Page 162 - Jolliffe I. Principal Component Analysis

6.1. How Many Principal Components?
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bootstrap rules, the results are viewed from a factor analysis rather than a
PCA perspective.
Franklin et al. (1995) compare 39 published analyses from ecology. They
seem to start from the unproven premise that ‘parallel analysis’ (see Sec-
tion 6.1.3) selects the ‘correct’ number of components or factors to retain,
and then investigate in how many of the 39 analyses ‘too many’ or ‘too few’
dimensions are chosen. Franklin et al. (1995) claim that 2 of the 39 analyses
3
retain too many dimensions. However, as with a number of other references
cited in this chapter, they fail to distinguish between what is needed for
PCA and factor analysis. They also stipulate that PCAs require normally
distributed random variables, which is untrue for most purposes. It seems
difficult to instil the message that PCA and factor analysis require differ-
ent rules. Turner (1998) reports a large study of the properties of parallel
analysis, using simulation, and notes early on that ‘there are important
differences between principal component analysis and factor analysis.’ He
then proceeds to ignore the differences, stating that the ‘term factors will
be used throughout [the] article [to refer] to either factors or components.’
Ferr´e (1995b) presents a comparative study which is extensive in its
coverage of selection rules, but very limited in the range of data for which
the techniques are compared. A total of 18 rules are included in the study,
as follows:
• From Section 6.1.1 the cumulative percentage of variance with four
cut-offs: 60%, 70%, 80%, 90%.

• From Section 6.1.2 Kaiser’s rule with cut-off 1, together with
modifications whose cut-offs are 0.7 and 2; the broken stick rule.

• From Section 6.1.3 the scree graph.
• From Section 6.1.4 Bartlett’s test and an approximation due to
Mardia.

• From Section 6.1.5 four versions of Eastment and Krzanowski’s cross-
validation methods, where two cut-offs, 1 and 0.9, are used and, for
each threshold, the stopping rule can be based on either the first or
ˆ
last occasion that the criterion dips below the threshold; Ferr´e’s f q ;
Besse and de Falguerolles’s approximate jackknife criterion.
• From Section 6.1.6 Velicer’s test.

The simulations are based on the fixed effects model described in Sec-
tion 6.1.5. The sample size is 20, the number of variables is 10, and each
simulated data matrix is the sum of a fixed (20 × 10) matrix Z of rank 8
and a matrix of independent Gaussian noise with two levels of the noise
variance. This is a fixed effects model with q = 8,sothatatfirstsightwe
might aim to choose m = 8. For the smaller value of noise, Ferre (1995b)
considers this to be appropriate, but the higher noise level lies between the