Common-Item (or Common Person) Equating with Different Test Discriminations

Figure 1. Exact common-item equating

Figure 2. Statistically-equivalent equating

Figure 3. Fahrenheit-to-Celsius Equating

If some items are the same in two different tests, then these may be chosen as "common items". When we perform a Rasch-analysis of the each test separately, we will obtain an estimate of the item difficulty of each common item on each test.

We expect that a scatterplot of the pairs of item estimates for the common items will have a trend line effectively parallel to the identity line. Figure 1 shows this situation in which the 12 common items lie along a trend line (dotted) parallel to the identity line (arrows). 0.0 logits for Form A (x-axis) corresponds to 1.1 logits for Form B (y-axis). To equate Form B with Form A, we subtract 1.1 logits from the Form B item and person measures.

But the trend line may not be parallel to the identity line. What do we do? Figure 2 shows a situation in which the trend line is not a line (arrow) parallel to the identity line. But the arrow falls with in the 95%-confidence error bands (black curves). We cannot reject the hypothesis that the trend line is statistically parallel to the identity line.

Under these circumstances, there are two ways forward.

1. We can choose the Form which we consider to be more definitive. Then obtain its item difficulties and Rasch-Andrich thresholds. Use these values as anchor (fixed) values in an analysis of the other Form. This puts the measures of the second form into the measurement frame-of-reference of the first form.

or 2. We can combine the data from the two Forms into one analysis (concurrent equating) in which all the items and persons are measured in the same frame-of-reference.

Figure 3 shows the situation in which the trend line (dotted line) is obviously and statistically not parallel to the identity line (arrow). Now we have tests of different discrimination, and a "Fahrenheit-to-Celsius" (F-C) equating situation, similar to that encountered with temperature conversions. The "Fahrenheit" measures on Form B needs to be rescale as "Celsius" measures compatible with Form A. An equation for doing this is:

Form B rescaled as Form A = (mean of Form A common items) + (Form B measure - mean of Form B common items) * S.D. of Form A common items / S.D. of Form B common items.

If we have two tests with common items that we know to be functionally different (such as from different test publishers, or in different presentation formats) then we expect to do F-C equating.

If we have two tests with common items that are supposed to be the same (such as alternate test forms, or pre-test and post-test forms), then we are reluctant to do F-C equating. We usually decide which form is the "correct" form (or combine the two forms) and use it as the basis for the equating.

(Suggested by Mg. Andrés Burga León, Ministerio de Educación del Perú)

Common-Item (or Common Person) Equating with Different Test Discriminations … Andrés Burga León, Rasch Measurement Transactions, 2008, 22:3, 1172

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

To be emailed about new material on
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 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,

Coming Rasch-related Events
June 29 - July 27, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps),
July 25 - July 27, 2018, Wed.-Fri. Pacific-Rim Objective Measurement Symposium (PROMS), (Preconference workshops July 23-24, 2018) Fudan University, Shanghai, China "Applying Rasch Measurement in Language Assessment and across the Human Sciences",
July 29 - August 4, 2018 Vth International Summer School `Applied Psychometrics in Psychology and Education`, Institute of Education at the Higher School of Economics, St. Petersburg, Russia,
July 30 - Nov., 2018Online Introduction to Classical and Rasch Measurement Theories (D.Andrich), University of Western Australia, Perth, Australia,
Aug. 10 - Sept. 7, 2018, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets),
August 25 - 28, 2018, Sat.-Tue.Análisis de Rasch introductorio (en español). (Agustín Tristán), Instituto de Evaluación e Ingeniería Avanzada. San Luis Potosí, México.
Sept. 3 - 6, 2018, Mon.-Thurs. IMEKO World Congress, Belfast, Northern Ireland,
Oct. 12 - Nov. 9, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps),


The URL of this page is