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A New Explanation and Proof of the Paradoxical Scoring Results in Multidimensional Item Response Models

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  • Pascal Jordan

    (University of Hamburg)

  • Martin Spiess

    (University of Hamburg)

Abstract

In multidimensional item response models, paradoxical scoring effects can arise, wherein correct answers are penalized and incorrect answers are rewarded. For the most prominent class of IRT models, the class of linearly compensatory models, a general derivation of paradoxical scoring effects based on the geometry of item discrimination vectors is given, which furthermore corrects an error in an established theorem on paradoxical results. This approach highlights the very counterintuitive way in which item discrimination parameters (and also factor loadings) have to be interpreted in terms of their influence on the latent ability estimate. It is proven that, despite the error in the original proof, the key result concerning the existence of paradoxical effects remains true—although the actual relation to the item parameters is shown to be a more complicated function than previous results suggested. The new proof enables further insights into the actual mathematical causation of the paradox and generalizes the findings within the class of linearly compensatory models.

Suggested Citation

  • Pascal Jordan & Martin Spiess, 2018. "A New Explanation and Proof of the Paradoxical Scoring Results in Multidimensional Item Response Models," Psychometrika, Springer;The Psychometric Society, vol. 83(4), pages 831-846, December.
    • Handle: RePEc:spr:psycho:v:83:y:2018:i:4:d:10.1007_s11336-017-9588-3
    DOI: 10.1007/s11336-017-9588-3
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    References listed on IDEAS

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    1. Matthew D. Finkelman & Giles Hooker & Zhen Wang, 2010. "Prevalence and Magnitude of Paradoxical Results in Multidimensional Item Response Theory," Journal of Educational and Behavioral Statistics, , vol. 35(6), pages 744-761, December.
    2. Pascal Jordan & Martin Spiess, 2012. "Generalizations of Paradoxical Results in Multidimensional Item Response Theory," Psychometrika, Springer;The Psychometric Society, vol. 77(1), pages 127-152, January.
    3. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities II. Multivariate reverse rule distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 499-516, December.
    4. Wim Linden, 2012. "On Compensation in Multidimensional Response Modeling," Psychometrika, Springer;The Psychometric Society, vol. 77(1), pages 21-30, January.
    5. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 467-498, December.
    6. Giles Hooker, 2010. "On Separable Tests, Correlated Priors, and Paradoxical Results in Multidimensional Item Response Theory," Psychometrika, Springer;The Psychometric Society, vol. 75(4), pages 694-707, December.
    7. Fumiko Samejima, 1974. "Normal ogive model on the continuous response level in the multidimensional latent space," Psychometrika, Springer;The Psychometric Society, vol. 39(1), pages 111-121, March.
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    Cited by:

    1. Pascal Jordan, 2023. "On Reverse Shrinkage Effects and Shrinkage Overshoot," Psychometrika, Springer;The Psychometric Society, vol. 88(1), pages 274-301, March.

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