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Item response theory and its applications in educational measurement Part I: Item response theory and its implementation in R

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Kazuki Hori, Hirotaka Fukuhara, Tsuyoshi Yamada

Published Online: Oct 13 2020

DOI: 10.1002/wics.1531

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Abstract Item response theory (IRT) is a class of latent variable models, which are used to develop educational and psychological tests (e.g., standardized tests, personality tests, tests for licensure, and certification). We review the theory and practices of IRT across two articles. In Part 1, we provide a broad range of topics such as foundations of educational measurement, basics of IRT, and applications of IRT using R. We focus particularly on the topics that the mirt package covers. These include unidimensional and multidimensional IRT models for dichotomous and polytomous items with continuous and discrete factors, confirmatory analysis and multigroup analysis in IRT, and estimation algorithms. In Part 2, on the other hand, we focus on more practical aspects of IRT, namely scoring, scaling, and equating. This article is categorized under: Applications of Computational Statistics > Psychometrics Software for Computational Statistics > Software/Statistical Software

Item information curves (IICs), test information curve (TIC), and the corresponding standard error of estimate. The four items have the following item parameters: Item 1: (a₁, b₁, c₁) = (1.0, 0.0, 0.0), Item 2: (a₂, b₂, c₂) = (1.0, 1.5, 0.0), Item 3: (a₃, b₃, c₃) = (1.5, − 1.5, 0.0), and Item 4: (a₄, b₄, c₄) = (1.0, 0.0, 0.2)

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Three item characteristic curves (ICCs) and the corresponding test characteristic curve (TCC). Item parameters: (a₁, b₁, c₁) = (1.0, 0.0, 0.2), (a₂, b₂, c₂) = (2.0, −1.0, 0.1), and (a₃, b₃, c₃) = (0.5, 0.5, 0.3)

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Item characteristic curves (ICCs) with different item parameter values. The subplots (a)–(c) have different values for discrimination parameter a, difficulty parameter b, and pseudo‐guessing parameter c, respectively. The other parameters are set at a = 1.0, b = 0.0, c = 0.0

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Example of a bifactor model with three specific factors. Errors are omitted

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A hypothetical example of an MIRT model with between‐item multidimensionality. Dashed lines represent paths of which coefficient is fixed at zero

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Item response surface (IRS) and contours of equal probabilities for compensatory two‐parameter logistic model (Item 1) and noncompensatory two‐parameter logistic model (Item 2). Note that both models have a_j = (1, 1)^′ and d_j = 0

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Category characteristic curves (CCCs) of generalized partial credit model with different sets of parameter values. Subplot (a) is a baseline. Subplot (b) has larger a_j value than (a) and has unequally distant categories. Subplot (c) has “reverse” categories (i.e., b₁ < b₃ < b₂)

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Further Reading

Lord,, F. M., & Novick,, M. R. (1968). Statistical theories of mental test scores, Reading, MA: Addison‐Wesley.
Markus,, K. A., & Borsboom,, D. (2013). Frontiers of test validity theory: Measurement, causation, and meaning, New York, NY: Routledge.

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