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.
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
|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|
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