IDEAS home Printed from https://ideas.repec.org/a/spr/psycho/v85y2020i2d10.1007_s11336-020-09712-7.html
   My bibliography  Save this article

Power Divergence Family of Statistics for Person Parameters in IRT Models

Author

Listed:
  • Xiang Liu

    (Columbia University
    Educational Testing Service)

  • James Yang

    (Columbia University)

  • Hui Soo Chae

    (Columbia University)

  • Gary Natriello

    (Columbia University)

Abstract

We generalize the power divergence (PD) family of statistics to the two-parameter logistic IRT model for the purpose of constructing hypothesis tests and confidence intervals of the person parameter. The well-known score test statistic is a special case of the proposed PD family. We also prove the proposed PD statistics are asymptotically equivalent and converge in distribution to $$\chi _{1}^2$$ χ 1 2 . In addition, a moment matching method is introduced to compare statistics and choose the optimal one within the PD family. Simulation results suggest that the coverage rate of the associated confidence interval is well controlled even under small sample sizes for some PD statistics. Compared to some other approaches, the associated confidence intervals exhibit smaller lengths while maintaining adequate coverage rates. The utilities of the proposed method are demonstrated by analyzing a real data set.

Suggested Citation

  • Xiang Liu & James Yang & Hui Soo Chae & Gary Natriello, 2020. "Power Divergence Family of Statistics for Person Parameters in IRT Models," Psychometrika, Springer;The Psychometric Society, vol. 85(2), pages 502-525, June.
    • Handle: RePEc:spr:psycho:v:85:y:2020:i:2:d:10.1007_s11336-020-09712-7
    DOI: 10.1007/s11336-020-09712-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11336-020-09712-7
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11336-020-09712-7?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    2. Yang Liu & Jan Hannig & Abhishek Pal Majumder, 2019. "Second-Order Probability Matching Priors for the Person Parameter in Unidimensional IRT Models," Psychometrika, Springer;The Psychometric Society, vol. 84(3), pages 701-718, September.
    3. Ogasawara, Haruhiko, 2012. "Supplement to the paper“ Asymptotic expansions for the ability estimator in item response theory”," 商学討究 (Shogaku Tokyu), Otaru University of Commerce, vol. 63(2/3), pages 329-336.
    4. Karl Klauer, 1991. "Exact and best confidence intervals for the ability parameter of the Rasch model," Psychometrika, Springer;The Psychometric Society, vol. 56(3), pages 535-547, September.
    5. Shaobo Jin & Måns Thulin & Rolf Larsson, 2017. "Approximate Bayesianity of Frequentist Confidence Intervals for a Binomial Proportion," The American Statistician, Taylor & Francis Journals, vol. 71(2), pages 106-111, April.
    6. Sandip Sinharay, 2016. "Asymptotically Correct Standardization of Person-Fit Statistics Beyond Dichotomous Items," Psychometrika, Springer;The Psychometric Society, vol. 81(4), pages 992-1013, December.
    7. Frederic Lord, 1983. "Unbiased estimators of ability parameters, of their variance, and of their parallel-forms reliability," Psychometrika, Springer;The Psychometric Society, vol. 48(2), pages 233-245, June.
    8. Ogasawara, Haruhiko, 2013. "Asymptotic properties of the Bayes and pseudo Bayes estimators of ability in item response theory," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 359-377.
    9. Yang Liu & Ji Seung Yang, 2018. "Bootstrap-Calibrated Interval Estimates for Latent Variable Scores in Item Response Theory," Psychometrika, Springer;The Psychometric Society, vol. 83(2), pages 333-354, June.
    10. Xiang Liu & Zhuangzhuang Han & Matthew S. Johnson, 2018. "The UMP Exact Test and the Confidence Interval for Person Parameters in IRT Models," Psychometrika, Springer;The Psychometric Society, vol. 83(1), pages 182-202, March.
    11. A. Felipe & P. Miranda & L. Pardo, 2015. "Minimum $$\phi $$ ϕ -Divergence Estimation in Constrained Latent Class Models for Binary Data," Psychometrika, Springer;The Psychometric Society, vol. 80(4), pages 1020-1042, December.
    12. Haruhiko Ogasawara, 2012. "Asymptotic expansions for the ability estimator in item response theory," Computational Statistics, Springer, vol. 27(4), pages 661-683, December.
    13. Thomas Warm, 1989. "Weighted likelihood estimation of ability in item response theory," Psychometrika, Springer;The Psychometric Society, vol. 54(3), pages 427-450, September.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Martin Biehler & Heinz Holling & Philipp Doebler, 2015. "Saddlepoint Approximations of the Distribution of the Person Parameter in the Two Parameter Logistic Model," Psychometrika, Springer;The Psychometric Society, vol. 80(3), pages 665-688, September.
    2. Sandip Sinharay, 2015. "The Asymptotic Distribution of Ability Estimates," Journal of Educational and Behavioral Statistics, , vol. 40(5), pages 511-528, October.
    3. Piero Veronese & Eugenio Melilli, 2021. "Confidence Distribution for the Ability Parameter of the Rasch Model," Psychometrika, Springer;The Psychometric Society, vol. 86(1), pages 131-166, March.
    4. Ogasawara, Haruhiko, 2013. "Asymptotic cumulants of ability estimators using fallible item parameters," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 144-162.
    5. Ogasawara, Haruhiko, 2013. "Asymptotic properties of the Bayes and pseudo Bayes estimators of ability in item response theory," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 359-377.
    6. Ying Cheng & Cheng Liu & John Behrens, 2015. "Standard Error of Ability Estimates and the Classification Accuracy and Consistency of Binary Decisions," Psychometrika, Springer;The Psychometric Society, vol. 80(3), pages 645-664, September.
    7. David Magis, 2016. "Efficient Standard Error Formulas of Ability Estimators with Dichotomous Item Response Models," Psychometrika, Springer;The Psychometric Society, vol. 81(1), pages 184-200, March.
    8. Maxwell Hong & Lizhen Lin & Ying Cheng, 2021. "Asymptotically Corrected Person Fit Statistics for Multidimensional Constructs with Simple Structure and Mixed Item Types," Psychometrika, Springer;The Psychometric Society, vol. 86(2), pages 464-488, June.
    9. Yang Liu & Jan Hannig & Abhishek Pal Majumder, 2019. "Second-Order Probability Matching Priors for the Person Parameter in Unidimensional IRT Models," Psychometrika, Springer;The Psychometric Society, vol. 84(3), pages 701-718, September.
    10. Xiang Liu & Zhuangzhuang Han & Matthew S. Johnson, 2018. "The UMP Exact Test and the Confidence Interval for Person Parameters in IRT Models," Psychometrika, Springer;The Psychometric Society, vol. 83(1), pages 182-202, March.
    11. Oberrauch, Luis & Kaiser, Tim, 2020. "Economic competence in early secondary school: Evidence from a large-scale assessment in Germany," International Review of Economics Education, Elsevier, vol. 35(C).
    12. Georg Gittler & Gerhard Fischer, 2011. "IRT-Based Measurement of Short-Term Changes of Ability, With an Application to Assessing the “Mozart Effectâ€," Journal of Educational and Behavioral Statistics, , vol. 36(1), pages 33-75, February.
    13. Sandip Sinharay, 2016. "Person Fit Analysis in Computerized Adaptive Testing Using Tests for a Change Point," Journal of Educational and Behavioral Statistics, , vol. 41(5), pages 521-549, October.
    14. Sandip Sinharay, 2016. "Asymptotically Correct Standardization of Person-Fit Statistics Beyond Dichotomous Items," Psychometrika, Springer;The Psychometric Society, vol. 81(4), pages 992-1013, December.
    15. Sandip Sinharay, 2017. "Detection of Item Preknowledge Using Likelihood Ratio Test and Score Test," Journal of Educational and Behavioral Statistics, , vol. 42(1), pages 46-68, February.
    16. Simon Grund & Oliver Lüdtke & Alexander Robitzsch, 2021. "On the Treatment of Missing Data in Background Questionnaires in Educational Large-Scale Assessments: An Evaluation of Different Procedures," Journal of Educational and Behavioral Statistics, , vol. 46(4), pages 430-465, August.
    17. Wim Linden, 1998. "Bayesian item selection criteria for adaptive testing," Psychometrika, Springer;The Psychometric Society, vol. 63(2), pages 201-216, June.
    18. Chun Wang, 2015. "On Latent Trait Estimation in Multidimensional Compensatory Item Response Models," Psychometrika, Springer;The Psychometric Society, vol. 80(2), pages 428-449, June.
    19. Seonghoon Kim, 2012. "A Note on the Reliability Coefficients for Item Response Model-Based Ability Estimates," Psychometrika, Springer;The Psychometric Society, vol. 77(1), pages 153-162, January.
    20. Anna Doebler & Philipp Doebler & Heinz Holling, 2013. "Optimal and Most Exact Confidence Intervals for Person Parameters in Item Response Theory Models," Psychometrika, Springer;The Psychometric Society, vol. 78(1), pages 98-115, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:psycho:v:85:y:2020:i:2:d:10.1007_s11336-020-09712-7. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.