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-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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