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

356 13. Principal Component Analysis for Special Types of Datain which q of the components span a common subspace, but there is norequirement that the individual components should be the same. Likelihoodratio tests are described by Flury (1988) for comparing the fit of models atdifferent levels of the hierarchy. Model selection can also be made on thebasis of Akaike’s information criterion (AIC) (Akaike, 1974).One weakness of the theory described in Flury (1988) is that it is onlyreally applicable to covariance-based PCA and not to the more frequentlyencountered correlation-based analysis.Lefkovitch (1993) notes that Flury’s (1988) procedure for fitting a commonPC model can be time-consuming for moderate or large data sets. Heproposes a technique that produces what he calls consensus components,which are much quicker to find. They are based on the so-called polar decompositionof a data matrix, and approximately diagonalize two or morecovariance matrices simultaneously. In the examples that Lefkovitch (1993)presents the consensus and common PCs are similar and, if the commonPCs are what is really wanted, the consensus components provide a goodstarting point for the iterative process that leads to common PCs.A number of topics that were described briefly by Flury (1988) in achapter on miscellanea were subsequently developed further. Schott (1988)derives an approximate test of the partial common PC model for G =2when the common subspace is restricted to be that spanned by the first qPCs. He argues that in dimension-reducing problems this, rather than anyq-dimensional subspace, is the subspace of interest. His test is extended toG>2 groups in Schott (1991), where further extensions to correlationbasedanalyses and to robust PCA are also considered. Yuan and Bentler(1994) provide a test for linear trend in the last few eigenvalues under thecommon PC model.Flury (1988, Section 8.5) notes the possibility of using models within hishierarchy in the multivariate Behrens-Fisher problem of testing equalitybetween the means of two p-variate groups when their covariance matricescannot be assumed equal. Nel and Pienaar (1998) develop this idea, whichalso extends Takemura’s (1985) decomposition of Hotelling’s T 2 statisticwith respect to principal components when equality of covariances isassumed (see also Section 9.1). Flury et al. (1995) consider the same twogroupset-up, but test the hypothesis that a subset of the p means is thesame in the two groups, while simultaneously estimating the covariancematrices under the common PC model. Bartoletti et al. (1999) considertests for the so-called allometric extension model defined by Hills (1982).In this model, the size and shape of organisms (see Section 13.2) are suchthat for two groups of organisms not only is the first PC common to bothgroups, but the difference in means between the two groups also lies in thesame direction as this common PC.In the context of discriminant analysis, Bensmail and Celeux (1996) usea similar hierarchy of models for the covariance matrices of G groups tothat of Flury (1988), though the hierarchy is augmented to include special