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The UMP Exact Test and the Confidence Interval for Person Parameters in IRT Models

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  • Xiang Liu

    (Teachers College of Columbia University)

  • Zhuangzhuang Han

    (Teachers College of Columbia University)

  • Matthew S. Johnson

    (Teachers College of Columbia University)

Abstract

In educational and psychological measurement when short test forms are used, the asymptotic normality of the maximum likelihood estimator of the person parameter of item response models does not hold. As a result, hypothesis tests or confidence intervals of the person parameter based on the normal distribution are likely to be problematic. Inferences based on the exact distribution, on the other hand, do not suffer from this limitation. However, the computation involved for the exact distribution approach is often prohibitively expensive. In this paper, we propose a general framework for constructing hypothesis tests and confidence intervals for IRT models within the exponential family based on exact distribution. In addition, an efficient branch and bound algorithm for calculating the exact p value is introduced. The type-I error rate and statistical power of the proposed exact test as well as the coverage rate and the lengths of the associated confidence interval are examined through a simulation. We also demonstrate its practical use by analyzing three real data sets.

Suggested Citation

  • Xiang Liu & Zhuangzhuang Han & Matthew S. Johnson, 2018. "The UMP Exact Test and the Confidence Interval for Person Parameters in IRT Models," Psychometrika, Springer;The Psychometric Society, vol. 83(1), pages 182-202, March.
    • Handle: RePEc:spr:psycho:v:83:y:2018:i:1:d:10.1007_s11336-017-9580-y
    DOI: 10.1007/s11336-017-9580-y
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    References listed on IDEAS

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    1. Karl Klauer, 1991. "Exact and best confidence intervals for the ability parameter of the Rasch model," Psychometrika, Springer;The Psychometric Society, vol. 56(3), pages 535-547, September.
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    4. Thomas Warm, 1989. "Weighted likelihood estimation of ability in item response theory," Psychometrika, Springer;The Psychometric Society, vol. 54(3), pages 427-450, September.
    5. Rizopoulos, Dimitris, 2006. "ltm: An R Package for Latent Variable Modeling and Item Response Analysis," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 17(i05).
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    7. Frederic Lord, 1983. "Unbiased estimators of ability parameters, of their variance, and of their parallel-forms reliability," Psychometrika, Springer;The Psychometric Society, vol. 48(2), pages 233-245, June.
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    Cited by:

    1. Xiang Liu & James Yang & Hui Soo Chae & Gary Natriello, 2020. "Power Divergence Family of Statistics for Person Parameters in IRT Models," Psychometrika, Springer;The Psychometric Society, vol. 85(2), pages 502-525, June.

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