IDEAS home Printed from https://ideas.repec.org/a/spr/psycho/v84y2019i3d10.1007_s11336-019-09675-4.html
   My bibliography  Save this article

Second-Order Probability Matching Priors for the Person Parameter in Unidimensional IRT Models

Author

Listed:
  • Yang Liu

    (University of Maryland)

  • Jan Hannig

    (The University of North Carolina at Chapel Hill)

  • Abhishek Pal Majumder

    (Stockholm University)

Abstract

In applications of item response theory (IRT), it is often of interest to compute confidence intervals (CIs) for person parameters with prescribed frequentist coverage. The ubiquitous use of short tests in social science research and practices calls for a refinement of standard interval estimation procedures based on asymptotic normality, such as the Wald and Bayesian CIs, which only maintain desirable coverage when the test is sufficiently long. In the current paper, we propose a simple construction of second-order probability matching priors for the person parameter in unidimensional IRT models, which in turn yields CIs with accurate coverage even when the test is composed of a few items. The probability matching property is established based on an expansion of the posterior distribution function and a shrinkage argument. CIs based on the proposed prior can be efficiently computed for a variety of unidimensional IRT models. A real data example with a mixed-format test and a simulation study are presented to compare the proposed method against several existing asymptotic CIs.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:psycho:v:84:y:2019:i:3:d:10.1007_s11336-019-09675-4
    DOI: 10.1007/s11336-019-09675-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11336-019-09675-4
    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/s11336-019-09675-4?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. 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.
    2. Ghosh, Malay, 2006. "Bayesian Nonparametrics via Neural Networks. Herbert K. H. Lee," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1313-1313, 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. David Magis & Gilles Raîche, 2012. "On the Relationships Between Jeffreys Modal and Weighted Likelihood Estimation of Ability Under Logistic IRT Models," Psychometrika, Springer;The Psychometric Society, vol. 77(1), pages 163-169, January.
    5. Elisabeth Deutskens & Ko de Ruyter & Martin Wetzels & Paul Oosterveld, 2004. "Response Rate and Response Quality of Internet-Based Surveys: An Experimental Study," Marketing Letters, Springer, vol. 15(1), pages 21-36, February.
    6. R. Darrell Bock, 1972. "Estimating item parameters and latent ability when responses are scored in two or more nominal categories," Psychometrika, Springer;The Psychometric Society, vol. 37(1), pages 29-51, March.
    7. Ying Cheng & Ke-Hai Yuan, 2010. "The Impact of Fallible Item Parameter Estimates on Latent Trait Recovery," Psychometrika, Springer;The Psychometric Society, vol. 75(2), pages 280-291, June.
    8. Weeks, Jonathan P., 2010. "plink: An R Package for Linking Mixed-Format Tests Using IRT-Based Methods," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 35(i12).
    9. Karl Klauer, 1991. "An exact and optimal standardized person test for assessing consistency with the rasch model," Psychometrika, Springer;The Psychometric Society, vol. 56(2), pages 213-228, June.
    10. L. Wasserman, 2000. "Asymptotic inference for mixture models by using data‐dependent priors," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(1), pages 159-180.
    11. Hua-Hua Chang, 1996. "The asymptotic posterior normality of the latent trait for polytomous IRT models," Psychometrika, Springer;The Psychometric Society, vol. 61(3), pages 445-463, September.
    12. Hua-Hua Chang & William Stout, 1993. "The asymptotic posterior normality of the latent trait in an IRT model," Psychometrika, Springer;The Psychometric Society, vol. 58(1), pages 37-52, March.
    13. David Magis, 2015. "A Note on Weighted Likelihood and Jeffreys Modal Estimation of Proficiency Levels in Polytomous Item Response Models," Psychometrika, Springer;The Psychometric Society, vol. 80(1), pages 200-204, March.
    14. 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.
    15. Haruhiko Ogasawara, 2012. "Asymptotic expansions for the ability estimator in item response theory," Computational Statistics, Springer, vol. 27(4), pages 661-683, December.
    16. 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)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.

    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. Sandip Sinharay, 2015. "The Asymptotic Distribution of Ability Estimates," Journal of Educational and Behavioral Statistics, , vol. 40(5), pages 511-528, October.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. Sandip Sinharay & Jens Ledet Jensen, 2019. "Higher-Order Asymptotics and Its Application to Testing the Equality of the Examinee Ability Over Two Sets of Items," Psychometrika, Springer;The Psychometric Society, vol. 84(2), pages 484-510, June.
    7. Ogasawara, Haruhiko, 2013. "Asymptotic cumulants of ability estimators using fallible item parameters," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 144-162.
    8. 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.
    9. 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.
    10. David Magis, 2015. "A Note on Weighted Likelihood and Jeffreys Modal Estimation of Proficiency Levels in Polytomous Item Response Models," Psychometrika, Springer;The Psychometric Society, vol. 80(1), pages 200-204, March.
    11. David Magis & Norman Verhelst, 2017. "On the Finiteness of the Weighted Likelihood Estimator of Ability," Psychometrika, Springer;The Psychometric Society, vol. 82(3), pages 637-647, September.
    12. Chun Wang & Gongjun Xu & Xue Zhang, 2019. "Correction for Item Response Theory Latent Trait Measurement Error in Linear Mixed Effects Models," Psychometrika, Springer;The Psychometric Society, vol. 84(3), pages 673-700, September.
    13. Maarten Marsman & Gunter Maris & Timo Bechger & Cees Glas, 2016. "What can we learn from Plausible Values?," Psychometrika, Springer;The Psychometric Society, vol. 81(2), pages 274-289, June.
    14. Ping Chen & Chun Wang, 2016. "A New Online Calibration Method for Multidimensional Computerized Adaptive Testing," Psychometrika, Springer;The Psychometric Society, vol. 81(3), pages 674-701, September.
    15. Alexander Robitzsch, 2021. "A Comprehensive Simulation Study of Estimation Methods for the Rasch Model," Stats, MDPI, vol. 4(4), pages 1-23, October.
    16. 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.
    17. Anders Skrondal & Sophia Rabe‐Hesketh, 2009. "Prediction in multilevel generalized linear models," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 172(3), pages 659-687, June.
    18. 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.
    19. Brzezińska Justyna, 2018. "Item Response Theory Models in the Measurement Theory with the Use of ltm Package in R," Econometrics. Advances in Applied Data Analysis, Sciendo, vol. 22(1), pages 11-25, March.
    20. Mia J. K. Kornely & Maria Kateri, 2022. "Asymptotic Posterior Normality of Multivariate Latent Traits in an IRT Model," Psychometrika, Springer;The Psychometric Society, vol. 87(3), pages 1146-1172, September.

    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:84:y:2019:i:3:d:10.1007_s11336-019-09675-4. 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.