Equating with Minimal Information

"I want to equate various forms of an examination. I know only the following information:

1. The ID numbers of each item on each form of the exam. Most items were used on multiple forms.

2. The Angoff ratings for the items on the original form of the exam.

3. The total number of candidates attempting each item on each form.

4. The total number of candidates answering each item correctly on each form."

Kurt T. Taube

Here's an equating procedure. Let's assume that the candidates attempting each item within a form have the same distribution. This is often not true with long tests - only the fast, usually more able, students take the last items. If so, drop out the last items on each form. Compute the logit difficulty of each item in each form:

Difficulty logit =
loge (count of candidates failing item/count of candidates succeeding)

Choose one form of the test (usually the one with the biggest sample or the most common items), we'll call it Form A. This form is used to define the metric.

Cross-plot the difficulty logits of common items between Form A and another form, Form B. Draw in the best fit line through the points. This slope and intercept of this line reflects the difference in the samples who took Forms A and B. The best fit line gives the "fahrenheit to celsius" conversion between the logit difficulties of the two forms. Convert difficulties on Form B to Form A metric. Continue this process through all forms, until all forms, and so all items, have been converted to Form A metric. This can be done either by direct comparison with Form A items or indirectly by comparison with another form already equated to Form A. The conversion is necessary to adjust for differences in candidates distributions across forms.

Finally convert the metric of Form A from its local logits to the "natural" logit, necessary for predicting individual candidate performance on individual items and for building score-to-measure tables. For this, you need one more piece of information, the standard deviation SD of the candidate raw scores on Form A. Then the conversion is:

where Pi is p-value of item i on Form A, i.e., (correctly/attempted).

Now you can transform form A logits to natural logits and compute "raw score"-to-"logit measure" tables for every form.

The Angoff ratings are completely useless for equating, but plotting them against the item difficulties might provide an interesting commentary on the validity and utility of the Angoff method.

Equating with Minimal Information Taube KT, Linacre J.M. … Rasch Measurement Transactions, 1999, 13:2 p. 697

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

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