# A Nonparametric Multidimensional Latent Class IRT Model in a Bayesian Framework

- 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

- 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 - Bacci, S., & Bartolucci, F. (2016). Two-tier latent class IRT models in R.
*The R Journal*,*8*, 139–166.Google Scholar - 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 - Bartolucci, F. (2007). A class of multidimensional IRT models for testing unidimensionality and clustering items.
*Psychometrika*,*72*, 141–157.CrossRefGoogle Scholar - 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 - 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. - Bartolucci, F., & Forcina, A. (2005). Likelihood inference on the underlying structure of IRT models.
*Psychometrika*,*70*, 31–43.CrossRefGoogle Scholar - 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 - 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 - 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 - Bock, R., Gibbons, R. D., & Muraki, E. (1988). Full-information item factor analysis.
*Applied Psychological Measurement*,*12*, 261–280.CrossRefGoogle Scholar - 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 - 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. - Christensen, K. B., Bjorner, J. B., Kreiner, S., & Petersen, J. H. (2002). Testing unidimensionality in polytomous Rasch models.
*Psychometrika*,*67*, 563–574.CrossRefGoogle Scholar - 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 - 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 - Goodman, L. A. (1974). Exploratory latent structure analysis using both identifiable and unidentifiable models.
*Biometrika*,*61*, 215–231.CrossRefGoogle Scholar - Graham, R. L., Knuth, D. E., & Patashnik, O. (1988).
*Concrete mathematics: A foundation for computer science*. Reading, MA: Addison-Wesley.Google Scholar - Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination.
*Biometrika*,*82*, 711–732.CrossRefGoogle Scholar - Green, P. J., & Richardson, S. (2001). Hidden Markov models and disease mapping.
*Journal of the American Statistical Association*,*97*, 1055–1070.CrossRefGoogle Scholar - Hambleton, R. K., & Swaminathan, H. (1985).
*Item response theory: Principles and applications*. Boston: Kluwer Nijhoff.CrossRefGoogle Scholar - 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 - Hurvich, C. M., & Tsai, C.-L. (1989). Regression and time series model selection in small samples.
*Biometrika*,*76*, 297–307.CrossRefGoogle Scholar - 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 - 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 - Kass, R. E., & Raftery, A. E. (1995). Bayes factors.
*Journal of the American Statistical Association*,*90*, 773–795.CrossRefGoogle Scholar - Klugkist, I., Kato, B., & Hoijtink, H. (2005). Bayesian model selection using encompassing priors.
*Statistica Nederlandica*,*59*, 57–69.CrossRefGoogle Scholar - Kuo, T., & Sheng, Y. (2015). Bayesian estimation of a multi-unidimensional graded response IRT model.
*Behaviormetrika*,*42*, 79–94.CrossRefGoogle Scholar - Lazarsfeld, P. F., & Henry, N. W. (1968).
*Latent structure analysis*. Boston: Houghton Mifflin.Google Scholar - 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 - Martin-Löf, P. (1973).
*Statistiska modeller*. Stockholm: Institütet för Försäkringsmatemetik och Matematisk Statistisk vid Stockholms Universitet.Google Scholar - 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 - 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 - Reckase, M. D. (2009).
*Multidimensional item-response theory*. New York: Springer.CrossRefGoogle Scholar - Schwarz, G. (1978). Estimating the dimension of a model.
*Annals of Statistics*,*6*, 461–464.CrossRefGoogle Scholar - Tierney, L. (1994). Markov chains for exploring posterior distributions.
*Annals of Statistics*,*22*, 1701–1762.CrossRefGoogle Scholar - 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 - 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 - Verhelst, N. D. (2001). Testing the unidimensionality assumption of the Rasch model.
*Methods of Psychological Research Online*,*6*, 231–271.Google Scholar - 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 - 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 - Zigmond, A. S., & Snaith, R. P. (1983). The hospital anxiety and depression scale.
*Acta Psychiatrika Scandinavica*,*67*, 361–370.CrossRefGoogle Scholar