Learning from
Principal Components Analysis of Residuals

A COMET Presentation by
J. Michael Linacre on 4-22-1999,
MESA Psychometric Laboratory

MESA Research Note #7

The Rasch model extracts from observational data the linear dimension best supported by those data. But each observation, to some degree, contains its own characteristic features. These contradict the dimension. The responses idiosyncratic characteristics are manifested as the differences between what the Rasch model predicts and what is observed, i.e., the observation residuals. Principal components analysis of these residuals identifies characteristics shared in common among items. These are often indications of secondary structures or sub-dimensions within the data.

In this presentation, an analysis of an Attitudes to Recreational Drugs survey is performed. This identifies "for" and "against" statements. Further, it confirms an earlier finding about that "for" and "against" are not always perceived as opposite. More subtle sub-dimensions are also identified in the instrument.

In this example, the power and utility of Principal Components analysis of Rasch residuals is demonstrated.

Background reading on Principal Components and Rasch Local dependency, correlations and principal components, Wright BD, 1996, RMT 10:3 p.509
Structure in Rasch residuals: why principal components analysis?, Linacre JM, RMT 12:2 p.636
Factor analysis and Rasch, 1996, RMT 9:4 p. 470, summary of Structural Equation Modeling 3:1, 1996, ed. Randall E. Schumacker
Detecting Multidimensionality: Which Residual Data-type Works Best? Linacre JM, 1998, Journal of Outcome Measurement 2:3, 266-283)
Background reading on the example Drug Survey Attitude to Drugs, Chapter 6 in Rating Scale Analysis, Wright BD, Masters GN, 1982, Chicago: MESA Press

1. The Drug Survey 20 items from Chapter 6 in Rating Scale Analysis (Pennsylvania State University Evaluation Scales).
2. The example data WINSTEPS control and data file for the 1999 drug survey
3. Item fit report - original scoring WINSTEPS Table 10.1 Item statistics: misfit order
4. Principal components analysis of residuals for items WINSTEPS Table 23.1 and 23.2
5. Principal components analysis of residuals for persons WINSTEPS Table 24.1 and 24.2
6. Item fit report - against items WINSTEPS Table 10.1 Item statistics: misfit order
7. Principal components analysis of "against" items WINSTEPS Table 23.1 and 23.2
8. Item fit report - for items WINSTEPS Table 10.1 Item statistics: misfit order
9. Principal components analysis of "for" items WINSTEPS Table 23.1 and 23.2
10. Plot of "for" person measures vs. "against" measures Person measures from PFILE=, plotted via EXCEL.
11. The example data with items reversed WINSTEPS control and data file for the 1999 drug survey
12. Principal components analysis with reversed items WINSTEPS Table 23.1 and 23.2

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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 Journal of Applied Measurement
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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