250 Structures of Personality Traits
relatively socially desirable), and/or a Situations effect (e.g.,
a personnel selection context gives rise to elevated scores),
and/or some interaction effect, but not a Persons or individual
differences effect. Thepcomponent concerns the latter.
Carrying out the slight rotation, if needed, to align the first
principal component in any particular data set with the desir-
ability variable should prove helpful in solving the vexing
problem of indeterminate rotational positions of components.
Saucier (2002a) has already documented that varimax rota-
tion does not help in this respect: Across data sets, the posi-
tions of unrotated principal components were at least as
replicable as were varimaxed components. Among the princi-
pal components, the first is by far the most replicable one.
Across differently composed sets of variables, however, part
of this stability gets lost (as discussed earlier). Anchoringpat
the desirability values, which are external to the studies,
should enhance replicability.
Thep-oriented model produces another taxonomic lever,
namely, a measure of the representativeness of a set of per-
sonality traits or items, in the shape of the correlation be-
tween the first principal component of the set and the
desirability variable. In a set overloaded with fairly neutral
extraversion-introversion items attracting the first principal
component, that correlation would be clearly below unity. In
a heterogeneous set of neutral items, the desirability variable
would be unstable, again lowering the correlation. In the
spirit of the lexical axiom, such sets would be judged
insufficiently representative. The proposed measure simu-
lates that judgment.
The Two-Dimensional Level
Upon extracting p, a residual remains in the shape of a matrix
of part scores. The first principal component of that residual
matrix comes close to the second principal component of the
original scores, at least in a representative set of variables.
Takingpas the ordinate, a 45-deg counterclockwise rotation
of the two components includingpwill produce an X struc-
ture, or a flat version of the double cone. The upper and lower
segments contain the most unambiguously positive and neg-
ative, or consonant, traits; the left and right segments contain
the most relatively neutral and discordant traits. The abridged
semicircumplex structure at this level contains two bipolar
facet vectors running from 11 o’clock to 5 o’clock and from
1 to 7 in addition to the 12 to 6pvector; the relatively neutral
traits are left unaccounted for by the model, as their projec-
tions on the vectors will be very low.
Substantively, the plane would resemble, but not be iden-
tical to, the interpersonal circumplex (Wiggins, 1980), the
III or AgreeablenessExtraversion slice of the 5-D
structure, Digman’s (1997) plane, and the like. In a
perfectly representative set of traits as defined earlier, the
model plane would be identical to the plane formed by the
first two (rotated) principal components; this property makes
it a good candidate for a canonical or reference structure. Its
suitability for that purpose is enhanced by the absence of
rotational freedom at this level: The positions of the model
vectors are indirectly prescribed by the desirability values of
the traits. Theoretical criteria, as in the interpersonal circum-
plex, or the simple-structure criterion as in 5-D models, are
insufficiently capable of serving that reference function.
In the rationale of the semicircumplex model, the transi-
tion from one pdimension to two dimensions means a
spreading of the desirability component, in the manner of the
unfolding of a fan. The primordial one becomes diluted in the
process, like the gfactor of intelligence does when it is
spread over two or more dimensions. Following elementary
rules of parsimony, the transition should not be made lightly;
the burden of proof is on those who take the step. Psycho-
metrics offers an adequate procedure for this proof: More-
dimensional assessments of personality should be shown to
have sufficient incremental validity over the pcomponent.
This requirement implies that an assessment of p, as a base-
line variable, would have to be part of any empirical study of
personality.
Incremental validity of variables other thanpwould nec-
essarily imply that variance orthogonal to it, thus neutral
variance, is valid. This implication bridges the present fam-
ily of models and those that capitalize on neutral variance,
like Peabody’s (1984) and Saucier’s (1994; Saucier et al.,
2001). In fact, the latter model is the complement of the
Semi-Circumplex, at the present and subsequent dimen-
sional levels; it fills in what the present model leaves empty.
Although the basic assumption—potential incremental valid-
ity of neutral variance—is thus necessarily the same, a
strategic difference remains at the executive level. In the
semicircumplex approach, neutral variance is assessed indi-
rectly, by suppressing thepvariance from consonant traits
rather than directly, as in Saucier (1994). The reason was
given earlier: Discordant personality concepts are difficult to
handle.Semicircumplex Spheres and HyperspheresThe three-dimensional member of the model family arises
as follows: Add the second principal component of the matrix
of residual scores (after removing p); retain the vertical ori-
entation so that a globe is formed with the positive traits on
the northern hemisphere and the negative traits on the south-
ern one; perform an orthogonal rotation of the three axes