm to sample binary matrices with fixed marginals. Psychometrika 73, 705–728 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 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
  45. Verhelst, N.D., Hatzinger, R., Mair, P.: The RaschSampler. J. Stat. Softw. 20, 1–14 (2007)CrossRefGoogle Scholar
  46. 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

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.University for Health and Life SciencesHallAustria

Personalised recommendations