Hierarchical and Circumplex Structures 243because many high-loading items have positive secondary
loadings on agreeableness.
The Alleged Broadness of Factors
Under conditions of actual simple structure, factors could be
called broad in a hierarchical sense, as they capture the com-
mon variance of a number of variables. Even then, factors are
not broad in a conceptual sense but rather more narrow than
variables, as their internal consistency is higher and their
angular position in the trait space is thus more fixed. Agfac-
tor of intelligence, for example, is not a broadband but a high-
fidelity measure of some latent trait. A fortiori, there is nothing
broad about a Big Five factor based on a particular domain of
trait variables. For lack of actual simple structure, it does not
encompass a sizable number of lower level items or scales.
The meaning of a factor, even if latent, is much more precise
in a psychometric sense than is the meaning of the variables
on which it is based. In that domain of variables, a set of five
rotated principal components covers more variance than does
any other set of five linear combinations, but “broadness” is an
inappropriate and misleading term for that.
In another terminology, to view the Big Five as broad fac-
tors is to treat them as a circumplex structure. In a regular
two-dimensional circumplex, the plane is sliced into a num-
ber of angular segments (e.g., 12 segments). Variables within
a segment form a homogeneous set. A special case is simple
structure, in which “mixed” segments are empty, as in
Figure 10.1, panel A. The actual situation, however, is closer
to panel B, amounting to a circumplex with four segments, of
which two are well filled. These segments contain very het-
erogeneous sets of variables; two of those variables may even
be orthogonal to each other. The very specific meaning of the
factor is thus not adequately captured by the broad array of
variables that have their primary loading on it.
Marker VariablesThe interpretation problem would be solved if stable marker
variables could be found, that is, trait terms that load exclu-
sively on one factor. Goldberg (1992a) presented such sets of
psychometric synonyms, for example, extraverted, talkative,
assertive, verbal, bold, and five other terms for the positive
pole of the extraversion factor. A minor problem with this
interpretation strategy is that markers for some factor poles
are difficult to find, for example, markers for emotional sta-
bility. A major problem is that marker sets appear to be no
longer orthogonal in fresh samples or upon translation. Any
two homogeneous sets of traits may be expected to correlate
positively if both are desirable or if both are undesirable,
negatively if they are opposite in that respect; neutral sets
hardly exist. Orthogonal sets may be selected in a sample, but
they will regress to obliqueness upon cross validation. On the
basis of a large-scale study, Saucier (2002b) has developed
marker scales that appear to be robustly orthogonal within his
several data sets and might thus defy the present analysis.
Still, one would have to wait and see how they do in another
laboratory, for example, when transported abroad.
The obliqueness problem (see, e.g., Block, 1995) cannot
be answered by the truism that varimax-rotated factors are by
definition orthogonal. The missed point is that they have no
interpretation—not because they are broad or fuzzy, but be-
cause any interpretation in terms of sets of variables is biased.
To interpret a Big Five factor properly, one would have to
perform and communicate a suppression operation, such as
the following: Factor I is what remains of extraversion after
suppressing any connotation of agreeableness or socialness
that may be associated with it, however firmly; Factor V is a
residue of intellect or openness to experience after subtract-
ing a virtually indissoluble tinge of energy, which rather
belongs to Factor I. That is a bit much to ask.Figure 10.1 Prototypical simple-structure (A) and semicircumplex (B) configurations.(A) (B)