Rating scale category boundaries can be conceptualized in a number of ways. L. L. Thurstone (1928) describes the computation of .50 cumulative proportions as "scale values." These scale values are now referred to as Thurstone thresholds. They are also the parameters in the "Graded Response" model.
Rasch rating scale structures are parameterized using the points of equal-probability of adjacent categories, rather than the points of equal probability of accumulated category probabilities. Nevertheless, in communicating Rasch findings, it can be convenient to represent Rasch rating scale functioning in terms of Thurstone- type thresholds.
Rasch polytomous models, such as Andrich "rating scale" or Masters "partial credit" models have the form:
1 |
with the usual notation conventions, and F_{g0} = 0. F_{gj} parameterizes the "Rasch-Andrich threshold" or "step", the point of equal probability of categories j-1 and j. The subscript "g" indicates the manner in which the set of {F_{gj}} parameters relates to the n or i parameters. For the Andrich "rating scale" model, "g" signifies all items. For the Masters' "Partial Credit" model, "g" signifies item i. For Ben Wright's "Style" model, "g" signifies person n. For an instrument in which different groups of items share common rating scales, "g" identifies the item groups.
Let the Rasch-Thurstone-type thresholds be identified as {T_{gj}} relative to item difficulty, D_{i.} Then
2 |
for j=1,m. So that, multiplying through by the normalizer,
3 |
Then let
4 |
so that, for each of j=1,m,
5 |
If the T_{gj} are specified, then the t_{j} are known, and the c_{k} can be obtained by solving the m simultaneous equations. From the c_{k}, the F_{gh} can be computed directly. Thus a polytomous Rasch model can be parameterized in terms of Thurstone-type thresholds using matrix notation and Cramer's rule.
On the other hand, if the F_{gh} are specified, then the c_{k} are known. Each of the m equations becomes a polynomial in t_{j}. The required root always exists. The lower bound of the search for t_{j} is zero (when the polynomial must be positive), and t_{j} can be increased until the polynomial becomes negative. When the value of t_{j} has been found for which the equation is well enough satisfied, then T_{gj} is computed.
A 3 category, so two threshold, item has Rasch-Andrich thresholds -0.85, 0.85. The lower Rasch-Thurstone-type Threshold is given by j=1:
6 |
so that t_{1} = 0.37, and T_{1} is -1.0. By symmetry, T_{2} is +1.0.
Working backwards for the Rasch-Andrich thresholds, if the Rasch-Thurstone-type thresholds are -1, +1, then
7 |
So that F_{1} = log_{e}(1/(e^{1} - e^{-1})) = -.85, and F_{2} = +.85.
John M. Linacre
Thurstone L.L. (1928) Attitudes can be measured. American Journal of Sociology, 33, 529-54.
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Estimating 50% Cumulative Probability (Rasch-Thurstone) Thresholds. Linacre JM. … Rasch Measurement Transactions, 2003, 16:4 p.901
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