244 Structures of Personality Traits
In conclusion, the PCA plus varimax set of operations
leads to an inadequate representation of personality. The
argument is not that traits are correlated, in any metaphysical
sense: For purely predictive purposes, linear regression of
criteria on orthogonal factors is a perfectly defensible ap-
proach. What was stressed is the conceptual risk of starting to
talk in Big Five terms, either among experts or with others.
Conceivably, we could keep our mouths shut, but in practice
that is too high a price to pay.
The PCA plus varimax model has been imported into per-
sonality from the domain of intelligence research. The ques-
tion arises whether it is appropriate in that domain. I
(Hofstee, 1994c) have argued that it is not. The empirical
structure of intelligence variables is an n-dimensional sim-
plex (the all-positive quadrant of an n-dimensional sphere)
characterized by positive manifold and lack of simple struc-
ture. Treating it as an orthogonal simple structure gives rise to
biased conceptualizations of the underlying dimensions and
inadequate representation of the domain. Essentially the
same objection holds for the domain of personality.
The Double Cone Model
A seminal attempt at a specific structure model of personality
in the 5-D framework is Peabody and Goldberg’s (1989) dou-
ble cone, based on Peabody’s (1984; see also De Boeck,
1978) work on separating descriptive and evaluative aspects
of trait terms. It focuses on the first three Factors; the smaller
factors IV, emotional stability, and V, intellect, are treated
as separate axes orthogonal to the sphere that is formed by
the bigger three: I Extraversion, II Agreeableness, and III
Conscientiousness.
The double cone model may be envisaged as follows:
Take a globe with desirability as its north-south axis, so that
all desirable traits are on the northern hemisphere and
their undesirables opposites are on the southern hemisphere
in the antipode positions. Apply an orthogonal rotation to the
Factors I, II, and III such that their angular distances to
the desirability axis become equal, namely, 54.7 deg with
cosine 1/3. Draw a parallel of latitude at 35.3 deg (close to
Kyoto and Oklahoma City) through the positive endpoints of
the Factors I, II, and III, and another one (close to Sydney and
Montevideo) through the negative endpoints. Connect each
possible pair of antipode points on the two circles by a vector.
Together, these vectors form the double cone. The model rep-
resents empirical trait variables by their projection on the
closest model vector.
The double cone was designed to embody a particular tax-
onomic principle, informally referred to by insiders as the
Peabody plot and named chiasmic structure by Hofstee and
Arends (1994). A classical example of a chiasm is
Thrifty Generous
Stingy Extravagant
In Peabody’s reasoning, this configuration arises by pitting a
content contrast (i.e., not spending vs. spending) against a so-
cial desirability contrast (thrifty and generous vs. stingy and
extravagant). In the double cone model, chiasmic structure
recurs in the shape of Xs that are formed by vertical slicings
through the center of the double cone. On the Northern circle,
we would have thrifty and generous at opposite longitudes;
on the southern hemisphere, stingy and extravagant. More
generally, descriptive and evaluative aspects are represented
by longitude and latitude, respectively.Evaluation of the Double Cone ModelThe model is readily generalized to five dimensions, although
it loses some of its aesthetic appeal in the process: Take all 10
subsets of 3 out of the 5 factors, that is, the IIIIII, I
IIIV, through IIIIVV subsets, and treat each of these
spheres in the manner just sketched. The generalized double
cone thus consists of a Gordian knot of 10 three-dimensional
double cones in the 5-D space sharing their vertical (desir-
ability) axis, or 10 pairs of latitude circles. There is no valid
reason why the range of the chiasmic structural principle
should be restricted to a particular subset of three dimensions.
But the model easily passes the generalizability test.
It is not entirely clear whether the algorithm for analyzing
the data as used by Peabody and Goldberg (1989) is consistent
with the model. Via Peabody (1984), the reader is referred
to an algorithm proposed by De Boeck (1978). De Boeck’s
procedure, however, sets the I, II, and III dimensions orthog-
onal to the desirability axis, rather than at 54.7 deg. Still, it
is certainly possible to design an algorithm that would be
consistent with the double-cone model.
The next question, however, concerns fit. That may be
tested by assessing the quality of chiasms that are generated
by the model. Hofstee and Arends (1994, Table 1) present
chiasms derived from Peabody and Goldberg’s (1989) mate-
rials. An example is
Forceful Peaceful
Quarrelsome Submissive
The content contrasts in this and in other examples are not
convincing. The reasons are not hard to find. First and most
important, the cone structure supposes an angular distance of
only 109.5 deg between terms that should form a content