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

Author

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  • Robert J. Zwitser

    (University of Amsterdam)

  • Gunter Maris

    (University of Amsterdam
    Cito Institute for Educational Measurement)

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.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:psycho:v:81:y:2016:i:1:d:10.1007_s11336-015-9481-x
    DOI: 10.1007/s11336-015-9481-x
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    References listed on IDEAS

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    1. 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.
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    4. 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.
    5. 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).
    6. 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.
    7. 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.
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