Comparing Rasch variables?

Question: "I want to compare each individual students' math and science achievements on the same scale. I want to be able say if this student did better on math than on science (after taking into account the different level of test difficulties)."

Response: In this type of situation it is always helpful to think of what you would do in a similar practical physical situation. Pretend your two tests are "weight" and "height" of children. How would you proceed? You would have to make an assertion about the relationship between height and weight for your students.

So, for your math and science tests, you need to make an assertion (assumption) about their relationship. Common assertions include:

a. The test items are equally difficult, on average, for both samples (with equal item difficulty dispersion).

b. The samples are equally able, on average, on both tests (with equal person measure dispersion).

c. Particular items on the math test have the same difficulty as particular items on the science test.

d. Particular persons or groups of persons on the math test have the same ability as particular (perhaps the same) persons or groups of persons on the science test.

An attractive short-cut might be to do a joint calibration of the math and science items. But imagine we are comparing the weight and height of children. If we tried to force them both into the same numerical variable, it would skew results for both. So what we might do instead is to match the mean and standard deviations of the sample's weights and heights in order to make weight/height comparisons.

Measure the math ability of each of the students. Measure the science ability of each of the students. Implement your assertion as to how the two ability distributions relate. Then you can report individual relative performances on math and science.

Comparing Rasch variable, … Rasch Measurement Transactions, 2004, 18:3 p. 994

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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