Dimensionality Assessment of Ordered Polytomous Items With
Parallel Analysis
Marieke E. Timmerman
Urbano Lorenzo-Seva
In this article, the authors use the most appropriate Parallel analysis (PA) procedure to assess the number of common factors underlying ordered polytomously scored variables.
They proposed minimum rank factor analysis (MRFA) as an extraction method, rather than the currently applied principal component analysis (PCA) and principal axes factoring.
The central idea of the original PA (Horn, 1965) is that true dimensions are associated with eigenvalues that are larger than the ones associated with dimensions derived from random data.
The simulated data were generated according to a linear CF model, where the resulting continuous variables were categorized to yield ordered polytomous observed variables.
There are 2 (number of major factors) *3 (number of
response categories) * 4 (sample size) *2 (number of variables
per factor) *2 (minor factors presence) *2 (distribution) *50
(replicates) = 9,600 simulated data set
The 18 PA procedures were applied to each simulated data
Matrix: For the Pearson-based PA procedures, all 12 possible combinations—2 (sampling technique) * 3 (extraction method) *2 (threshold)—were considered.
For the polychoric based PA procedures, random categorical data are needed, and therefore we used only permutation as a sampling technique, yielding six combinations—3 (extraction method) * 2 (threshold).
To assess the quality of the PA analyses, we considered the degree of recovery of the correct number of common factors (both the recovery of the number of major factors and of the total number of factors)
Results：
The results show that of the three extraction methods examined,
PA-MRFA performed best in indicating the correct number of major factors, followed very closely by Horn’s PA, and a distant third was PA-PAFA. The PA-PAFA procedures generally showed poor performance in the recovery of the number of factors.
A further comparison of Horn’s PA and PA-MRFA showed that the use of polychoric rather than Pearson correlations appeared to reduce the correctly indicated number of major factors considerably.
The polychoric-based procedures generally outperform their Pearson-based variants, provided that convergence is reached
What I have learnt：
The Horn’s PA is the comparison of the eigenvalues of the two types of correlation matrices (those of the observed data and those of the randomly generated data)
While, the difference of The PA based on principal axes factor analysis from Horn’s PA is that its eigenvalues are obtained from the reduced correlation matrix, with estimates of the communalities on its diagonal.
The idea of the PA based on minimum rank factor analysis is to compare the proportions of explained common variance (ECV) of successive CFs of observed data with the ECV distributions obtained from randomly generated data.
Some didn't make sense for me：
With increasing sample sizes, the eigenvalues of random correlation matrices approach one, whereas the eigenvalues of random reduced correlation matrices approach zero. Why?
Such a matrix has negative eigenvalue(s), implying that its highest eigenvalue(s) must be larger than the highest eigenvalue(s) of the unreduced correlation matrix. Why?