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

7.4. An Example of Factor Analysis 161are actually derived from correlation matrices corresponding exactly to theunderlying model. In practice, the model itself is unknown and must beestimated from a data set. This allows more scope for divergence betweenthe results from PCA and from factor analysis. There have been a numberof studies in which PCA and factor analysis are compared empirically ondata sets, with comparisons usually based on a subjective assessment ofhow well and simply the results can be interpreted. A typical study ofthis sort from atmospheric science is Bärring (1987). There have also beena number of comparative simulation studies, such as Snook and Gorsuch(1989), in which, unsurprisingly, PCA is inferior to factor analysis in findingunderlying structure in data simulated from a factor model.There has been much discussion in the behavioural science literature ofthe similarities and differences between PCA and factor analysis. For example,114 pages of the first issue in 1990 of Multivariate Behavioral Researchwas devoted to a lead article by Velicer and Jackson (1990) on ‘Componentanalysis versus common factor analysis ...,’ together with 10 shorter discussionpapers by different authors and a rejoinder by Velicer and Jackson.Widaman (1993) continued this debate, and concluded that ‘...principalcomponent analysis should not be used if a researcher wishes to obtainparameters reflecting latent constructs or factors.’ This conclusion reflectsthe fact that underlying much of the 1990 discussion is the assumption thatunobservable factors are being sought from which the observed behaviouralvariables can be derived. Factor analysis is clearly designed with this objectivein mind, whereas PCA does not directly address it. Thus, at best,PCA provides an approximation to what is truly required.PCA and factor analysis give similar numerical results for many examples.However PCA should only be used as a surrogate for factor analysiswith full awareness of the differences between the two techniques, and eventhen caution is necessary. Sato (1990), who, like Schneeweiss and Mathes(1995) and Schneeweiss (1997), gives a number of theoretical comparisons,showed that for m = 1 and small p the loadings given by factor analysisand by PCA can sometimes be quite different.7.4 An Example of Factor AnalysisThe example that follows is fairly typical of the sort of data that are oftensubjected to a factor analysis. The data were originally discussed by Yuleet al. (1969) and consist of scores for 150 children on ten subtests of theWechsler Pre-School and Primary Scale of Intelligence (WPPSI); there arethus 150 observations on ten variables. The WPPSI tests were designedto measure ‘intelligence’ of children aged 4 1 2–6 years, and the 150 childrentested in the Yule et al. (1969) study were a sample of children who enteredschool in the Isle of Wight in the autumn of 1967, and who were tested