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-Related Resources: Rasch Measurement YouTube Channel
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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

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