Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

156 7. Principal Component Analysis and Factor AnalysisFigure 7.2. Factor loadings for two factors with respect to original and obliquelyrotated factors.An interesting point is that factor loadings found by maximum likelihoodfor a correlation matrix are equivalent to those for the correspondingcovariance matrix, that is, they are scale invariant. This is in completecontrast to what happens for PCA (see Sections 2.3 and 3.3).A potential problem with the maximum likelihood approach is thatit relies on the assumption of multivariate normality, which may not bejustified, and Everitt and Dunn (2001, Section 12.7) caution against usingsuch estimates when the data are categorical. However, it can beshown (Morrison, 1976, Section 9.8; Rao, 1955, which is also reproducedin Bryant and Atchley (1975)) that the maximum likelihood estimators(MLEs) also optimize two criteria that make no direct distributional assumptions.If the factor model (7.1.2) holds exactly, then the partialcorrelations between the elements of x, given the value of f, are zero(see also Section 6.1.6), as f accounts for all the common variation inthe elements of x. To derive the criterion described by Morrison (1976),