Rasch Model derived from Campbell Concatenation: additivity, interval scaling

Can mental testing meet physicist Norman Campbell's "concatenate heads" requirement for fundamental measurement, namely that the combination of X and Y be more than X or Y, and that X and Y concatenate to produce an X+Y result, interpretable as a physical addition operation? Campbell's concatenation follows strict rules. When a rod of length X is "added" (in a specific way denoted by Ð ) to a rod of length Y, then their combined length is X+Y, i.e.,

In psychological measurement, what happens to the probability of a right answer when two persons, m and n of abilities B_n and B_m, join forces to agree on an answer to item i of difficulty D_i?

Let P_ni be the probability that person n succeeds on item i. Then

also

Are two heads better than one when both are smarter than the item, i.e., both X and Y are positive, so that P_ni>0.5 and P_mi>0.5? Yes, because, since they agree on the answer, the probability space is limited to both succeeding or both failing:

This consensus, "team-work", approach to problem solving is often used by committees. It succeeds admirably provided that each individual committee member has a better than 50% chance of coming to a correct decision.

Can person abilities be concatenated? Yes, because the probability that persons n and m succeed or fail on item i when combined as a team produces:

Thus the Rasch probability of a right answer when two persons work together depends on the concatenations of the difficulty of item D_i with the abilities of the two persons, B_n and B_m. The relation

is a concatenation of abilities on the interval scale defined by

What happens when one person is faced with two items, i and j, combined in such a way that the person either succeeds on both or fails on both? Is the combination of two hard items harder than one? Yes, because P_ni<0.5 and P_nj<0.5, so that:

Can item difficulties be concatenated? Yes, in the same way as person abilities. The Rasch probability of getting two items correct depends on the concatenations of the ability of the person, B_n, with the difficulties of the two items, D_i and D_j, producing

Can item difficulties be added to equate person abilities? Yes, because, when P_ni follows the Rasch model, the addition necessary to make the performance of person n on item i appear stochastically equivalent to the performance of person m on the same item i is:

1) pair up another item j with each item i for person n.

2) choose item j such that its difficulty is

3) consider for person n only those cases in which person n succeeds or fails on both i and j, the probability of success (or failure) of person n on items i and j combined becomes

i.e., equivalent to person m on item i.

Benjamin D. Wright

Rasch model derived from Campbell concatenation: additivity, interval scaling. Wright B.D. … Rasch Measurement Transactions, 1988, 2:1 p. 16.

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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