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

3.7. Inference Based on Sample Principal Components 53been used to assess the stability of subspaces defined by a subset of thePCs, and hence to choose how many PCs to retain. In these circumstancesthere is more than one plausible way in which to conduct the bootstrap(see Section 6.1.5).3.7.3 Hypothesis TestingThe same results, obtained from (3.6.1)–(3.6.3), which were used aboveto derive confidence intervals for individual l k and a kj ,arealsousefulforconstructing tests of hypotheses. For example, if it is required to test H 0 :λ k = λ k0 against H 1 : λ k ≠ λ k0 , then a suitable test statistic isl k − λ k0,τλ k0which has, approximately, an N(0, 1) distribution under H 0 , so that H 0would be rejected at significance level α if∣l k − λ k0τλ k0∣ ∣∣∣≥ z α/2 .Similarly, the result (3.7.5) can be used to test H 0 : α k = α k0 vs. H 1 :α k ≠ α k0 . A test of H 0 against H 1 will reject H 0 at significance level α if(n − 1)α ′ k0(l k S −1 + l −1k S − 2I p)α k0 ≥ χ 2 (p−1);α .This is, of course, an approximate test, although modifications can be madeto the test statistic to improve the χ 2 approximation (Schott, 1987). Othertests, some exact, assuming multivariate normality of x, are also available(Srivastava and Khatri, 1979, Section 9.7; Jackson, 1991, Section 4.6). Detailswill not be given here, partly because it is relatively unusual that aparticular pattern can be postulated for the coefficients of an individualpopulation PC, so that such tests are of limited practical use. An exceptionis the isometry hypothesis in the analysis of size and shape (Jolicoeur(1984)). Size and shape data are discussed briefly in Section 4.1, and inmore detail in Section 13.2.There are a number of tests concerning other types of patterns in Σ andits eigenvalues and eigenvectors. The best known of these is the test ofH 0q : λ q+1 = λ q+2 = ···= λ p , that is, the case where the last (p−q) eigenvaluesare equal, against the alternative H 1q , the case where at least twoof the last (p − q) eigenvalues are different. In his original paper, Hotelling(1933) looked at the problem of testing the equality of two consecutiveeigenvalues, and tests of H 0q have since been considered by a number ofauthors, including Bartlett (1950), whose name is sometimes given to suchtests. The justification for wishing to test H 0q is that the first q PCs mayeach be measuring some substantial component of variation in x, but thelast (p − q) PCs are of equal variation and essentially just measure ‘noise.’Geometrically, this means that the distribution of the last (p − q) PCs has