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Bayes Factors for Evaluating Latent Monotonicity in Polytomous Item Response Theory Models

Author

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  • Jesper Tijmstra

    (Tilburg University)

  • Maria Bolsinova

    (ACTNext)

Abstract

The assumption of latent monotonicity is made by all common parametric and nonparametric polytomous item response theory models and is crucial for establishing an ordinal level of measurement of the item score. Three forms of latent monotonicity can be distinguished: monotonicity of the cumulative probabilities, of the continuation ratios, and of the adjacent-category ratios. Observable consequences of these different forms of latent monotonicity are derived, and Bayes factor methods for testing these consequences are proposed. These methods allow for the quantification of the evidence both in favor and against the tested property. Both item-level and category-level Bayes factors are considered, and their performance is evaluated using a simulation study. The methods are applied to an empirical example consisting of a 10-item Likert scale to investigate whether a polytomous item scoring rule results in item scores that are of ordinal level measurement.

Suggested Citation

  • Jesper Tijmstra & Maria Bolsinova, 2019. "Bayes Factors for Evaluating Latent Monotonicity in Polytomous Item Response Theory Models," Psychometrika, Springer;The Psychometric Society, vol. 84(3), pages 846-869, September.
    • Handle: RePEc:spr:psycho:v:84:y:2019:i:3:d:10.1007_s11336-019-09661-w
    DOI: 10.1007/s11336-019-09661-w
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    References listed on IDEAS

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    1. 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.
    2. van der Ark, L. Andries, 2007. "Mokken Scale Analysis in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 20(i11).
    3. 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.
    4. Geoff Masters, 1982. "A rasch model for partial credit scoring," Psychometrika, Springer;The Psychometric Society, vol. 47(2), pages 149-174, June.
    5. Bas Hemker & Klaas Sijtsma & Ivo Molenaar & Brian Junker, 1996. "Polytomous IRT models and monotone likelihood ratio of the total score," Psychometrika, Springer;The Psychometric Society, vol. 61(4), pages 679-693, December.
    6. Klaas Sijtsma & Bas Hemker, 1998. "Nonparametric polytomous IRT models for invariant item ordering, with results for parametric models," Psychometrika, Springer;The Psychometric Society, vol. 63(2), pages 183-200, June.
    7. David Andrich, 1978. "A rating formulation for ordered response categories," Psychometrika, Springer;The Psychometric Society, vol. 43(4), pages 561-573, December.
    8. Hartmann Scheiblechner, 1995. "Isotonic ordinal probabilistic models (ISOP)," Psychometrika, Springer;The Psychometric Society, vol. 60(2), pages 281-304, June.
    9. 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).
    10. 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.
    11. Paul Rosenbaum, 1984. "Testing the conditional independence and monotonicity assumptions of item response theory," Psychometrika, Springer;The Psychometric Society, vol. 49(3), pages 425-435, September.
    12. 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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