Rater-mediated assessments are used extensively in a variety of educational contexts (Engelhard, 2002). In evaluating the quality of ratings obtained in these contexts, the idea of rater-mediated operating characteristic functions (rm-OCFs) has not been systematically explored. OCFs can be used to enhance the substantive interpretations of rater behaviors. For example, the substantive interpretation of crossing item response functions (IRFs) is fairly well known (Wright, 1997), and Perkins and Engelhard (2009) have discussed crossing person response functions (PRFs). Similar ideas can be used to develop rater-mediated domain response functions (rm-DRFs), as well as rater-mediated person response functions (rm-PRFs). Just as crossing IRFs or PRFs create differential ordering of item difficulty and person performance, crossing rm-DRFs and rm-PRFs have implications for the substantive interpretation of rater behavior. When rm-DRFs cross, the interpretation of the domains across the latent variable is not invariant above and below the intersection points. This note provides an illustration of crossing rm-DRFs, and demonstrates the substantive interpretation of this situation.
Both Rasch (1960/1980) and Birnbaum (1968) propose operating characteristic functions for dichotomous responses that can be used to model dichotomous ratings. For example, a Rasch model for dichotomous ratings can be written as follows:
where φnmi is the probability when θn is the judged location of person n on the latent variable (e.g., writing proficiency) by rater m with a severity of λm on domain i with a judged difficulty of δi .
A Birnbaum Model for dichotomous ratings can be written as
where αi is a scale parameter that varies across domains, and ci is the lower asymptote of the function that represents rater reluctance to assign low ratings to persons (a comparable upper asymptote can also be introduced for rater reluctance to assign high scores).
In the context of rater-mediated assessments, the rm-DRF for a Rasch rater (λR) on domain one (δ1) rated dichotomously (fail/pass) can be written as:
and for a Birnbaum rater (lB):
The general requirements for invariant measurement are summarized by Engelhard and Perkins (2011), and these requirements can be extended for raters (Wind & Engelhard, 2011):
The measurement of persons must be independent of the particular raters that happen to be used for measuring: Rater-invariant measurement of persons.
Figure 1 illustrates the effects of crossing rm-DRFs for two raters who are rating writing proficiency using three domains: Mechanics (M), Content (C), and Organization (O). Panel A is a Rasch rater with non-crossing DRFs, while Panel B is a Birnbaum rater with crossing DRFs. Panel C shows a substantive interpretation for non-crossing DRFs that produce comparable judged domain difficulties over subgroups of persons. The ordering of the three domains is invariant with the mechanics (M) domain judged easiest and organization (O) domain judged as hardest across the latent variable of writing proficiency. Non-crossing DRFs result in equivalent ordering of domains across subsets of persons, and yields invariant measurement from the Rasch rater.
Panel D shows the substantive interpretation of crossing DRFs based on a Birnbaum rater. The meaning of person performance on domains varies as a function of person subgroup locations on the latent variable of writing proficiency. The Rasch rater interprets the domains in a comparable way over subgroups with domains ordered as M < C < O, while the domain difficulties are variant for the Birnbaum rater. The Birnbaum rater rates the organization (O) domain easiest for persons with low writing proficiency, while organization (O) is rated hardest for persons with high writing proficient.
In practice, model-data fit and the requirements of invariant measurement can be usefully visualized with OCFs. This note highlights the need for researchers to examine differential domain functioning as an additional aspect of model-data fit within the context of rater-mediated assessments. It is recognized that domains may function differently over subgroups of persons (differential domain functioning).
Stefanie A. Wind & George Engelhard, Jr.
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability, Part 5. In F.M. Lord and M.R. Novick (Eds.), Statistical theories of mental test scores. Reading, MA: Addison-Wesley Publishing Company, Inc.
Engelhard, G. (2002). Monitoring raters in performance assessments. In G. Tindal and T. Haladyna (Eds.), Large-scale Assessment Programs for ALL Students: Development, Implementation, and Analysis, (pp. 261-287). Mahwah, NJ: Erlbaum.
Engelhard, G, & Perkins, A.F. (2011). Person response functions and the definition of units in the social sciences. Measurement: Interdisciplinary Research and Perspectives, 9, 40-45.
Perkins, A., & Engelhard, G. (2009). Crossing person response functions. Rasch Measurement Transaction, 23(1), 1183-1184.
Rasch (1960/1980). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research. (Expanded edition, Chicago: University of Chicago Press, 1980).
Wind, S.A., & Engelhard, G. (2011, July). Evaluating the quality of ratings in writing assessment: Rater agreement, precision, and accuracy. Paper presented at the Pacific Rim Objective Measurement Seminar (PROMS) in Singapore.
Wright, B.D. (1997). A history of social science measurement. Educational Measurement: Issues and Practice, Winter, 33- 45, 52.
|Rater Invariant Measurement||Rater Variant Measurement|
Panel A: Rasch Rater
Panel B: Birnbaum Rater
Figure 1. Impact of Crossing Rater-Mediated Domain Response Functions.|
The domains are Mechanics (M), Content (c), and Organization (O).
Rater-Mediated Domain Response Functions, Stefanie A. Wind & George Engelhard, Jr. ... Rasch Measurement Transactions, 2011, 251:2, 1321-2
|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|
|Forum||Rasch 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|
|March 21, 2019, Thur.||13th annual meeting of the UK Rasch user group, Cambridge, UK, http://www.cambridgeassessment.org.uk/events/uk-rasch-user-group-2019|
|April 4 - 8, 2019, Thur.-Mon.||NCME annual meeting, Toronto, Canada,https://ncme.connectedcommunity.org/meetings/annual|
|April 5 - 9, 2019, Fri.-Tue.||AERA annual meeting, Toronto, Canada,www.aera.net/Events-Meetings/Annual-Meeting|
|April 12, 2019, Fri.||On-line course: Understanding Rasch Measurement Theory - Master's Level (G. Masters), https://www.acer.org/au/professional-learning/postgraduate/rasch|
|May 24 - June 21, 2019, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|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 email@example.com|
|June 17-19, 2019, Mon.-Wed.||In-person workshop, Melbourne, Australia: Applying the Rasch Model in the Human Sciences: Introduction to Rasch measurement (Trevor Bond, Winsteps), Announcement|
|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|
|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|
|Nov. 3 - Nov. 4, 2019, Sun.-Mon.||International Outcome Measurement Conference, Chicago, IL,http://jampress.org/iomc2019.htm|
|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|
The URL of this page is www.rasch.org/rmt/rmt252a.htm