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

352 13. Principal Component Analysis for Special Types of Datator. In an example on tree seedlings, he finds that ANOVAs carried out onthe first five PCs, which account for over 97% of the variation in the originaleight variables, give significant differences between treatment means(averaged over blocks) for the first and fifth PCs. This result contrastswith ANOVAs for the original variables, where there were no significantdifferences. The first and fifth PCs can be readily interpreted in Jeffers’(1962) example, so that transforming to PCs produces a clear advantage interms of detecting interpretable treatment differences. However, PCs willnot always be interpretable and, as in regression (Section 8.2), there is noreason to expect that treatment differences will necessarily manifest themselvesin high variance, rather than low variance, PCs. For example, whileJeffer’s first component accounts for over 50% of total variation, his fifthcomponent accounts for less than 5%.Jeffers (1962) looked at ‘between’ treatments and blocks PCs, but thePCs of the ‘within’ treatments or blocks covariance matrices can also provideuseful information. Pearce and Holland (1960) give an example havingfour variables, in which different treatments correspond to different rootstocks,but which has no block structure. They carry out separate PCAsfor within- and between-rootstock variation. The first PC is similar in thetwo cases, measuring general size. Later PCs are, however, different for thetwo analyses, but they are readily interpretable in both cases so that thetwo analyses each provide useful but separate information.Another use of ‘within-treatments’ PCs occurs in the case where there areseveral populations, as in discriminant analysis (see Section 9.1), and ‘treatments’are defined to correspond to different populations. If each populationhas the same covariance matrix Σ, and within-population PCs based onΣ are of interest, then the ‘within-treatments’ covariance matrix providesan estimate of Σ. Yet another way in which ‘error covariance matrix PCs’can contribute is if the analysis looks for potential outliers as suggested formultivariate regression in Section 10.1.A different way of using PCA in a designed experiment is describedby Mandel (1971, 1972). He considers a situation where there is only onevariable, which follows the two-way modelx jk = µ + τ j + β k + ε jk ,j=1, 2,...,t; k =1, 2,...,b, (13.4.1)that is, there is only a single observation on the variable x at each combinationof treatments and blocks. In Mandel’s analysis, estimates ˆµ, ˆτ j ,ˆβ k are found for µ, τ j , β k , respectively, and residuals are calculated ase jk = x jk − ˆµ − ˆτ j − ˆβ k . The main interest is then in using e jk to estimatethe non-additive part ε jk of the model (13.4.1). This non-additive part isassumed to take the formm∑ε jk = u jh l h a kh , (13.4.2)h=1