When we used PCA or FA to explore the dimensionality of our set of variables, the question first is how many number of dimensions in this set. That means how could we set the reasonable eigenvalue criterion for determine the number of dimensions. Parallel Analysis is a method to help us find the number of the component of variable set. This main concept is that the eigenvalues of true dimensions are larger than the eigenvalues of the dimension form random data. In this study three PA method were compared, including the Horn's approach, PA-PAFA, and PA-MRFA. Furthermore, this study manipulated many different conditions in PA procedure, such as the type of correlation matrix (pearson and polychoric), random sample techniques (permutation and normal distribution), and thresholds (95% and mean). The result showed that the PA-MRFA was the best, and the Horn's PA was just slightly lower than PA-MRFA. When we use the appropriate polychoric correlation for polytomous data, the performance was more reliable than Pearson correlations. However, the convergence problem of the polychoric approach resulted in the Pearson are much needed in empirical data.
1) If we use the multidimensional PCM to replace the simulation data, how about the performance of those PA procedure ? I expect the result will not change too much.
2) In PA-PAFA, the reduced correlation matrices are often non-Gramian. It cause the matrix has strange negative eigencalues. It is fundamental problem with PA-PAFA so that I think PA-PAFA is not appropriate regardless of the performance.