Advertisement

Psychometrika

, Volume 82, Issue 4, pp 952–978 | Cite as

A Nonparametric Multidimensional Latent Class IRT Model in a Bayesian Framework

  • Francesco Bartolucci
  • Alessio Farcomeni
  • Luisa Scaccia
Article
  • 240 Downloads

Abstract

We propose a nonparametric item response theory model for dichotomously-scored items in a Bayesian framework. The model is based on a latent class (LC) formulation, and it is multidimensional, with dimensions corresponding to a partition of the items in homogenous groups that are specified on the basis of inequality constraints among the conditional success probabilities given the latent class. Moreover, an innovative system of prior distributions is proposed following the encompassing approach, in which the largest model is the unconstrained LC model. A reversible-jump type algorithm is described for sampling from the joint posterior distribution of the model parameters of the encompassing model. By suitably post-processing its output, we then make inference on the number of dimensions (i.e., number of groups of items measuring the same latent trait) and we cluster items according to the dimensions when unidimensionality is violated. The approach is illustrated by two examples on simulated data and two applications based on educational and quality-of-life data.

Keywords

cluster analysis encompassing priors item response theory unidimensionality Markov chain Monte Carlo reversible-jump algorithm stochastic partitions 

References

  1. Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In B. N. Petrov & F. Csaki (Eds.), Second international symposium of information theory (pp. 267–281). Budapest: Akademiai Kiado.Google Scholar
  2. Bacci, S., & Bartolucci, F. (2016). Two-tier latent class IRT models in R. The R Journal, 8, 139–166.Google Scholar
  3. Bacci, S., Bartolucci, F., & Gnaldi, M. (2014). A class of multidimensional latent class IRT models for ordinal polytomous item responses. Communication in Statistics - Theory and Methods, 43, 787–800.CrossRefGoogle Scholar
  4. Bartolucci, F. (2007). A class of multidimensional IRT models for testing unidimensionality and clustering items. Psychometrika, 72, 141–157.CrossRefGoogle Scholar
  5. Bartolucci, F., Bacci, S., & Gnaldi, M. (2015). Statistical analysis of questionnaires: A unified approach based on Stata and R. Boca Raton, FL: Chapman and Hall/CRC Press.Google Scholar
  6. Bartolucci, F., Bacci, S., & Gnaldi, M. (2016). MultiLCIRT: multidimensional latent class item response theory models. In R package version, version 2.10. https://cran.r--project.org/web/packages/MultiLCIRT/index.html.
  7. Bartolucci, F., & Forcina, A. (2005). Likelihood inference on the underlying structure of IRT models. Psychometrika, 70, 31–43.CrossRefGoogle Scholar
  8. Bartolucci, F., Scaccia, L., & Farcomeni, A. (2012). Bayesian inference through encompassing priors and importance sampling for a class of marginal models for categorical data. Computational Statistics and Data Analysis, 56, 4067–4080.CrossRefGoogle Scholar
  9. Béguin, A. A., & Glas, C. A. W. (2001). MCMC estimation and some model-fit analysis of multidimensional IRT models. Psychometrika, 66, 541–561.CrossRefGoogle Scholar
  10. Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee’s ability. In F. M. Lord & M. R. Novick (Eds.), Statistical theories of mental test scores (pp. 395–479). Reading, MA: Addison-Wesley.Google Scholar
  11. Bock, R., Gibbons, R. D., & Muraki, E. (1988). Full-information item factor analysis. Applied Psychological Measurement, 12, 261–280.CrossRefGoogle Scholar
  12. Bolt, D. M., & Lall, V. F. (2003). Estimation of compensatory and noncompensatory multidimensional item response models using Markov chain Monte Carlo. Applied Psychological Measurement, 27, 395–414.CrossRefGoogle Scholar
  13. Chalmers, P., Pritikin, J., Robitzsch, A., Zoltak, M., Kim, K., Falk, C. F., & Meade, A. (2017). MIRT: Multidimensional item response theory. In R package version, version 1.23. https://cran.r--project.org/web/packages/mirt/index.html.
  14. Christensen, K. B., Bjorner, J. B., Kreiner, S., & Petersen, J. H. (2002). Testing unidimensionality in polytomous Rasch models. Psychometrika, 67, 563–574.CrossRefGoogle Scholar
  15. Costantini, M., Musso, M., Viterbori, P., Bonci, F., Del Mastro, L., Garrone, O., et al. (1999). Detecting psychological distress in cancer patients: Validity of the Italian version of the hospital anxiety and depression scale. Support Care Cancer, 7, 121–127.CrossRefPubMedGoogle Scholar
  16. Diebolt, J., & Robert, C. (1994). Estimation of finite mixture distributions through Bayesian sampling. Journal of the Royal Statistical Society, Series B, 56, 363–375.Google Scholar
  17. Goodman, L. A. (1974). Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika, 61, 215–231.CrossRefGoogle Scholar
  18. Graham, R. L., Knuth, D. E., & Patashnik, O. (1988). Concrete mathematics: A foundation for computer science. Reading, MA: Addison-Wesley.Google Scholar
  19. Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82, 711–732.CrossRefGoogle Scholar
  20. Green, P. J., & Richardson, S. (2001). Hidden Markov models and disease mapping. Journal of the American Statistical Association, 97, 1055–1070.CrossRefGoogle Scholar
  21. Hambleton, R. K., & Swaminathan, H. (1985). Item response theory: Principles and applications. Boston: Kluwer Nijhoff.CrossRefGoogle Scholar
  22. Hojtink, H., & Molenaar, I. W. (1997). A multidimensional item response model: Constrained latent class analysis using the Gibbs sampler and posterior predictive checks. Psychometrika, 62, 171–189.CrossRefGoogle Scholar
  23. Hurvich, C. M., & Tsai, C.-L. (1989). Regression and time series model selection in small samples. Biometrika, 76, 297–307.CrossRefGoogle Scholar
  24. Junker, B. W., & Sijtsma, K. (2001). Nonparametric item response theory in action: An overview of the special issue. Applied Psychological Measurement, 25, 211–220.CrossRefGoogle Scholar
  25. Karabatsos, G. (2001). The Rasch model, additive conjoint measurement, and new models of probabilistic measurement theory. Journal of Applied Measurement, 2, 389–423.PubMedGoogle Scholar
  26. Kass, R. E., & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90, 773–795.CrossRefGoogle Scholar
  27. Klugkist, I., Kato, B., & Hoijtink, H. (2005). Bayesian model selection using encompassing priors. Statistica Nederlandica, 59, 57–69.CrossRefGoogle Scholar
  28. Kuo, T., & Sheng, Y. (2015). Bayesian estimation of a multi-unidimensional graded response IRT model. Behaviormetrika, 42, 79–94.CrossRefGoogle Scholar
  29. Lazarsfeld, P. F., & Henry, N. W. (1968). Latent structure analysis. Boston: Houghton Mifflin.Google Scholar
  30. Lindley, D. V. (1957). A statistical paradox. Biometrika, 44, 187–192.CrossRefGoogle Scholar
  31. Lindsay, B., Clogg, C., & Grego, J. (1991). Semiparametric estimation in the Rasch model and related exponential response models, including a simple latent class model for item analysis. Journal of the American Statistical Association, 86, 96–107.CrossRefGoogle Scholar
  32. Martin-Löf, P. (1973). Statistiska modeller. Stockholm: Institütet för Försäkringsmatemetik och Matematisk Statistisk vid Stockholms Universitet.Google Scholar
  33. Pan, J. C., & Huang, G. H. (2014). Bayesian inferences of latent class models with an unknown number of classes. Psychometrika, 79, 621–646.CrossRefPubMedGoogle Scholar
  34. Rasch, G. (1961). On general laws and the meaning of measurement in psychology. Proceedings of the IV Berkeley Symposium on Mathematical Statistics and Probability, 4, 321–333.Google Scholar
  35. Reckase, M. D. (2009). Multidimensional item-response theory. New York: Springer.CrossRefGoogle Scholar
  36. Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461–464.CrossRefGoogle Scholar
  37. Tierney, L. (1994). Markov chains for exploring posterior distributions. Annals of Statistics, 22, 1701–1762.CrossRefGoogle Scholar
  38. Tuyl, F., Gerlach, R., & Mengersen, K. (2009). Posterior predictive arguments in favor of the Bayes–Laplace prior as the consensus prior for binomial and multinomial parameters. Bayesian Analysis, 4, 151–158.CrossRefGoogle Scholar
  39. Van Onna, M. J. H. (2002). Bayesian estimation and model selection in ordered latent class models for polytomous items. Psychometrika, 67, 519–538.CrossRefGoogle Scholar
  40. Verhelst, N. D. (2001). Testing the unidimensionality assumption of the Rasch model. Methods of Psychological Research Online, 6, 231–271.Google Scholar
  41. Vermunt, J. K. (2001). The use of restricted latent class models for defining and testing nonparametric and parametric item response theory models. Applied Psychological Measurement, 25, 283–294.CrossRefGoogle Scholar
  42. von Davier, M. (2008). A general diagnostic model applied to language testing data. British Journal of Mathematical and Statistical Psychology, 61, 287–307.CrossRefGoogle Scholar
  43. Zigmond, A. S., & Snaith, R. P. (1983). The hospital anxiety and depression scale. Acta Psychiatrika Scandinavica, 67, 361–370.CrossRefGoogle Scholar

Copyright information

© The Psychometric Society 2017

Authors and Affiliations

  1. 1.Dipartimento di EconomiaUniversità di PerugiaPerugiaItaly
  2. 2.Dipartimento di Sanità Pubblica e Malattie InfettiveSapienza - Università di RomaRomeItaly
  3. 3.Dipartimento di Economia e DirittoUniversità di MacerataMacerataItaly

Personalised recommendations