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.

AspectDetails
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 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 2nd. 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

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