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Nonparametric Item Response Function Estimates with the EM Algorithm

Author

Listed:
  • Natasha Rossi
  • Xiaohui Wang
  • James O. Ramsay

Abstract

The methods of functional data analysis are used to estimate item response functions (IRFs) nonparametrically. The EM algorithm is used to maximize the penalized marginal likelihood of the data. The penalty controls the smoothness of the estimated IRFs, and is chosen so that, as the penalty is increased, the estimates converge to shapes closely represented by the three-parameter logistic family. The one-dimensional latent trait model is recast as a problem of estimating a space curve or manifold, and, expressed in this way, the model no longer involves any latent constructs, and is invariant with respect to choice of latent variable. Some results from differential geometry are used to develop a data-anchored measure of ability and a new technique for assessing item discriminability. Functional data-analytic techniques are used to explore the functional variation in the estimated IRFs. Applications involving simulated and actual data are included.

Suggested Citation

  • Natasha Rossi & Xiaohui Wang & James O. Ramsay, 2002. "Nonparametric Item Response Function Estimates with the EM Algorithm," Journal of Educational and Behavioral Statistics, , vol. 27(3), pages 291-317, September.
    • Handle: RePEc:sae:jedbes:v:27:y:2002:i:3:p:291-317
    DOI: 10.3102/10769986027003291
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    Citations

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    Cited by:

    1. Christian Genest & Johanna G. Nešlehová, 2014. "A Conversation with James O. Ramsay," International Statistical Review, International Statistical Institute, vol. 82(2), pages 161-183, August.
    2. Tomasz Górecki & Mirosław Krzyśko & Waldemar Wołyński, 2015. "Classification Problems Based On Regression Models For Multi-Dimensional Functional Data," Statistics in Transition New Series, Polish Statistical Association, vol. 16(1), pages 97-110, March.
    3. Fabrizio Maturo & Francesca Fortuna & Tonio Di Battista, 2019. "Testing Equality of Functions Across Multiple Experimental Conditions for Different Ability Levels in the IRT Context: The Case of the IPRASE TLT 2016 Survey," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 146(1), pages 19-39, November.
    4. Tomasz Górecki & Mirosław Krzyśko & Waldemar Wołyński, 2019. "Variable Selection In Multivariate Functional Data Classification," Statistics in Transition New Series, Polish Statistical Association, vol. 20(2), pages 123-138, June.
    5. Marie Wiberg & James O. Ramsay & Juan Li, 2019. "Optimal Scores: An Alternative to Parametric Item Response Theory and Sum Scores," Psychometrika, Springer;The Psychometric Society, vol. 84(1), pages 310-322, March.

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