On the Detection of the Correct Number of Factors in Two-Facet Models by Means of Parallel Analysis.

Methods for optimal factor rotation of two-facet loading matrices have recently been proposed. However, the problem of the correct number of factors to retain for rotation of two-facet loading matrices has rarely been addressed in the context of exploratory factor analysis. Most previous studies were based on the observation that two-facet loading matrices may be rank deficient when the salient loadings of each factor have the same sign. It was shown here that full-rank two-facet loading matrices are, in principle, possible, when some factors have positive and negative salient loadings. Accordingly, the current simulation study on the number of factors to extract for two-facet models was based on rank-deficient and full-rank two-facet population models. The number of factors to extract was estimated from traditional parallel analysis based on the mean of the unreduced eigenvalues as well as from nine other rather traditional versions of parallel analysis (based on the 95th percentile of eigenvalues, based on reduced eigenvalues, based on eigenvalue differences). Parallel analysis based on the mean eigenvalues of the correlation matrix with the squared multiple correlations of each variable with the remaining variables inserted in the main diagonal had the highest detection rates for most of the two-facet factor models. Recommendations for the identification of the correct number of factors are based on the simulation results, on the results of an empirical example data set, and on the conditions for approximately rank-deficient and full-rank two-facet models.