Hierarchical and Circumplex Structures 247relatively pure factor markers, that is, traits assigned to the 10
diagonal (IIto VV) cells of the table. If simple struc-
ture would in fact materialize, most if not all of the variables
would be found in those cells. If, on the other hand, the em-
pirical structure is essentially circumplex, only 11.1% of
the variables would find their way to the diagonal cells. In
Hendriks’s (1997) analysis of 914 items, 105 (11.5%) ended
up in those cells. That illustration is as dramatic as is the per-
centage of variables that would have to be discarded in a
proper application of the simple-structure model.
In the discussion of the person-centered approach, I intro-
duced a distinction between the contexts of prediction and
communication. Against that background, it should be noted
that the predictive gain of the off-diagonal AB5C facets over
the five principal components is nil, as the facets are linear
combinations of the components. However, they do serve
conceptual, interpretive, and communicative purposes. An
individual’s profile of scores on the FFPI, for example, may
be typified by that person’s single most characteristic facet;
thus, for example, a person whose highest score is on factor
Vand whose second highest score is on IIImay be char-
acterized by the cluster of expressions and adjectives that
form the VIIIfacet (Knows what he/she is talking about,
Uses his/her brains, Sees through problems, and the many
other items listed by Hendriks, 1997, for this “Tight Intelli-
gence” facet). One or more of these catch phrases should be
more effective than presenting a 5-D profile or even the
subset based on the scores in question (“This person is pri-
marily someone who Thinks quickly [V], and secondarily
someone who Does things according to plan [III]”).
Furthermore, at the theoretical level the AB5C model ac-
counts for a large number of concepts that do not coincide
with the five Factors but are quite adequately reconstructed as
their mixtures.
Another way to document the flexibility of the AB5C
design is in noting that it incorporates features of both
oblique rotation and cluster analysis on an orthogonal basis.
Oblique rotation as such does not solve the simple-structure
problem when the configuration of variables is essentially
circumplex. However, the insertion of oblique model vectors
enables one to capture relatively homogeneous clusters of
traits. That function is also served by cluster analysis proce-
dures, but they lose sight of the dimensional fabric of the
structure and the recursive definitions of clusters.
With respect to predictive purposes, the loss incurred by
adopting the AB5C model is quite limited. First, the princi-
pal components base maximizes the internal consistency of
the facets (Ten Berge & Hofstee, 1999), which should be
beneficial to their validity. Second, if factors beyond the
Big Five are needed to increase validity, the model is easily
extended to include those factors. That would be more
efficient than including separate scales for each additional
specific concept.Undoing HierarchiesThe traditional design of questionnaires is hierarchical: Items
are grouped into subscales, subscales into scales. From the
manuals of such questionnaires (see, e.g., Costa & McCrae,
1992) it is easily verified that subscales actually form a
network; they have substantial secondary correlations with
scales other than the one they are assigned to. Upon analyzing
the single items of a questionnaire, a similar tissue pattern
would arise; items would appear to have all sorts of promiscu-
ous relationships, inviting circumplex analysis of the data.
Generalized (beyond two dimensions) circumplex analy-
sis would proceed as follows: First, the item scores are
subjected to PCA. The maximum number of principal com-
ponents would be the number of subscales or facets (e.g., 30
in the case of the NEO-PI-R). Note that these 30 principal
components extract more variance by definition than tradi-
tional scoring does. (In practice, it would soon become
apparent that only a part of these principal components
should be retained because the redundancies in the item
tissue are captured by fewer components than the number of
subscales.)
At the scale level, the optimal strategy is to proceed from
the firstmprincipal components, mbeing the number of
superordinate scales (e.g., 5), as they make more efficient use
of the data than do traditional scale scores. If, for reasons of
continuity, the original interpretations of the scales are to be
simulated, target rotations of the mprincipal components to-
ward these scales could be carried out. If the original scales
are conceived to be orthogonal, as in 5-D questionnaires, the
optimal approximation procedure would be a simultaneous
orthogonal target rotation of the mprincipal components to-
wards the set of mscales. That procedure conserves internal
consistency (Ten Berge & Hofstee, 1999); consequently, the
average coefficient alpha of the rotated principal components
is maximal. Most notably, it is automatically higher than the
average alpha of the original scales.
Subscales of traditional questionnaires are very short;
therefore, they are unreliable or consist of asking essentially
the same question over and again, which is annoying to re-
spondents and introduces unintended specific variance. If
they are to be retained, their quality can be improved to a con-
siderable extent by estimating subscale scores on the basis of
(maximally) as many principal components as are postulated