There are numerous ways to conceptualize rating scales. One useful conceptualization is to imagine that the rating scale is equivalent to a set of dichotomous items. Huynh Huynh investigated this: Huynh H. (1994) On equivalence between a partial credit item and a set of independent Rasch binary items. Psychometrika, 59, 111-119, and Huynh H. (1996) Decomposition of a Rasch partial credit item into independent binary and indecomposable trinary items. Psychometrika, 61, 31-39.
A crucial finding is that the Rasch-Andrich thresholds must advance (i.e., not exhibit "threshold disordering") for a polytomy to have the mathematical properties of a set of dichotomies. But merely advancing is not enough.
Consider a polytomy with m+1 ordinally advancing categories. There are m transition points, so this could be conceptualized as m dichotomies. As the Rasch-Andrich thresholds for the polytomy become further apart then the set of dichotomous items would have a wider difficulty range. The boundary condition is that the m dichotomies be of equal difficulty. Then a score of k on the polytomous item would be equivalent to scoring k on m equally-difficulty dichotomies.
A set of equally difficulty dichotomies constitute a set of Bernoulli (binomial) trials. The polytomous Rasch model for this is (with the familiar notation):
This provides the lower limits by which Rasch-Andrich thresholds must advance in order that a polytomy have the same mathematical properties as a set of dichotomies. A useful rule-of-thumb is "thresholds must advance by one-logit". The exact values are tabulated below.
John Michael Linacre
Minimum Rasch-Andrich threshold advances for a polytomy to be equivalent to a set of dichotomies | |||||||||
---|---|---|---|---|---|---|---|---|---|
Thresholds:------Categories: | 1 to 2 | 2 to 3 | 3 to 4 | 4 to 5 | 5 to 6 | 6 to 7 | 7 to 8 | 8 to 9 | 9 to 10 |
3 | 1.39 | ||||||||
4 | 1.10 | 1.10 | |||||||
5 | .98 | .81 | .98 | ||||||
6 | .92 | .69 | .69 | .92 | |||||
7 | .88 | .63 | .58 | .63 | .88 | ||||
8 | .85 | .59 | .51 | .51 | .59 | .85 | |||
9 | .83 | .56 | .47 | .45 | .47 | .56 | .83 | ||
10 | .81 | .54 | .44 | .41 | .41 | .44 | .54 | .81 | |
11 | .80 | .52 | .42 | .38 | .36 | .38 | .42 | .52 | .80 |
Dichotomous Equivalents to Rating Scales, Linacre J.M. Rasch Measurement Transactions, 2006, 20:1 p. 1052
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