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Principal Component Analysis of Smoothed Tetrachoric Correlation Matrices as a Measure of Dimensionality

The application of principal component analysis and parallel analysis to smoothed tetrachoric correlation matrices was investigated in a simulation study. To evaluate the effect of several smoothing algorithms, 360 different types of data sets were simulated. Under each simulated condition, two item sets, each fitting a unidimensional two-parameter logistic model, were combined with each other. The simulations differed in the size of the simulated item sets, the size of the person samples, the distribution of the difficulty and discrimination parameters, and the correlation between the person parameters. In general, the application of a smoothing algorithm led to an improved performance in the assessment of dimensionality, but minor differences between the three investigated smoothing algorithms were found. Procedures to apply two of the three investigated smoothing algorithms via R software packages are presented.