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On the Bayesian Nonparametric Generalization of IRT-Type Models

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Abstract

We study the identification and consistency of Bayesian semiparametric IRT-type models, where the uncertainty on the abilities’ distribution is modeled using a prior distribution on the space of probability measures. We show that for the semiparametric Rasch Poisson counts model, simple restrictions ensure the identification of a general distribution generating the abilities, even for a finite number of probes. For the semiparametric Rasch model, only a finite number of properties of the general abilities’ distribution can be identified by a finite number of items, which are completely characterized. The full identification of the semiparametric Rasch model can be only achieved when an infinite number of items is available. The results are illustrated using simulated data.

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References

  • Agresti, A., Caffo, B., & Ohman-Strickland, P. (2004). Examples in which misspecification of a random effects distribution reduces efficiency. Computational Statistics and Data Analysis, 47, 639–653.

    Article  Google Scholar 

  • Antoniak, C.E. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. The Annals of Statistics, 2, 1152–1174.

    Article  Google Scholar 

  • Bechger, T.M., Verhelst, N.D., & Verstralen, H.H.F.M. (2001). Identifiability of nonlinear logistic test models. Psychometrika, 66, 357–372.

    Article  Google Scholar 

  • Borsboom, D., Mellenbergh, G.J., & van Heerden, J. (2003). The theoretical status of latent variables. Psychological Review, 110, 203–219.

    Article  PubMed  Google Scholar 

  • Burr, D., & Doss, H. (2005). A Bayesian semiparametric model for random-effects meta-analysis. Journal of the American Statistical Association, 100, 242–251.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Chandra, S. (1977). On the mixture of probability distributions. Scandinavian Journal of Statistics, 4, 105–112.

    Google Scholar 

  • Conway, J.B. (1985). A course in functional analysis. New York: Springer.

    Google Scholar 

  • De Boeck, P., & Wilson, M. (2004). Explanatory item response models. a generalized linear and nonlinear approach. New York: Springer.

    Google Scholar 

  • De Leeuw, J., & Verhelst, N. (1986). Maximum likelihood estimation in generalized Rasch models. Journal of Educational Statistics, 11, 183–196.

    Article  Google Scholar 

  • Doob, J.L. (1949). Applications of the theory of martingales. Colloques Internationaux du Centre National de le Recherche Scientifique, 13, 23–27.

    Google Scholar 

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

    Article  Google Scholar 

  • Escobar, M.D., & West, M. (1995). Bayesian density estimation and inference using mixtures. Journal of the American Statistical Association, 90, 577–588.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Ferguson, T.S. (1974). Prior distribution on the spaces of probability measures. The Annals of Statistics, 2, 615–629.

    Article  Google Scholar 

  • Florens, J.-P., Mouchart, M., & Rolin, J.-M. (1990). Elements of Bayesian statistics. New York: Dekker.

    Google Scholar 

  • Florens, J.-P., & Rolin, J.-M. (1984). Asymptotic sufficiency and exact estimability. In Florens, J.-P., Mouchart, M., Raoult, J.-P., & Simar, L. (Eds.), Alternative approaches to time series analysis (pp. 121–142). Bruxelles: Publications des Facultés Universitaires Saint-Louis.

    Google Scholar 

  • Geisser, S., & Eddy, W. (1979). A predictive approach to model selection. Journal of the American Statistical Association, 74, 153–160.

    Article  Google Scholar 

  • Ghosh, M. (1995). Inconsistent maximum likelihood estimators for the Rasch model. Statistical and Probability Letters, 23, 165–170.

    Article  Google Scholar 

  • Hanson, T. (2006). Inference for mixtures of finite Polya tree models. Journal of the American Statistical Association, 101, 1548–1565.

    Article  Google Scholar 

  • Hanson, T., & Johnson, W.O. (2002). Modeling regression error with a mixture of Polya trees. Journal of the American Statistical Association, 97, 1020–1033.

    Article  Google Scholar 

  • Ishwaran, H. (1998). Markov-chain Monte Carlo: some practical implications of theoretical results. Discussion. The Canadian Journal of Statistics, 26(1), 20–27.

    Article  Google Scholar 

  • Jansen, M.G.H., & van Dujin, M.A.J. (1992). Extensions of Rasch’s multiplicative Poisson model. Psychometrika, 57, 405–414.

    Article  Google Scholar 

  • Jara, A. (2007). Applied Bayesian non- and semi-parametric inference using DPpackage. Rnews, 7(3), 17–26.

    Google Scholar 

  • Jara, A., Hanson, T., & Lesaffre, E. (2009). Robustifying generalized linear mixed models using a new class of mixtures of multivariate Polya trees. Journal of Computational and Graphical Statistics, 18, 838–860.

    Article  Google Scholar 

  • Kadane, J. (1975). The role of identification in Bayesian theory. In Fienberg, S., & Zellner, A. (Eds.), Studies in Bayesian econometrics and statistics (pp. 175–191). Amsterdam: North-Holland.

    Google Scholar 

  • Karabatsos, G., & Walker, S. (2009). Coherent psychometric modelling with Bayesian nonparametrics. British Journal of Mathematical and Statistical Psychology, 62, 1–20.

    Article  PubMed  Google Scholar 

  • Kiefer, J., & Wolfowitz, J. (1956). Consistency of the maximum likelihood estimators in the presence of infinitely many incidental parameters. The Annals of Mathematical Statistics, 27, 887–906.

    Article  Google Scholar 

  • Kleinman, K.P., & Ibrahim, J.G. (1998a). A semi-parametric Bayesian approach to generalized linear mixed models. Statistics in Medicine, 17, 2579–2596.

    Article  PubMed  Google Scholar 

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

    Article  PubMed  Google Scholar 

  • Lavine, M. (1992). Some aspects of Polya tree distributions for statistical modeling. The Annals of Statistics, 20, 1222–1235.

    Article  Google Scholar 

  • Lavine, M. (1994). More aspects of Polya tree distributions for statistical modeling. The Annals of Statistics, 22, 1161–1176.

    Article  Google Scholar 

  • Li, Y., Lin, X., & Müller, P. (2009). Bayesian inference in semiparametric mixed models for longitudinal data. Biometrics, 66, 70–78.

    Article  PubMed  Google Scholar 

  • Lindley, D.V. (1971). Bayesian statistics: a review. Montpelier: Society for Industrial and Applied Mathematics.

    Google Scholar 

  • Lindsay, B.G., Clogg, C.C., & Grego, J. (1991). Semiparametric estimation in the Rasch model and related exponential response models, including a simple latent class model in item analysis. Journal of the American Statistical Association, 86, 96–107.

    Article  Google Scholar 

  • Maris, G., & Bechger, T.M. (2004). Equivalent MIRID models. Psychometrika, 69, 627–639.

    Article  Google Scholar 

  • Miyazaki, K., & Hoshino, T. (2009). A Bayesian semiparametric item response model with Dirichlet process priors. Psychometrika, 74, 375–393.

    Article  Google Scholar 

  • Mouchart, M., & Rolin, J.M. (1984). A note on conditional independence with statistical applications. Statistica, 44, 557–584.

    Google Scholar 

  • Mouchart, M., & San Martin, E. (2003). Specification and identification issues in models involving a latent hierarchical structure. Journal of Statistical Planning and Inference, 111, 143–163.

    Article  Google Scholar 

  • Mukhopadhyay, S., & Gelfand, A.E. (1997). Dirichlet process mixed generalized linear models. Journal of the American Statistical Association, 92, 633–647.

    Article  Google Scholar 

  • Müller, P., & Rosner, G.L. (1997). A Bayesian population model with hierarchical mixture priors applied to blood count data. Journal of the American Statistical Association, 92, 1279–1292.

    Article  Google Scholar 

  • Neyman, J., & Scott, E. (1948). Consistent estimates based on partially consistent observations. Econometrica, 16, 1–32.

    Article  Google Scholar 

  • Pfanzagl, J. (1993). Incidental versus random nuisance parameters. The Annals of Statistics, 21, 1663–1691.

    Article  Google Scholar 

  • Picci, G. (1977). Some connections between the theory of sufficient statistics and the identifiability problem. SIAM Journal on Applied Mathematics, 33, 383–398.

    Article  Google Scholar 

  • Rao, M.M. (1984). Probability theory with applications. New York: Academic Press.

    Google Scholar 

  • Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: The Danish Institute for Educational Research.

    Google Scholar 

  • Revuelta, J. (2009). Identifiability and estimability of GLLIRM models. Psychometrika, 74, 257–272.

    Article  Google Scholar 

  • Revuelta, J. (2010). Estimating difficulty from polytomous categorial data. Psychometrika, 75, 331–350.

    Article  Google Scholar 

  • Roberts, G.O., & Rosenthal, J. (1998). Markov-chain Monte Carlo: some practical implications of theoretical results. The Canadian Journal of Statistics, 26(1), 5–20.

    Article  Google Scholar 

  • San Martín, E. (2000). Latent structural models: specification and identification problems. Belgium: Ph.D. Dissertation, Institute of Statistics, Université Catholique de Louvain.

  • San Martin, E., Del Pino, G., & De Boeck, P. (2006). IRT models for ability-based guessing. Applied Psychological Measurement, 30, 183–203.

    Article  Google Scholar 

  • San Martín, E., & Mouchart, M. (2007). On joint completeness: sampling and Bayesian versions, and their connections. Sankhyā, 69, 780–807.

    Google Scholar 

  • Teicher, H. (1961). Identifiability of mixtures. The Annals of Statistics, 32, 244–248.

    Article  Google Scholar 

  • Walker, S.G., & Mallick, B.K. (1997). Hierarchical generalized linear models and frailty models with Bayesian nonparametric mixing. Journal of the Royal Statistical Society, Series B, 59, 845–860.

    Article  Google Scholar 

  • Wasserman, L. (1998). Asymptotic properties of nonparametric Bayesian procedures. In Dey, D., Müller, P., & Sinha, D. (Eds.), Developments in statistical inference and data analysis (pp. 293–304). New York: Springer.

    Google Scholar 

  • Woods, C.M. (2006). Ramsay-curve item response theory (RC-IRT) to detect and correct for nonnormal latent variables. Psychological Methods, 11, 253–270.

    Article  PubMed  Google Scholar 

  • Woods, C.M. (2008). Ramsay-curve item response theory for the three-parameter logistic item response model. Applied Psychological Measurement, 32, 447–465.

    Article  Google Scholar 

  • Woods, C.M., & Thissen, D. (2006). Item response theory with estimation of the latent populational distribution using spline-based densities. Psychometrika, 71, 281–301.

    Article  Google Scholar 

  • Yang, M., & Dunson, D.B. (2010a). Bayesian semiparametric structural equation models with latent variables. Psychometrika, 75, 675–693.

    Article  Google Scholar 

  • Yang, M., & Dunson, D.B. (2010b). Semiparametric Bayes hierarchical models with mean and variance constraints. Computational Statistics and Data Analysis, 54, 2172–2186.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Statistics & Measurement Center MIDE UC, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile

    Ernesto San Martín

  2. Department of Statistics, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile

    Alejandro Jara

  3. Institut de Statistique, Biostatistique et Sciences Actuarielles, Université catholique de Louvain, Louvain-la-Neuve, Belgium

    Jean-Marie Rolin & Michel Mouchart

Authors
  1. Ernesto San Martín
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  2. Alejandro Jara
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  3. Jean-Marie Rolin
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  4. Michel Mouchart
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Correspondence to Ernesto San Martín.

Additional information

The first author was partially supported by CEPPE CIEO-01-CONICYT grant and PUENTE grant 08/2009 from the Pontificia Universidad Católica de Chile. The second author is supported by FONDECYT 3095003 and 11100144 grants. The last two authors were partially supported by the Interuniversity Attraction Poles Program P6/03 from the Belgian State Federal Office for Scientific, Technical and Cultural Affairs.

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San Martín, E., Jara, A., Rolin, JM. et al. On the Bayesian Nonparametric Generalization of IRT-Type Models. Psychometrika 76, 385–409 (2011). https://doi.org/10.1007/s11336-011-9213-9

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  • DOI: https://doi.org/10.1007/s11336-011-9213-9

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