# Dichotomous Equivalents to Rating Scales

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 22 to 33 to 44 to 55 to 66 to 77 to 88 to 99 to 10
31.39
41.101.10
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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