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

13.2. Analysis of Size and Shape 343the techniques of reciprocal averaging and dual (or optimal) scaling arewidely used in ecology and psychology, respectively. The rationale behindeach technique is different, and differs in turn from that given above forcorrespondence analysis, but numerically all three techniques provide thesame results for a given table of data.The ideas of correspondence analysis can be extended to contingencytables involving more than two variables (Greenacre, 1984, Chapter 5), andlinks with PCA remain in this case, as will now be discussed very briefly.Instead of doing a correspondence analysis using the (r × c) matrix N, itispossible to carry out the same type of analysis on the [n·· ×(r+c)] indicatormatrix Z =(Z 1 Z 2 ). Here Z 1 is (n·· × r) and has (i, j)th element equalto1iftheith observation takes the jth value for the first (row) variable,and zero otherwise. Similarly, Z 2 is (n·· × c), with (i, j)th element equalto1iftheith observation takes the jth value for the second (column)variable and zero otherwise. If we have a contingency table with morethan two variables, we can extend the correspondence analysis based onZ by adding further indicator matrices Z 3 , Z 4 ,..., to Z, one matrix foreach additional variable, leading to ‘multiple correspondence analysis’ (seealso Section 14.1.1). Another alternative to carrying out the analysis onZ =(Z 1 Z 2 Z 3 ...) is to base the correspondence analysis on the so-calledBurt matrix Z ′ Z (Greenacre, 1984, p. 140).In the case where each variable can take only two values, Greenacre(1984, p. 145) notes two relationships between (multiple) correspondenceanalysis and PCA. He states that correspondence analysis of Z is closelyrelated to PCA of a matrix Y whose ith column is one of the two columnsof Z i , standardized to have unit variance. Furthermore, the correspondenceanalysis of the Burt matrix Z ′ Z is equivalent to a PCA of the correlationmatrix 1n··Y ′ Y. Thus, the idea of correspondence analysis as a form ofPCA for nominal data is valid for any number of binary variables. A finalrelationship between correspondence analysis and PCA (Greenacre, 1984,p. 183) occurs when correspondence analysis is done for a special type of‘doubled’ data matrix, in which each variable is repeated twice, once in itsoriginal form, and the second time in a complementary form (for details,see Greenacre (1984, Chapter 6)).We conclude this section by noting one major omission, namely the nonlinearprincipal component analyses of Gifi (1990, Chapter 4). These aremost relevant to discrete data, but we defer detailed discussion of them toSection 14.1.1.13.2 Analysis of Size and ShapeIn a number of examples throughout the book the first PC has all its coefficientsof the same sign and is a measure of ‘size.’ The orthogonalityconstraint in PCA then demands that subsequent PCs are contrasts be-