More about Critical Eigenvalue Sizes (Variances) in Standardized-Residual Principal Components Analysis (PCA)

Gilles Raîche in RMT 19:1 p. 102 reports eigenvalues in the range 1.4 to 2.1 for the first component in a PCA of inter-item correlations of standardized residuals of Rasch-fitting data. Test lengths were in the range 20 to 60 items.

Here those findings are extended to dichotomous and polytomous data with test lengths from 3 to 1000 items. The generating person sample of 1000 persons has a normal distribution with a mean of 0 logits and a standard deviation of 2 logits. The generating item distribution is uniform from -2 to +2 logits. For the 5-category polytomous data, the generating Rasch-Andrich thresholds are: -2.53, -0.35, 0.56, 2.32 logits. The Figures shows the eigenvalues sizes of the first components (contrasts) in a PCA of the standardized-residual item-correlation matrices.

For the dichotomous simulations, the eigenvalue increases from 1.3 for 3 items to 4.0 for 1000 items. For 5-category polytomous items, the eigenvalues have the same range.

John Michael Linacre

"Monte Carlo PCA for Parallel Analysis" is Marley Watkins' free software for performing this type of investigation using simulated random-normal deviates, which standardized residuals approximate. For 200 items (variables) and 1000 persons (subjects), that software reports that the first PCA component in the random-normal deviates has an eigenvalue of 2.05 which accords with the findings above.

Alan Tennant

1. PCA: Data Variance: Explained, Modeled and Empirical
2. Critical Eigenvalue Sizes (Variances) in Standardized Residual Principal Components Analysis (PCA)
3. More about Critical Eigenvalue Sizes (Variances) in Standardized-Residual Principal Components Analysis (PCA)
4. Data Variance Explained by Rasch Measures
5. PCA: Variance in Data Explained by Rasch Measures

Linacre J.M., Tennant A. (2009) More about Critical Eigenvalue Sizes (Variances) in Standardized-Residual Principal Components Analysis (PCA), Rasch Measurement Transactions, 2009, 23:3, 1228

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Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn. George Engelhard, Jr. & Jue Wang	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
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	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	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
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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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

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