Assessing Dimensionality in Non-Positive Definite Tetrachoric Correlation Matrices: Does Matrix Smoothing Help?

We performed two simulation studies that investigated dimensionality recovery in NPD tetrachoric correlation matrices using parallel analysis. In each study, the NPD matrices were rehabilitated by three smoothing algorithms. In Study 1, we replicated the work by Debelak and Tran on the assessment of dimensionality in one- or two-dimensional common factor models. In Study 2, we extended the Debelak and Tran design in three important ways. Specifically, we investigated: (a) a wider range of factors; (b) models with varying amounts of model error; and (c) models generated from more realistic population item parameters. Our results indicated that matrix smoothing of NPD tetrachoric correlation matrices improves the performance of parallel analysis with binary data. However, these improvements were modest and often of trivial size. To demonstrate the effect of matrix smoothing on an empirical data set, we applied parallel analysis and factor analysis to Adjective Checklist data from the California Twin Registry.