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 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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