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

4.1. Anatomical Measurements 67spaces, as in Table 4.2. Some such simple representation is often helpfulin interpreting PCs, particularly if a PCA is done on a large number ofvariables.Sometimes a simplification such as that given in Table 4.2 may be rathertoo extreme, and it is therefore advisable to present the coefficients roundedto one or two decimal places as well. Principal components with roundedcoefficients will no longer be optimal, so that the variances of the first fewwill tend to be reduced, and exact orthogonality will be lost. However, ithas been shown (Bibby, 1980; Green, 1977) that fairly drastic roundingof coefficients makes little difference to the variances of the PCs (see Section10.3). Thus, presentation of rounded coefficients will still give linearfunctions of x with variances very nearly as large as those of the PCs, whileat the same time easing interpretations.It must be stressed that interpretation of PCs is often more subtle thanis generally realised. Simplistic interpretation can be misleading. As well astruncation or rounding of PC coefficients, a number of other ideas are availableto aid interpretation. Some of these involve manipulation of the PCcoefficients themselves, whilst others are based on alternative, but similar,techniques to PCA. In this chapter we concentrate on simple interpretation.Its dangers, and various alternative ways of tackling interpretation,are discussed in Chapter 11.Turning now to the interpretation of the PCs in the present example,the first PC clearly measures overall ‘size’ for both sexes, as would be expected(see Section 3.8), as all the correlations between the seven variablesare positive. It accounts for 53% (women) or 60% (men) of the total variation.The second PC for both sexes contrasts hand and wrist measurementswith height, implying that, after overall size has been accounted for, themain source of variation is between individuals with large hand and wristmeasurements relative to their heights, and individuals with the converserelationship. For women, head and chest measurements also have some contributionto this component, and for men the forearm measurement, whichis closely related to height, partially replaces height in the component.This second PC accounts for slightly less than 20% of the total variation,for both sexes.It should be noted that the sign of any PC is completely arbitrary. Ifevery coefficient in a PC, z k = a ′ k x, has its sign reversed, the variance of z kis unchanged, and so is the orthogonality of a k with all other eigenvectors.For example, the second PC for men as recorded in Tables 4.1 and 4.2 haslarge positive values for students with large hand and wrist measurementsrelative to their height. If the sign of a 2 , and hence z 2 , is reversed, thelarge positive values now occur for students with small hand and wristmeasurements relative to height. The interpretation of the PC remains thesame, even though the roles of ‘large’ and ‘small’ are reversed.The third PCs differ more between the sexes but nevertheless retainsome similarity. For women it is almost entirely a contrast between head