# Thurstone Thresholds and the Rasch Model

A Thurstone threshold is a location on the latent variable with a precise definition. It is the point on the variable (in the context of a particular item) at which the probability of being observed below a given category is equal that of being observed in or above that category. The Thurstone threshold corresponding to category k for a rating scale with categories 0,m is given by

In the Thurstone approach, the category boundaries are merely convenient markers along the latent variable. One set of categories is as good as another. Choice of categorization is arbitrary. Dichotomizing ages, weights, temperatures or percentages often involves decisions in which there are a range of equally defensible "Thurstone" choices.

 Table I. Adding a category under Thurstone conditions Cat. Count Thurstone threshold Rasch step threshold 0 1 2 10 10 10 - loge(10/20)=-.7 loge(20/10)=+.7 - loge(10/10)=0 loge(10/10)=0 0 1 2 3 10 10 5 5 - loge(10/20)=-.7 loge(20/10)=+.7 loge(25/5)=+1.6 - loge(10/10)=0 loge(10/5)=1.3 loge(5/5)=0

In the Rasch approach, each category represents a distinct qualitative advance over its predecessor. The Rasch threshold is given by

Pk = Pk-1

There is a distinct "Rasch" difference between a "right" answer and a "wrong" answer. For dichotomous items, the Thurstone threshold and the Rasch item difficulty are at the same place, but with polytomous items, the relationship between the Rasch step "thresholds" and the Thurstone thresholds is convoluted.

 Table II. Adding a category under Rasch conditions Cat. Count Thurstone threshold Rasch step threshold 0 1 2 10 10 10 - loge(10/20)=-.7 loge(20/10)=+.7 - loge(10/10)=0 loge(10/10)=0 0 1 2 3 8 8 8 6 - loge(8/22)=-1.0 loge(16/14)=+.1 loge(24/6)=+1.4 - loge(8/8)=0 loge(8/8)=0 loge(8/6)=+.3

Thurstone thresholds appear to implement a virtue absent from Rasch rating scale step thresholds. They are independent of the number of categories in the rating scale. Imagine that an extra category, m+1 is defined at the top of the scale. From the Thurstone perspective, this will reduce the probability of category m, but only by Pm+1. In principle, no other category probabilities are affected. Table I illustrates 30 responses by persons exactly targeted on an item. Initially the scale has 3 categories, then a fourth category is defined at the top of the scale in accordance with Thurstone's approach. The Thurstone thresholds always advance, as is seen for both the 3 and 4 category scales.

In this example, the Rasch step thresholds for the 3 category scale are equal. Adding the additional category introduces step disordering. The step thresholds become 0, 1.3, 0. A change to the top of the rating scale raises doubt about the functioning of the whole scale.

Table II illustrates adding a category while maintaining the Rasch step thresholds. The new category doesn't merely subtract from its neighboring category, it subtracts from all other categories. In doing so, it changes the Thurstone thresholds. They are still ordered but now at new locations on the variable. Adding an additional category has changed the definition of the rating scale. Previously 3 levels defined the entire continuum. Now there is a fourth level to be reached. Adding a level at the top requires that all levels below be squashed down to make room. Categories are not separately defined, as in the Thurstone case, but jointly defined. Conceptually, the respondent interacts with the whole set of categories, not the categories one at a time.

 Table III. Analyzing Thurstone-style rating data under Rasch conditions Thurstone observations Rasch dichotomies Additional category X = 0 X = 1 X = 2 X = 3 X = 4 Item: ABCD {0} = 0000 {1} = 1000 {2} = 1100 {3} = 1110 {4} = 1111 Item: ABCDE {0} = 00000 {1} = 10000 {2} = 11000 {3} = 11100 {4} = 11110 {5} = 11111

Rasch measures with a Thurstone scales

If data are collected using Thurstone's rating scale specifications, they can easily be reformulated to support Rasch measurement. Each Thurstone threshold is implemented as a Rasch dichotomy. Then the thresholds/dichotomies are as independent as possible, maintaining their relative difficulties as new categories are added.

Table III shows what a Thurstone observation on a rating scale from 0-4 implies. The rating scale is a set of arbitrary cut-points, so that the 5 category scale represents 4 cut-points. These 4 cut-points can be implemented as 4 dichotomous items (represented by columns ABCD in the Table cells). "1" is recorded if the cut-point is crossed, "0" otherwise. Adding a higher category adds an additional item E, but does not change the rest of the data.

The dichotomies in Table III are hierarchically correlated, a consequence of the Thurstone approach. This nesting of one category within another introduces dependencies that produce overfit to the Rasch model. Nevertheless, when data are collected on a Thurstone scale, the dichotomizations (or even the original rating scale categories) usually approximate Rasch specifications closely enough to produce useful measures. Frequently Thurstone cut-points have substantive meanings, such as an age of 21, or a temperature of 100C. But even when qualitative advances are not obvious, the manner in which the scale has been used by respondents often indicates that qualitative advances are implied. Rasch rating scale statistics assist with identifying these advances, and in reformulating arbitrary Thurstone data into data that better support meaningful Rasch measurement.
John Michael Linacre

Andrich, D. (1995). Distinctive and incompatible properties of two common classes of IRT modules for graded responses. Applied Psychological Measurement. 19, (1) 101-119.

Andrich, D. (1996). Measurement criteria for choosing among models for graded responses. In A. von Eye and C. C. Clogg (Eds.), Analysis of categorical variables in developmental research. Orlando, Florida: Academic Press. (Chapter 1, pp 3-35.)

Thurstone thresholds and the Rasch model.Linacre J.M. … Rasch Measurement Transactions, 1998, 12:2 p. 634-5

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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