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

13.4. Principal Component Analysis in Designed Experiments 35113.4 Principal Component Analysis in DesignedExperimentsIn Chapters 8 and 9 we discussed ways in which PCA could be used as a preliminaryto, or in conjunction with, other standard statistical techniques.The present section gives another example of the same type of application;here we consider the situation where p variables are measured in the courseof a designed experiment. The standard analysis would be either a set ofseparate analyses of variance (ANOVAs) for each variable or, if the variablesare correlated, a multivariate analysis of variance (MANOVA—Rencher,1995, Chapter 6) could be done.As an illustration, consider a two-way model of the formx ijk = µ + τ j + β k + ɛ ijk ,i=1, 2,...,n jk ; j =1, 2,...,t; k =1, 2,...,b,where x ijk is the ith observation for treatment j in block k of a p-variatevector x. The vector x ijk is therefore the sum of an overall mean µ, atreatment effect τ j , a block effect β k and an error term ɛ ijk .The most obvious way in which PCA can be used in such analyses issimply to replace the original p variables by their PCs. Then either separateANOVAs can be done on each PC, or the PCs can be analysed usingMANOVA. Jackson (1991, Sections 13.5–13.7) discusses the use of separateANOVAs for each PC in some detail. In the context of analysinggrowth curves (see Section 12.4.2) Rao (1958) suggests that ‘methods ofmultivariate analysis for testing the differences between treatments’ can beimplemented on the first few PCs, and Rencher (1995, Section 12.2) advocatesPCA as a first step in MANOVA when p is large. However, as notedby Rao (1964), for most types of designed experiment this simple analysisis often not particularly useful. This is because the overall covariancematrix represents a mixture of contributions from within treatments andblocks, between treatments, between blocks, and so on, whereas we usuallywish to separate these various types of covariance. Although the PCs areuncorrelated overall, they are not necessarily so, even approximately, withrespect to between-group or within-group variation. This is a more complicatedmanifestation of what occurs in discriminant analysis (Section 9.1),where a PCA based on the covariance matrix of the raw data may proveconfusing, as it inextricably mixes up variation between and within populations.Instead of a PCA of all the x ijk , a number of other PCAs havebeen suggested and found to be useful in some circumstances.Jeffers (1962) looks at a PCA of the (treatment × block) means ¯x jk ,j=1, 2,...,t; k =1, 2,...,b, where¯x jk = 1n∑ jkx ijk ,n jkthat is, a PCA of a data set with tb observations on a p-variate random vec-i=1