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

116 6. Choosing a Subset of Principal Components or VariablesFigure 6.1. Scree graph for the correlation matrix: blood chemistry data.the plotted points are ‘steep’ to the left of k, and ‘not steep’ to the right.This value of k, defining an ‘elbow’ in the graph, is then taken to be thenumber of components m to be retained. Its name derives from the similarityof its typical shape to that of the accumulation of loose rubble, orscree, at the foot of a mountain slope. An alternative to the scree graph,which was developed in atmospheric science, is to plot log(l k ), rather thanl k , against k; this is known as the log-eigenvalue (or LEV) diagram (seeFarmer (1971), Maryon (1979)).In introducing the scree graph, Cattell (1966) gives a somewhat differentformulation from that above, and presents strong arguments that when itis used in factor analysis it is entirely objective and should produce the‘correct’ number of factors (see Cattell and Vogelmann (1977) for a largenumber of examples). In fact, Cattell (1966) views the rule as a means ofdeciding upon an upper bound to the true number of factors in a factoranalysis after rotation (see Chapter 7). He did not seem to envisage its usein PCA, although it has certainly been widely adopted for that purpose.The way in which Cattell (1966) formulates the rule goes beyond a simplechange of slope from ‘steep’ to ‘shallow.’ He looks for the point beyond