Brazilian Journal of Probability and Statistics

Extensions of the skew-normal ogive item response model

Jorge Luis Bazán, Márcia D. Branco, and Heleno Bolfarine

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We introduce new applications of the skew-probit IRT model by considering a flexible skew-normal distribution for the latent variables and by extending this model to include an additional random effect for modeling dependence between items within the same testlet. A Bayesian hierarchical structure is presented using a double data augmentation approach. This can be easily implemented in WinBUGS or SAS by considering MCMC algorithms. Several Bayesian model selection criteria, such as DIC, EAIC and EBIC, have been considered; in addition, we use posterior sum of squares of the latent residuals as a global discrepancy measure to model comparison. Two applications illustrate the methodology, one data set related to a mathematical test and another related to reading comprehension, both applied to Peruvian students. Results indicate better performance of the more flexible models proposed in this paper.

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Braz. J. Probab. Stat., Volume 28, Number 1 (2014), 1-23.

First available in Project Euclid: 5 February 2014

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Bayesian estimation item response models latent variables skew-normal ogive model skew-normal distribution skew-probit link testlet


Bazán, Jorge Luis; Branco, Márcia D.; Bolfarine, Heleno. Extensions of the skew-normal ogive item response model. Braz. J. Probab. Stat. 28 (2014), no. 1, 1--23. doi:10.1214/12-BJPS191.

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  • Albert, J. H. (1992). Bayesian estimation of normal ogive item response curves using Gibbs sampling. Journal of Educational Statistics 17, 251–269.
  • Albert, J. H. and Chib, S. (1995). Bayesian residual analysis for binary response regression models. Biometrika 82, 747–759.
  • Albert, J. H. and Ghosh, M. (2000). Item response modeling. In Generalized Linear Models: A Bayesian Perspective (D. K. Dey, S. K. Ghosh and B. F. Mallick, eds.) 173–193. New York: Marcel Dekker.
  • Arellano-Valle, R. B., Branco, M. D. and Genton, M. G. (2006). A unified view on skewed distributions arising from selections. The Canadian Journal of Statistics 34, 581–601.
  • Azzalini A. (1985). A class of distributions which includes the normal ones. Scandinavian Journal Statistical 12, 171–178.
  • Azevedo, C. L. N., Bolfarine, H. and Andrade, D. F. (2010). Bayesian inference for a skew-normal IRT model under the centred parameterization. Computational Statistics and Data Analysis 55, 353–365.
  • Baker, F. B. and Kim, S.-H. (2004). Item Response Theory—Parameter Estimation Techniques, 2nd ed. New York: Marcel Dekker.
  • Bazán, J. L., Branco, D. M. and Bolfarine, H. (2006). A skew item response model. Bayesian Analysis 1, 861–892.
  • Bazán, J. L., Bolfarine, H. and Branco, D. M. (2004). A new family of asymmetric models for item response theory: A Skew-Normal IRT Family. Technical report RT-MAE-2004-17, Dept. Statistics, Univ. São Paulo.
  • Bazán, J. L., Bolfarine, H. and Branco, D. M. (2010). A framework for skew-probit links in Binary regression. Communications in Statistics—Theory and Methods 39, 678–697.
  • Bolfarine, H. and Bazán, J. L. (2010). Bayesian estimation of the logistic positive exponent IRT model. Journal of Educational Behavioral Statistics 35, 693–713.
  • Bradlow, E., Wainer, H. and X. Wang (1999). A Bayesian random effect model for testlets. Psychometrika 64, 153–168.
  • Brooks, S. P. (2002). Discussion on “Bayesian measures of model complexity and fit” by Spiegelhalter, Best, Carlin and van der Linde (2002). Journal of the Royal Statistical Society, Ser. B 64, 616–618.
  • Camilli G. (1994). Origin of the scaling constant $d=1.7$ in item response theory. Journal of Educational and Behavioral Statistics 19, 293–295.
  • Carlin, B. P. and Louis, T. A. (2000). Bayes and Empirical Bayes Methods for Data Analysis, 2nd ed. Boca Raton, FL: Chapman & Hall.
  • Chen, M. H., Shao, Q. M. and Ibrahim, J. G. (2000). Monte Carlo Methods in Bayesian Computation. New York: Springer-Verlag.
  • Chincaro, O. (2010). Dichotomous Rasch model with application to education. M.Sc. thesis, Pontificia Universidad Católica del Perú (in Spanish).
  • Duncan, K. and MacEachern, S. (2008). Nonparametric Bayesian modelling for item response. Statistical Modeling 8, 41–66.
  • Ghosh, M., Ghosh, A., Chen, M.-H. and Agresti, A. (2000). Noninformative priors for one parameter item response models. Journal of Statistical Planning and Inference 88, 99–115.
  • Hashimoto, Y. (2002). Motivation and willingness to communicate as predictors of reported l2 use: The Japanese ESL context. Second Language Studies 20, 29–70.
  • Holland, P. and Rosenbaum, P. (1986). Conditional association and unidimensionality in monotone latent variable models. The Annals of Statistics 14, 1523–1543.
  • Henze, N. (1986). A probabilistic representation of the “skew-normal” distribution. Scandinavian Journal Statistical 13, 271–275.
  • Holmes, C. and Held, L. (2006). Bayesian auxiliary variable models for binary and multinomial regression. Bayesian Analysis 1, 145–168.
  • Jackman, S. (2004). Bayesian analysis for political research. Annual Review of Political Science 7, 483–505.
  • Johnson, T. (2003). On the use of heterogeneous thresholds ordinal regression models to account for individual differences in response style. Psychometrika 68, 563–583.
  • Junker, B. W. and Ellis, J. L. (1997). A characterization of monotone unidimensional latent variable models. The Annals of Statistics 25, 1327–1343.
  • Lunn, D. J., Thomas, A., Best, N. and Spiegelhalter, D. (2000). WinBUGS—A Bayesian modelling framework: Concepts, structure, and extensibility. Statistics and Computing 10, 325–337.
  • Micceri, T. (1989). The unicorn, the normal curve, and other improbable creatures. Psychological Bulletin 105, 156–166.
  • Patz, R. J. and Junker, B. W. (1999). A straightforward approach to Markov chain Monte Carlo methods for item response models. Journal of Educational and Behavioral Statistics 24, 146–178.
  • Poleto, F. Z., Paulino, C. D., Molenberghs, G. and Singer, J. M. (2011). Inferential implications of over-parametrization: A case study in incomplete categorical data. International Statistical Review 79, 92–113.
  • Rannala, B. (2002). Identifiability of parameters in MCMC Bayesian inference of phylogeny. Systematic Biology 51, 754–760.
  • Riddle, A. S., Blais, M. R. and Hess, U. (2002). A multi-group investigation of the CES-D’s measurement structure across adolescents, young adults and middle-aged adults. Scientific Series 2002s-36, Centre Interuniversitaire de recherche eit analysis des organizations, Montreal.
  • Rupp, A., Dey, D. K. and Zumbo, B. (2004). To Bayes or not to Bayes, from whether to when: Applications of Bayesian methodology to item response modeling. Structural Equations Modeling 11, 424–451.
  • Sahu, S. K. (2002). Bayesian estimation and model choice in item response models. Journal of Statistical Computation and Simulation 72, 217–232.
  • Samejima, F. (1997). Departure from normal assumptions: A promise for future psychometrics with substantive mathematical modeling. Psychometrika 62, 471–493.
  • Samejima, F. (2000). Logistic positive exponent family of models: Virtue of asymmetric item characteristics curves. Psychometrika 65, 319–335.
  • SAS Institute Inc. (2009). The MCMC Procedure, SAS/STAT Help Documentation. Cary, NC: SAS Institute Inc.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society, Ser. B 64, 583–639.
  • Tuerlinckx, F. and De Boeck, P. (2001). The effects of ignoring item interactions on the estimated discrimination parameters in item response theory. Psychological Methods 6, 181–195.
  • Wainer H. and Kiely, G. L. (1987). Item clusters and computerized adaptive testing: A case for testlets. Journal of Educational Measurement 24, 185–201.
  • Wainer, H. and Wang, X. (2000). Using a new statistical model for testlets to score TOEFL. Journal of Educational Measurement 37, 203–220.
  • Wainer, H., Bradlow, E. T. and Wang, X., eds. (2007). Testlet Response Theory and Its Applications. New York: Cambridge Univ. Press.
  • Wainer, H., Brown, L. M., Bradlow, E. T., Wang, X., Skorupski, W. P., Boulet, J. and Mislevy, R. J. (2006). An application of testlet response theory in the scoring of a complex certification examination. In Automated Scoring of Complex Tasks in Computer Based Testing (D. M. Williamson, R. J. Mislevy and I. I. Bejar, eds.) Chapter 6, 169–200. Hillsadle, NJ: Lawrence Erlbaum Associates.
  • Wang, X., Baldwin, S., Wainer, H., Bradlow, E. T., Reeve, B. B., Smith, A. W., Bellizzi, K. M. and Baumgartner, K. B. (2010). Using testlet response theory to analyze data from a survey of attitude change among breast cancer survivors. Statistics in Medicine 29, 2028–2044.
  • Wang, W.-C. and Wilson, M. (2005). The Rasch testlet model. Applied Psychological Measurement 29, 126–149.
  • Zaider, T. I., Heimberg, R. G., Fresco, D. M., Schneier, F. R. and Liebowitz, M. R. (2003). Evaluation of the Clinical Global Impression Scale among individuals with social anxiety disorder. Psychological Medicine 33, 611–622.