Facets, Factors, Elements and Levels

1. Guttman's "Facet" Theory

Early test analysis was based on a simple rectangular conception: people encounter items. This could be termed a "two-facet" situation, loosely borrowing a term from Guttman's (1959) "Facet Theory". From a Rasch perspective, the person's ability, competence, motivation, etc., interacts with the item's difficulty, easiness, challenge, etc., to produce the observed outcome. In order to generalize, the individual persons and items are here termed "elements" of the "person" and "item" facets.

2. The Facets "many-facets" approach

Paired comparisons, such as a Chess Tournament or a Football League, are one-facet situations. The ability of one player interacts directly with the ability of another to produce the outcome. The one facet is "players", and each of its elements is a player. This can be extended easily to a non-rectangular two-facet design in order to estimate the advantage of playing first, e.g., playing the white pieces in Chess. The Rasch model then becomes:


where player n of ability Bn plays the white pieces against player m of ability Bm, and Aw is the advantage of playing white.

A three-facet situation occurs when a person encountering an item is rated by a judge. The person's ability interacting with the item's difficulty is rated by a judge with a degree of leniency or severity. A rating in a high category of a rating scale could equally well result from high ability, low difficulty, or high leniency.

Four-facet situations occur when a person performing a task is rated on items of performance by a judge. For instance, in Occupational Therapy, the person is a patient. The rater is a therapist. The task is "make a sandwich". The item is "find materials".

A typical Rasch model for a four-facet situation is:


where Di is the difficulty of item i, and Fik specifies that each item i has its own rating scale structure, i.e., the "partial credit" model.

And so on, for more facets. In these models, no one facet is treated any differently from the others. This is the conceptualization for "Many-facet Rasch Measurement" (Linacre, 1989) and the Facets computer program.

Of course, if all judges are equally severe, then all judge measures will be the same, and they can be omitted from the measurement model without changing the estimates for the other facets. But the inclusion of "dummy" facets, such as equal-severity judges, or gender, age, item type, etc., is often advantageous because their element-level fit statistics are informative.

3. The "Generalizability" approach

Multi-facet data can be conceptualized in other ways. In Generalizability theory, one facet is called the "object of measurement". All other facets are called "facets", and are regarded as sources of unwanted variance. Thus, in G-theory, a rectangular data set is a "one-facet design".

4. The LLTM "Linear Logistic Test Model" approach

In Gerhard Fischer's Linear Logistic Test Model (LLTM), all non-person facets are conceptualized as contributing to item difficulty. So, the dichotomous LLTM model for a four-facet situation (Fischer, 1995) is:


where p is the total count of all item, task and judge elements, and wil identifies which item, task and judge elements interact with person n to produce the current observation. The normalizing constraints are indicated by {c}. In this model, the components of difficulty are termed "factors" instead of "elements", so the model is said to estimate p factors rather than 4 facets. This is because the factors were originally conceptualized as internal components of item design, rather than external elements of item administration. Operationally, this is a two-facet analysis combined with a linear decomposition.

5. The RUMM2020 "Factor" approach

David Andrich's Rasch Unidimensional Measurement Models (RUMM) takes a fourth approach. Here the rater etc. facets are termed "factors" when they are modeled within the person or item facets, and the elements within the factors are termed "levels". Our four-facet model is expressed as a two-facet person-item model, with the item facet defined to encompass three factors. The "rating scale" version is:


where Di is an average of all δmij for item i, Am is an average of all δmij for task m, etc.

This approach is particularly convenient because it can be applied to the output of any two-facet estimation program, by hand or with a spreadsheet program. Operationally, this is a two-facet analysis followed by a linear decomposition.. Missing δmij may need to be imputed. With a fully-crossed design, a robust averaging method is standard-error weighting (RMT 8:3 p. 376). With some extra effort, element-level quality-control fit statistics can also be computed.

John M. Linacre

Fischer, G.H., & Molenaar, I.W. (Eds.) (1995) Rasch Models: Foundations, Recent Developments and Applications. New York: Springer.

Guttman, L. (1959) A structural theory for intergroup beliefs and action. American Sociological Review, 24, 318-328.

Facets, factors, elements and levels. Linacre, JM. … 16:2 p.880


Facets, factors, elements and levels. Linacre, JM. … Rasch Measurement Transactions, 2002, 16:2 p.880



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
Rasch Books and Publications: Winsteps and Facets
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 Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang 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
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

To be emailed about new material on www.rasch.org
please enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from Rasch.org

www.rasch.org welcomes your comments:

Your email address (if you want us to reply):

 

ForumRasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.

Coming Rasch-related Events
Oct. 4 - Nov. 8, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 17 - Feb. 21, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 16 - June 20, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 20 - July 18, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Further Topics (E. Smith, Facets), www.statistics.com
Oct. 3 - Nov. 7, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com

 

The URL of this page is www.rasch.org/rmt/rmt162h.htm

Website: www.rasch.org/rmt/contents.htm