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Estimating the Latent Trait Distribution with Loglinear Smoothing Models

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New Developments in Quantitative Psychology

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 66))

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Abstract

This research provides a framework for specifying the latent trait distribution using the family of loglinear smoothing models. Connections between loglinear smoothing models and the standard normal, normal, and direct estimation of the distribution are the beginning of this framework. The framework is important because it gives the connection between the standard approaches and loglinear smoothing models so that parsimonious (smoothing) models providing adequate representation of the distribution can be identified. Future extensions will include additional complex models for estimating the latent trait distribution.

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Notes

  1. 1.

    Note that in the present paper, we label estimation methods based on the model for the latent distribution only; while it is acknowledged that the parametric form of the item response function may change, it will be restricted to having a parametric form throughout this discussion.

  2. 2.

    LLSEM: LogLinear Smoothing Expectation Maximization (Casabianca and Lewis 2011) is a software available upon request by Jodi M. Casabianca (jodicasa@andrew.cmu.edu).

  3. 3.

    The maximum absolute difference between the estimated item characteristic curves (ICCs) and the true ICCs was used to assess overall recovery of item parameters. That is, the absolute difference between the ICC using estimated item parameters and the ICC using the true item parameters was computed for each item across the Q quadrature points. Within condition, item, and replication, the maximum of these absolute differences (over the Q quadrature points) was determined. The mean of the absolute maximum difference was taken across the 50 items, and the mean was also taken across replications.

References

  • Bock, R. D., & Aitkin, M. (1981). MML estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443–459.

    Article  MathSciNet  Google Scholar 

  • Bock, R. D., & Lieberman, M. (1970). Fitting a response model for n dichotomously scored items. Psychometrika, 35, 179–197.

    Article  Google Scholar 

  • Boulet, J. R. (1996). The effect of nonnormal ability distributions on IRT parameter estimation using full-information and limited-information methods (IRT, nonlinear factor analysis). Dissertation abstracts online, University of Ottawa, Canada.

    Google Scholar 

  • Cai, L. (2010). A two-tier full-information item factor analysis model with applications. Psychometrika, 75, 581–612.

    Article  MathSciNet  MATH  Google Scholar 

  • Casabianca, J. M. (2011). Loglinear smoothing for the latent trait distribution: A two-tiered evaluation (Doctoral Dissertation). ProQuest Dissertations and Theses (Accession Order No. AAT 3474125).

    Google Scholar 

  • Casabianca, J. M., & Lewis, C. (2011). LLSEM: A computer program for loglinear smoothing in an expectation maximization algorithm (Unpublished software). Bronx, NY

    Google Scholar 

  • Casabianca, J. M., & Lewis, C. (2012). Loglinear smoothing for the latent trait distribution in the marginal maximum likelihood estimation of 3PL item parameters (Unpublished manuscript).

    Google Scholar 

  • Casabianca, J. M., Xu, X., Jia, Y., & Lewis, C. (2010). Estimation of item parameters when the underlying latent trait distribution of test takers is nonnormal. In Annual meeting of the National Council for Measurement in Education, Denver, Colorado.

    Google Scholar 

  • Dempster, A., Laird, N., & Rubin, D. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1–38.

    MathSciNet  MATH  Google Scholar 

  • Haberman, S. J. (2005). Identifiability of parameters in item response models with unconstrained ability distributions (Research Report 05-24). Princeton, NJ: Educational Testing Service.

    Google Scholar 

  • Haberman, S. J., von Davier, M., & Lee, Y. H. (2008). Comparison of multidimensional item response models: Multivariate normal ability distributions versus multivariate polytomous distributions (Research Report 08-45). Princeton, NJ: Educational Testing Service.

    Google Scholar 

  • Heinen, T. (1996). Latent class and discrete latent trait models. Thousand Oaks, CA: Sage.

    Google Scholar 

  • Holland, P. W., & Thayer, D. T. (1987). Notes on the use of log-linear models for fitting discrete probability distributions (Research Report 87-31). Princeton, NJ: Educational Testing Service.

    Google Scholar 

  • Holland, P. W., & Thayer, D. T. (1998). Univariate and bivariate loglinear models for discrete test score distributions (Research Report 98-1). Princeton, NJ: Educational Testing Service.

    Google Scholar 

  • Holland, P. W., & Thayer, D. T. (2000). Univariate and bivariate loglinear models for discrete test score distributions. Journal of Educational and Behavioral Statistics, 25, 133–183.

    Article  Google Scholar 

  • Laird, N. M. (1978). Nonparametric likelihood estimation of a mixing distribution. Journal of American Statistical Association, 73, 805–811.

    Article  MathSciNet  MATH  Google Scholar 

  • Mislevy, R. J. (1984). Estimating latent distributions. Psychometrika, 49, 359–381.

    Article  MATH  Google Scholar 

  • Mislevy, R. J., & Bock, R. D. (1985). Implementation of the EM Algorithm in the estimation of item parameters: The BILOG computer program. In D. J. Weiss (Ed.), Proceedings of the 1982 IRT and computerized adaptive testing conference (pp. 189–202). Minneapolis: University of Minnesota, Department of Psychology, Computerized Adaptive Testing Laboratory.

    Google Scholar 

  • Rost, J., & von Davier, M. (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, NY: Springer Verlag.

    Chapter  Google Scholar 

  • Stone, C. A. (1992). Recovery of MML estimates in the two-parameter logistic response model: An evaluation of MULTILOG. Applied Psychological Measurement, 16, 1–16.

    Article  Google Scholar 

  • Swaminathan, H., & Gifford, J. (1983). Estimation of parameters in the three-parameter latent trait model. In D. J. Weiss (Ed.), New horizons in testing: Latent trait test theory and computerized adaptive testing (pp. 13–30). New York: Academic.

    Google Scholar 

  • von Davier, M. (2005). A general diagnostic model applied to language testing data (Research Report 05-16). Princeton, NJ: Educational Testing Service.

    Google Scholar 

  • Woods, C. M., & Lin, N. (2009). IRT with estimation of the latent density using Davidian curves. Applied Psychological Measurement, 33, 102–117.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Xu, X., & Jia, Y. (2011). The sensitivity of parameter estimates to the latent ability distribution (Research Report 11-40). Princeton, NJ: Educational Testing Service.

    Google Scholar 

  • Xu, X., & von Davier, M. (2008). Fitting the structured general diagnostic model to NAEP data (Research Report 08-27). Princeton, NJ: Educational Testing Service.

    Google Scholar 

  • Yamamoto, K., & Muraki, E. (1991, April). Non-linear transformation of IRT scale to account for the effect of nonnormal ability distribution on the item parameter estimation. Paper presented at the annual meeting of the American Educational Research Association, Chicago, IL.

    Google Scholar 

  • Zimowski, M. F., Muraki, E., Mislevy, R. J., & Bock, R. D. (2003). BILOG-MG 3 for windows: Multiple-group IRT analysis and test maintenance for binary items [Computer software]. Lincolnwood, IL: Scientific Software International, Inc.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Statistics, Carnegie Mellon University, 8119 Wean Hall, 5000 Forbes Avenue, Pittsburgh, PA, 15213, USA

    Jodi M. Casabianca

  2. Department of Statistics, Carnegie Mellon University, 232 Baker Hall, 5000 Forbes Avenue, Pittsburgh, PA, 15224, USA

    Brian W. Junker

Authors
  1. Jodi M. Casabianca
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  2. Brian W. Junker
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Corresponding author

Correspondence to Jodi M. Casabianca .

Editor information

Editors and Affiliations

  1. Department of Psychology, Arizona State University, Tempe, AZ, USA

    Roger E. Millsap

  2. Department of Methodology and Statistics, Tilburg University, Tilburg, The Netherlands

    L. Andries van der Ark

  3. Department of Educational Psychology, University of Wisconsin, Madison, WI, USA

    Daniel M. Bolt

  4. Department of Psychology, University of Kansas, Lawrence, KS, USA

    Carol M. Woods

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Casabianca, J.M., Junker, B.W. (2013). Estimating the Latent Trait Distribution with Loglinear Smoothing Models. In: Millsap, R.E., van der Ark, L.A., Bolt, D.M., Woods, C.M. (eds) New Developments in Quantitative Psychology. Springer Proceedings in Mathematics & Statistics, vol 66. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9348-8_27

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