Equating Constants with Mixed Item Types

Question: I am doing non-equivalent-groups common-item equating of two tests using the Rasch model. My common items are 20 multiple-choice items (worth 1 point) and 1 partial credit (essay) item worth 4 points. How do I compute an equating constant?

Response: Let's assume a scatter-plot of the difficulties of the 21 items on the two tests indicates that their pattern approximates an identity line. (If it does not, you need to investigate whether the "common items" really are common.) You now need to compute a defensible equating constant.

Check if the structure of the partial credit item has changed between tests. The default "item difficulty" for a partial credit item is the point at which top and bottom categories are equally probable. The bottom category of a long scale tends to be relatively rarely and idiosyncratically used. So, for equating purposes, it may be more robust to define the item difficulty to the be point at which the two most frequent categories (across the two tests) are equally probable. This provides a more stable, and statistically more secure, "item difficulty" for the polytomous item.

If your software does not report one overall partial-credit item difficulty in each analysis, but instead a set of threshold difficulties, then average the threshold difficulties for an overall difficulty.

A. The examination board may assume that the one 4-point partial credit item is equivalent to 4 dichotomous items. If so the equating constant between the two tests is:

((Sum of MCQ common-item difficulty differences) + 4*(partial credit difficulty difference)) / (20 + 4)

B. The examination board may want the one 4-point partial credit item to have the same influence as all 20 dichotomous items. If so the equating constant is:

((Sum of MCQ common-item difficulty differences) + 20*(partial credit difficulty difference)) / (20 + 20)

C. An option beloved of statisticians is "information-weighting" (i.e., weighting by the inverse of the item-difficulty-standard-error-squared). This will give the 4-point partial credit item about 6 times the influence of a 1-point dichotomous item.

D. Inverse-standard-error weighting ("effect-size" weighting) of the items is more consistent when combining subtests, see RMT 8:3, p. 376. This will give the 4-point partial credit item about 2.5 times the influence of a 1-point dichotomous item.

In this example, to specify that for 1 partial-credit item = 4 dichotomous items would also be to opt for a compromise between the two statistical viewpoints.

Equating constants with mixed item types, … Rasch Measurement Transactions, 2004, 18:3 p. 992

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

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