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

3.7. Inference Based on Sample Principal Components 55having equal variances, a very restrictive assumption. The test with q =0reduces to a test that all variables are independent, with no requirementof equal variances, if we are dealing with a correlation matrix. However, itshould be noted that all the results in this and the previous section are forcovariance, not correlation, matrices, which restricts their usefulness stillfurther.In general, inference concerning PCs of correlation matrices is morecomplicated than for covariance matrices (Anderson, 1963; Jackson, 1991,Section 4.7), as the off-diagonal elements of a correlation matrix are nontrivialfunctions of the random variables which make up the elements ofa covariance matrix. For example, the asymptotic distribution of the teststatistic (3.7.6) is no longer χ 2 for the correlation matrix, although Lawley(1963) provides an alternative statistic, for a special case, which does havea limiting χ 2 distribution.Another special case of the test based on (3.7.6) occurs when it isnecessary to test⎡⎤1 ρ ··· ρH 0 :Σ=σ 2 ⎢ ρ 1 ··· ρ⎥⎢⎣..ρ ρ ··· 1against a general alternative. The null hypothesis H 0 states that all variableshave the same variance σ 2 , and all pairs of variables have the samecorrelation ρ, in which caseσ 2 [1 + (p − 1)ρ] =λ 1 >λ 2 = λ 3 = ···= λ p = σ 2 (1 − ρ)(Morrison, 1976, Section 8.6), so that the last (p − 1) eigenvalues are equal.If ρ, σ 2 are unknown, then the earlier test is appropriate with q = 1, but ifρ, σ 2 are specified then a different test can be constructed, again based onthe LR criterion.Further tests regarding λ and the a k can be constructed, such as thetest discussed by Mardia et al. (1979, Section 8.4.2) that the first q PCsaccount for a given proportion of the total variation. However, as stated atthe beginning of this section, these tests are of relatively limited value inpractice. Not only are most of the tests asymptotic and/or approximate, butthey also rely on the assumption of multivariate normality. Furthermore, itis arguable whether it is often possible to formulate a particular hypothesiswhose test is of interest. More usually, PCA is used to explore the data,rather than to verify predetermined hypotheses.To conclude this section on inference, we note that little has been donewith respect to PCA from a Bayesian viewpoint. Bishop (1999) is an exception.He introduces prior distributions for the parameters of a modelfor PCA (see Section 3.9). His main motivation appears to be to providea means of deciding the dimensionality of the model (see Section 6.1.5).⎥⎦