A Family Model of Trait Structure 251includingpso that they become equidistant (the angles being
54.7 deg, with cosine
√
1 / 3 ; further constraints are discussed
later) from the vertical axis. All this is in correspondence with
the double cone model. Now form three slices (circumplexes)
by taking two rotated axes at a time. The projection ofpon
these tilted planes has the 12 o’clock to 6 o’clock direction,
and the 3 o’clock and 9 o’clock positions are on the equator.
Additional model vectors are constructed running from 11
to 5 and from 1 to 6, as in the two-dimensional member of the
model family.
The central positions in this structure are taken by the 12
to 6 vectors—to be labeled I/II, I/III, and II/III—that are the
bisectrices of the right angle between the two rotated princi-
pal components forming the circumplex. The I, II, and III
axes themselves merely guard the boundaries of the model
structure; as such, they have no place or name in the model.
The central model vectors appear to be close to p, namely, at
a distance of 35.3 deg (with cosine .816 or
√
√^2 /^3 ; generally,
2 /n,wherenis the number of dimensions). Note that this
oblique structure arises as a side effect of an orthogonal rota-
tion, not through some more liberal oblique rotation proce-
dure as such. The central model vectors are thus much more
saturated with desirability than are the factors themselves; at
all dimensional levels of the model, they share exactly
√
2 as
much variance withpas do the orthogonal factors.
What is new about this structure is that mixtures or blends
of factors have stolen the central place that used to belong to
the factors. Instead of being derivatives, the bisectrices of the
factor pairs have become the central concepts. This play of
musical chairs comes about because of the closer association
of the central vectors with p, which entitles them to their po-
sition. In passing, the model resolves the uneasiness of
inserting orthogonal axes into an essentially oblique struc-
ture; it rigorously defines oblique axes without giving up the
convenience of an orthogonal base. The only price is that the
number of musical chairs has to be increased, from four di-
mensions onward: There are n(n−1)/ 2 central vectors, with
nthe number of dimensions or factors. However, that exten-
sion will be welcomed by those who have always wondered
whether five is all there is. The model has shaken off the last
remnants of simple-structure thinking. Parenthetically, I note
that the model is equally appropriate in other domains,
notably, intelligence.
With four dimensions, the rotated factors are at an angle of
60 deg with respect to thepfactor; the central model vectors
are at 45 deg from that pivot. With five dimensions, the fac-
tors are at 63.4 deg, and the central vectors are at 50.7 deg.
Still, the model rotation maximizes the sum of the correla-
tions of the central axes with p, and in that sense minimizes
their average neutralness. Conversely, any other orthogonal
rotation of these dimensions (e.g., varimax) is inferior in this
respect: It takes in more neutral traits, which are less repre-
sentative of the domain.
With three or more dimensions, the model leaves freedom
of spin. A three-dimensional structure, for example, may be
rotated around its vertical p-axis without violating the model.
For reasons of continuity, this freedom may be used for max-
imizing the correspondence of the rotated factors with the
current varimax factors, particularly, the 5-D model factors.
This amounts to some lowering of the positive poles of the
current dimensions I and III toward the hyperequator, and
some lifting of the others. One may speculate, for example,
that the American lexical extraversion factor loses its aggres-
sive connotation and moves in the direction of sociability.
However, it is difficult to gauge what the substantive effects
of the joint rotation will be on all the versions in all the dif-
ferent languages (see, e.g., Saucier et al., 2000) that have
been proposed. The labels of the 5-D model are probably
used in a manner vague enough to permit this twisting.
(Agreeableness and conscientiousness, in particular, do not
even fit their present axes; see Hofstee et al., 1992.)
From the three-dimensional level on, there is some redun-
dancy between model vectors at different levels. At the top
level, there is the one vector. At the second level, two addi-
tional bipolar vectors appear, which satisfy the requirement
of being 30 deg removed from p. At the third, we find three
semicircumplexes with three model vectors each; at the
fourth, there are 63 at the fifth, the AB5SC model with
30 vectors appears; in general, at the nth level from 3 on,
there are 1. 5 n(n−1)vectors specific to that level. In succes-
sively adding levels, the cumulative number of model vectors
thus becomes 1, 3, 12, 30, and 60. From the third level on, it
appears impossible to rotate the central vectors in such a way
that all the additional vectors stay at least 30 deg away from
the ones at the second level. Thus some vectors would have
indistinguishable interpretations.
One strategy would be to settle for a particular dimension-
ality of the trait space. That would prevent overlap and would
simplify things in general. The foremost drawback is that
from three dimensions onward the most central trait concepts
would be missed. Furthermore, that strategy would only stir
up the debate on the dimensionality of the trait space, to
which there is no cogent solution; it would thus frustrate the
attainment of a canonical structure rather than contribute to
it. The other, preferable, strategy is to adopt the model family
as a whole, including as many (or as few) levels as will ap-
pear to be needed, and deleting concepts at lower levels that
are virtual clones of those at higher levels. The foreseeable
result of this strategy is maximal convergence of structures at
each level, and maximal efficiency in communicating about