## Fuzzy Logic and the Rasch Model

"Fuzzy logic" was devised in the 1960s as a technique for analyzing situations in which precise probabilities cannot be determined, and in which the combination of probabilities does not follow the rules of probability theory. One implementation is the "Fuzzy Logic Model of Perception" (FLMP) discussed by Crowther, Batchelder and Hu (CBH, 1995). It relates to the recognition of patterns as belonging to particular categories.

CBH note that the "fuzzy truth" values for two factors, ci for category i and oj for object j, are combined according to Massaro & Friedman's (1990) "relative goodness rule", which is a version of Luce's (1959) choice rule:

pij = ci*oj/(ci*oj + (1-ci)(1-oj))

This FLMP equations has the form of a Team (Wright, RMT 9:2 432- 433) with two members. Moreover, CBH show the effect of adding more factors. This corresponds exactly to adding members to a Team.

CBH note that equation (1) becomes the Rasch model when

ci = 1 + exp(-Di)
oj = 1 + exp(Bj)

CBH reject the linear Di metric for the non-linear ci metric because the latter resembles 0-1 closed interval probability scaling.

Though equation (1) is intended to model a physical process, CBH perceive that conceptualizing it as a Rasch model endows it with precise properties, far superior to those of necessarily nebulous fuzzy logic. Nevertheless, CBH perceive hindrances to adoption of the Rasch model in psychophysics. Here are what CBH consider hindrances:

1) "Participants are generally drawn from a population at random."
Though this is sometimes true, as in the NAEP studies, the majority of subjects analyzed by the Rasch model are intentionally selected or self-selected individuals concerned about the implications of their own performances.

2) "The Rasch model does not include constraints on the rank ordering of persons or items."
Imposing such ordering a priori inhibits the data from challenging the validity of the externally imposed ranking. Subsequent plotting of Rasch measures against presumed rankings, physical measures or the like, reveals the empirical nature of the relationship.

3) "The Rasch model concerns the case where only one observation is obtained for each combination of participant and item."
This is true for Rasch analysis programs using rectangular data matrices (such as BIGSTEPS). But programs with more flexible data formats (such as Facets) allow multiple observations in each participant-item cell.

4) "Fortunately, there is a way to reformulate the Rasch model into useful log-linear forms. A variety of standard, readily available software exists to analyze such models."
It may seem convenient to perform analysis with familiar standard statistical packages. But readily available special-purpose Rasch software simplifies analysis and reporting, enables more parameters to be estimated, allows more flexible handling of missing data, and permits data with different Rasch parameterizations to be included in the same analysis.

CBH conclude "We hope our analysis of the measurement scale properties of Equation (1) of FLMP and its relation to the well- understood Rasch model will facilitate its use and, in particular, encourage more work designed to uncover the underlying processing events that give rise to the success of the equation in fitting data from factorial classification tasks" (p. 407). Ben Wright speculates that all mental processes will ultimately be discovered to be joint ventures of brain cells working as packs or teams. He expects that the convenient accident that many psychophysical processes are usefully described by the Rasch model will emerge as a theoretical necessity for the successful functioning of an organism.

Crowther C.S., Batchelder W.H., Hu X. (1995) A measurement- theoretic analysis of the fuzzy logic model of perception. Psychol Review 102(2) 396-408.

Luce R.D. (1959) Individual Choice Behavior. New York: Wiley.

Massaro D.W., Friedman D. (1990) Models of integration given multiple sources of information. Psychological Review 97 225-252.

Fuzzy Logic and the Rasch Model. Fisher WP Jr, Luce RD. … Rasch Measurement Transactions, 1995, 9:3 p.442

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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June 25 - July 23, 2021, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com