Rasch Model derived from Thurstone's Scaling Requirements

A popular "hierarchy" of psychometric models is

"1-parameter"

"2-parameter"

"3-parameter"

These models are said to be a "hierarchy" because when C_i = 0, then 3p becomes 2p, and when A_i=1, then 2p becomes 1p. This is sometimes interpreted to imply that the "best" model must be the one with the most parameters, i.e., 3p. The purpose of these models, however, is to construct scales on which measures can be made. The fundamental question, then, is do these models fulfill that purpose?

According to L.L. Thurstone (American Journal of Sociology, 1928, 33, 529-554, reprinted in The Measurement of Values, 1959, 228):

"The scale must transcend the group measured... One crucial experimental test must be applied to our method of measuring attitudes before it can be accepted as valid. A measuring instrument must not be seriously affected in its measuring function by the object of measurement. To the extent that its measuring function is so affected, the validity of the instrument is impaired or limited. If a yardstick measured differently because of the fact that it was a rug, a picture, or a piece of paper that was being measured, then to that extent the trustworthiness of that yardstick as a measuring device would be impaired. Within the range of objects for which the measuring instrument is intended, its function must be independent of the object of measurement."

What kind of model is necessary to meet Thurstone's requirement for a scale?

What model is required for the comparison and hence calibration of questions i and j to be independent of the persons used to elicit evidence of their relative standing on the scale? Questions i and j appear different only when they are answered differently. Realizing a comparison of i and j requires comparing how often i is answered "yes" when j is simultaneously answered "no" with how often the reverse happens.

1) The estimation of a comparison of i and j from these frequencies requires a comparison of their probabilities for some person n,

But this comparison must be the same regardless of which persons are involved. To satisfy Thurstone's requirement for scale validity, it must be the case that the comparison of questions i with j is always the same whoever the person.

2) Thus it must follow for any other person m that

3) For simplicity let j=o and m=o become the reference origins of the question and person scales so that the calibration of question i becomes its difference from "standard" item j=o and the measure of person n becomes its difference from "standard" person m=o, and align these scale origins so that P_mj = P_oo = ½, then

which defines a ratio scale for the measure of person n and the calibration of item i.

4) This ratio scale can be linearized by

which when solved for P_ni shows that

is the model necessary for Thurstone scale validity.

Step 2 can be rewritten to address Thurstone's concomitant 1926 requirement that the individual measure not depend on which particular items are used so that it becomes "possible to omit several test questions at different levels of the scale without affecting the individual score." This requires that the comparison of any pair of persons n and m be invariant with respect to the particular items employed as in

which is equivalent to step (2) and so leads to step (4).

Thus of the popular "hierarchy", only the "1-parameter" Rasch model meets Thurstone's requirement for scale validity. The "2-parameter" and "3-parameter" models do not. Their use does not, and cannot, produce a scale which is independent of the persons used to obtain it.

Benjamin D. Wright

Rasch model derived from Thurstone's scaling requirements. Wright B.D. … Rasch Measurement Transactions, 1988, 2:1 p. 13-4.

1. The Rasch Model derived from E. L. Thorndike's 1904 Criteria, Thorndike, E.L.; Linacre, J.M. … 2000, 14:3 p.763
2. Rasch model derived from consistent stochastic Guttman ordering, Roskam EE, Jansen PGW. … 6:3 p.232
3. Rasch model derived from Counts of Right and Wrong Answers, Wright BD. … 6:2 p.219
4. Rasch model derived from counting right answers: raw Scores as sufficient statistics, Wright BD. … 1989, 3:2 p.62
5. Rasch model derived from Thurstone's scaling requirements, Wright B.D. … 1988, 2:1 p. 13-4.
6. Rasch model derived from Campbell concatenation: additivity, interval scaling, Wright B.D. … 1988, 2:1 p. 16.
7. Dichotomous Rasch model derived from specific objectivity, Wright BD, Linacre JM. … 1987, 1:1 p.5-6

Rasch Books and Publications
Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn. George Engelhard, Jr. & Jue Wang	Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene	Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver	Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone	Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	Statistical Analyses for Language Testers (Facets), Rita Green	Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind	Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind	Rasch Measurement: Applications, Khine	Winsteps Tutorials - free Facets Tutorials - free	Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre	Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan
Other Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free	An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse	Rasch Measurement Theory Analysis in R, Wind, Hua	Applying the Rasch Model in Social Sciences Using R, Lamprianou	El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar	Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch	Rasch Models for Measurement, David Andrich	Constructing Measures, Mark Wilson	Best Test Design - free, Wright & Stone Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias	Diseño de Mejores Pruebas - free, Spanish Best Test Design	A Course in Rasch Measurement Theory, Andrich, Marais	Rasch Models in Health, Christensen, Kreiner, Mesba	Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen

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