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 Ci = 0, then 3p becomes 2p, and when Ai=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 Pmj = Poo = 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 Pni 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 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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May 22 - 30, 2019, Wed.-Thu. Measuring and scale construction (with the Rasch Model), University of Manchester, England, https://www.cmist.manchester.ac.uk/study/short/intermediate/measurement-with-the-rasch-model/
June 4 - 7, 2019, Tue.-Fri.In-Person Italian Rasch Analysis Workshop based on RUMM (Fabio La Porta and Serena Caselli; entirely in Italian). Prof David Andrich from Western Australia University will be hosted by the workshop. For enquiries and registration email to workshop.rasch@gmail.com
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June 20-21, 2019, Thurs.-Fri. In-person workshop, Melbourne, Australia: Applying the Rasch Model in the Human Sciences: Advanced Rasch measurement with Facets (Trevor Bond, Facets), Announcement
June 28 - July 26, 2019, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
July 2-5, 2019, Tue.-Fri. 2019 International Measurement Confederation (IMEKO) Joint Symposium, St. Petersburg, Russia,https://imeko19-spb.org
July 11-12 & 15-19, 2019, Thu.-Fri. A Course in Rasch Measurement Theory (D.Andrich), University of Western Australia, Perth, Australia, flyer - http://www.education.uwa.edu.au/ppl/courses
Aug 5 - 10, 2019, Mon.-Sat. 6th International Summer School "Applied Psychometrics in Psychology and Education", Institute of Education at HSE University Moscow, Russia.https://ioe.hse.ru/en/announcements/248134963.html
Aug. 9 - Sept. 6, 2019, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Aug. 14 - 16, 2019. Wed.-Fri. An Introduction to Rasch Measurement: Theory and Applications (workshop led by Richard M. Smith) https://www.hkr.se/pmhealth2019rs
August 25-30, 2019, Sun.-Fri. Pacific Rim Objective Measurement Society (PROMS) 2019, Surabaya, Indonesia https://proms.promsociety.org/2019/
Oct. 11 - Nov. 8, 2019, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
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Jan. 24 - Feb. 21, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 22 - June 19, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 26 - July 24, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 7 - Sept. 4, 2020, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 9 - Nov. 6, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 25 - July 23, 2021, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com