Abstract
Modern test theory models focus on the interaction of a person with an item rather than upon a total scoreas in CTT. Rasch specified a two-way frame of reference of items and persons. In the dichotomous Rasch model, the probability of a person’s response to an item is a function of the difference between two model parameters, the item’s location (δ—difficulty in assessment of proficiency) and the person’s location (\( \beta \)—proficiency in assessment of proficiency). The dichotomous Rasch model is the simplest modern test theory model. An item characteristic curve (ICC) shows the probability of a correct response to an item with location \( \delta \) according to the model for persons of different locations \( \beta \) on the variable. It is a requirement fulfilled by the Rasch model that comparisons between persons in terms of their parameters and between items in terms of their parameters, within a frame of reference, are invariant.
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References
Andrich, D. (2004). Controversy and the Rasch model: A characteristic of incompatible paradigms? Medical Care,42(1), i7–i16.
Andrich, D. (2005). Rasch, George. In K. Kempf-Leonard (Ed.), Encyclopedia of social measurement (Vol. 3, pp. 299–306). Amsterdam: Academic Press.
Rasch, G. (1960/1980). Foreword and introduction. In Probabilistic models for some intelligence and attainment tests (pp. 3–12, pp. ix–xix). Copenhagen, Danish Institute for Educational Research. Expanded edition (1980) with foreword and afterword by B. D. Wright. Chicago: The University of Chicago Press. Reprinted (1993) Chicago: MESA Press.
Further Reading
Andrich, D. (1988). Chapters 3 and 4. Rasch models for measurement. Newbury Park, CA: Sage.
Ryan, J. P. (1983). Introduction to latent trait analysis and item response theory. In W. E. Hathaway (Ed.), Testing in the schools: New directions for testing and measurement (Vol. 19, pp. 49–64). San Fransisco: Jossey-Bass.
Wright, B. D., & Stone, M. H. (1979). The measurement model. In Best test design: Rasch measurement (pp. 1–17). Chicago: MESA Press.
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Exercises
Exercises
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1.
In the Rasch model, we denote the proficiency of person n by \( \beta_{n} \).
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(a)
What is the corresponding parameter in CTT?
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(b)
When we refer to the parameter \( \beta_{n} \) as the proficiency of person n, in what sense is this a person’s proficiency?
[Use no more than two sentences to answer this question]
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(c)
Why is a model such as the Rasch model referred to as a unidimensional model?
[Use no more than two sentences to answer this question]
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(a)
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2.
Suppose person n has the proficiency \( \beta_{n} \) = 1.2 and that this person attempts three items with difficulties \( \delta_{\,1} \) = −1.0, \( \delta_{2} \) = 1.2, and \( \delta_{\,3} \) = 2.0.
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(a)
What is the probability that this person will answer each item correctly?
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(b)
Describe in no more than two sentences what you understand by the term probability of answering an item correctly.
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(a)
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3.
Suppose five persons with proficiencies \( \beta_{1} \) = −1.9, \( \beta_{2} \) = −0.9, \( \beta_{3} \) = 0.1, \( \beta_{4} \) = 1.1, and \( \beta_{5} \) = 2.1 attempt an item with difficulty \( \delta \) = 0.3.
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(a)
What is the probability that each person will answer the item correctly?
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(b)
Draw a pair of axes for a graph with the proficiency as the horizontal axis and the probability of a correct response as the vertical axis. On this graph, plot the probability for the five persons attempting the item. Mark the proficiency values of the persons and the difficulty value of the item on the horizontal axis.
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(a)
For further exercises see Exercise 1: Interpretation of RUMM2030 printout in Appendix C.
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Andrich, D., Marais, I. (2019). The Dichotomous Rasch Model—The Simplest Modern Test Theory Model. In: A Course in Rasch Measurement Theory. Springer Texts in Education. Springer, Singapore. https://doi.org/10.1007/978-981-13-7496-8_6
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