Abstract
Ordered categories are taken as analogous to physical measurement with a continuum partitioned by successive thresholds. However, unlike physical measurements, the thresholds which form the categories are not assumed to be equidistant. Thresholds are defined by minimum proficiencies required to succeed at the thresholds. The threshold form of the polytomous Rasch model involves the hypothesis that the thresholds are ordered. Threshold estimates from data may be disordered, in which case they reflect a problem with the empirical ordering of the categories.
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References
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Exercises
Exercises
Suppose the minimum proficiencies of \( \beta_{P},\beta_{C},\beta_{D} \) for achieving Pass (P), Credit (C) and Distinction (D), respectively, on an item i with four ordered categories are \( - 0.50, - 0.10,0.60. \)
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(a)
What are the threshold values \( \delta_{iP},\delta_{iC},\delta_{iD} \)?
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(b)
Assume a person has a proficiency of \( \beta_{n} = 0.0 \). Complete the probabilities \( \Pr \{y_{niP} |\Omega \},\Pr \{y_{niC} |\Omega \} \,{\text{and}}\Pr \{y_{niD} |\Omega \} \) of Table 27.2.
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(c)
Calculate the probabilities \( \Pr \{(y_{niP},y_{niC},y_{niD})|\Omega \} \) for the subset of Guttman patterns of Table 27.4.
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(d)
Normalize the subset of probabilities calculated in (c) to give the probabilities \( \Pr \{x;\beta_{n},(\delta_{i})|\Omega ^{G} \},x = 0,1,2,3 \) (that is, ensure their sum is 1).
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(e)
Calculate \( \Pr \{x;\beta_{n},(\delta_{i})|\Omega _{x - 1,x}^{G} \} \) for \( x = 1,2,3 \).
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(f)
Which of the probabilities you calculated in (e) above are, respectively, identical to the probabilities you calculated in (b) above.
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Andrich, D., Marais, I. (2019). Derivation of the Threshold Form of the Polytomous Rasch Model. In: A Course in Rasch Measurement Theory. Springer Texts in Education. Springer, Singapore. https://doi.org/10.1007/978-981-13-7496-8_27
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DOI: https://doi.org/10.1007/978-981-13-7496-8_27
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