IDEAS home Printed from https://ideas.repec.org/a/spr/psycho/v81y2016i1p39-59.html
   My bibliography  Save this article

Ordering Individuals with Sum Scores: The Introduction of the Nonparametric Rasch Model

Author

Listed:
  • Robert Zwitser
  • Gunter Maris

Abstract

When a simple sum or number-correct score is used to evaluate the ability of individual testees, then, from an accountability perspective, the inferences based on the sum score should be the same as the inferences based on the complete response pattern. This requirement is fulfilled if the sum score is a sufficient statistic for the parameter of a unidimensional model. However, the models for which this holds true are known to be restrictive. It is shown that the less restrictive nonparametric models could result in an ordering of persons that is different from an ordering based on the sum score. To arrive at a fair evaluation of ability with a simple number-correct score, ordinal sufficiency is defined as a minimum condition for scoring. The monotone homogeneity model, together with the property of ordinal sufficiency of the sum score, is introduced as the nonparametric Rasch model. A basic outline for testable hypotheses about ordinal sufficiency, as well as illustrations with real data, is provided. Copyright The Psychometric Society 2016

Suggested Citation

  • Robert Zwitser & Gunter Maris, 2016. "Ordering Individuals with Sum Scores: The Introduction of the Nonparametric Rasch Model," Psychometrika, Springer;The Psychometric Society, vol. 81(1), pages 39-59, March.
    • Handle: RePEc:spr:psycho:v:81:y:2016:i:1:p:39-59
    DOI: 10.1007/s11336-015-9481-x
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s11336-015-9481-x
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s11336-015-9481-x?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. Klaas Sijtsma & Bas T. Hemker, 2000. "A Taxonomy of IRT Models for Ordering Persons and Items Using Simple Sum Scores," Journal of Educational and Behavioral Statistics, , vol. 25(4), pages 391-415, December.
    2. Paul R. Milgrom, 1981. "Good News and Bad News: Representation Theorems and Applications," Bell Journal of Economics, The RAND Corporation, vol. 12(2), pages 380-391, Autumn.
    3. Sekhon, Jasjeet S., 2011. "Multivariate and Propensity Score Matching Software with Automated Balance Optimization: The Matching package for R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 42(i07).
    4. Gunter Maris, 2008. "A Note on “Constant Latent Odds-Ratios Models and the Mantel–Haenszel Null Hypothesis” Hessen, 2005," Psychometrika, Springer;The Psychometric Society, vol. 73(1), pages 153-157, March.
    5. David Hessen, 2005. "Constant latent odds-ratios models and the mantel-haenszel null hypothesis," Psychometrika, Springer;The Psychometric Society, vol. 70(3), pages 497-516, September.
    6. Huynh Huynh, 1994. "A new proof for monotone likelihood ratio for the sum of independent bernoulli random variables," Psychometrika, Springer;The Psychometric Society, vol. 59(1), pages 77-79, March.
    7. D. Grayson, 1988. "Two-group classification in latent trait theory: Scores with monotone likelihood ratio," Psychometrika, Springer;The Psychometric Society, vol. 53(3), pages 383-392, September.
    8. Bas Hemker & Klaas Sijtsma & Ivo Molenaar & Brian Junker, 1997. "Stochastic ordering using the latent trait and the sum score in polytomous IRT models," Psychometrika, Springer;The Psychometric Society, vol. 62(3), pages 331-347, 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. Robert J. Zwitser & Gunter Maris, 2016. "Ordering Individuals with Sum Scores: The Introduction of the Nonparametric Rasch Model," Psychometrika, Springer;The Psychometric Society, vol. 81(1), pages 39-59, March.
    2. Klaas Sijtsma & Rob Meijer, 2001. "The person response function as a tool in person-fit research," Psychometrika, Springer;The Psychometric Society, vol. 66(2), pages 191-207, June.
    3. Johnson, Matthew S., 2007. "Modeling dichotomous item responses with free-knot splines," Computational Statistics & Data Analysis, Elsevier, vol. 51(9), pages 4178-4192, May.
    4. Jesper Tijmstra & David Hessen & Peter Heijden & Klaas Sijtsma, 2013. "Testing Manifest Monotonicity Using Order-Constrained Statistical Inference," Psychometrika, Springer;The Psychometric Society, vol. 78(1), pages 83-97, January.
    5. Wim Linden, 1998. "Stochastic order in dichotomous item response models for fixed, adaptive, and multidimensional tests," Psychometrika, Springer;The Psychometric Society, vol. 63(3), pages 211-226, September.
    6. L. Ark & Wicher Bergsma, 2010. "A Note on Stochastic Ordering of the Latent Trait Using the Sum of Polytomous Item Scores," Psychometrika, Springer;The Psychometric Society, vol. 75(2), pages 272-279, June.
    7. repec:jss:jstsof:20:i11 is not listed on IDEAS
    8. Bas Hemker & L. Andries van der Ark & Klaas Sijtsma, 2001. "On measurement properties of continuation ratio models," Psychometrika, Springer;The Psychometric Society, vol. 66(4), pages 487-506, December.
    9. Rudy Ligtvoet, 2012. "An Isotonic Partial Credit Model for Ordering Subjects on the Basis of Their Sum Scores," Psychometrika, Springer;The Psychometric Society, vol. 77(3), pages 479-494, July.
    10. Rudy Ligtvoet, 2022. "Incomplete Tests of Conditional Association for the Assessment of Model Assumptions," Psychometrika, Springer;The Psychometric Society, vol. 87(4), pages 1214-1237, December.
    11. Samantha Bouwmeester & Klaas Sijtsma, 2004. "Measuring the ability of transitive reasoning, using product and strategy information," Psychometrika, Springer;The Psychometric Society, vol. 69(1), pages 123-146, March.
    12. Jesper Tijmstra & Herbert Hoijtink & Klaas Sijtsma, 2015. "Evaluating Manifest Monotonicity Using Bayes Factors," Psychometrika, Springer;The Psychometric Society, vol. 80(4), pages 880-896, December.
    13. L. Ark, 2005. "Stochastic Ordering Of the Latent Trait by the Sum Score Under Various Polytomous IRT Models," Psychometrika, Springer;The Psychometric Society, vol. 70(2), pages 283-304, June.
    14. van der Ark, L. Andries, 2012. "New Developments in Mokken Scale Analysis in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 48(i05).
    15. Nicolas Jacquemet & Stéphane Luchini & Julie Rosaz & Jason F. Shogren, 2019. "Truth Telling Under Oath," Management Science, INFORMS, vol. 65(1), pages 426-438, January.
    16. J. Straat & L. Ark & Klaas Sijtsma, 2013. "Comparing Optimization Algorithms for Item Selection in Mokken Scale Analysis," Journal of Classification, Springer;The Classification Society, vol. 30(1), pages 75-99, April.
    17. Bertoli-Barsotti, Lucio, 2003. "An order-preserving property of the maximum likelihood estimates for the Rasch model," Statistics & Probability Letters, Elsevier, vol. 61(1), pages 91-96, January.
    18. Bas Hemker & Klaas Sijtsma & Ivo Molenaar & Brian Junker, 1997. "Stochastic ordering using the latent trait and the sum score in polytomous IRT models," Psychometrika, Springer;The Psychometric Society, vol. 62(3), pages 331-347, September.
    19. van der Ark, L. Andries, 2007. "Mokken Scale Analysis in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 20(i11).
    20. Guirong Li & Jiajia Xu & Liying Li & Zhaolei Shi & Hongmei Yi & James Chu & Elena Kardanova & Yanyan Li & Prashant Loyalka & Scott Rozelle, 2020. "The Impacts of Highly Resourced Vocational Schools on Student Outcomes in China," China & World Economy, Institute of World Economics and Politics, Chinese Academy of Social Sciences, vol. 28(6), pages 125-150, November.
    21. Ginger Zhe Jin & Andrew Kato & John A. List, 2010. "That’S News To Me! Information Revelation In Professional Certification Markets," Economic Inquiry, Western Economic Association International, vol. 48(1), pages 104-122, 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:81:y:2016:i:1:p:39-59. 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.