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Testing Manifest Monotonicity Using Order-Constrained Statistical Inference

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  • Jesper Tijmstra
  • David Hessen
  • Peter Heijden
  • Klaas Sijtsma

Abstract

Most dichotomous item response models share the assumption of latent monotonicity, which states that the probability of a positive response to an item is a nondecreasing function of a latent variable intended to be measured. Latent monotonicity cannot be evaluated directly, but it implies manifest monotonicity across a variety of observed scores, such as the restscore, a single item score, and in some cases the total score. In this study, we show that manifest monotonicity can be tested by means of the order-constrained statistical inference framework. We propose a procedure that uses this framework to determine whether manifest monotonicity should be rejected for specific items. This approach provides a likelihood ratio test for which the p-value can be approximated through simulation. A simulation study is presented that evaluates the Type I error rate and power of the test, and the procedure is applied to empirical data. Copyright The Psychometric Society 2013

Suggested Citation

  • 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.
    • Handle: RePEc:spr:psycho:v:78:y:2013:i:1:p:83-97
    DOI: 10.1007/s11336-012-9297-x
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    References listed on IDEAS

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    1. Jesper Tijmstra & David Hessen & Peter Heijden & Klaas Sijtsma, 2011. "Invariant Ordering of Item-Total Regressions," Psychometrika, Springer;The Psychometric Society, vol. 76(2), pages 217-227, April.
    2. Paul Rosenbaum, 1987. "Comparing item characteristic curves," Psychometrika, Springer;The Psychometric Society, vol. 52(2), pages 217-233, June.
    3. 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.
    4. L. Ark & Marcel Croon & Klaas Sijtsma, 2008. "Mokken Scale Analysis for Dichotomous Items Using Marginal Models," Psychometrika, Springer;The Psychometric Society, vol. 73(2), pages 183-208, June.
    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. 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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    Cited by:

    1. 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.
    2. 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.
    3. Jules L. Ellis & Klaas Sijtsma, 2023. "A Test to Distinguish Monotone Homogeneity from Monotone Multifactor Models," Psychometrika, Springer;The Psychometric Society, vol. 88(2), pages 387-412, June.

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