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Bayesian Estimation of a Multi-Unidimensional Graded Response IRT Model

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Abstract

Unidimensional graded response models are useful when items are designed to measure a unified latent trait. They are limited in practical instances where the test structure is not readily available or items are not necessarily measuring the same underlying trait. To overcome the problem, this paper proposes a multi-unidimensional normal ogive graded response model under the Bayesian framework. The performance of the proposed model was evaluated using Monte Carlo simulations. It was further compared with conventional polytomous models under simulated and real test situations. The results suggest that the proposed multi-unidimensional model is more general and flexible, and offers a better way to represent test situations not realized in unidimensional models.

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References

  • Albert, J.H. (1992). Bayesian estimation of normal ogive item response curves using Gibbs sampling. Journal of Educational Statistics, 17, 251–269.

    Article  Google Scholar 

  • Albert, J. H. & Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88, 669–679.

    Article  MathSciNet  Google Scholar 

  • Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 561–573.

    Article  Google Scholar 

  • Beguin, A. A. & Glas, C. A. W. (2001). MCMC estimation and some model-fit analysis of multidimensional IRT models. Psychometrika, 66, 541–562.

    Article  MathSciNet  Google Scholar 

  • Birnbaum, A. (1969). Statistical theory for logistic mental test models with a prior distribution of ability. Journal of Mathematical Psychology, 6, 258–276.

    Article  Google Scholar 

  • Buchanan, R. D. (1994). The development of the Minnesota Multiphasic Personality Inventory. Journal of the History of the Behavioral Sciences, 30, 148–161.

    Article  Google Scholar 

  • Carlin, B. P. & Louis, T. A. (2000). Bayes and empirical Bayes methods for data analysis (2nd ed.). London: Chapman & Hall.

    Book  Google Scholar 

  • Cowels, M. K. (1996). Accelerating Monte Carlo Markov chain convergence for cumulative-link generalized linear models. Statistics and Computing, 6, 101–111.

    Article  Google Scholar 

  • Ferero, C. G. & Maydeu-Olivares, A. (2009). Estimation of IRT graded response models: limited versus full Information methods. Psychological Methods, 14, 275–299.

    Article  Google Scholar 

  • Fox, J. P. (2010). Bayesian item response modeling: Theory and applications. New York, NY: Springer-Verlag.

    Book  Google Scholar 

  • Fu, Z. H.; Tao, J. & Shi, N. Z. (2010). Bayesian estimation of the multidimensional graded response model with nonignorable missing data. Journal of Statistical Computation and Simulation, 80, 1237–1252.

    Article  MathSciNet  Google Scholar 

  • Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (2003). Bayesian data analysis (2nd ed.). London: Chapman & Hall.

    MATH  Google Scholar 

  • Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97–109.

    Article  MathSciNet  Google Scholar 

  • Henley, N. M., Meng, K., O’Brien, D., McCarthy, W. J., & Sockloskie, R. J. (1998). Developing a Scale to Measure the diversity of feminist attitudes. Psychology of Women Quarterly, 22, 317–348.

    Article  Google Scholar 

  • Hoijtink, 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.

    Article  Google Scholar 

  • Kang, T. & Cohen, A. S. (2007). IRT model selection methods for dichotomous itemss. Applied Psychological Measurement, 31, 331–358.

    Article  MathSciNet  Google Scholar 

  • Kelderman, H. (1996). Multidimensional rasch models for partial-credit scoring. Applied Psychological Measurement, 20, 155–168.

    Article  Google Scholar 

  • Lee, H. (1995). Markov chain Monte Carlo methods for estimating multidimensional ability in item response theory (Doctoral dissertation). University of Missouri, Columbia, MO.

    Google Scholar 

  • Lee, S. Y. (2007). Structural equation modeling: A Bayesian approach. Chichester: John Wiley and Sons.

    Book  Google Scholar 

  • Likert, R. (1932). A technique for the measurement of attitudes. Archives of Psychology, 22, 5–55.

    Google Scholar 

  • Linacre, J. M. (2002). Optimizing rating scale category effectiveness. Journal of Applied Measurement, 3, 85–106.

    Google Scholar 

  • Lord, F. M. (1980). Applications of item response theory to practical testing problems (2nd ed.). New Jersey, NJ: Hillsdale.

    Google Scholar 

  • Lord, F. M. & Novick, M. R. (1968). Statistical theories of mental test scores (1st ed.). Maryland, MA: Addison-Wesley.

    MATH  Google Scholar 

  • Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 147–174.

    Article  Google Scholar 

  • Metropolis, N. & Ulam, S. (1949). The Monte Carlo method. Journal of the American Statistical Association, 44, 335–341.

    Article  MathSciNet  Google Scholar 

  • Molenaar, I. W. (1995). Estimation of item parameters (2nd ed.). New York, NY: Springer-Verlag.

    MATH  Google Scholar 

  • Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16, 159–176.

    Article  Google Scholar 

  • Muraki, E. (1990). Fitting a polytomous item response model to Likert-type data. Applied Psychological Measurement, 14, 59–71.

    Article  Google Scholar 

  • Patz, R. J., & Junker, B. W. (1999a). A straightforward approach to Markov chain Monte Carlo methods for item response models. Journal of Educational and Behavioral Statistics, 24, 146–178.

    Article  Google Scholar 

  • Plummer, M. (2008). Penalized loss functions for Bayesian model comparison. Biostatistics, 9, 523–539.

    Article  Google Scholar 

  • Patz, R. J. & Junker, B.W. (1999b). Applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses. Journal of Educational and Behavioral Statistics, 24, 342–366.

    Article  Google Scholar 

  • Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests (2nd ed.). Danmark: Danmarks Paedagogiske Institute.

    Google Scholar 

  • Reckase, M. (2009). Multidimensional item response theory (2nd ed.). New York, NY: Springer- Verlag.

    Book  Google Scholar 

  • Rizopoulos, D. (2006). ltm: An R package for latent variable modeling and item response theory analyses. Journal of Statistical Software, 17, 1–25.

    Article  Google Scholar 

  • Rubio, V. J., Aguado, D., Hontangas, P. M., & Hernandez, J. M. (2007). Psychometric properties of an emotional adjustment measure. European Journal of Psychological Assessment, 23, 39–46.

    Article  Google Scholar 

  • Sahu, S. K. (2002). Bayesian estimation and model choice in item response models. Journal of Statistical Computation and Simulation, 72, 217–232.

    Article  MathSciNet  Google Scholar 

  • Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika, 35, 139–139.

    Article  Google Scholar 

  • Sheng, Y. (2010). A sensitivity analysis of Gibbs sampling for 3PNO IRT models: Effects of prior =specific on parameter estimates. Behaviormetrika, 37, 87–110.

    Article  Google Scholar 

  • Sheng, Y. (2008). A MATLAB package forMarkov chain Monto Carlo with a multi-unidimensional IRT model. Journal of Statistical Software, 28, 1–20.

    Article  Google Scholar 

  • Sheng, Y. & Headrick, T. C. (2012). A Gibbs sampler for the multidimensional item response model. ISRN Applied Mathematics, 2012, 1–14.

    Article  MathSciNet  Google Scholar 

  • Sheng, Y. & Wikle, C. K. (2009). Bayesian IRT models in incorporating general and specific abilities. Behaviormetrika, 36, 27–48.

    Article  MathSciNet  Google Scholar 

  • Sheng, Y. & Wikle, C. K. (2008). Bayesian multidimensional IRT models with a hierarchical structure. Educational and Psychological Mesurement, 68, 413–430.

    Article  MathSciNet  Google Scholar 

  • Sheng, Y. & Wikle, C. K. (2007). Comparing multiunidimensional and unidimensional Item Response theroy models. Educational and Psychological Mesurement, 67, 899–919.

    Article  Google Scholar 

  • Sinharay, S. & Stern, H. S. (2003). Posterior predictive model checking in hierarchical models. Journal of Statistical Planning and Inference, 111, 209–221.

    Article  MathSciNet  Google Scholar 

  • Spiegelhalter, D. J., Best, N., Carlin, B., & van der Linde, A. (2002). Bayesian measures of model complexity and fit (with discussion). Journal of the Royal Statistical Society, Series B, 64, 583–640.

    Article  MathSciNet  Google Scholar 

  • Yao, L. (2003). BMIRT: Bayesian multivariate item response theory. Monterey, CA: CTB/McGraw-Hill.

    Google Scholar 

  • Yao, L. & Schwarz, R. (2006). A multidimensional partial credit model with associated item and test statistics: An application to mixed-format tests. Applied Psychological Measurement, 30, 469–492.

    Article  MathSciNet  Google Scholar 

  • Yao, L. & Boughton, K. A. (2007). A multidimensional item response modeling approach for improving subscale proficiency estimation and classification. Applied Psychological Measurement, 31, 83–105.

    Article  MathSciNet  Google Scholar 

  • Zhu, X. & Stone, C. A. (2011). Assessing fit of unidimensional graded response models using Bayesian methods. Journal of Educational Measurement, 48, 81–97.

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Educational Psychology and Special Education, Southern Illinois University Carbondale, Carbondale, IL, 62901-4618, USA

    Tzu-Chun Kuo & Yanyan Sheng

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  1. Tzu-Chun Kuo
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  2. Yanyan Sheng
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Correspondence to Tzu-Chun Kuo.

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Kuo, TC., Sheng, Y. Bayesian Estimation of a Multi-Unidimensional Graded Response IRT Model. Behaviormetrika 42, 79–94 (2015). https://doi.org/10.2333/bhmk.42.79

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  • DOI: https://doi.org/10.2333/bhmk.42.79

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