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A Didactic Presentation of Snijders’s lz* Index of Person Fit With Emphasis on Response Model Selection and Ability Estimation

Author

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  • David Magis

    (Department of Mathematics, University of Liège, Belgium)

  • Gilles Raîche

    (Département d’éducation et pédagogie, Université du Québec à Montréal, Canada)

  • Sébastien Béland

    (Département d’éducation et pédagogie, Université du Québec à Montréal, Canada)

Abstract

This paper focuses on two likelihood-based indices of person fit, the index l z and the Snijders’s modified index l z *. The first one is commonly used in practical assessment of person fit, although its asymptotic standard normal distribution is not valid when true abilities are replaced by sample ability estimates. The l z * index is a generalization of l z , which corrects for this sampling variability. Surprisingly, it is not yet popular in the psychometric and educational assessment community. Moreover, there is some ambiguity about which type of item response model and ability estimation method can be used to compute the l z * index. The purpose of this article is to present the index l z * in a simple and didactic approach. Starting from the relationship between l z and l z *, we develop the framework according to the type of logistic item response theory (IRT) model and the likelihood-based estimators of ability. The practical calculation of l z * is illustrated by analyzing a real data set about language skill assessment.

Suggested Citation

  • David Magis & Gilles Raîche & Sébastien Béland, 2012. "A Didactic Presentation of Snijders’s lz* Index of Person Fit With Emphasis on Response Model Selection and Ability Estimation," Journal of Educational and Behavioral Statistics, , vol. 37(1), pages 57-81, February.
    • Handle: RePEc:sae:jedbes:v:37:y:2012:i:1:p:57-81
    DOI: 10.3102/1076998610396894
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    References listed on IDEAS

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    1. Tom Snijders, 2001. "Asymptotic null distribution of person fit statistics with estimated person parameter," Psychometrika, Springer;The Psychometric Society, vol. 66(3), pages 331-342, September.
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    4. Rizopoulos, Dimitris, 2006. "ltm: An R Package for Latent Variable Modeling and Item Response Analysis," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 17(i05).
    5. Ivo Molenaar & Herbert Hoijtink, 1990. "The many null distributions of person fit indices," Psychometrika, Springer;The Psychometric Society, vol. 55(1), pages 75-106, March.
    6. C. Glas & Anna Dagohoy, 2007. "A Person Fit Test For Irt Models For Polytomous Items," Psychometrika, Springer;The Psychometric Society, vol. 72(2), pages 159-180, June.
    7. Kikumi Tatsuoka, 1984. "Caution indices based on item response theory," Psychometrika, Springer;The Psychometric Society, vol. 49(1), pages 95-110, March.
    8. Thomas Warm, 1989. "Weighted likelihood estimation of ability in item response theory," Psychometrika, Springer;The Psychometric Society, vol. 54(3), pages 427-450, September.
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    Cited by:

    1. Tendeiro, Jorge N. & Meijer, Rob R. & Niessen, A. Susan M., 2016. "PerFit: An R Package for Person-Fit Analysis in IRT," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 74(i05).

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