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A Bayesian Semiparametric Item Response Model with Dirichlet Process Priors

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Abstract

In Item Response Theory (IRT), item characteristic curves (ICCs) are illustrated through logistic models or normal ogive models, and the probability that examinees give the correct answer is usually a monotonically increasing function of their ability parameters. However, since only limited patterns of shapes can be obtained from logistic models or normal ogive models, there is a possibility that the model applied does not fit the data. As a result, the existing method can be rejected because it cannot deal with various item response patterns.

To overcome these problems, we propose a new semiparametric IRT model using a Dirichlet process mixture logistic distribution. Our method does not rely on assumptions but only requires that the ICCs be a monotonically nondecreasing function; that is, our method can deal with more types of item response patterns than the existing methods, such as the one-parameter normal ogive models or the two- or three-parameter logistic models.

We conducted two simulation studies whose results indicate that the proposed method can express more patterns of shapes for ICCs and can estimate the ability parameters more accurately than the existing parametric and nonparametric methods. The proposed method has also been applied to Facial Expression Recognition data with noteworthy results.

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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 

  • Ansari, A., & Iyengar, R. (2006). Semiparametric Thurstonian models for recurrent choices: A Bayesian analysis. Psychometrika, 71(4), 631–657.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Bock, R.D. (1997). The nominal categories model. In W.M. van der Linden & R.K. Hambleton (Eds.), Handbook of modern item response theory (pp. 33–49). Berlin: Springer.

    Google Scholar 

  • Bock, R.D., & Zimowski, M.F. (1997). Multiple group IRT. In W.M. van der Linden & R.K. Hambleton (Eds.), Handbook of modern item response theory (pp. 433–448). Berlin: Springer.

    Google Scholar 

  • Bush, C.A., & MacEashern, S.N. (1996). A semiparametric Bayesian model for randomised block designs. Biometrica, 83, 275–285.

    Article  Google Scholar 

  • Diebolt, J., & Robert, C.P. (1994). Estimation of finite mixture distributions through Bayesian sampling. Journal of the Royal Statistical Society, Series B, 56(2), 363–375.

    Google Scholar 

  • Duncan, K.A., & MacEachern, S.N. (2008). Nonparametric Bayesian modelling for item response. Statistical Modelling, 8(1), 41–66.

    Article  Google Scholar 

  • Dunson, D.B. (2006). Bayesian dynamic modeling of latent trait distributions. Biostatistics, 7, 551–568.

    Article  PubMed  Google Scholar 

  • Dunson, D.B., Pillai, N., & Park, J. (2007). Bayesian density regression. Journal of the Royal Statistical Society, Series B, 69, 163–183.

    Article  Google Scholar 

  • Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Annals of Statistics, 1, 209–230.

    Article  Google Scholar 

  • Hoshino, T. (2009). Dirichlet process mixtures of structural equation modeling and direct calculation of posterior probabilities of the numbers of components. Psychometrika (accepted for publication).

  • Ishwaran, H., & James, L.F. (2001). Gibbs sampling methods for Stick-Breaking priors. Journal of the American Statistical Association, 96, 161–173.

    Article  Google Scholar 

  • Ishwaran, H., & James, L.F. (2002). Approximate Dirichlet process computing in finite normal mixtures: Smoothing and prior information. Journal of Computational and Graphical Statistics, 11, 508–532.

    Article  Google Scholar 

  • Ishwaran, H., & Zarepour, M. (2000). Markov Chain Monte Carlo in approximate Dirichlet and beta two-parameter process hierarchical models. Biometrika, 87, 371–390.

    Article  Google Scholar 

  • Ishwaran, H., & Zarepour, M. (2002). Dirichlet prior sieves in finite normal mixtures. Statistica Sinica, 12, 941–963.

    Google Scholar 

  • Ishwaran, H., James, L.F., & Sun, J. (2001). Bayesian model selection in finite mixtures by marginal density decompositions. Journal of the American Statistical Association, 96, 1316–1332.

    Article  Google Scholar 

  • Junker, B.W., & Sijtsma, K. (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory. Applied Psychological Measurement, 25(3), 258–272.

    Article  Google Scholar 

  • Kleinman, K.P., & Ibrahim, J.G. (1998). A semiparametric Bayesian approach to the random effects model. Biometrika, 54, 921–938.

    Google Scholar 

  • Ramsey, J.O. (1991). Kernel smoothing approaches to nonparametric item characteristic curve estimation. Psychometrika, 56, 611–630.

    Article  Google Scholar 

  • Ramsey, J.O., & Winsberg, S. (1991). Maximum marginal likelihood estimation for semiparametric item analysis. Psychometrika, 56, 365–379.

    Article  Google Scholar 

  • Reckase, M.D. (1985). The difficulty of test items that measure more than one ability. Applied Psychological Measurement, 9(4), 401–412.

    Article  Google Scholar 

  • Reckase, M.D., & Mackinly, R.L. (1991). The discriminating power of items that measure more than one dimension. Applied Psychological Measurement, 15(4), 361–373.

    Article  Google Scholar 

  • Robert, C.P., & Mengersen, K.L. (1999). Reparameterisation issues in mixture modelling and their bearing on MCMC algorithms. Computational Statistics and Data Analysis, 29, 325–343.

    Article  Google Scholar 

  • Rost, J. (1990). Rasch models in latent classes: An integration of two approaches to item analysis. Applied Psychological Measurement, 14(3), 271–282.

    Article  Google Scholar 

  • Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statistica Sinica, 4, 639–650.

    Google Scholar 

  • Sijtsma, K., & Molenaar, I.W. (2002). Introduction to nonparametric item response theory. Sage.

  • Stephens, M. (2000). Dealing with label switching in mixture models. Journal of the Royal Statistical Society, Series B, 62(4), 795–809.

    Article  Google Scholar 

  • Suzuki, A., Hoshino, T., & Shigemasu, K. (2006). Measuring individual differences in sensitivities to basic emotions in faces. Cognition, 99, 327–353.

    Article  PubMed  Google Scholar 

  • von Davier, M. (2006). Introduction to Rasch measurement. Applied Psychological Measurement, 30(5), 443–446.

    Article  Google Scholar 

  • von Davier, M., & Rost, J. (1995). Mixture distribution Rasch models. In G.H. Fischer & I.W. Molenaar (Eds.), Rasch models—Foundations, recent developments, and applications (pp. 257–268). New York: Springer.

    Google Scholar 

  • Walker, S.G., Damien, P., Laud, P.W., & Smith, A.F.M. (1999). Bayesian nonparametric inference for random distributions and related functions. Journal of the Royal Statistical Society, Series B, 61, 485–527.

    Article  Google Scholar 

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Correspondence to Takahiro Hoshino.

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Miyazaki, K., Hoshino, T. A Bayesian Semiparametric Item Response Model with Dirichlet Process Priors. Psychometrika 74, 375–393 (2009). https://doi.org/10.1007/s11336-008-9108-6

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