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Note: This is an unpublished ms. written in 1996-7. This was after I had published
the discovery of the Structural Model (Hammond 1994) but before I discovered the
scientific proof of God.
The unification with Cattell referred to here is the identification of his popularly
known 7-2nd order factors (Krug & John’s 1986). This ms. also predates my
discovery that all 13 of Cattell’s 2nd orders (Cattell 1973) represent symmetry axes of
the cube. I discovered this shortly after writing this initial draft. It was this latter
discovery that led me forward into the investigation of his 3rd order factors, the
inclusion of “g” as the 4th dimension of the Metric, and ultimately to the scientific
proof of God in 1997.
This ms. is presented mainly for it’s coverage of many interesting details from
Psychometry and Factor Analysis. No attempt has been made to edit it for journal
publication as the paper is only of historical interest now.
THE CARTESIAN THEORY: UNIFICATION
OF CATTELL, EYSENCK AND THE BIG-5
GEORGE E. HAMMOND
Abstract- This paper announces the emergence of the discovery of the Structural
Model of Personality. In a previous paper the Cartesian Theoryunified the
biological data of Personality (Hammond 1994). In this paper the 3 leading
psychometric schools of Personality are unified by the discovery of Heymans' cube
in Cattell's 7-2nd order factors and by the discovery of the physical origin of
Thurstone's theory of Simple Structure. The same geometrical structure is
numerically confirmed in the Big-5 in Peabody and Goldberg's Lexical Double Cone.
The 2,3,4,5,6,7 factor models are unified as simply geometric redactions of the 7-
Factor, Cattell General Solution. The Cartesian Theory has unified the entire field
of Personality and substantially proven that the long sought for Structural Model
originates in the 3-axis Cartesian structure of the brain.
Section I: The Cartesian Theory
The Cartesian Theory is a physics theory of the biological origin of the
Structural Model of Personality and was first reported in a recent paper
(Hammond 1994). A brief synopsis of the Cartesian Theory will be given here.
The Cartesian Theory advances that the origin of the Structural Model is
the "Physics Brain", Fig. 1a.
Figure 1
In my 1994 paper it is demonstrated that this physics brain originates in
the first 3 orthogonal cleavages of the egg in embryology and is clearly
identified in embryology, gross anatomy and neuropsychology. Moreover
"axiomatic" physics dictates that this Cartesian structure has to exist.
The 3 major neurological divisions of the brain are then identified with
Eysenck's E,N,P. In Fig. 1b Gray's diagonal Personality model is unified
with this Cartesian structure and is seen to produce 4 additional diagonal
dimensions of Personality, which represent bipolar neuropsychological
contrasts between the diagonally situated lobes; the origin of Personality
conflict.
Fig. 1c then is the psychometric result of Fig. 1a plus 1b and shows the
psychometric relation between the 7 fundamental axes of the complete
Structural Model of Personality. Fig. 1c I have named the Cattell General
Solution because as we will see presently, Cattell is the first scientist in
the field to confirm the complete structure empirically. This structure is
also known as Heymans' cube after the (now historical) work of G. Heymans
(Heymans 1929; see Hofstee 1994 for a review).
In my 1994 paper the biological and physics basis of the theory has been
elaborated. In this paper we wish to show how the Cartesian Theory unifies and
explains all of the psychometric results of Personality research,
particularly the 3 leading results, those of Cattell, Eysenck and the Big-5.
We turn then to the only complete, unitary confirmation of Fig. 1c,
Cattell's 7-2nd order factors as reported in a definitive 1986 study (Krug &
Johns 1986).
According to Eysenck, Samuel E. Krug (as of 1981) was "official spokesman for
Cattell's Institute of Personality and Ability Testing" (Eysenck 1991, p.779) and has
coauthored numerous papers with R.B. Cattell. The 1986 study is a high quality
study using "a large and representative sample" (N=17,381) (Eysenck 1991,p.778).
According to Krug & Johns the results "agree with 14 previous studies by Cattell
and others" (p. 687).
The item of interest is the oblique factor intercorrelations, Table 2, p.
689 (see Fig. 2 this paper). Since Cattell's 16PF is known to be a broad
assessment of primary personality characteristics, and we have here an
accurate determination of the 2nd order trait factors, what we suspect is
that these 7 factors must be the 7 psychometric factors of Fig. 1c. It turns
out that a simple trigonometric (circumplex) calculation of this table
confirms that they are.
Before turning to this calculation some preliminary remarks are necessary.
The first question is why, if the Structural Model originates in a simple
structure in 3D space, didn't psychometric research discover this years ago? The
answer is that the correlation of personality dimensions depends not only on
geometrical brain structure (the Physics Brain) but also on the biological (Nature)
and environmental (Nurture) covariation of the neuropsychological systems
embedded in the geometrical structure of the brain. So the purely geometric
"structural" intercorrelations are significantly reduced by both the nature and
nurture variance of the population sample. It is only the small residual correlations
that are left over that allow us to detect the existence of Heymans' cube as the
physical origin of the Structural Model. Low accuracy in the early years of
research of course further obscured the structure. In fact, it is only after the
discovery of the Cartesian Theory; knowing what to look for, that the Cattell result
has been discovered.
So it turns out that the (large) correlations indicated by Fig. 1c (see
Fig. D1, App. D) are simply reduced in an overall manner, but that the
geometrical relation among the small residual correlations remains. A
circumplex calculation (see App. A) so called, is easily able to reconstruct
the underlying geometry. This then, is the psychometric proof of the theory.
Limitations of space do not allow an elaborate description of the
calculation so the details have been put in an appendix (App. B) for those
who wish to verify it.
For the general reader we may outline the procedure briefly. A
Correlation coefficient is also the cosine of an angle. Therefore the numbers in
K&J's table represent a table of angles, that could represent a 7-axis structure in 3D
space. Appendix A describes circumplex calculations in general and appendix B is
the actual circumplex calculation for K&Js' table. Fig. 2 is the graphical result of the
calculation of appendix B. If you make a photocopy of this figure and cut and fold
where indicated, what you obtain is the direct empirical confirmation of the
theoretical Cattell
General Solution of Fig. 1c. Given that not only is the geometry
impressively confirmed, but that also the resulting "cross correlations"
mathematically agree with the data (see App. B)- we can see that this result leaves
little doubt that Cattell's 7-2nd order factors truly are, an
elementary geometrical structure in 3D (E,N,P) space. We certainly must
attribute this scientific milestone in psychology to the enormous empirical
labor of Raymond B. Cattell in 60 years of Personality research.
Figure 2
Finally of course, the usual analysis of the statistical significance must
be made. As stated above K&Js' table 2 might represent a structure in 3D
space and it might not. The question is, what is the chance that a random
table would produce a model like Fig. 2? Both theoretical calculation and
random number computer analysis confirm that the odds against such an
accident are well in excess of 20 to 1. Appendix C contains this probability
calculation as well as a discussion of the computer verification.
This result then, Fig. 2, is a powerful and convincing empirical proof of
the fact that Cattell's 7-2nd order factors absolutely do, form a simple
geometric structure in 3D space, exactly as predicted by the Cartesian
Theory.
As a result of this discovery, we are now able to resolve one of the most
perplexing debates in modern Personality psychometry. In the literature
there are many 2,3,4,5,6,7 factor models all claiming to be valid structural
models of personality. The debate over how many dimensions there are in the
Structural Model is very much alive (Eysenck 1991, 1992; Costa & McCrae 1992).
What we see now is that the Cartesian Theory and the Cattell General Solution
provide a simple resolution of this problem. All of the lower dimensioned models
are simply cruder and cruder geometrical approximations of Heymans' cube, Fig. 3.
Figure 3
These figures are just the simplest symmetric approximations of Heymans' cube
that can be drawn with the designated number of factors. How do we know that
they are correct? Well, the 7F is confirmed by K&Js' data. The 2F and 3F
models have been confirmed by Eysenck (Eysenck 1947; Eysenck H.J. & Eysenck
S.B.G 1969; Eysenck H.J. & Eysenck M.W. 1985). The structure of the 4F model
has been confirmed by Clarke and Merenda (Merenda 1987; Merenda & Clarke
1959). As we will see in the next section the 5F structure has been
numerically confirmed by Peabody and Goldberg's Lexical Double Cone model.
Five out of the 6 figures have direct empirical confirmation.
Now finally, we must mention the possible biological light that the theory
sheds on this debate. The dispute over how many factors are in the
Structural Model has presently centered itself on Eysenck's theoretical model
the Gigantic-3 and those of various personality schools such as Cattell and
the Big-5, and others. Eysenck argues that the so called 4,5,6,7th factors
may not be truly independent, orthogonal dimensions. He suggests they may be
components of E,N,P or primaries or possibly hybrid 1st/2nd order factors, or
even worse possibly heterogenous factors ("chimeras") manifesting a pseudo
orthogonality to E,N,P (see Eysenck 1991, 1993). Personality psychologists
maintain that these factors are clinically real, homogenous and orthogonal.
They point to the fact that eigenvalue scree plots universally show 6 to 9
factors before the "elbow" in the curve, not 3.
Now the present theory points to a possible resolution of this dilemma.
First, the physics brain is an octupole structure with 3 "hemispheric"
dichotomies superimposed on it; the Bell-Magendie, Sperrian and Macleanian
(see App. D). This tells us that the 3 normal factors (E,N,P) are generated
by the action of 4-lobes vs. 4-lobes acting front-back, left-right and up-
down respectively, while the diagonals are simply dipole factors; 1-lobe vs.
1-lobe acting diagonally. This would explain why the 1st 3 factors are so
much broader (Extraversion, Neurosis, Psychosis) in concept than the
4,5,6,7th factors bearing such narrow psychoanalytical labels as Control,
Anxiety, Impulsivity, Socialization, Conscientiousness, or even Openness to
Experience. Secondly the physics brain offers an explanation of why it is,
that these latter factors with fairly manifest psychoanalytical locations in
E,N,P space (cf. Eysenck 1992b, p.669), should have so much "uncorrelated
variance" as to appear in experimental data as orthogonal factors. The
physics brain now indicates that this is simply because the dipole structures
vary independently (neurologically) from the 3 superimposed neurological
systems generating E,N,P. The biological variance between the diagonal lobes
(dipole) is significantly independent (uncorrelated) with the biological
variation existing in the Bell-Magendie, Sperrian, Macleanian (hemispheric)
dichotomies. The result is that the 7-geometric axes in the physics brain
vary, in substantial measure, independently. This, even though
psychoanalytically speaking the factors manifest diagonal character in E,N,P
space as Eysenck has observed.
Now, at present psychometric data is not accurate enough to numerically
confirm this, hence the controversy, but if the Cartesian Theory is basically
correct- and that N and P are bipolar and not unipolar (see Sec II and App.
E), and this problem is corrected in psychometry, and the evaluation problem
is corrected vis a vis P&G's double cone discovery, I believe that the
accuracy of psychometric data will dramatically improve and that this
neuroscientific question can soon be answered. That it is answered of
course, is of crucial importance to the future of neuroscientific research in
Personality.
In conclusion then, we see that not only has the Cartesian Theory unified
the leading schools of Cattell, Eysenck and the Big-5, it has unified in
principle all of the existing Personality psychometry results.
Section II The Big-5 Model
Cattell has argued that the Big-5 is "a distorted approximation to the 2nd
order factors" (Smith 1988). In Fig. 3c we have seen what this distortion
is; a projection of the 4 diagonals of the General Solution onto the E-N
plane, hence 3+2=5.
Now it turns out that there is a long history of these two robust
diagonals
in Eysenck's E-N plane; a history independent of the discovery of the
Cartesian Theory. Historically their first detection was an intriguing
result of Ferguson's in Social Attitudes (Ferguson 1939, 1973; Eysenck
1954,1971; Eysenck & Wilson 1978) and proceeds now to the monumental diagonal
Personality model of Professor Jeffrey A. Gray (Gray 1970, 1972, 1981, 1982,
1987a, 1987b, 1987c, 1988, 1991a, 1991b, 1991c).
Now, implicit to the Cartesian Theory is that the Structural Model of
Social Attitudes is simply a social manifestation of the underlying
Structural Model of Personality and I advanced this in my 1994 paper.
Eysenck has advanced this connection in many research studies over the
years.
Soon after his celebrated 2-axis Structural Model of Social Attitudes
appeared (Eysenck 1944a) it was recognized that there was also a "diagonal"
model, rotated at 45 degrees, discovered by Ferguson. Brand points out that
these "normal" and "diagonal" dimensions in the E-N plane appear both in
Social Attitudes and Personality (Brand 1981, p. 22).
Moving to Personality then, this diagonal structure is clearly shown in
Gray's 1970 theory Fig. 4a. Later as Gray's work became more sophisticated
it was discovered they don't actually lie in the E-N plane (this a distorted
approximation to use Cattell's phraseology) but that they are actually in 3D
E,N,P space, and we have Fig. 4b. This diagram we can see, is now well on its
way towards Heymans' cube, Fig. 1c, even though it contains only 2 of the 4
diagonals of Heymans' cube.
Figure 4 a,b
Now, does the empirical data support this 3+2=5 theoretical explanation of
the Big-5? Well, the effort in Big-5 research is not as integrated and
comprehensive as Cattell. Most effort involves small academic studies
(N=300) and many different researchers. Furthermore as we can see from Fig.
3c there is a geometric problem unique to the Big-5. We can project the 4
diagonals not only to the E-N plane, but onto the E-P or N-P planes also.
This effect will make itself felt in the vicissitudes of "blind" computerized
factor analysis. It undoubtedly contributes to the difference for instance
between the numerous Big-5 models currently in the literature (Costa & McCrae
1992, Costa, McCrae & Dye 1991; Goldberg 1990, 1992, 1993; John 1990;
Zuckerman et al. 1988, 1991, 1993 and others).
As Fig. 3c indicates then, the Cartesian Theory says that Big-5 dimensions
I, IV and II are identified with Eysenck's E, -N and -P. This leaves the
problem of identifying the 2 remaining dimensions III and V with 2 above
mentioned diagonals of the E-N plane.
Figure 5 a,b
Now, in my 1994 paper I introduced the terminology "Right and Left" for
these 2 diagonals, Fig. 5b, and this is based on their historic role in
social psychology (Adorno et al. 1950, Ferguson 1939, 1973; Eysenck & Wilson
1978, p. 4; Ray 1986 p. 157; Eysenck 1954, 1971) as well as their important
appearance in the Roll Call voting statistics of our BI/2P system (Hammond
1994, p. 158). In Personality theory quadrant 2 is classically the SrEgo (or
conscience) quadrant, Fig. 5a (Hammond 1994, p. 163), so that we identify
"Conscientiousness" or FFM III with the R-diagonal. This leaves the L-
diagonal or its obverse in this case for factor V. That this is the case is
currently a topic of debate, a "..scientific embarrassment" of sorts
(Goldberg 1993, p. 27). On theoretical grounds however, I put it in the 3rd
quadrant, the obverse pole of the L-diagonal (see point 3, App. E).
Although it is outside the scope of this discussion allow me to mention
that all the social conflicts of history can be framed in terms of these L &
R diagonals. WWI and the French revolution were R-diagonal conflicts while
WWII was left diagonal. Evident for this is the fact that Adorno's
antisemitism, Fascism scale is congruent with the left diagonal for instance
(Ray 1986, Fig. 1 p. 157).
Here however we are dealing with the location of these diagonals in
empirical Personality research and we must turn to the Big-5 data. It turns
out that the same technique of circumplex analysis can be applied to FFM
results to locate these two diagonals. What can be done is to inspect FFM
data, identify E,N,P among the factors and then use circumplex trigonometry
to locate the two remaining factors. Fig. 6 presents the graphical result for
two of the leading Big-5 Models.
Figure 6 a,b
These results offer only sketchy support for our theory of FFM III and V
being the Right and Left E-N plane diagonals and also demonstrates that there
is more variation in the Big-5 field than in Cattell's time tested result.
Despite the atheoretical nature of Big-5 research and disputed convergence,
there is an extrodinarily high level of activity on this model (Digman 1990,
John 1990). What is the explanation for this?
The answer is to be found in a comparison of the models of Fig. 3. The 5-
Factor model is unique in that it is almost a planar model, with only a
single axis in the 3rd dimension. The plane consists of the 4-classic
personalities (4-Humors see Sec. V) as well as the two major axes of
conflict. It is a first step beyond Eysenck's primary 3-axis model- it adds
the two primary (Left and Right) diagonal axes of personality and social
conflict. As such it becomes the simplest "complete" model of personality
structure and function. The plane in this model represents a complete working
model of the BI/2P system for instance (see Sec. V). All this points to the
Big-5 as the simplest, compact approximation of the General Solution that
contains all the basic features of Personality. It is this fact that has
been sensed by Personality psychologists and explains the enormous, indeed
international effort currently under way in this field.
Now while we have demonstrated only sketchy agreement so far with our
theoretical or "canonical" Big-5 of Fig. 3c, it turns out that Peabody &
Goldberg have made a fundamental theoretical advance in Big-5 research which
now clearly shows the identity between Fig 3c and the general empirical
results of the Big-5 field. This result is known as the Lexical Double Cone,
a result that Hofstee has characterized as "a major intellectual tour de
force" (Hofstee 1994a, p.30).
While Costa & McCrae have now established Big-5 research in the
questionnaire domain (Costa & MaCrae 1985, 1992; Costa, McCrae & Dye 1991)
the Big-5 originated in Lexical, adjective based, research (see John 1990 for
a review). Professor Goldberg is particularly visible for his attempt to
push the "Lexical Hypothesis" to a conclusion (Goldberg 1981, Saucier &
Goldberg in press). Making use of Peabody's longstanding research on the
problem of "evaluation vs. description" (Peabody 1970, 1984, 1987), Peabody
& Goldberg (1989) have discovered the Double Cone structure in Lexical
psychometry. This model attempts to finally solve the problem of
Evaluation. What the model does is rotate the entire factor solution to a
"General Evaluation" axis (see App. E), and then computes two orthogonal
circumplex planes one for positive Evaluation and one for negative
evaluation. Two orthogonal axes lying in these planes are then labeled T-L
and A-U. Factors I, II and III are then located in this 3D "double cone"
space (see P&G 1989, Fig. 1, p. 558 also Fig E1 App. E). They have used
exactly the same mathematical technique for computing the angular locations
of vectors from the ratios of residual correlations, that I have used in App.
B. I would therefore highly recommend a reading of their landmark 1989 paper
describing the discovery of the Double Cone (see particularly p.556).
Now what I have found by examining P&G's data, is that their 3D space is
identical to Eysenck's E,N,P space. "General Evaluation" turns out to be
collinear with Eysenck's P, and T-L and A-U turn out to be, once again, the
R and L diagonals of the E-N plane. Determining the location of Eysenck's
E,N,P in their Double Cone space yields a unified version of the Double Cone
, Fig. 7. What this demonstrates is that the Lexical Double cone is
identical to Fig. 3c, the canonical Big-5 model as predicted by the Cartesian
Theory. The psychometric details of how the E,N,P axes have been located in
P&G's model are given in Appendix E, but the main results may be stated here.
Figure 7
1. "General Evaluation" in lexical research is actually collinear with
Eysenck's P axis.
2. Once Big-5 dimensions are "unconfounded" with General Evaluation, the
structure resolves itself into the "canonical" Big-5, Fig. 3c.
3. It is immediately seen that the N-axis in this model differs
substantially from Eysenck's classical formulation of N, in that it is
bipolar not unipolar, it has pathology at both ends and normality in the
middle.
Finally then, we have here a conclusive confirmation of the "theoretical"
Big-5 (Fig.3c) from empirical five factor research. The Lexical Double Cone
is identical to the canonical Big-5 as predicted by the Cartesian Theory.
Now point 3 above is a fundamental result and has considerable import for
Personality research and Big-5 research in particular. Historically Eysenck
formulated N as a dimension running from "normal to neurotic", a unipolar
dimension, Fig 8a (Eysenck 1947). Although the Cartesian Theory predicts
bipolarity for N from the outset, in fact advances the Sperrian
Lateralization dichotomy as the biological origin of N, the Lexical Double
Cone is the first psychometric result to clearly indicate this bipolar
structure.
Here I will introduce a new terminology for these positive and negative
poles of N. We are indebted to (Reverend) Professor Leslie J. Francis for
the terminology shown in fig 8b (Francis 1993).
It turns out that Francis has discovered a gender anomaly in Eysenck's
Neuroticism scales which leads to a dual scale, named Neuroa and Neuros. The
key items of these scales correspond closely to the adjective lists derived
from the positive and negative poles of N in the Double Cone (see app. E).
The discovery of the bipolar nature of Eysenck's N-dimension indicates a
major revision in psychometric research since N appears in nearly every
Figure 8 a,b
psychometric model in the literature. In Big-5 research the effects of this
are immediately apparent.
It has been long observed that there are few adjectives describing IV and
many describing II (Goldberg 1990). If Eysenck has formulated N with the
negative pole missing, folded over double on the positive pole, or mistakenly
assigned to P, as appendix E indicates (see also Sec. III, Fig. 11), we see
that perhaps as many as 25% of the adjectives assigned to -II actually belong
to the missing pole of factor IV- hence a direct explanation of this
longstanding anomaly.
The basic effect then of the "missing negative pole of N" is to warp the
entire Big-5 out of shape. It is presently in a space somewhat reminiscent
of a flat tire, round but flat on one side. In order to fix this flat tire
the missing spoke, the negative half of the factor IV axis, has to be
reinserted back into the hub. This N-axis problem is a significant part of
the problem now preventing a convergence of research in that field. This
suggests an immediate effort to devise an instrument to measure true bipolar
N.
After straightening out this problem in Big-5 research generally, an
effort should be made to unconfound evaluation and description in Lexical research
by rotating to Peabody's "General Evaluation" and this should be then
identified with factor II. It is forecast here that these two major
corrections will lead rapidly to an identification of figure 3c as the
Canonical Big-5 Model which the field has been looking for for 30 years,
since the pioneering work of Tupes & Crystal (1961) and Norman (1963).
In conclusion then, it is seen that the circumplex construction of the
Lexical Double Cone provides yet another powerful confirmation of the
psychometric predictions of the Cartesian Theory.
It is now evident from the direct empirical results cited in this paper,
that the vast empirical achievements of Cattell, Eysenck and the Big-5 school
have been reconciled, explained and unified.
Section III Eysenck's Personality Model
and the Structure of Mental Pathology
Clearly the most influential work in contemporary Personality research is
that of Hans Eysenck. As we have now seen ENP has proven to be the central
construct of the Structural Model of Personality.
Unarguably the best way to comprehend Eysenck's fifty years, 800 papers
and dozens of volumes on Personality is a historical approach, therefore my
interpretation of that history will be presented here.
Born in Berlin in 1916, Hans Eysenck escaped Germany in the 30's after the
rise of Hitler. Eysenck took his Ph.D. at the U. of London in the psychology
department headed by Cyril Burt. Eysenck was familiar with the concepts of E
and N by 1940; inasmuch as Ray (1986 p. 155) cites an early paper in which
Eysenck is explaining poetry appreciation in terms of Extraversion and
Neuroticism (Eysenck 1940). Eysenck's first scientific effort was in Social
Attitudes. In a now famous paper (Eysenck 1944a) he advanced that Social
Attitudes could be described in two dimensions, the classic Right-Left and
another named Practical vs. Theoretical (later Toughmindedness, T). This
achieved considerable attention because Eysenck proposed it explained the
difference between Fascism and Communism.
Shortly after this Eysenck turned to Personality in another seminal paper
in which he factor analyzed the testing of 700 neurotic servicemen (Eysenck
1944b). In this paper Eysenck first advanced E and N. This theory, "first
put forward in 1947" (Eysenck 1964 p.286), was well received and apparently
moved Eysenck to press on and posit psychosis, P, as a third dimension;
Eysenck says that the concept was "first mooted" (Eysenck 1976, p.1) in 1952
(Eysenck 1950, 1952).
Meanwhile, Eysenck's 1944 theory in Social Attitudes appeared in The
Psychology of Politics (Eysenck 1954). At this point, Eysenck began to run
into trouble. As Ray puts it (Ray 1986, p. 158) the Zeitgeist of the times
was being overtaken by The Authoritarian Personality (Adorno et al. 1950),
and this school was not at all happy with Eysenck's more scientifically
neutral theory of Social Attitudes. The Frankfurt school was intent on
showing one thing- that the political extreme right has an Authoritarian
personality (de facto psychotic in their opinion). What Eysenck's two-
dimensional theory showed was that authoritarianism lay in a second dimension
orthogonal to the R-L axis. The Zeitgeist of the times, post WWII, was not
ready to listen. In the prestigious (APA) Psychological Bulletin (1956),
Rokeach, Hanley and Christie mounted "scathing attacks on Eysenck and his
work in social attitude and political research" and "..which appears to have
caused even Eysenck to abandon the field for many years." (Ray 1986 p. 160).
At this point Eysenck settled in for a long siege operation.
Between 1955 and 1967, Eysenck attempted to discover a credible biological
basis for his Structural Model- proof as it were. Claridge says it was
"..with the publication of his book, Dynamics of Anxiety and Hysteria-
..actually his 1955 paper in the Journal of Mental Science- that Eysenck took
the step which most drastically altered contemporary Western thought about
personality.....his attempt to ground individual differences in their
biological roots,.." (Claridge 1986 p.74). This culminated with the advance
of his famous biological (neurological) model for E and N (Eysenck 1967).
This model stimulated an unprecedented surge forward in neuroscientific
research in Personality.
It is at this point that Eysenck returned to his inspirations of 1952 and
decided to "revive" as Claridge puts it, his third dimension Psychoticism,
and here, we must pause to distill the preceding developments as they relate
to the subject of "neurosis and psychosis" in personality theory.
In traditional knowledge and vernacular usage, first of all, there is a
latent L-R dichotomy lurking just below the surface of the terms neurosis and
psychosis. Traditionally a neurotic is a cry baby and a worry wart, while a
psychotic is something more sinister; he may be a wife beater, and ax
murderer or a mad bomber. On the other hand, in the popular mind, the Left
wing with its radicalism, naive utopianism, bleeding heart liberalism etc. is
perceived generally to be neurotic. On the Right on the other hand is every
conservative tyrant since Caligulia and Nero to Simon Legree and Hitler;
these people in the popular mind are psychotics. The case may be made I
submit, that the terms neurotic and psychotic emerged, to a certain degree,
in psychology as epitaphs designed by the Right and the Left to characterize
the opposition.
There is even a formal outcropping of this sentiment in academic and
clinical psychology. The classic separation of manic depression from
dementia praecox by Kraepelin in 1897 the case in point. This theory was
elaborated subsequently by Kretschmer (Kretschmer 1946, 1948) using the terms
cyclothymia-schizothymia. While manic depression and dementia praecox are
both psychosis, dementia praecox traditionally has a more neurotic halo;
while manic depression has an irritable and frustrative component. Dementia
Praecox might signify someone who is simply demented, mania on the other hand
could easily signify a maniac.
The problem for Eysenck as apologist for the Right then, is to head off
the incipient identification of psychosis with the Right, by showing that it is
orthogonal to Neurosis; much as he showed Authoritarianism to be orthogonal
to the L-R axis in Social Attitudes. However, his catastrophic foray into
social attitudes has taught him that there may be stiff opposition to this;
should his adversaries ever tumble to the fact that N is synonymous with the
R-L axis. So far then, this eventuality has been mooted in that we only have
a unipolar N (Left-N; classic neurosis). Unrecognized by anyone so far is
that there is also -N (e.g. a Right-N, actually Kraepelin and Kretschmer's
mania), which because of its frustrative (manic) propensity could easily
become saddled with the label "psychotic". It is Eysenck's challenge to
separate psychosis from neurosis before there is yet another stampede
blocking the road to a pure scientific truth. Consequently, we never see the
appearance of bipolar N, and we see a long tautology by Eysenck, using
discriminant function analysis, proving that psychosis and neurosis are
orthogonal and that Kraepelin and Kretschmer' theories just can't be correct
(Eysenck 1970a, 70b). Furthermore, although Eysenck does attempt to identify
E and/or P with "T" in Social Attitudes, he never takes the obvious other
step, which is to identify N with "R", the Right-Left political axis (Eysenck
& Wilson 1978 p. 306; Ray 1986 p.167); this obvious inference simply remains
moot.
By 1976 then (Eysenck & Eysenck 1976), Eysenck has succeeded in
establishing psychoticism as a 3rd dimension orthogonal to N and E-I.
Eysenck, in a desperate end run maneuver has succeeded in establishing the
scientific truth, without getting waylaid by the political problem. However,
Both N and P are now formulated as unipolar dimensions, running from "normal
to pathological". Shortly after this Eysenck returns to the Social Attitudes
battleground with Wilson; publishing the Psychological Basis of Ideology
(Eysenck & Wilson 1978), a final vindication of his ignominious defeat by
Adorno in that field 25 years earlier.
Now this history is a roadmap as it were, of Eysenck's scientific strategy
and technical accomplishments, and it is presented here largely to address
the following major scientific dilemma that currently exists in Personality
research.
As explained above Eysenckian theory has arrived at a unipolar formulation
of N and P for the historical reasons stated. According to Eysenck both N
and P run from "normal to pathological" (Eysenck 1947, 1992b).
The Cartesian Theory indicates however that both these dimensions are
"bipolar" in the same sense that E (Extroversion-Introversion) is. The
Physics brain identifies N with the bipolar neuropsychological dichotomy of
Sperrian Lateralization for instance. P is seen to originate in a primary
bipolar cell cleavage in embryology the same as E and N; the third cleavage
(Hammond 1994).
In Section II we have discussed the effect of this on psychometric
research notably on the Big-5. It turns out however that this dilemma has an impact
on an even more fundamental area of psychology research and that is in the
realm of mental illness classification itself, and we will turn for a moment
to that subject here.
While there exist many theories and models of the structure of mental
illness, including those of Kraepelin, Kretschmer and more recently Eysenck
and many others- none of these theories explicitly identifies Personality
Type as the sine qua non of the theory of mental illness category. What I am
here to advance is this; that in the E-N plane there exist four basic
personality types and that if and when, through environmental stress or other
causes, they become mentally disturbed, they develop "characteristic"
illnesses specific to these personality types. Further, that the names of
these illnesses are already well known to psychology, as are the personality
types. In Fig. 9 four major Personality types, in the E-N plane are shown
and the four major categories of mental disturbance associated with these
personality types is shown:
SrEgo | Id psychosis | Neurosis
_______|_________ N => __________|____________ N
| |
Ego | Libido Mania | Phobia
E E
Fig. 9a Fig. 9b
The 4-Personalities The 4-Mental Illnesses
of the E-N Plane of the E-N Plane
Figure 9a shows the Structural Model in the E-N plane in Classical (Freudian)
terminology. These 4 designations may be used to refer to the 4-Canonical
Personalities of the Structural Model. The 4 corresponding pathological
terms in 9b are not meant to be rigorously clinically accurate, but rather
more nearly representative of vernacular, or historical usage; e.g. more or
less comparable to the common usage understanding of the 4-titles in fig. 9a.
In full three dimensional Personality space, Fig. 10a, we have the completely
expanded delineation of eight mental illness types and here I have availed
myself of several contemporary (DSM) labels:
Figure 10 a,b
Now, what evidence do we have that Fig. 10a is, in principle, correct?
The support for this identification of mental illness types with Personality
types, in fact, is most strongly supported by the work of Hans Eysenck.
Figure 10b (Eysenck 1970, p. 29), it can be seen, bears a very close
resemblance to Fig. 10a.
Now, Fig. 10b is an early attempt by Eysenck to accommodate Kraepelin's
classic (1897) separation of manic depressive illness from dementia praecox
(schizophrenia), and Kretschmer's (1946, 1948) theory that this distinction
is based on a difference in Extroversion (cyclothymia-schizothymia)- with his
orthogonal separation of N and P (see Eysenck 1970 pp. 24-30). We can see
now, in view of the present theory, that the main thing separating Fig. 10b
from 10a, is the fact that N is not bipolar but unipolar in Eysenck's 1970
(cube) diagram. Had Eysenck established N as a bipolar dimension, we
obviously would have had Fig. 10a in 1970 instead of 1995.
As a final demonstration of the "folded" nature of N and P, I cite Fig.
11a which is taken from Eysenck & Eysenck 1976, p. 12, and dates from him in
1960. Note the four points labeled Phobic Reactions, Dementia Praecox,
Affective Psychosis and Catatonia. According to the bipolar theory of N and
P advanced here, this diagram should actually be unfolded into a 4-quadrant
diagram, by "unfolding" both the N and P axes, Fig. 11b. In Fig. 11b we have
mirror reflected Affective Psychosis over to -N, Phobic over to -P and
Catatonic over to -N and -P. Not shown of course is the E dimension, which
would actually show Phobia and Catatonia to be +E and Dementia Praecox and
Affective Psychosis to be -E. At any rate, this exercise serves to
demonstrate how a bipolar N converts Eysenck's 1970 cube, Fig. 10b, over into
our proposed model of mental illness classification, Fig. 10a.
Figure 11 a,b
Now, the lack of a bipolar N has hampered Eysenck's identification of
psychotic typology with personality typology vis a vis E and N, but
Eysenckian theory is, as we have seen, not very far removed from the
solution. We have Fig. 10b in which Eysenck virtually hit the nail on the
head in 1970. Elsewhere, he has continually supported the notion that E, if
not N, is the differentiator of Psychotic types e.g. "..he would argue that
'psychoticism' is a general dimension of personality predisposing to all
forms of psychotic illness, the particular direction in which disorder occurs
depending on the individuals weighting on other personality characteristics,
mainly introversion-extroversion." (Claridge 1985 p. 161). Both E and N are
implicated in 1992 (Eysenck 1992b p. 768): "The possibility certainly exists,
and should be investigated, that the major difference between functional
psychoses are related to the other major dimensions of personality, i.e. E
and N.".
Finally, we have Eysenck's formal identification of E,N and P with the
classification of Personality Disorders as defined by the DSM-IV (APA 1994).
Axis II of the DSM contains 11 personality disorders grouped into three
clusters dubbed the "dramatic", "anxious" and "odd" categories (Reich &
Thompson 1987; Kass et al. 1985). Eysenck (Eysenck 1987, recent paper 199X)
has suggested that the three clusters (dramatic, anxious, and odd), resemble
his dimensions of E,N, and P. This result has been given further empirical
confirmation recently (O'Boyle & Holzer 1992). This tells us that
Personality Disorders are simply milder forms of mental syndromes found
further out toward the poles of the axes of the Structural Model. These more
extreme syndromes (e.g. manic depression and schizophrenia) are separated in
the DSM from the less extreme "Personality Disorders" and are listed under
Axis-I, Clinical Disorders. In view of this diagnostic organization, we may
expect that the discovery of the Structural Model will have a significant and
immediate impact on the future structural organization of the DSM categories.
Given this 45 year history of Eysenck's juggling of the existing data and
theories, we must conclude that, save for the late discovery of the
bipolarity of N vis a vis Sperrian Lateralization and the Physics Brain
(Hammond 1994), Eysenck's formulation of the problem would clearly have gone
over by now, from Fig. 10b to Fig. 10a. In view of the fact that it is
Eysenck who has identified neurosis and psychosis with two of the 3-primary
dimensions of the Structural Model of Personality, it is my opinion that it
is nothing less than pedantic to try and argue that Eysenck has not
established that the classification of mental disorder is congruent with the
Structural Model of Personality.
In closing this review of Eysenckian personality theory I will mention
here a matter which transcends ordinary science. Nowhere in the above is there an
answer to why the two leading terms of mental pathology, neurosis and
psychosis should be identified with the primary dimensions of Personality
structure. It can only be argued that they are because Hans Eysenck has
theorized and proven that they are. This development in Eysenckian theory is
then the scientific equivalent of Freud's formulation of the Ego and the Id
a generation ago, and in the opinion of this author, clearly places Eysenck
on a plane with Freud in the scientific evolution of Psychology. What we see
is that it is in fact Eysenck who has scientifically equated mental illness
category with the Structural Model of Personality.
With agreement then, here achieved, between Physics and Psychology, on the
basic structuring of mental illness types, Fig. 10a, we certainly must be
writing the concluding chapter of the long historical quest for the
Structural Model. It is the final resolution of this basic and leading
problem in Psychology, that qualifies now an inspiration to great confidence,
that the Cartesian Theory is correct.
Section IV The Physical Origin
of Thurstonian Simple Structure
Students of Factor Analysis are aware that, basically, all that factor
analysis can tell you is "how many factors are in the data". It turns out
that if factor analysis determines that there are n factors, that these
factors have to be "rotated" in n-dimensional space (i.e. with respect to the
data points) until they line up with some psychologically interpretable
dimensions. This is because the location of the n-factors with respect to
the data points is different for every different method of factoring the
intercorrelation matrix, and there are several different methods. This
"rotational problem" in factor analysis has a long history and a considerable
literature.
The founding principle in rotation is due to Thurstone, who was the first
one to discover "Simple Structure" so called (Thurstone, 1933, 1947). Now
obviously, if n=7 and it is found that all of the data points are located in
7 distinct clusters, the 7 axes should be rotated until one axis runs through
each cluster. This is a simple example of Simple Structure and was well
known before Thurstone. What Thurstone discovered is that there are degrees
of Simple Structure. Clustering is an example of where each test in a
battery contains only one factor. Psychometricians say in that case that all
of the tests have "complexity" one. What Thurstone discovered is that some
tests have complexity 2, or 3 or even 4 or 5. If all the tests had
complexity 2 for instance, this would mean that all of the test vectors would
be arranged in "fans" (or pinwheels) in the n(n-1)/2 2-factor planes of the
n-dimensional space. This would be 2nd degree simple structure. There is
3rd, 4th.... etc. degree Simple Structure. The weakest, most general Simple
Structure is (n-1)th Simple Structure, and this Thurstone called "hyperplane
Simple Structure". Thurstone discovered all of this experimentally, when he
first discovered experimentally, the existence of 2nd degree "fans" in many
cases of experimental data, including Personality data. Thurstone argued
very forcefully that all factor solutions of factor analysis, must be rotated
to a Simple Structure orientation before the factors would have any
psychological meaning.
Despite a raging debate over the years about the meaning of Simple
Structure (remember Thurstone said it had to do with test complexity) it
appears that Thurstone's intuition has survived the test of time. The
evidence for this is the fact that the modern computer statistical packages
have come down to us with a mathematical version of Thurstone's rotation to
Simple Structure built into them (e.g. Varimax-Promax etc.)
The development of these mathematical techniques has a very interesting
history, most of it recorded in the volumes of Psychometrika, 1935-1970.
Thurstone working in the 30's and 40's did not have computers, and had to
rely principally on graphical methods of rotation. This was comprised mainly
of eyeball criteria for the visual rotation of the axes among the data
points. These visual criteria later became the stepping off point for the
mathematical formulation of the problem for high speed electronic computers
in the 50's and 60's, notably by Horst (1941), Carroll (1953), Kaiser (1958),
Ferguson (1954), Neuhaus & Wrigley (1954), Hendrickson & White (1964) and
others (see Harman (1976) or Gorsuch (1983) for summaries).
Now, the basic visual criteria used (or discovered) by Thurstone, and the
one basically incorporated in the modern mathematical formulations, is the
criteria of "hyperplane count", or hyperplane density. This is simply the
percentage of the data points that lie in the n-1 hyperplanes of the n-
dimensional personality space. This number, it is found, usually runs 60-80%
in good personality data. When the factor axes have been rotated to the
position where this percentage is a maximum, the psychometricians say that
the solution is a Simple Structure solution.
Now at this point we wish to address the question of what the discovery of
the Cartesian Theory and the Physics Brain has to say about the matter of
"Simple Structure". First of all, we have to ask "what is the cause of
Simple Structure". As a physicist, it is my intuition that Simple Structure
has a physical (biological) cause, and I think that Thurstone probably
suspected this in 1947. He also knew that any explanation of it was well
beyond the reach of 1947 science, so he simply offered the argument about
"test complexity" as a gloss on the subject. Now however, with the discovery
of the Physics Brain and the Cartesian Theory, there is a very real
probability that we have finally discovered the physical origin of Simple
Structure, 50 years after Thurstone first discovered it experimentally and
named it.
The Physics Brain offers this explanation of Simple Structure. The
Physics Brain is an octupole Structure (one brain lobe located at each of the 8
corners of Heymans' cube). However, this becomes a "face centered cube" to
use a physics term from crystallography, with the addition or superposition
of the Sperrian, Bell-Magendie and MaCleanian dichotomies, so that there are
actually 14 poles (7-axes of symmetry). If you start drawing connecting
lines between all these poles, sort of a web structure, you begin to see what
kind of field structure (or vector structure in psychometry) this
configuration causes. The first thing you notice is that the "density" of
lines is much higher in the symmetry planes of the cube, the planes
containing the symmetry axes. This leads one to believe that the Hyperplanes
of Personality space are to be identified with the Symmetry Planes of
Heymans' Cube!
Now there should be a way to mathematically demonstrate the truth of this
proposition. First, we have seen in Appendix B how the 7-Factors of Krug &
Johns' table 2 can be identified as the 7-symmetry axes of Heymans' cube.
The next thing that we notice is that these axes lie on the intersection of
2 or 3 of the symmetry planes of the cube. This fact and the fact that the
density distribution has higher density in the symmetry planes tells us that
the symmetry planes of Heymans' Cube are probably the physical origin of the
increased "hyperplane count" in the n-dimensional hyperplanes of Personality
space.
Now, to go one step further in mathematically proving this, we have to go
back to Thurstone's original technique of graphically rotating axes in
Personality space. What Thurstone did is consider the "plane" that is normal
to each of the original Factor axes derived from the factorization process.
The original factors he called the "reference axes" and the n-1 planes normal
to these axes he called "hyperplanes". He then graphically plotted two of
the reference axes at a time and rotated them visually until their respective
normal planes contained as many points as possible (maximum hyperplane count,
so called). In the case of 2nd degree simple structure each of these planes
would contain a "fan" of test vectors for instance. He then identified the
intersection of these planes as the final "primary factors" of the solution.
These for instance, are the 7-factors listed in Krug & Johns' table 2 from
which the Cattell General Solution is constructed- they are the 7 symmetry
axes of Heymans' Cube.
Now, if we are able to locate the 7-primaries in Heymans' Cube, we should
also be able to locate the 7 corresponding reference axes too. Since these
axes are normal to the hyperplanes in Personality space, it would be of great
interest to find out their orientation to the symmetry planes of Heymans'
Cube!
Now, students of the mathematics of all this, know that Thurstone's
primary and reference factors are just two different coordinate systems in N
dimensional personality space, and moreover, that they are mathematically
related. In fact, it turns out that these two systems are "reciprocals" of
each other, that is, that the matrix of one is the inverse of the matrix of
the other. In mathematical physics by the way, they are universally referred
to as the covariant and contravariant coordinate systems of the space.
What this means is that while Krug & Johns' table 2 is the
intercorrelation among the primaries, the matrix inverse of that table is the
intercorrelation among the reference axes. From the geometry of this matrix we
should be able to figure out the geometrical configuration of the reference axes
using the same technique we used to figure out the geometry of the primaries, in
Appendix B.
It turns out that if you invert Table B1, Appendix B, what you get is
approximately the same table with all the signs reversed (except for the
diagonals). This is generally true, i.e., that the correlation between the
primaries is about equal to the correlation between the reference axes with
the opposite sign (Gorsuch, 1983, p. 225 states "Second, the correlation
between a factor's reference vector and another factor's reference vector is
opposite in sign and approximately the same magnitude as the actual
correlation of those two factors."). What this means is that the reference
vector geometry in Heymans' cube is identical to the Factor vector geometry,
except that the 4-diagonals are reversed. This is not surprising, since
their is only one symmetric configuration of 7 axes in 3D space; if 3 of them
are to be orthogonal.
We now know then, where Thurstone's reference vectors end up being
situated in the Cattell General Solution. Do these locations maximize the "symmetry
plane count" in Heymans' cube, parallel to the fact that they maximize the
"hyperplane count" in Personality space? I think it is rather obvious that
they do.
First, we know that the density is highest in the symmetry planes of
Heymans' cube, so, what are the orientations of the reference vectors to
these planes? Direct observation tells us that there are 9 symmetry planes-
3 containing E,N and P plus two more each (diagonally) through E,N, and P and
containing opposite diagonals. Inspection shows that D1-D4 each lie in 3 of
the latter 6 while being 35.26 degrees off of the normal to the other 3.
This maximizes the variance along these reference vectors. The E,N,P
reference vectors are precisely 90 degrees normal to the 3 Cartesian (E,N,P)
planes. Thus the reference vectors are situated so as to maximize "symmetry
plane count" in 3 dimensions in the Physics Brain (Heymans' Cube) and this
translates directly into the empirically observed "maximum hyperplane count"
observed in the Cattell General Solution. Note that Krug & Johns report that
their data "showed a high degree of simple structure (p.683, see abstract);
reporting that the "Hyperplane counts obtained from the Promax rotations were
74% in each matrix" (Note: "each matrix" refers to the male, female and
combined population samples).
We note here that Heymans' cube is a 3D structure and therefore can only
possess 1st, 2nd and 3rd degree Simple Structure. In fact, since, by
observation the majority of the points are either at the poles or in the
symmetry planes, we may say for all practical purposes that it only has 1st
and 2nd degree Simple Structure. This agrees well with the general findings
of Hofmann in studies on variable complexity, that the average complexity in
most Personality studies seems to lie between 1 and 2, say around 1.5 (see
Gorsuch 1983 p. 230, table 10.3.1). Now as we have seen Heymans' Cube is
mainly composed of 1st and 2nd degree Simple Structure, indicating, providing
a physical explanation of Hofmann's average complexity figure of 1.5.
The conclusion from this remarkable demonstration of the
mathematical/geometrical correspondence between the symmetry planes of the
Physics Brain and the well established Hyperplanes of psychometry, is
certainly that this is a powerful supporting argument for the correctness of
the Cartesian Theory and the Physics Brain.
Section V Conclusion: The Discovery of
the Structural Model of Personality
The most profound and immediate impact of the discovery of the Structural
Model is a realization that the BI/2P system is the axiomatic structure of
Social Psychology. Since the axiomatic nature of the Cartesian Theory leaves
little room for doubt about this we may expect a renewed interest in this
area in an attempt to confirm it. The first step would be an empirical
analysis of the Roll Call voting statistics from the BI/2P legislatures world
wide. This data is on magnetic tape for the U.S. Congress for instance.
This analysis should confirm a universal "X" structure in voting statistics,
Fig. 12 (the R and L diagonals of conflict, Fig. 5b, Sec. II). An empirical
confirmation of this would go a long way toward confirming the Cartesian
Theory in social attitudes (Hammond 1994, p. 158).
Figure 12
The importance of this discovery is exemplified for instance, by the
conflict that has existed between the BI/1P system of Communism for the past
70 years, and the BI/2P system of western democracy. An axiomatic proof of
the BI/2P system would provide a tremendous impetus for Russia to formalize
the institution a BI/2P system. The benefits to global political unification
and stability can hardly be elaborated here.
Above the political level in human affairs of course, comes the matter of
religion, and particularly the conflicts between the major branches of
divinity; Christianity, Islam, Judaism, Hinduism, Buddhism &c. The conflict
between these religious systems has historically been the benchmark
underlying the psychological anxiety that separates one culture from another in
historical terms.
At this point it is perhaps appropriate to assemble in one place the
projection of the Structural Model of Personality, formally Eysenck's E-N
plane, on the fields of Theology, Social Psychology and Psychology, Fig. 13:
Figure 13 a,b,c
The discovery of the hard science, axiomatic physics origin of the
Structural Model, it is now evident, provides us at once with not only an
axiomatic proof of the BI/2P System of representative government, but
provides us for the first time in the history of the world, with a hard
scientific proof of a major structural element of religion; the quadrature of
the 4-Gospel Canon. The discovery of the Structural Model is not only to
have an impact on mans progress toward stable world government, it is to have
an immediate impact on the ecumenical unification of world religion. The
Gospel quadrature exists and can be identified in the canon of every major
religion and provides a formal path towards ecumenicalism. The enormity of
this is exemplified for instance by the prospect of half a billion Hindis
liberated from the bondage of the Caste system by a scientific proof that the
4-Varnas (Castes) are actually the BI/2P system.
It is the general view of this author therefore that the discovery of the
Structural Model, which has been a 3,000 year effort in scientific thought,
since the age of Hippocrates, that it is one of the most significant
achievements in the history of civilization.
There is little doubt that its purpose is to effect a dramatic and
enormous unification and mobilization of world society unleashing a cataclysmic
advance in social progress and the standard of living unprecedented in world
history. As a scientist, this author can only speculate that this
mobilization is necessary in order to undertake a solution to the global
technological challenges of energy, population and ecology now threatening
human progress.
APPENDICES
APPENDIX A Circumplex calculations
Circumplex calculations are based on the fact that a correlation
coefficient is also the cosine of an angle. The name circumplex was
originally coined by Timothy Leary and later developed by Wiggins (Leary
1957, Wiggins 1980) and more recently by De Raad & Hofstee (1993), Hofstee,
De Raad & Goldberg (1992), Johnson & Ostendorf (1993) and others and is
currently a familiar psychometric technique in the literature.
Ordinarily the term circumplex refers to a 2D diagram, or circle, showing
the vector locations of data points each of which has a correlation, rx and
ry, with 2 orthogonal factor axes X and Y. The length, or magnitude, of the
vector is given by M=(rx2+ry2)1/2 and its angular location by 0=cos-1(rx/M).
Generally M=1; they are not unit vectors (the physical significance of this
is discussed in Appendix D). However, the vectors may be arbitrarily
extended to the unit circle C, for convenience, so that the names of the
factors can be written in a ring around the edge of the circle, hence the
origin of the term "circumplex".
Figure A1
Although circumplex diagrams are normally 2D, the method may be easily
extended to 3D. If we have 3 orthogonal axes, say E,N,P, and factor
correlations rE,rN,rP then we may graphically draw that factor as a vector in
E,N,P space with "vector components" of rE,rN,rP (Fig. A1). Elementary
trigonometry allows us to calculate the "azimuth" and "elevation" of the
vectors in the E-N plane, which is useful for the construction of a
mechanical model in 3 dimensions.
AZM=tan-1(rE/rN) ELV=sin-1(rP/M)
where M=sqrt(rE2+rN2+rP2)
In the case of the Cattell General Solution (App. B), we have 4 vectors
D1,D2,D3,D4 in E,N,P space. The azimuth and elevation for each vector is
calculated by this method to construct the mechanical model (Fig. 2, Sec. I).
In the above circumplex method it is assumed that the coordinate axes are
orthogonal to begin with. In the case of the Cattell General Solution the
factor loadings are originally given in an oblique system. There a
transformation to an orthogonal system must first be made before the
circumplex calculation can be carried out.
In connection with this oblique transformation there is a fundamental
structural thesis employed in the calculation of Appendix B; one of
specifically psychometric origin. While the correlations between D1,D2,D3,D4
and E,N,P are considered to be actually quite large, but "reduced in
magnitude" by the Nature/Nurture variance of the population (App. D), the
intercorrelations of E,N and P are not considered to be the same thing, but
are taken as experimental error angles causing the coordinate axes to be
slightly oblique. The rationale for this is the overwhelming independent
proof (Eysenck loc. cit.) that E,N and P actually are orthogonal.
APPENDIX B Krug & Johns calculation
CONSTRUCTION OF THE 7PF CATTELL GENERAL SOLUTION
OF THE STRUCTURAL MODEL
If the male-female values in Krug & Johns (1986) Table 2 are averaged
(.53M+.47F), the following symmetric intercorrelation matrix for the 7-
factors is obtained. (note: decimal points have been omitted, and the order
of the factors has been rearranged, note also that the signs have been
reversed in the rows and columns headed -Control and -Socialization, since
the obverse pole of these factors is used in the model.)
Table B1
E N P D1 D2 D3 D4
Ext Anx Ind -Con GA -Soc TP
E | 1 -267 231 -190 -0171 135 105 |
| |
N | -267 1 -167 325 -0912 -0759 132 |
| |
P | 231 -167 1 305 127 330 058 |
| |
D1 | -190 325 305 1 0800 214 -0684 |
| |
D2 | -0171 -0912 127 0800 1 161 -141 |
| |
D3 | 135 -0759 330 214 161 1 002 |
| |
D4 | 105 132 0580 -0684 -141 002 1 |
The intercorrelations of Cattell's E,N,P axes (in the square above; the
"metric") are not zero and shows us that the 4 vectors D1,D2,D3,D4 (circled
above) are given in an oblique coordinate system. In order to locate them,
and construct a model, we must transform them to an orthogonal system- here
called e,n,p. N is set collinear with n, E is set in the e-n plane and,
consequently, P tilts into the octant bounded by e,-n and p (Fig.B1).
Figure B1
Simple trigonometry allows us to express E,N,P (the oblique axes) in terms
of e,n,p, the new orthogonal axes:
E = 0.9637e - 0.2671n + 0.0000p
Eqns.B1 N = 0.0000e + 1.0000n + 0.0000p
P = 0.1930e - 0.1671n + 0.9669p
From the first equation we note that E is oblique to e by an angle tan-
1(.2671/.9637)=15.49 degrees. Similarly, trigonometry applied to the third
equation tells us that P tilts at an angle tan-1[.9669/sqrt(.19302+.16712)]=14.8
degrees off the normal towards azimuth 180+tan-1(.1930/.1671)=229.1 degrees.
Fig. B1
The matrix of coefficients here, T, is the transformation matrix for
transforming a vector from e,n,p to E,N,P. Since we wish to transform vectors
in the opposite direction, from E,N,P to e,n,p, we simply invert this matrix:
1.0377 .27716 0
Eqn.B2 T-1 = 0 1 0
-.20713 .11749 1.03423
This roundabout method is used because writing the matrix in the orthogonal
system is much simpler than writing it in the oblique system.
Because T is expressed in "coordinates" which are in fact "contravarient
components", T-1 it turns out, is the "contravarient" transformation matrix
which will properly transform the "covarient" vectors circled in K & J's
table, into the orthogonal coordinate system (note: direction cosines are
"covarient" components in an oblique coordinate system).
By simply multiplying the matrix T-1 times the column vectors circled in
the
table, we obtain the same 4 vectors, now expressed in the orthogonal system
e,n,p, Eqns. B3. Since we are now in an orthogonal coordinate system we may
easily calculate the Altitude and Azimuth of these vectors, needed to
construct the model, Eqns. B4:
Eqns d1=-.1072e+.3247n+.3926p Eqns. ALT=sin-1(dp/M)
B3 d2=-.0431e-.0912n+.1237p B4 AZM=tan-1(de/dn)
d3=.11873e-.0759n+.3045p M=sqrt(de2+dn2+dp2)
d4=.14572e+.1322n+.0537p
These ALTs and AZMs are: Table B2
Vector ALT AZM M
d1 48.9 -18.3 .521
d2 50.8 -25.3 .159
d3 65.2 57.4 .336
d4 15.3 47.8 .204
We now have everything we need to construct the graphical model shown in
figure 5. These angles are printed in the appropriate locations in the
model.
Inspection of the model readily shows that d1,d2 and d4 are well within
their respective octants. Because of the tilt of P into the 3rd octant
however, we see that d3 narrowly misses the E-P plane. This clearance angle
may be calculated from vector analysis using (E P) d3 and turns out to be
1.04 degrees. Thus the model is shown to be rigorously and exactly correct
(note: the probability result of 20:1 allows the vectors to fall anywhere
inside the octants.). Here is not the place to argue that, obviously, a
minor rotation of the axes could dramatically "symmetrize" the model (this is
a task for future research); for now it is sufficient to rigorously
demonstrate that the model exists!
Finally, there are the 6 remaining numbers in the table in the lower right
triangle. These are the cross correlations between the diagonals. Now, the
angles between the diagonal vectors in the model can easily be calculated
from:
Eqn.B5 d1 dot d2
cos(ang) = -----------
|d1||d2|
and using equations B3. These angles are listed in Table B3 and are, of
course, the angles actually observed between the vectors in the mechanical
model, Fig. 5.
Now the cosines of these angles are not expected to match the cross
correlation coefficients in K&J's table, because as we have seen, there is a
multiplication factor (mathematically the "normalization") multiplying the
calculated direction cosines which reduces them to the values of K&J's table.
For the individual vectors in relation to E,N and P, we have seen that this
factor is simply M (the magnitude), given in eqn 4. From the same
mathematical relation it is easy to show that in the case of comparing two
diagonals with different magnitudes (M1 and M2) that the multiplication factor
is sqrt(M1M2). What we expect then, if the 3D model is correct, is that
these theoretically computed values will be the same as the experimentally
measured values appearing in the triangle in K&J's table. Agreement of these
two quantities would of course represent a "triangulation" type of proof as
it were, that the 7 factors are a figure in 3D space.
The next question that arises is how exactly to compare the numbers, or
rather, how to express the accuracy with which they agree or disagree. The
problem is that cosines are not linear quantities so that a simple
"percentage accuracy" really is not too meaningful. The only thing "linear"
involved is the underlying angle. So, instead of multiplying our theoretical
Diagonal-Diagonal direction cosines by sqrt(M1M2) and comparing them to K&J's
experimental values, what we do is divide K&J's values by sqrt(M1M2). We then
take the inverse cosines and compare the angles. In short, what we are doing
is comparing the actual angles that can be seen in the cut & fold model, Fig.
5, with cos-1[K&J/sqrt(M1M2)] where "K&J" refers to the experimentally
measured
elements appearing in the triangle in Table B1. These two angles, the
"theoretical" and the "experimental" are listed in Table 3B. The experimental
angle is on top and the theoretical angle is subtracted from it.
From this table we can see that the average angular error between the
theoretical and the experimental values is only 16 degrees, a remarkable
result.
Now, we notice that the agreement is very close for D1,D2 and D3, less than
4 degrees. For the correlations with Tough Poise we observe larger errors
and it is soon found that the cause of this can be traced to the gender
unreliability of this factor. Simple inspection of K&J's table shows that
the average difference between the M-F correlations for TP is three times as
great as it is for any of the other six factors and this error range is
mathematically sufficient to account for the angular errors of TP in Table
B3. Moreover, while all 6 of the other factors are well known to
psychometric theory, Tough Poise is unique to Cattellian theory. Because of
the peculiarly low P-component of this factor and the significantly larger M-
F correlation coefficient differences, we suspect that Tough Poise is
probably a facet of some more fundamental vector running through this
quadrant (probably connected with Impulsivity); a matter for future research.
Finally, we consider the statistical significance of this result. At first
glance an agreement of 16 degrees seems an awesome result but there are
reasons that it is not as awesome as it seems. When considering the overall
statistical significance of the model we have to consider first the
probability that 4 vectors will fall into 4 quadrants (1 chance in 20) and
secondly that the cross correlations will agree. It turns out however that
these two probabilities are not mutually exclusive in the sense of
statistical probability. Restricting the location of the vectors to 4
individual quadrants markedly increases the probability that the cross
correlations will agree. In unpartitioned space the vectors could correlate
anywhere from 0 to 180 degrees (+90). Restricting each to lie "within a
quadrant" and given the range of variation exhibited in K&J's numerical data,
we see that this number is reduced to something like +40 degrees.
Given that a similar representative variation in the experimental cross
correlation data (and when divided by sqrt(M1M2)avg and the cos-1 is taken)
we find a representative variation of about +30 degrees. Subtraction of
these two quantities yields a value with an error +70 degrees or simply 70
degrees absolute error. The average absolute error would then be about half
this or 35 degrees.
Thus by saying nothing more than 4 vectors must fall in 4 quadrants and
given the numerical variation existing in K&J's table, we are guaranteed not
to find an average error exceeding 35 degrees in the cross correlations. The
fact that we find an error less than half this (16 degrees), nevertheless of
course, is still statistically significant, if not awesomely so.
Finally of course we have to take into account the fact that we have very
close agreement for 3 of the vectors, almost all of the error coming from the
4th vector and that there is an immediately obvious reason for this in the
large M-F error range in the data for this vector, and this vector only
(note: we have averaged M-F data to construct the model). Obviously if this
error range were not abnormally high the agreement would be significantly
better. All told then, it would be very difficult to argue that the 16
degree agreement is not a statistically significant result, although just how
significant it is, is debatable.
It is concluded here then, that the agreement of the cross correlation
coefficients as actually achieved, provides us with a classically compelling
piece of experimental evidence supporting the correctness of the underlying
theory.
GA -SOC TP
exper 73.89 59.20 102.11
theor -73.52 -61.94 -62.89 -CON
error ------ ------ ------
0.37 -2.74 39.23
45.92 141.40
-42.52 -111.05 GA
------ ------
3.40 30.35
89.56
-69.81 -SOC
------
19.75
Table B3 Angular error between the theoretical
and experimental cross correlations.
The average error is 15.97 degrees.
APPENDIX C Probability calculation- Cattell General Solution
Limitations of space permit only a broad outline of this calculation
(calculation of statistical significance). A complete exposition of this
analysis is available from the author (request document B5#-1).
A circumplex calculation of Krug & Johns' intercorrelation table has
produced Heymans' cube, as predicted by the Cartesian Theory. The question
then arises: What are the odds against the result simply being an accident?
In this appendix the odds are first calculated theoretically and then
verified experimentally by direct computer simulation. Both confirm that the
odds against chance are longer than 20 to 1.
The theoretical calculation begins with a survey of the result, Fig. 2,
Sec.
I. We presumed Cattell's Anxiety and Exvia are Eysenck's E and N. Trial and
error have selected IND as P (its the only factor that will work). We assume
the intercorrelations for these 3 are simply error angles (since Eysenck has
proven they are orthogonal). We then treat the correlations of D1,D2,D3,D4
(see App. B, Table 1) as vector components of the diagonals. This circumplex
calculation, as we have demonstrated in App. B, results in Fig. 2. This
mechanical model may be described as follows:
1. 4 vectors fall, one each, into the 4 quadrantal
positions of E,N,P space (D1,D2,D3,D4).
2. There are only 3 possible candidates for P, namely
TP, IND or -CON (the 6th and 7th factors are too
small to be acceptable, see Eysenck 1991, p. 778).
It turns out that IND=P yields a solution.
3. TP, IND and -CON all correlate positively with P,
which they must on independent psychometric grounds
(see Eysenck 1991 p.778).
4. Of these 3 factors, IND is 2nd in size,
variancewise.
These 4 criteria embody first, the description of the fundamental
probability event itself, item 1, that the 4 factors after E,N,P will fall in
the 4 "diagonal locations" of E,N,P space. Items 2,3,4 embody (the only)
obvious psychometric criteria we must meet vis a vis the "commonly known"
psychometric character of these particular dimensions (a review of Krug &
Johns result is given by Eysenck, 1991 p.778).
The question then is simply this, what are the odds that a random
correlation table might produce a structure which meets the above criteria?
To calculate an answer to this mathematically a probability pn is computed for
each of the criteria; the total probability for chance then becomes
pT=p1p2p3p4.
Now, the chance that 4 snowflakes will fall in 4 coffeecups, one in each
cup, is simply p1=(4/4)(3/4)(2/4)(1/4)=6/64. Since we have 3 choices for P,
p2=3.
Item 3 says that after P is selected the other 2 eligible factors must
correlate positively with it. Since the chance of this is 50-50 for each,
p4=(1/2)(1/2)=1/4. Item 4 is the probability that P would be the 2nd in
variance of 3 possible choices, hence p4=2/3. The total probability for
chance is then, pT=(6/64)(3)(1/4)(2/3)=3/64=.046875 or approximately 20 to 1
odds against chance.
Now, the above items can be used to experimentally confirm this probability
calculation on a computer. By using the RND number generator in a desktop
computer you can generate a random 7x7 matrix and then apply the 4 "criteria"
operations on it. I have done this and the program only runs to 25 lines
written in BASIC. I have run this program 500,000 cycles (500,000 random
tables) and the result converged to pT=.0471177, again 20 to one odds against
chance (this program is included in doc. B5-#1).
Now, this summary does not mention the computation of a 2,064 path "event
tree" to confirm that items 1,2,3,4 are mutually exclusive probability events
(incl. in doc. B5-#1) and elaborate computer confirmation of all aspects of
the theoretical calculation using the experimental computer program.
Finally, beyond this probability is the additional probability that the
cross correlations should agree (see App. B). The statistical significance
of this agreement has not been calculated, but whatever the odds are, they
would then be multiplied by existing odds of 20:1. Obviously the final odds
against chance are even longer than 20 to 1.
Given the remarkable symmetry of the result, Fig. 2, and the fact there is
only 1 chance in 20 (or less) that the result is an accident, we are faced
with no choice but to conclude that Cattell's 7-2nd order factors do,
actually, form Heymans' cube in 3D space.
APPENDIX D Nature-Nurture
Figure D1
Simply taking the cosines of the actual geometric angles in an ordinary
cube results in a "theoretical" correlation table for the 7 axes of Heymans' cube,
Fig. D1.
Now we have proven that K&J's correlation table is geometrically similar to
this table, it produces the same geometric figure, even though all the
numbers are uniformly reduced in size. What is causing this reduction? The
answer to that question, of course, is the reason the existence of Heymans'
cube in psychometric data has apparently
eluded detection for so long. As we will now see, a simple study of this
question leads to yet another profound neuropsychological proposition in
Personality structure.
We have several well established psychometric facts to reconcile: First,
the hundreds of published eigenvalue scree plots in the literature universally
show that there are at least 7 factors before the "elbow" in the curve is
reached, i.e. there are at least 7 factors with significant variance.
Second, these factors are always nearly orthogonal, even in oblique
solutions (ravg=.15). Finally the new fact- we have demonstrated that these
7 factors form a geometric structure in 3D space (Fig. 2) where 4 of the
factors are "diagonal" with respect to E,N,P.
Our first intuition would be that "diagonals" should correlate
cos(57.5)=.577 with E,N,P but as the above facts confirm this is just not
found to be the case in empirical research. How can the "diagonals" be
virtually uncorrelated with E,N,P if they are lying at an angle of 57.5
degrees from these axes. That is the mystery that now needs to be explained.
This mystery then, leads us to a new scientific thesis concerning the
neuropsychological operation of the Physics Brain: The neuropsychological
system generating the diagonals, must be "quasi-independent" of the
neuropsychological system producing E,N,P! This result then leads to the
following deduction concerning the neuropsychological operation of the
Physics Brain. First there is an octupole (8-lobed) structure, superimposed
on which are 3 orthogonal neuropsychological gradients- the Bell-Magendie,
the Sperrian, and the MaCleanian. These 3 gradients (or dichotomys) produce
E,N,P. Underlying this however is the original (primary) octupole field
itself caused by the 8-lobed structure. This is known from physics, where it
is a very familiar structure, to be predominantly a diagonally oriented field
structure. This structure, we must conclude, varies independently of E,N,P,
controlled only by the (biological) variance existing among the 8-lobes. The
net result is 7 quasi-independent (hence nearly orthogonal) factors whose
"residual" oblique correlations still actually reproduce Heymans' cube in 3
dimensions.
Since the 7 dimensions of the General Solution are nearly independent, the
6,5,4,3,2 factor solutions which are merely geometric approximations of this,
will also be nearly orthogonal. This phenomena explains the "near
orthogonality" reported by all researchers whether they are reporting a
7,6,5,4,3,2 factor model. We have shown in Fig. 3 is that these models
simply represent cruder and cruder geometrical approximations of Heymans'
cube- all in 3D space. If you decide to extract a 7F solution, what you are
doing is tapping Fig. 3a in 3D space where the biological (and environmental)
variance of the population randomizes the intercorrelations to near
orthogonality. If you decide on a 5F solution you are tapping Fig. 3c in 3D
space, where again the biological variance of the population sample
randomizes the structural intercorrelations to near orthogonality. Bear in
mind that the residual intercorrelations (in an oblique solution) will
always, circumplexly, reproduce the underlying geometry of the model in 3D
space. This is the empirical proof of the Cartesian theory of the brain.
This has been demonstrated dramatically in the case of the 7F General
Solution (Fig. 2, Sec. I; App. B). It has also been confirmed in the 5F
(Big-5) case by Peabody & Goldberg's Lexical Double Cone (Fig 3c, Sec. I;
Fig. 7, Sec. II). To be noted here is that Eysenck's 3 factor result (E,N,P)
is the simplest known confirmation of the Cartesian Theory, since the Physics
Brain shows these 3 factors to have zero structural intercorrelation.
Eysenck's well established finding that these 3 factors are orthogonal to a
high degree of precision, further confirms the theory. These three major
empirical results then, are the core of our claim to the empirical
"unification of Cattell, Eysenck and the Big-5".
APPENDIX E Lexical Double Cone analysis
In this appendix it is demonstrated that Peabody & Goldberg's Big-5 Lexical
Double Cone confirms identically Fig. 3c, the theoretical Big-5 as predicted
by the Cartesian Theory. One should refer to P & G's original paper (1989,
Fig. 1, p. 558). A partial reproduction of their Fig. 1 is shown in Fig. E1.
Figure E1
The thrust of the Double Cone approach is to come to grips with the problem
of "evaluation" (see footnote 4) in lexical research. The confounding of
evaluation with description has been a longstanding problem.
The magnitude of this problem is perhaps nowhere so vividly demonstrated as
in the AB5C model where 8 out of the 10 circumplexes show a "SW vs. NE"
dispersion of the data (Hofstee et al. 1992 p. 155). This is because all
Big-5 factors are bipolar with respect to evaluation, so that the data is
dispersed along an axis leading from the middle of the +I,+II,+III,+IV,+V
quadrant to the middle of the -I,-II,-III,-IV,-V in Big-5 space. I have
dubbed this axis "The Axis of the Universe". Peabody, meanwhile, has
discovered this axis in his 3D solution and called it the "General
Evaluation" axis. In the Double Cone this axis is the cone axis.
Fig. E1 Double Cone
from P&G 1989.
Now, it turns out, that the key to showing that the Double Cone is
identical to Fig. 3c (the "canonical" Big-5), lies in the fact that this General
Evaluation axis is actually none other than Hans Eysenck's celebrated P-axis
[Point 1 below]. General Evaluation then, is a lexical facet of P, and
Eysenck's Psychoticism it turns out, is "The Axis of the Universe".
Once we find out that the cone axis is P; one of Eysenck's Giant-3 (E,N,P),
and since the Double Cone is in 3D space, we immediately suspect that the
circumplex planes must be Eysenck's E-N plane. It turns out that T-L and A-U
the two axes in this plane are not E and N however, but the familiar R and L
diagonals (cf. Fig. 5b, Sec. II); shifted 45 degrees from E and N.
Finally, P & G's data allow us to locate E and N, precisely at 45 degrees
between T-L and A-U [point 2 below], and furthermore confirms T-L and A-U to
be factors III and V (conscientiousness and openness),as predicted [point 3
below]. Hence, what we see, is that the Double Cone, when completely
unravelled, Fig. 7, Sec. II, is identical to Fig. 3c, Sec. II, the
theoretical (canonical) Big-5 as derived from the Cartesian Theory. Not only
has Cattell's research confirmed the Cartesian Theory, Lexical Big-5 research
has now confirmed it also.
Now, as a byproduct of this second major empirical confirmation of the
Cartesian Theory; from the field of Big-5 research, we have also uncovered a
psychometric discovery of fundamental import to Personality theory. We have
discovered that Eysenck's historical N-axis, is actually bipolar and not
unipolar, as he has always believed [point 4 below]. It is predicted that
this discovery is the key to effecting a rapid convergence of psychometric
results in the field of Big-5 research.
In the remainder of this appendix then, the technical details of points
[1,2 and 3] will be presented, as well as a discussion of the discovery of the
"other missing half of Eysenck's N-axis" in point [4].
[Point 1] That Peabody's General Evaluation (GE) dimension; the cone
axis, is
identical to Eysenck's P is evidenced by several facts. First, GE is located
very close to factor II (Hofstee 1994 p. 30) and factor II (reversed) is
known to correlate highly with Eysenck's P, it has even been suggested that
II is simply the obverse of P (Eysenck 1992, p. 672). This alone confirms
that GE is close to P. Secondly, Eysenck's P is de facto one of the most
socially undesirable factor poles in Personality; explaining why it should
lie at the negative pole of the GE axis. There are more adjectives in
language describing II than any other factor, most of them highly negative,
and the same can be said of Eysenck's P. All of this adds up to an
overwhelming case for identifying "General Evaluation" with Eysenck's P-axis.
[Point 2] Referring to Fig.1 of their 1989 paper, P&G state (p.564):
"Extraversion could correspond to a descriptive dimension unconfounded with
evaluation, running between the upper left (Impulse Expression) and the lower
right (Impulse Control) in either or both of the circles in Figure 1.". I
simply agree, and thus locate the E-axis, as they indicate, in Fig 7.
Having now located P and E and knowing the cone is in 3D space, it is
immediately obvious where N must lie; orthogonal to E. From their Fig. 1 we
see that N falls on the projection of II in the circumplex planes. What this
means is that II has bifurcated into P and N. The larger component of II is
on the cone axis as explained in point 1, and the modest transverse component
has become N. This is further supported by the statement found on p. 558
that "When the Emotional Stability factor becomes relatively large, it picks
up variance from scales that are usually related to Factor II
(Agreeableness).". Thus N is located along the projection of II, in the
circumplex planes, in Fig. 7. The polarity of the N-axis will be discussed
in point 4.
[Point 3] From their Fig. 1 it can be seen that III is located in the
T-L
plane, and here it is argued that like E, when unconfounded with evaluation
it will lie in the E-N plane; precisely along the T-L axis. That V lies
along the A-U axis can be deduced from the factor loadings in the 9th column
of their Table 2, p. 557. Note the first six numbers in that column:
12,13,20,07,15,13. Compared to all the other sets of numbers in the (V)
column, these are clearly the largest (consistent) loadings. Thus factor V
is located at the positive pole of the A-U axis as indicated. Again, when
unconfounded with evaluation it will lie flat in the plane. Here we see that
Peabody has quite possibly solved the "scientific embarrassment" of Factor V
by indicating that its true name is Assertiveness.
[Point 4] Although the location of N in the Double Cone is easy to
establish,
determining the polarity turns up something of a dilemma. Returning to P&G's
data again, in table 4, p 559, we have loadings for factor IV (Eysenck's N
reversed). In the III column we see the core adjectives of IV, "Calm-
Excitable" and "Stable-Unstable" loading .40 and .52 on +III. Now Calm and
Stable are at the positive pole of IV, and +III is located on the right hand
side of their Fig. 1. So IV must run from left to right in their Fig. 1, and
likewise in Fig. 7. Since IV is the obverse of N, then N must run from right
to left, i.e. +N is on the left side of Fig. 7. Therefore N is labeled
accordingly in Fig. 7.
Now point 4 leads us to a very interesting discovery; that Eysenck's
N-axis is actually bipolar and not unipolar. Eysenck's N-axis, as classically
conceived is supposed to run from "normal to Neurotic", with neurotic being
the positive pole. But if you look at P&G's lower circle (Fig 1, p. 558),
instead of anything approaching normality at the -N pole, what you find is
behavior characterized as: selfish, unkind, harsh, irritable, distrustful,
inflexible and stingy, while at the +N (neurotic) pole we find a large blank
bounded by Lazy and Unambitious. However, once again from Table 4 (p. 559),
we can fill in this blank space with Discontented, Excitable and Unstable,
which lie on the left hand side of the circle as pointed out previously, and
as we see in the table have significant -II loadings (+P or "undesirable
pole"; lower circle, their Fig. 1). Thus the +N pole is characterized by:
Lazy, Discontented, Excitable, Unstable, Unambitious.
Thus while +N definitely appears to be neurotic, -N appears to be anything
but normal. What we see is that N must be bipolar like all of the other
dimensions in the Structural Model (like Introversion-Extroversion for
instance). There is emotional pathology at both ends of the N-axis, and
"normality" is somewhere in the middle.
(See Fig. E2, appended at end of paper)
Instead of "calm-excitable" being the lexical core of Eysenck's N (FFM
-IV) as has always been assumed, we seee that "irritable-excitable" is the lexical
core of N, with "calm" somewhere in the middle of the axis. We have, in
fact, discovered the other missing half of Eysenck's N-axis; the missing
negative pole of N.
Finally we see that the bipolar adjectives for N derived from the Double
Cone are virtually identical to the bipolar adjectives drawn from Francis'
Neuros-Neuroa (see Francis' questionnaire items for the two tests 1993,
p.59), Fig. E2.
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