Rater-Mediated Domain Response Functions

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.

Emory University

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 MeasurementRater Variant Measurement

Panel A: Rasch Rater

Panel B: Birnbaum Rater

Panel C

Writing ProficiencyLowMediumHigh

Panel D

Writing ProficiencyLowMediumHigh
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

Please help with Standard Dataset 4: Andrich Rating Scale Model

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
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