Factor analysis : healing an ailing model

Ertel Suitbert

2013

Universitätsverlag Göttingen

Exploratory factor analysis (EFA) is a statistical tool for digging out hidden factors which give rise to the diversity of manifest objectives in psychology, medicine and other sciences. EFA had its heyday as psychologist Leon Thurstone (1935 and 1948) based EFA on what he called the “principle of simple structure” (SS). This principle, however, was erroneous from the beginning what remained unrecognized despite subsequent inventions of more sophisticated statistical tools such as confirmatory analysis and structural equation modeling. These methods are highly recommended today as tolerable routes to model complexities of observation. But they did not remove the harmful errors that SS had left behind. Five chapters in this book demonstrate and explain the trouble. In chapter 2 the ailment of SS is healed by introducing an unconventional factor rotation, called Varimin. Varimin gives variables of an analysis an optimal opportunity to manifest functional interrelations underlying correlational observations. Ten applications of Varimin (in chapter 2) show that its results are superior to results obtained by the conventional Varimax procedure. Further applications are presented for sports achievements (chapter 3), intelligence (chapter 4), and personality (chapter 5). If Varimin keeps on standing the tests new theoretical building blocks will arise together with conceptual networks promoting a better understanding of the domains under study. Readers may check this prognosis by themselves using the statistical tool (Varimin) which is provided by open access in the internet.

Psychology

English

9783863951337

E

xploratory factor analysis (EFA) is a statistical tool for digging out hidden factors which
give rise to the diversity of manifest objectives in psychology, medicine and other sciences. EFA had its heyday as psychologist Leon Thurstone (1935 and 1948) based EFA on
what he called the “principle of simple structure” (SS). This principle, however, was erroneous from the beginning what remained unrecognized despite subsequent inventions of
more sophisticated statistical tools such as confirmatory analysis and structural equation
modeling. These methods are highly recommended today as tolerable routes to model
complexities of observation. But they did not remove the harmful errors that SS had left
behind. Five chapters in this book demonstrate and explain the trouble. In chapter 2 the
ailment of SS is healed by introducing an unconventional factor rotation, called Varimin.
Varimin gives variables of an analysis an optimal opportunity to manifest functional interrelations underlying correlational observations. Ten applications of Varimin (in chpter 2)
show that its results are superior to results obtained by the conventional Varimax procedure. Further applications are presented for sports achievements (chapter 3), intelligence
(chapter 4), and personality (chapter 5). If Varimin keeps on standing the tests new theoretical building blocks will arise together with conceptual networks promoting a better
understanding of the domains under study. Readers may check this prognosis by themselves using the statistical tool (Varimin) which is provided by open access in the internet.

Suitbert Ertel

Factor Analysis

Suitbert Ertel

Factor Analysis

Healing an Ailing Model

ISBN: 978-3-86395-133-7

Universitätsverlag Göttingen

Universitätsverlag Göttingen

Suitbert Ertel
Factor Analysis
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Published in 2013 by Universitätsverlag Göttingen

Suitbert Ertel

Factor Analysis
Healing an Ailing Model

Universitätsverlag Göttingen
2013

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Suitbert Ertel
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ISBN: 978-3-86395-133-7

Dedicated to Elisabeth
With Gratitude

Contents

Abbreviations ........................................................................................................... 7
Foreword ................................................................................................................... 9
Preface .....................................................................................................................11
Chapter 1 Critique of the simple structure doctrine..................................13
Introduction ............................................................................................................13
01. The present state of factor analytical research ............................................15
02. The doctrine of simple structure (SS) ...........................................................19
03. The fallacy’s consequences.............................................................................21
04. Detailed error analysis.....................................................................................23
05. Reorientation ....................................................................................................27
06. Where did we go wrong? ................................................................................32
07. Unheeded critical voices .................................................................................34
08. Can non-factorial procedures take us forward? ..........................................36

4

Contents

Discussion of chapter 1 and outlook ................................................................. 41
Chapter 2 Finding complex structures ......................................................... 43
Introduction ........................................................................................................... 43
Questions and Test Runs ..................................................................................... 45
Test run 1: Evaluating phoneme similarities ..................................................... 49
Test run 2: Similarity judgments of British coins ............................................. 53
Test run 3: Differentiation of response styles at responding to
questionnaires. ....................................................................................................... 59
Test run 4: Semantic features of kinship terms ................................................ 62
Test run 5: Intellectual development in childhood .......................................... 66
Test run 6: Body size and body shape in cattle ................................................. 69
Test run 7: Intelligence tests and performance tests ........................................ 71
Test run 8: Psychophysiological activity indicators (Data: Köhler &). ........ 73
Test run 9: Knowledge test with varying test methods. .................................. 75
Test run 10: Self-assessment and external assessment of children ................ 77
Discussion of chapter 2 ........................................................................................ 79
Chapter 3 Decathlon data under analysis .................................................... 85
Introduction ........................................................................................................... 85
Description of data ............................................................................................... 86
Results: Factor analyses ........................................................................................ 88
1. Interpreting Varimin factors............................................................................ 90
2. Attempt at an interpretation of Varimax factors .......................................... 92
3. Attempt at an interpretation of initial factors ............................................... 94
4. Expert rankings for validating Varimin factors ............................................ 95
Discussion of chapter 3 ........................................................................................ 97
Chapter 4 Intelligence data under analysis ............................................... 101
Study I: Varimin analysis of IST factors .......................................................... 102
Data analysis......................................................................................................... 104
Results with comments....................................................................................... 105

Contents

5

Summary of Study I ............................................................................................ 108
Study II: Validations of Varimin-rotated IST factors .................................... 109
Objective .............................................................................................................. 109
Validation 1: School performance .................................................................... 109
Validation 2: Culture-free IQ Test CFT, spelling and numeracy test ........ 110
Validation 3: Culture-free IQ test FRT ........................................................... 115
Summary of Study II .......................................................................................... 116
Discussion of chapter 4 ..................................................................................... 117
Chapter 5 Varimin factors from Big Five personality data .................. 119
Point of departure ............................................................................................... 119
Material and analysis ........................................................................................... 120
Towards an interpretation of Varimin factors of personality ...................... 123
More preliminaries .............................................................................................. 123
Varimin factor interpretations .......................................................................... 126
Concluding remarks............................................................................................ 129
The interpretations of Varimin factors on probation ................................... 130
Re-interpreting Varimax factors by profiles of Varimin factors .................. 132
Discussion of chapter 5 ..................................................................................... 133
Insight and Outlook ........................................................................................ 135
Review of chapters 1 to 5................................................................................ 137
Appendix ............................................................................................................ 141
References .......................................................................................................... 145

Abbreviations
CFA

Confirmatory factor analysis

CFT

Culture-free intelligence test

CS

Complex structure

CSM

Complex structure modelling

DV

Dependent variable

EFA

Exploratory factor analysis

ERS

Empirical relations system

ESEM Exploratory structural equation modelling
FA

Factor analysis

FRS

Formal relations system

FRT

Figure Reasoning Test

g

General factor

IST

Intelligenz-Struktur-Test

IV

Independent variable

MDS

Multidimensional scaling

MTMM Multitrait-multimethod (analysis)
NMDS Nonmetric multidimensional scaling
PCA

Principal component analysis

SD

Social desirability

SEM

Structural equation modelling

SS

Simple structure

SSM

Simple structure modelling

Foreword
Paul Barrett
This book is about factor analysis. It explains what happens when you model covariances among sets of variables with a method sensitive to underling complex
relations. An idealized model that forces a structural simplicity onto such relations,
is considered as leading us astray. Thus stripping away all the technicalities, it
comes down to a simple question: Do you force a model onto data irrespective of
reality, or do you let the data speak for itself?
Make no mistake: This question contrasts a more realistic perception of human behaviours, cognitions and attributes against a hypothetical statistical ideal.
The statistical technique of factor extraction is considered healthy. The ailment
arises when the investigator chooses to construct a model of factor loadings corresponding to Thurstone’s Simple Structure.
The defining principle of Simple Structure is that variables should load highly
on one single factor and near-zero (or zero) on all other factors. This kind of solution produces clusters of homogenous variables and interpretations of factorial
meanings constructed from the content of variables within clusters. Variables that
possess one or more sizeable loadings in a factor analysis are selected/rejected
from solutions on the basis of their complexity. They are usually rejected completely from a solution if they possess “cross-loadings” or at least two highloading values across two factors. The attraction of such solutions is that they
appear to be easily interpretable. However, the drawback is that they may bear no
relation to the reality of the underlying explanatory processes. Removing the
complexity inherent in many psychological attribute interrelations is perhaps the
very opposite of what the social scientist must now begin to consider.
This is the fundamental thesis of Suitbert Ertel’s propositions: Do not try to
force simplicity on what is (or is observed to be) complex. Instead, model the
complexity itself (if present) and work with structured variables that account for
that complexity. To achieve this goal, the methodology created and set out in this
book is called Varimin. If there is no complexity in the covariance patterns among
variables, a simple solution will be found. But if complexity among variable relations is present, Varimin will produce factors that account for the inherent complexity.
As Suitbert shows in several chapters devoted to analysing several kinds of
variables (including those within the fields of cognitive ability and personality), the
consequences of using Varimin are theoretically profound. Varimin factors are no
longer conceived as ultimate dimensions, but as components of multifactorial
variables. They thus seem to align more with what we know and observe within

Foreword

10

other areas of psychological investigation, far more so than what is generally expected from seemingly homogeneous factors of Simple Structure.
Clearly, if you work within Simple Structure-constrained analyses (whether using exploratory or confirmatory factor analysis), you are likely to be highly sceptical of Varimin’s logic and approach. However, it is interesting to ask yourself the
reasons for that scepticism: Are these based more on scientific considerations and
observations of the phenomenon of interest, or just habit and ‘status-quo’ recommendations?
Thurstone created Simple Structure at a time when complexity within data relations was seen as “a problem in need of a solution”, partly because there were
no methods or technologies to deal computationally with complexity. Just look
how that view has changed in recent years, as methods across many sciences now
routinely deal with complexity as a feature of multivariable interrelations. The
entire field of complex systems theory is founded on systems approaches to understanding phenomena . as in systems biology, business dynamics and the newer
network models for epidemiological and psychological concepts, such as mental
disease comorbidity and personality.
While a single book cannot, by itself, change an entire field of thinking and
endeavour, it can pose and answer the big questions. Within these pages, there is
enough content matter and avenues of investigation to kick-start several Masters
and PhD theses, exploring the consequences of Varimin in areas where investigators are more familiar with representing data structures. One cannot help being
curious about Varimin because of its impact upon how we might theorise about
the nature of factors in the future. Even if you are sceptical this is a fascinating
volte face proposition and methodology in its own right, which just might, over
time, become the new established method for factor-analytic investigations.
Thurstone’s Simple Structure was the 20th-century response of a pioneering
psychologist to the challenge of reducing complexity within the factor analysis of
questionnaire items and other kinds of variables. Varimin and Complex Structure
is the 21st-century response from another pioneering psychologist, namely Suitbert
Ertel, to the challenge of the complexities inherent in the functioning of human
cognitive and other multivariable systems.

Preface
Why is factor analysis considered an ailing model in this book? I deem Simple
Structure, a basic principle for factor rotation, introduced by Thurstone, as mistaken. In his foreword, Paul Barrett provides grounds for my view. I should add
that two detrimental conditions have prolonged the methodical ailment: Not being able to diagnose the actual reasons for the sickly symptoms under which this
model has suffered and the widespread belief that Simple Structure is an indispensable ideal. Thurstone’s tenet states that factors are uninterpretable without rotating them. By rotation towards simplicity − he holds – individual variables should
obtain as few factorial loadings as possible. Users of his procedure have never
questioned this. Varimax rotation is the widely preferred technique, but the outcomes are misleading, which as Barrett has intimated in his preface, is the fundamental thesis of this book.
Readers who are trained by Thurstone’s verbally impressive principle may be
irritated that I dare reject it in the first place and may demand an alternative. This
will be provided after replacing the ideal of simplicity – which cannot be achieved
by Thurstone’s rotation anyway – with complexity, the aim of rotating extracted
factors by letting individual variables display as many factorial components as are
suggested by empirical data. The alternative is called Varimin.
By doing the opposite of what Thurstone considered necessary for grasping
factorial meanings, it may appear that we arrive at a bewildering quandary. In
chapters #1 and #2 an alternative method called “minimal pair comparison” is
introduced, a procedure imported from linguistics. It will be shown that the meaning of Varimin factors can safely be discerned. Two variables whose loadings are
equal (or nearly equal) for N-1 factors, are successively paired while the loadings
on only one factor are extremely different or, ideally, opposite in sign. A difference in meaning between the two paired variables must then be considered as due
only to the factor whose loadings on the two variables are extremely different.
Varimin factors eventually turn out to be latent components. Factor rotation? Yes.
For improving interpretability? Yes. However, this is only achievable by Varimin
and is impeded by Varimax, as will be demonstrated with ten test runs #2, in
chapter #3 with sports data, #4 with intelligence data, and #5 with personality
data. The benefit of the new paradigm of factor analysis may also be discovered
with non-psychological multivariate data. I do hope that the reader will attempt to
replicate findings as illustrated in my book so that he/she may also discover that
the future of factor analysis has indeed a hopeful prognosis.

Preface

12

Acknowledgements
I was encouraged and inspired in my renegade methodical search by the advice
and constructive criticism and comments of Paul Barrett, André Beauducel,
Elisabeth Cott, Herbert Götzl, Gerd Lüer, Pierre Sachse, and Tatjana Schnell.
Uwe Engeland implemented my ideas of a program that models latent complexity.
Jürgen Guthke and his assistant, Barbara Seiwald, provided the first opportunity
to test this program in their research. They had failed after applying conventional
SSM methods, but using CSM they obtained the predicted solution. Two monographs in German (Ertel, 2011a,b) preceded this English translation which has
been recommended by inspired readers of my German publications. The translation was initiated and encouraged by Matthias Bellmann who supported this project with valuable editorial and time-consuming technical improvements. Jürgen
Hecker and Werner Rawe translated my German into English.

Chapter 1
Critique of the simple structure doctrine

There is no place for dogma in science. The scientist is free to ask any question, to seek
any evidence, to correct any error. ... Dogmatism has found itself incompatible with the
progress of science. J. Robert Oppenheimer. (1904–1967)

Introduction
In chapter 1 I shall examine what I deem a serious error by factor analysts adhering to Thurstone’s lead. Ever since its creation in 1935/47, his simple structure
(SS) principle has been as problematic as it is attractive. How did methodologists
deal with this ambivalence? Were there no critics who recognised the fallacy that
SS had introduced? Yes, there were a few, but they were largely ignored. Why was
that?
Researchers eventually changed track, i.e., abandoned exploratory research that
kept on producing questionable results. They switched to confirmatory methods,
above all to structural equation modelling. Was that progress? Hardly, because
even structural equation users stuck to simple structure modelling, SSM, which I
believe to be the main source of errors. Why did they cling to SSM? Because no-

14

Chapter 1 – Critique of the simple structure doctrine

body dared touch SSM which had become a doctrine of statistical reasoning that
nobody saw through. How could this happen?
Readers who have never doubted the validity of the SS principle may dislike
my attempt to prove them wrong. I am aware that theoretical considerations alone
in this first chapter will hardly change deep-rooted convictions, not least because
creating truly simple structures seems to be an undeniable goal of all science, not
only of factor analysis.
I hope, nevertheless, that my criticism will make sense to you as you read chapters
2 to 5. I want to show what can be achieved by factorial analysis when it is freed
from the constraints of SSM and when complexity is revealed by appropriate statistical decisions.
Chapters 1, 2 and 4 were first published in 2009 and 2010 in the little known
Psychologie des Alltagshandelns and were reworked for a German monograph Ertel
(2011a). The content of chapter 3 was published in Personality and Individual Differences (Ertel, 2011c) and revised for this book. Chapter 5 (on personality) is based
on another German monograph (Ertel, 2011b), condensing and adapting the message of that more extended work.
Chapter 1 gives a theoretical overview. Obvious weaknesses of previous methodical reasoning are discussed. They could have been avoided if analytical procedures had been applied more prudently and if common sense had also been
given a say in the matter. Common sense tells us that complexity of conditions of
manifest behaviour is self-evident. I do not only voice my own critique but also
quote the supporting opinions of many others which unfortunately have been and
are being studiously ignored by the majority of experts in this field..
In chapter 2, complex structure modelling, CSM, is presented as an alternative to
conventional simple structure modelling, SSM. Varimax rotation is replaced with
Varimin rotation. The use of Varimin is explained by giving ten empirical examples
using data from published factor analyses.
In chapter 3, data from decathlon (ten physical events performed by Olympic
athletes) are subjected to Varimin analysis. I considered that an interpretation of
factors of physical sports events was easier compared with the interpretation of
factors of mental performance which are more commonly analysed, but more
liable to controversy.
In chapter 4, intelligence data obtained from a study using the well-known
German test of general intelligence are analysed by Varimin and for comparison,
also by Varimax. Does the commonly accepted SS-based distinction between fluid
and crystallised intelligence remain valid if identical data are subjected to Varimin
rotation?
Chapter 5 conveys the gist of my book by a CS-based analysis of personality
data (Ertel, 2011b), where the Big Five personality data are selected as the main
focus. An interpretation of factors obtained from such data is more demanding
and cannot avoid first attempts at theoretical reasoning. Pertinent discussions

Chapter 1 – Critique of the simple structure doctrine

15

expounded in the German monograph (Ertel, 2011b) might be helpful and will be
more comprehensively translated in a future publication.
The five chapters of this book may be read independently. There are some
overlaps among chapters where basic issues are viewed from complementary perspectives. At the end of the book is an abstract for each chapter. Readers may use,
by clicking URL http://www.varimin.com, Dr Uwe Engeland’s statistical program
“factor analysis” online which allows application of principal component analysis
with Varimin and Varimax rotations. A user manual is available on this website.

01. The present state of factor analytical research
Detrimental characterisations and metaphors employed by disappointed authors
are symptomatic of the chronic anomalies encountered in factor analytical research: “uneasiness in factor analysis” (Kallina, 1967); “alarming lack of commitment”;
“subjectivity in factor analysis” (Horn, 1967); “ambivalence of factorial research” (Meili,
1968); “destruction of generality” (Davies, 1971); “product of chance and imaginary evidence”
(Greif, 1972); “nonsensical effort” (Revenstorf, 1978); “ambiguity of factorial rotation”
(Buse & Pawlik, 1978); “dubious legacy” (Schönemann, 1981); “faktoranalytis” (Jäger
& Hörmann, 1981); “myth of factor analysis” (Lenk, 1983); “factors are fictions” (Revelle,
1983); “morass of factor analysis” (Eysenck, 1992); “psychopathology of factor indeterminacy”
(Schönemann, 1996)1; “pathology of psychometrics” (Borsboom, 2003).
The shortcomings of FA, however, are trivialised by most users and quite
happily buried under optimism; they thumb their nose at critics and maintain that
there exist, after all, “significant results”. This is opined, for example, by Pawlik
(1977) in a comprehensive German overview of the first decades of FA research.
But, say critics prepared to face the dilemma, FA research has miscarried:
“Exploratory factor analysis has never been developed to anything approaching its full promise
and potential, despite the eighty-year history of its efforts …” (Yates, 1987, p. 325). In an
overview of “fifty years of test theory”, Blinkhorn (1997) concludes that neither
the “considerable technical strides” made during the past decades nor the “wellknown contributions of Jöreskog and McDonalds” basically changed the dilemma:
1

Schönemann (1981) and Steiger (Steiger & Schönemann, 1975), after Guttman (1955), belong to
the middle generation of methodologists critical of FA. Their criticism was harsh (“theoretical
problems”, “users are generally uninformed about the defects of this model”, pp. 175, 188), but
they did not focus on the simple structure principle. Instead they confined themselves to the “indeterminacy” of factorial dimensions and their “lack of identifiability”. Following the re-analysis
of 13 published FA studies that resulted in devastating criticism of these studies (Schönemann
& Wang, 1972), Schönemann and Steiger (1976) developed an alternative method for multivariate data reduction (Regression Component Decomposition, RCD). It promised greater conceptual clarity and computational efficiency plus the possibility of model falsification. But this approach remained unnoticed given the success of Thurstone’s “multiple factor analysis”. Admittedly, the
alternative approach did not offer new insights into the transformation of “components” determined by RCD. Moreover, the results of RCDs did not seem much different from those supplied by Thurstone’s factor analysis.

16

Chapter 1 – Critique of the simple structure doctrine

“How curious … that we are so little further forward in our understanding of the psychology of
individual differences as a result of these advances … Can anyone identify a single publication in
the last 50 years in which the use of factor analysis has led to counter-intuitive, or surprising, or
genuinely enlightening outcomes?” (Blinkhorn, 1997, p. 181). Already 50 years ago one
could and should have noted what Schönemann reported retrospectively (1994)
about Louis Guttman, who had delivered a “eulogy” for multiple factor analysis in
1955: “It was left to Louis Guttman to read the eulogy (p. 209, p. 406): The era of Multiple
Factor Analysis had come to an end – for knowledgeable people at any rate. … It was logical,
then, to ask: What lies ahead for Factor Analysis? (Guttman, 1958). He answered it with a
vision that challenged habits of thought that had led nowhere. This vision he kept pursuing for
the rest of his life.” Schönemann and Borg (1996, p. 249) took stock: “Today we know
that the explorative factor analysis era that Thurstone heralded brought very few lasting insights.”2
Two calamitous results of exploratory factor analysis (EFA) are particularly deplorable:
 EFA research engendered a myriad of constructs in psychology and
thus produced the opposite of what it set out to achieve
EFA was supposed to describe the multitudes of correlating manifest variables
parsimoniously and advantageously. This was thought to be achieved by extracting
from them a small number of factorial variables which were assigned the role of
latent dimensions3.
Decades of EFA research produced an inexhaustible number of latent dimensions supposedly underlying the observable variables. At the 11th European Conference on Personality (2002) in Jena, Lee Sechrest pointed out the glut of construct
variables in psychology, citing an author who had counted 7800. Many are new
creations of EFAs. Is Sechrest’s number unrealistic? An internet search of article
titles containing the word “scale” from the PubMed database provided 889 differ2

3

Moosbrugger and Hartig (2002) and Fabrigar et al. (1999) delivered survey papers on EFA
research methods with an implementation orientation – most of them without seminal criticism.
Their papers were preceded by articles with similar objectives: Stevenson (1993), Tinsley &
Tinsley (1987), Ford et al. (1986), Glass & Taylor (1966), Cattell (1965), Peel (1953). The text
books most often quoted on EFA are: Comrey & Lee (1992), Child (2006), Gorsuch (1983),
Harman (1976), Mulaik (1972), Weber (1978), Revenstorf (1976), Überla (1971). Lienert (1969)
has an introduction as an appendix in a textbook, and Bortz (2005, 1977) devotes one textbook
chapter (chapter15) to factor analysis (FA).
By applying the term “dimension”, a claim is staked for a metric that was never challenged. Just
as the three dimensions of Euclidean space serve to locate objects in space, it was thought that
the primary variables of psychological observation could be positioned with factorially acquired
“dimensions”. This is overtaxing of the limits of the space dimension metaphor (the same goes
for the phrase “semantic space” by C. E. Osgood). Thus the term “dimension”, while legitimate
in mathematics, is misleading and superfluous when merely a naming of sources of variance is
required.

Chapter 1 – Critique of the simple structure doctrine

17

ent scale denominations ranging from the Abel and Becker Cognition Scale to the Zung
Self Rating Anxiety Scale. Most scales were compiled or adapted by FA. Thus, if one
scale delivers on average, say, three or four factors, some 3000 factorial constructs
were generated in clinical psychology and medicine alone.
Moreover, many scales and derived constructs have been developed in nonclinical differential psychology. Every issue of the journal Personality and Individual
Differences offers new material, so that the guestimate made earlier seems realistic:
“The idle practice of producing new personality scales continues unabated, making it less likely
that they will ever arrive in the promised land of the paradigm which alone would endow our
efforts with scientific respectability” (Eysenck, 1992, p. 672).4
 SS-based constructs identified as dimensions lack
theoretical connections
Factorial constructs obtained from SSM-oriented analyses are unrelated and thus
isolated from one another; i.e., they form mere aggregates. Only if they enter into
relationships may constructs be conceived as components of some processual
whole. As early as 1956 Stephenson wrote: “… simple structure may have resulted in an
analysis into too many unrelated, and UNRELATABLE, primaries [primary factors]”
(Stephenson, 1956, p. 6; emphasis by S.E.). Andresen’s (1998) comprehensive
critical and historic overview of EFA personality research leaves behind a chaotic
impression.
The Big Five factor model, developed since the1990s in personality research,
was welcomed enthusiastically and soon achieved reputation. Did it remedy the
theoretical shortcomings? No, it merely showcased five middling invariant dimensions in the “chaotic plethora of personality constructs” (Funder, 2001, p. 200).
The invariance of constructs, however, does not signify validity since inferior constructs may be as invariant as high quality ones.
Some proponents of the Big Five model believe that their factors were analogous to chemical elements (this idea seems to have started with Goldberg (1981)).
Such optimism is out of place. The discovery of chemical elements in the nineteenth century introduced a scientific revolution5. Advocates of the Big Fiver
4

5

Ruttkowski (1974) tried to capture the totality of typological constructions in differential psychology, and not only those of FA origin. Sponsel (1998) commented as follows: “Worldwide,
there are more than 1,000 personality or character typologies (Ruttkowski, 1974). Most of them are probably
… contentious. Many have disappeared in cultural or scientific history. Many overlap. It seems as if a random
number of constructions are possible – depending on differing goals and purposes.” Gigerenzer and Strube
(1987, p. 85) arrived at a similar conclusion: “It is the crux of factor analytical research to have come up
with so many ‘accepted’ personality factors that even simple dichotomisation of dimensions leaves us with a number for the resulting available high-order quadrants of approximately 250 … which is around four hundred thousand times the population of Earth.”
Blinkhorn (1997, p. 180) criticises the excessive hopes held by the pioneers of FA: “The words
they use, for example ‘primary mental abilities’ (Thurstone) or ‘source traits’ (R. B. Cattell), are witness to the

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Chapter 1 – Critique of the simple structure doctrine

model claimed that complex differential psychological constructs, like molecules
made of chemical elements, could be put together using five element-like dimensions. The notion arose that future extractions of factors in the domain of personality would only be legitimate if they correlated with the Big Five6.
However, “this comparison [with chemical elements] did not hold water” (Lukesch &
Kleiter, 1974, p. 294). H, He, C, Ca, N, etc. have a functionally definable place
within the periodic table. Atoms form molecules because of bonding properties
caused by the number of protons in the nuclei, the density of electrons, etc. In
brief, chemical elements are related by their components and compositions. The
Big Five personality “dimensions”, however, do not exhibit components that
would allow an assessment of similarities and differences. Very few observers take
exception to this general belief (for example Briggs, 1989, and Block, 19957).
H. J. Eysenck was irritated by EFA research that lacked theoretical underpinnings and accused Big Five researchers of not transcending superficial taxonomic
goals. In so doing, said Eysenck, they remained at the psychometric surface instead of developing biologically interpretable models of relatedness (Eysenck,
1992, 1997).
Eysenck attempted to conjoin the three dimensions of his PEN model (psychoticism, extraversion, neuroticism). He postulated differential cortical areas
assigning them neuro-psychological roles that were supposed to have functional
relationship. As welcome as Eysenck’s aim may have been in principle, he did not
achieve it8. He did not recognise the true cause of the lamented “morass of factor
analysis” (1992, p. 672), it could not be found where he was looking for it.
Factor analytical data analysis has also been conducted in numerous nonpsychological disciplines (cf. Rummel, 1970, Reyment & Jöreskog, 1993, and Figure 1.01), and it is not uncommon that discomfort is also voiced there. Earth scientist Davies, for example, who methodically utilised SSM-orientated EFA, comments: “By Varimax] we may be butchering our results; cutting up the body of generality into a

6
7

8

faith and trust placed in factor analysis as revealing the psychological analogue of the periodic table of elements, or
the list of subatomic particles.”
Ozer and Reise (1994) “characterized the Big Five as the ‘latitude and longitude’ along which any new
personality construct should be routinely mapped” (Funder, 2001, p. 200).
Briggs (1989): “… a coherent and falsifiable explanation for the five factors has yet to be put forward. There
is no theoretical reason why it should be these five rather than some other five.” (p. 249). “The structure of trait
attributions may not correspond straightforwardly to the deep structure or neurophysiological basis of human
tendencies.” (p. 250). “Perhaps the critical step in elucidating these concepts [interpreting the five factors] … is
the specification of their exact nature: What are the elements or components of each factor? How are they interrelated?” (p. 253). Block (1995) quotes Briggs and criticises more specifically: “No functioning psychological ‘system’, with its rules and bounds, is designated or implied by the ‘Big Five’ formulation; it does not offer
a sense of what goes on within the structured, motivation-processing, system-maintaining individual.” (p. 188).
“How should the Big-5-or-6 be understood in psychological terms? Sadly, despite many years of research –
especially into extraversion – the picture is still very unclear (see e.g., A. Gale & M. W. Eysenck, 1992,
Handbook of Individual Differences: Biological Perspectives; G. Matthews, 1993, in A. Smith & D. Jones,
Factors Affecting Human Performance.). Here are some possibilities that still look viable, yet falsifiable.” .

Chapter 1 – Critique of the simple structure doctrine

19

set of unrelated fragments without ever realizing that these fragments can ever be considered as
part of a larger entity” (Davies, 1971: p. 113). Davies repeatedly characterises the
effect the Varimax rotation has on factorised data as “destructive”. This will be
examined more closely in the following section.

Figure 1.01: Papers on factor analysis, by discipline, identified by Kaplunowsky (2007).

02. The doctrine of simple structure (SS)
The above account of the situation of factor analytical research helps understand
where the calamity comes from. An “unease in factor analysis” is generally ascribed to an arbitrariness of procedural decision taking. Arbitrariness occurs when
variables for correlations are selected, when samples of individuals are formed,
when the number factors to be extracted are determined, when the choice between orthogonal or oblique rotation is made, and when one rotation procedure is
selected from among a large number of options (cf. Finch & West 1997, p. 464 et
sqq.). To me the effect of such arbitrariness on the results of FA appears negligible compared with what caused EFA’s most serious defect.9
9

Velicer (1977) found that extraction procedures of maximum likelihood, image analysis, and
principal components analysis had “extremely similar” (p.18) results when tested in nine sets of
test data. Using a representative data set of trait descriptions, Goldberg (1990) was able to deliver an almost invariant reproduction of the Big Five factor model, regardless of whether he varied the method of factor extraction (principal components, principal factors, alpha-factoring,

20

Chapter 1 – Critique of the simple structure doctrine

The idea of SS is widely known and is considered “intuitively compelling” (Kaiser,
1958, p. 188). In the coordinate system of initial factors, clouds of points represent variables (cf. Figure 1.02A). Statisticians draw regression lines through such
clouds so that squared distances between regression lines and variable points are
reduced to a minimum. To achieve this, the system’s coordinates are rotated in
such a way that they coincide with the regression vector (Figure 1.02B). Since two
coordinates are always rotated simultaneously, the convergence of both coordinates will be optimised simultaneously. Hard-core SSM methodologists are even
more radical than proponents of orthogonal rotation because they opt for disadvantageous oblique rotations just to come closer to SS10.

Figure 1.02: Two clusters of variables in an initial factor system F1, F2 (A) and following orthogonal
simple structure rotation (B).

SS was, incidentally, considered an inevitable continuation of the simplification
strategy that reduced the large diversity of variables to a smaller number of fac-

10

image-factoring, maximum-likelihood), or whether he applied orthogonal rotation (Varimax), or
oblique rotation (Oblimin). But without exception these methodical variations were conducted
with SS orientation. Alternatives to SS were not considered, although at least Goldberg seemed
to consider a renunciation of SS, according to an Eysenck quote (1981): “Correlational psychology
cannot in the nature of things come up with objective, universally agreed dimensions or categories; there are innumerable, mathematically equivalent ways of rotating factors, for instance, and no statistical magic key (not even
simple structure) can close the door on alternative solutions (p. 43).”
In an overview of factor rotation’s analytical methods, Warburton (1963) describes the logic of
mathematical parsimony in simple terms: “He [Ferguson] approached the problem of parsimony by considering a single variable, represented by a point, and asking himself what was its most parsimonious description.
He suggested that, intuitively, the most parsimonious description results when one of the axes passes through the
point. It is seen that when the reference frame is rotated so that one of the axes approaches the point, the product
of the two co-ordinates grows smaller …” (p. 169).

Chapter 1 – Critique of the simple structure doctrine

21

tors: “In the factor problem, striving for simplicity aimed at coming up with the smallest possible
number of factors … The rotation problem … seeks to design the correlation of variables and
factors … as simply as possible within the … predetermined … factor space” (Überla, 1971,
p. 176). The simplification principle of “less is better”, which factor extraction
rightfully employs, is carried over to subsequent factor rotations, where it is, however, no longer valid or legitimate. It is tantamount to transferring a principle that
helped solve one problem to all other problems.

03. The fallacy’s consequences
The simplicity enforced on factorial data modelling is the result of formal reasoning but with damaging consequences for the representation of reality that the
model should bring forth. Inadvertently, Überla (1971) provides an instructive
example: 90 men had their systolic and diastolic blood pressure measured six
times within half an hour. The initial results of a FA of the 2 x 6 = 12 variables are
shown in Figure 1.03A. Figure 1.03B shows the Varimax results for these factors,
orthogonally optimised by SS. Überla claims the solution depicted in 1.03B is the
only sensible one, arguing that the measurement variations are “basically determined
by two parameters, the systolic and the diastolic blood pressure” (p. 265).

Figure 1.03: Initial factor solution of blood pressure data (Überla, 1971) (A) and rotation to simple
structure (B) _____ orthogonal - - - oblique.

22

Chapter 1 – Critique of the simple structure doctrine

But in fact, “systolic” and “diastolic” are not “parameters”, but merely two different measuring instances reflecting varying conditions of pressure. Blood pressure
is a unit that varies with location and time. At the occurrence of systole and diastole, blood pressure varies just as atmospheric pressure varies with geographical
altitudes. Regulation of blood pressure by therapeutic drugs does not specify systolic or diastolic “parameters”, they aim at improving blood pressure as a systemic
feature (cf. Journal of Human Hypertension and Hypotension)11. A FA of Überla’s twelve
blood pressure variables would reflect the physiological facts best if the highly
correlating variance of “systolic-diastolic”, i.e., the generality of inter- and intraindividual differences of blood pressure, were represented by a general factor. The
first layout of the required general factor is already provided by the initial solution,
as shown in Figure 1.03A, but disappears by an SSM rotation (1.03B).
The second initial factor, however, suggests some difference between “systolic” and “diastolic” measures. Physiological reasons give rise to a second initial
factor, because the two blood pressure measurements show additional variance
under particular conditions. Systolic data generally react somewhat more strongly
to external influences than diastolic data. With circadian measurement readings,
the systolic variance is greater than the diastolic variance (Halberg, 1980, Fig. 8, p.
552). For age-related hypertension measures, systolic values are generally larger
than diastolic values, etc. Obviously, this variance of the two measurements is
present in the initial solution by a second factor explaining, as is to be expected, a
smaller percentage than the first factor.
A similar situation arises with factorial intelligence research. For intelligence,
too, a general factor g is theoretically demanded. In most instances a notable approximation to g exists already in an initial solution. SSM makes g disappear. But
since a g-factor is expected, g is often reconstructed in a cumbersome way12, for
example, by orthogonalising obliquely rotated primary factors using a particular
procedure designed by Schmid-Leiman (1957). Thus “second-order” factors are

11

12

When internists employed a common SS-oriented EFA to determine the insulin resistance
syndrome (ISR), of which blood pressure is a part, much to their chagrin the researchers discovered that systolic and diastolic blood pressure did not load on the same factor, although “… systolic and diastolic blood pressure are more strongly associated with each other than they are with other components
of the insulin resistance syndrome, something which most clinicians would expect” (Lawlor et al., 2004, p. 6).
They got around this problem by only leaving one of the two blood pressure variables in the set
of ISR variables to avoid the methodical artefact: “Therefore, the evidence for inclusion or exclusion of
hypertension in the definition of the syndrome is based on whether one or two blood pressure measurements are included in the model rather than on any sound clinical or pathophysiological reasoning” (Lawlor et al., 2004, p.
1016).
Even in 1958 this procedure was caricatured by British authors who did not feel bound by the
Thurstone doctrine: “It has been said that what was thus thrown out of the door [the general factor] returned
through the window: for correlated factors in turn give rise to a second order factor, and this is virtually the general
factor of the centroid … under another name” (Hamilton, 1958, p. 167).

Chapter 1 – Critique of the simple structure doctrine

23

derived and a hierarchy of factors is introduced while theoretical follow-up problems and hazards of consequences are ignored13.
Jensen very succinctly describes the effect of an SSM rotation on the initial
first factor in intelligence test analyses: “The tests are all positively correlated and therefore
all had some factor in common - a general factor, or g. The general factor that was so prominent
in the analysis depicted in Figure 3.3 [showing the initial solution] seems to have disappeared
from Figure 3.4 [showing the simple structure-rotated solution] as a result of rotating the factor
axes. Actually, it has simply been dispersed (or redistributed) among the rotated factors … So if
you ask where g went, the answer is that it has been divided up and lies ‘hidden’ among all of the
tests’ smaller loadings on all of the orthogonally rotated factors. Its variance has not disappeared;
it has simply been obscured by being dispersed throughout the whole factor matri” (Jensen,
1998, p. 66).
Pett et al. (2003, p. 143) quote authors dealing with this problem: “Nunnally and
Bernstein (1994) … warn against prematurely concluding that, based on a Varimax solution, a
general factor is absent, because Varimax is designed to eliminate general factors (Gorsuch,
1983). Comrey and Lee (1992) suggest that the researcher avoid including too many factors in a
Varimax rotation solution because it tends to overinflate the importance of lesser factors. Although the authors do not indicate how many factors are ‘too many’ they point out that trial and
error is the only way to arrive at the appropriate number of factors.”
The fact that an identified error is readily corrected by haphazard trial and error operations shows how constricting the predicament has become.

04. Detailed error analysis
 The demotion of initial factor structures
Thurstone and all those adhering to his “American school of thought” considered
an initial orthogonal FA solution to be “generally uninterpretable” (Überla, 1971, p.
175) and unstable. Regarding the alleged instability of initial factors, Überla remarks: “[The initial factors] … change from sample to sample … because new variables often
shift weights significantly and, in so doing, change the position of the axes” (p. 175).
But the claim that initial factors lack invariance is unsubstantiated. Literature
on FA offers no evidence that rotated factors are more invariant than unrotated
factors (Andresen (1998) bemoans this empirical deficit 14), whereas there are
13

14

“Hierarchy” suggests a classifying order with subordinate elements “contained in” superordinate
units. But bio-psychological reality does not proceed in such a manner. What reality does show
are different concurring influences. A more comprehensive factor, for instance general intelligence ‘g’, is not made up of smaller factorial units. Intelligence may become functionally manifest without including, say, specialised tools of expression, such as verbal tools (Revenstorf,
1976, p. 313, refers to a similar observation).
“For construct exploration tasks in personality questionnaire scales, final proof still needs to be delivered of the
superiority of the simple structure optimising method with regard to definitely improved cross-variable sample
structure replications.” (Andresen, 1998, p. 74).

24

Chapter 1 – Critique of the simple structure doctrine

definitely complaints about a lack of invariance in rotated factors (Fittkau, 1968,
Butler, 1969)15.
The idea that initial factors are not interpretable must also be discounted as illfounded. Although variables loading on only one factor, after SSM rotation, might
immediately make more sense than variables loading on multiple initial factors,
more comprehensible interpretations cannot indiscriminately be presumed to be
more valid than less comprehensible interpretations. One might as well continue
to believe that the universe revolves around the Earth, and reject the Copernican
view.
The claim that initial factors have to be transformed by SS rotation was occasionally also inspired by the argument that a most prominent initial “general factor” must be a methodical artefact, because initially general factors are, as a rule,
extracted from greatly differing data sets. Überla voiced this widely held notion:
“The [initial] factors … depict an arbitrary distribution of variance. The variance distribution is
not derived from data, but inherent to the method.”. (1971, p. 175). Here, a mathematical
regularity is deemed “arbitrary” or coincidental. One should rather consider that a
FA of sources of variance underlying some set of variables might generally reveal
one dominating source of variance. Sources of variance of factors 2, 3, 4 … on the
other hand, generally reveal less variance what should already be expected from
ranked eigenvalues of respective vectors16.
Despite warnings by the authors of textbooks, researchers often accept the
first unrotated factor of factor analyses as an expected final result. Many intelligence researchers discover an expected general intelligence factor (explicated by
Jensen, 1998, pp. 65-68) as an unrotated first factor. To conduct a Varimax rotation of intelligence factors, Maxwell (1972) limited factor rotations to factors F2 to
F5 and did not rotate the initial factor (F1). Dealing with questionnaire data, personality researchers often recognise hypothetically preferred factors operationalised in initial first factors (starting with Hamilton, 1960, Lumsden, 1961, Leventhal
& Stedman, 1970).
By an SSM rotation, however, an initial F1 very often disappears. To prevent a
first initial factor from disappearing, some FA exponents resort to some kind of a
trick: They repeat a first analysis after eliminating variables with unwelcome high
loadings on a second and third factor, so as to get the second or third etc. eigenvalues falling under the Kaiser and Guttman (= 1.0) criterion of exclusion. This
way, extractions of second and third factors are avoided and the ostensible obliga15

16

Fittkau (1968, p. 110): “… the results of all the analytical rotation processes [are] … not invariant to adding
other variables or replacing some variables with others.” Butler (1969, p. 13): “the simple structure concept does
not solve one of the most crucial and fundamental problems of factor analysis, the problem of the likelihood of factorial invariance.”
The assertion that the position of axes in an initial solution was arbitrary and thus had no interpretative value was Thurstone’s (1934): “A characteristic of the multiple factor problem is that the location
of the axes is arbitrary and that hence the factorial components are to that extent arbitrary and without fundamental psychological significance.”

Chapter 1 – Critique of the simple structure doctrine

25

tion to conduct an SS rotation is circumvented without violating the convention.
This is a formalistic approach more akin to exploiting legal loopholes than to facing facts. The trick as such is obvious, but rarely does anyone take offence at it17.
Gibbons and Hedeker (1992) proceeded using a tidier method by utilising a socalled bi-factor model. In this model a general factor (F1) with all variables participating is accepted, while each variable is allowed to load one additional factor. The
bi-factor rotation model that conserves the general factor fitted significantly better
than SSM rotations when data from knowledge tests and from depression questionnaires were analysed18.
Quite early on, special rotation techniques were developed to rescue the general factor as it disappeared as a result of SS rotations. To a certain extent, the
orthogonal Quartimax rotation by Neuhaus and Wrigley (1954) manages to do
this. There are, however, obvious weaknesses (Gorsuch, 1974, p. 191) that Quartimax does not eliminate.
Nonetheless, Überla and all those sharing his textbook views believe that the
allegedly “incomprehensible” and “arbitrary” initial factor solutions must be transformed into allegedly more comprehensible and more stable solutions. Simple
structure is taken as the only justifiable or even the only possible guide-line. Alternatives are not debated.
At the dawn of FA, Thurstone’s “American school of thought” was being opposed and criticised by the British school headed by Cyril Burt. Burt had recognised the presence of valuable information in initial factorial solutions, or in the
initial bi-polar structures as he called them: “… to the experienced factorist both the
regularities and irregularities [of the pattern of signs] will yield considerable insight into the data
he is analysing, even without any further rotation or analysis” (Burt, 1954, p. 16). Burt also
routinely transformed bi-polar initial factors into uni-polar solutions that he called
“group factor solutions”. But he did not depart as far from initial solutions as did
17

18

Thalbourne (1998) is a typical example of the popular exclusion of items to sidestep the obligatory SS rotation. For reasons of interpretation, Groner and Groner (1991) even go so far as to
limit themselves to just the first of 17 (!) initial factors with eigenvalues greater than 1.
Gangestad and Snyder (1985) and Snyder and Gangestad (1985, 1986) present an illuminating
argument for saving the concept of “self-monitoring”, which would be lost if SS rotations were
applied. The dilemmas are increasingly neutralised by methodical compromises. These include
the extraction of second-order factors (causing hierarchical models) and the monitoring of various model possibilities by confirmatory and structural equation procedures (Undheim and Gustafsson (1987) aim at “Restoring general intelligence”). The problem’s real sources are hereby
disguised.
Hamilton (1958) employed the then still topical method of “simple summation” to find a general anxiety factor (F1) in an anxiety questionnaire. He also found a weaker factor F2, indicating
the centre of anxiety (psychic vs. somatic symptom dominance). Benactyzin, an antidepressant
which was given to patients participating in the experiment, caused changes in F 1, as expected,
and not in F2. The author therefore criticised Thurstone’s SS procedure which did not reveal actual general anxiety reduction. A Varimin re-analysis corroborated Hamilton’s criticism and confirmed, using a published correlation matrix, his factor interpretation (results of an unpublished
re-analysis by the author).

26

Chapter 1 – Critique of the simple structure doctrine

the American factorists. Burroughs and Miller (1961, p. 35-37) also had reservations about misleading SS rotations: “… the subsequent rotations are apt to obscure [the
objective and dichotomous classification based directly on the data]” (p. 36). Had the competition between British and American factorists continued a little longer – unfortunately it did not – mistakes and failures caused by Thurstone’s position might have
been delayed or possibly even prevented.
 The “positive manifold” is misinterpreted
Correlation coefficients of intelligence tests are usually positive (called “positive
manifold”). Hence, initial factor solutions from test intercorrelations like blood
pressure measurements, exhibit a unidirectional, positive general factor F1 (= g).
As a rule, however, they also exhibit additional bi-polar factors F2, F3, etc., i.e.,
quite a few variables possess negative loadings. Now, early intelligence researchers
expected factors representing intellectual abilities, which by definition could not
have negative manifestations. They were convinced that intelligence test factors
could only have positive loadings or at best almost-zero loadings, meaning no
ability. Therefore it seemed necessary to transform bi-polar loadings to positiveonly loadings. SS transformations, bringing this about, were therefore considered
even more justified (Thurstone, 1947, p. 341)19.
In his blood pressure FA, Überla (1971) had assumed, like researchers of intelligence, that only positive loadings were admissible because the smallest value of
the blood pressure on a mmHg measurement scale is zero and not negative. Here
the properties of a metric that is suited for manifest observations (blood pressure
and intelligence measurements use ratio scales) are transferred to the metric suitable for latent conditions (showing properties of an interval scale). “… rigorous
measures, such as direct counts, latency, or duration, are excellent measures if used as descriptions
of behaviour but may become arbitrary metrics if they are used to infer some psychological construct” (Kazdin, 2006).
Following the publication of five contributions to the discussion about “arbitrary metrics” (American Psychologist, 61, 2006), agreement may soon be reached
about the metric resulting at the latent level of factor loadings. It may be shown
19

“It is … natural to postulate that when a unique simple structure is found for a battery of tests of mental abilities, then the non-vanishing entries in the factorial matrix are positive” (Thurstone, 1947, p. 341). Thurstone says that the limiting condition of the “positive manifold” was not required for applying
the simple structure principle. But SS rotation would be appropriate especially when “the factor
loadings shall be positive or zero”. (p. 23) Berneyer (1957) reports: “The different methods of [factor] analysis [of mental aptitudes] yield factors which have negative loadings … Such factors, so Thurstone contends, must
be devoid of ‘scientific meaning’. They do not permit us to ‘interpret the various tests as functions of the mental aptitudes which those tests elicit’” (p. 23). C. Burt also shares Thurstone’s “positive manifold” view: “…
an ability for x is by definition a dispositional property that facilitates doing x, i.e., it denotes a positive and never
a negative tendency. Hence we shall be compelled to seek factors with positive saturations only.” (Burt, 1954, p.
18).

Chapter 1 – Critique of the simple structure doctrine

27

that arbitrariness of zero-points and signs is not only permitted but indeed required. Vukovich (1967) argued at an early date for liberal scaling decisions: “You
can treat measurements any which way you like as long as you do not compromise their illustrative character. If empirical comparisons do not contain evidence about the zero-point or the size of
measurement units, they can be chosen freely and they will conveniently be determined to facilitate
the lucid description of larger contextual conditions” (p. 114).
In short: An initial general factor must not be downgraded to a methodical artefact (Überla, 1971) and need not be removed at all. It may actually be the rule
that one source of variance out of several sources emerges as predominant. Additional minor sources of variance (factors) might orthogonally modify the main
effect with bipolar distributions.
As a rule, a general factor is extracted first which serves as a reference for variance sources of subsequently extracted factors. This is equivalent to a standardised
distribution of values whose mean, a zero point, is the reference for values deviating from the mean in positive or negative direction. If negative loadings occur
together with positive loadings for a second, third, etc. factor, this signifies that
additional sources of variance are set in relation to the main source. Similar views
were voiced by Thompson (1963), which, however, apparently have not been
taken up by other factor analysts20.
The present criticism of Thurstone and his followers’ assessment of positive
manifolds is not meant to consider initial factor extractions as final. Initial solutions often require improvements by Complex Structure (CS) rotation. This will
be elaborated in sections 05 and 08.

05. Reorientation
 Natural processes are complex
The FA of blood pressure measures showed that the variance total of variables
can be split into two latent sources. One represents blood pressure independent of
heart beat, the other yields proportions of systolic-diastolic variance. This result
adds empirical reasons supporting my SSM criticism.
It may sound trivial, but in almost all domains of nature observed variables are
dependent on multiple conditions. SSM-orientated FA ignores this phenomenon.
The economic pay-off of the first step of FA, achieved by assigning large numbers
of manifest variables to smaller numbers of latent sources of variance (factorial
“building blocks”), is not disputed. But this achievement is gambled away by ap20

“If they are able to choose one or the other … psychologists tend to prefer unidirectional to bi-polar measurement,
probably as a result of the prestige of such measures as the standard metre in physics. … Thurstone expressed a
preference for a positive manifold (without, in the writer’s opinion, giving a fully convincing explanation) … Bipolar and unidirectional measurement are both needed in psychology.” (Thompson, 1963, p.22). The latter
statement is justified at length in Thompson (1962).

28

Chapter 1 – Critique of the simple structure doctrine

plying factor transformations that leave just one latent factor to each manifest
variable, or as few as possible. The very opposite should be required (see additional support for this view in section 08).
It is remarkable that the SS fallacy did not lead to doubts and reflections earlier
because the SS ideal was almost never fulfilled. All the agonised wrestling about
SS was a Don Quixote-like tilting at windmills. Even Harry Harman, a leading
author on FA, had to admit: “An orthogonal uni-factor solution is practically impossible
with empirical data and not very likely even when the factors are permitted to be oblique. Nonetheless … it is towards that end that the simple structure principles are proposed for the multiple
factor solution” (Harman, 1968, p. 99).21
FA may be compared with multivariate analysis of variance (MANOVA). Factor analysts, however, generally deal with independent variables (IVs, sources of
variance, “factors”, for example, abilities) that are latent or hypothetical and merely assumed to generate manifest, observed, dependent variables (DVs, for example, test measures). For conducting a FA, IVs need not be known, they are
deemed to manifest themselves through factors. ANOVA researchers, however,
do not only deal with manifest dependent variables (DVs), but also with manifest
independent variables (IVs) that need to be manipulated experimentally. Their
focus is not on underlying causes that make measurable variables become manifest.
Another difference between conventional SSM factor analysis and MANOVA
may be pointed out: MANOVA does take into account “structural” conditions of
the investigated DVs, i.e., their interactions. However, MANOVA is only interested in interactions among manifest variables. SSM-orientated FA excludes latentlevel interactions.
Had common elementary knowledge been impartially considered, the SS principle would have appeared suspect from the outset. Variables investigated in science are generally based on components interrelated on lower levels. Atoms consist of protons, electrons, and neutrons, salt consists of sodium and chloride, and
21

Criticism of the SS principle was voiced on rare occasions: “Deciding which of many possible mathematical solutions to use depended on a formal rule, the simple structure principle, which lacks any theoretical substantiation. Freed of all necessity to conduct more elementary deliberations, (researchers) now delivered bulk work
… The result of this method, as could have been predicted, was a surplus production of superficially defined factors” (Meili, 1969, p. 278). Similarly, Revenstorf (1976) deemed SS “generally unlikely … [especially]
in large feature compilations (questionnaires)”, because “features can increasingly depict an endless range of
combinations of factorial measures. In this case, the features of the configuration of variables are scattered across
the entire factor space, and a simple structure in a Thurstone sense … can no longer be discerned” (p. 321).
Countless examples of doubtful SS-factor interpretations exist: “It is obvious that the simple structure
is lacking” (Bierhoff, 2000); “… despite the comparatively high intercorrelations we cannot assume a simple
structure” (Schaper & Baumgart, 2002); “Generally, simple structure does not appear to be succinct “
(Beauducel, Strobel, & Brocke, 2003), “… not a simple structure according to the Bargmann test” (Herzberg, 2002); “An acceptable simple structure for the factor loading matrix could not be achieved by an orthogonal or an oblique rotation of the three main axes” (Schmitt, 2000); “The result does not indicate a simple
structure” (Lambert et al., 2002).

Chapter 1 – Critique of the simple structure doctrine

29

genes have an effect with complex diversity only (“polygenic” effects)22. Perception of an individual colour is the result of electrical excitation of three cone pigments. To continue: clauses consist of words, words of morphemes, morphemes
of phonemes. Each phoneme is based on several phonetic features. Applied to
FA, it should be expected that manifest variables in a study are based on concurring variance sources.
Complexity, i.e., the concurrence of conditional components, is in accordance
with Occam’s principle of economy, indeed it is an immediate consequence thereof. When available units of operation can be combined, new adaptive processes
result, the cumbersome production of particular units for additional purposes
becomes superfluous. Evolution did not develop additional receptors for the perception of purple, ochre, ultramarine etc.
A recent investigation into the evolution of language culminated in the conclusion that even the highest mental attainments are the result of successfully combining process-related resources. Initiating the evolution of an extra programme
for establishing linguistic communication of hominids would have amounted to a
waste of resources, say Bates et al. (1992)23 and Gould et al. (2002).
Wolf Singer (2003) does not tire of describing the human cerebral cortex as
having the ability “to play a combinatory game” (p. 84). Elementary brain structures
have “a similar format” and are of a “surprisingly monotonous build” (p. 44), “nature is
very conservative …” (p. 46). When centralisation is lacking, brain activity produces
its “performance through constructivist bonds” (i.e., combinatorics) (p. 75). “[In the] brain’s
functional architecture [it is] crucial, who gets in touch with whom, how intensely, and whether it
happens in an inhibitive or in an agitating manner” (p. 38).
Tens of thousands of dendrites facilitate contacts among brain cells. Transferring Thurstone’s principle of SS onto brain activity based on neural units as solitary pegs, would contradict the principle of free bonds. Singer regards the bond
phenomenon as “the core problem of neuroscience” (p. 57). Apparently processes of
living nature and even of inanimate nature are subject to combinatorics of elementary building blocks. The biologist Humberto Maturana (1998, pp. 158-189) refers
to the universality of “structural determination” of nature’s activities24.

22
23

24

Simple anthropometric characteristics like height and hair colour are based on the concurrence
of multiple genes.
“Language learning appears to be based on a relatively plastic mix of neural systems that also serve other functions. I believe that this conclusion renders the mysteries of language evolution … somewhat more tractable. That
is, the continuities that we have observed between language and other cognitive systems make it easier to see how
this capacity came about in the first place.” (Bates, E. Modularity, domain specificity, and the development of language. URL: http://www.ecs.soton.ac.uk/~harnad/Papers/Py104/ bates-1994.html, no
longer accessible).
Numerous quotable statements support this view, e.g.: “A living system is a structural determinant
system and everything in this system happens as a result of relations of its constituent parts …” (Maturana,
1998, p. 184).

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Chapter 1 – Critique of the simple structure doctrine

The complexity issue deserves consideration from a more encompassing perspective, too. Malcolm Forster (1998) examines the terms “parsimony and simplicity” and
concludes: “The paucity of parameters is a limited notion … There are no compelling ideas
about why such properties should count in favour of one theory being closer to the truth than
another […] Whether a simple curve is preferable to some more complex alternative, or the reverse is true, has nothing to do with simplicity and everything to do with predictive accuracy”
(Forster & Sober, 1994, p. 28).
In Complexity - a Philosophical Overview, Nicholas Rescher states: “… In the development of knowledge – as elsewhere in the domain of human artifice – progress is always a matter of complexification. An inherent impetus towards greater complexity pervades the entire realm
of human creative effort.” (Rescher, 1998, p. 58). “There really are no adequate grounds for
supposing the ‘simplicity’ of the world’s make-up. Instead, the so-called ‘principle of simplicity’ is
really a principle of complexity-management.” (p. 61). By the same token one finds positive resonance in Bechtel and Richardson (1993): “It is ultimately the pattern of connections in the system, and not the jobs performed by specific units in the system …, that is critical
to the behavioural systems”.
Gawronski (2000) qualifies the simplicity posit for epistemological reasons:
Simplicity is a “vague criterion”. “If clusters of variables are to be described by a mathematical function, there may still be a certain consensus about the ‘simplest’ graph. But if verbal scientific theories are concerned, judgments about simplicity can vary greatly … depending on structures of knowledge. In this sense, the demand for simplicity becomes relative.” (p. 10).
 The new guideline of factor transformation is complex structure modelling or
CSM
Factor analytical research needs a new guideline which turns Thurstone’s simple
structure principle upside down. This is it: complex structure modelling.
Initial EFA factors should be transformed in such a way so as to optimise the
simultaneous presence of extracted factors with individual variables. In chapter 2 I
show that this can be done, supported by maths and statistics, and that it prevails
over conventional procedures.
It will also be shown that initial factors already engender complex solutions,
even without factor rotation, provided the data sets are based on only small numbers of sources of variance (factors), as was the case with blood pressure measurements. When working with more than two substantial factors, however, a rotation procedure is called for to improve the complexity of their representations.
The procedure should assign factorial sources of variance, established for the
respective domain, to all individual variables, if possible, and then have the results
tested against empirical boundaries25.
25

An extensive German-language criticism of factor analysis (Holz-Ebeling, 1995) takes exception
to the vagueness of factor interpretations. The author is right in stating that investigated variables (DV) are generally dependent on multiple conditions (“multi-IV conditionality”). She adds

Chapter 1 – Critique of the simple structure doctrine

31

CSM aims to show actual complexity, while SSM tries to enforce non-existent
simplicity. The ideal of solitary variables (each variable is allegedly explained by
only one factor or by as few factors as possible) actually generates non-simplicity
and non-parsimony.
Employing CSM for factor transformation is no rebuff of Newton’s “natura est
simplex” or Occam’s razor. Occam’s razor just needs to be applied in such a way as
to reveal the strategy of parsimony that nature itself brings forth (by utilising its
bonding capacities as shown earlier)26.
Incidentally, the complex structure model does not expect to be taken as an innovation. It transfers a view generally held in science to a particular field of methodical operations where until now it has not gained a foothold. An opposite view of
enforced simplification, associated with SS, has survived to this day in a protected
corner of statistical methodology and has been leading research astray27 28.

26

27

28

that factor analysis does not do justice to the multi-IV conditionality of DVs. She suggests using
a procedure methodically related to variance analysis to “replace or complement” factor analysis.
Multi-IVs influencing singular DVs are accessible by variance analysts. The author’s impractical
(her words) suggestion is interesting in as much as it attempts to correct errors caused by the anti-complexity SS doctrine. She is aiming in the right direction but does not touch the dilemma’s
real causes.
Time and again, factor analysts confronted by unwieldy and complex variables have encountered the limits of the SS principle. Guilford and Zimmermann (1963) risked a liberalisation of
the parsimony principle by admitting complexity: “In general, investigators need to relax somewhat their
drive to achieve parsimony. Long ago, psychology should have progressed beyond the stage in which investigators
continue to look for the ‘philosopher’s stone’” (p. 299). When modelling complexity, parsimony cannot
be curtailed. Nature applies different economic strategies than those that Thurstone used for
depicting nature’s variables geometrically.
Oblique rotation, devised by Thurstone and vehemently propagated by Cattell, is comprehensible from a CSM perspective. Oblique rotation is actually a dubious balancing act between obtaining monofactorial variables, while also having to take into account the complexity of variables as witnessed in the natural universe. “It is unreasonable to expect that a great variety of influences operating and interacting in the same universe would be completely uncorrelated” (Cattell & Dickman, 1962, p.
390). The factor analyst applying oblique rotation will thus allow for factor correlations in a
non-orthogonal factor space. By permitting factors to correlate, a back door is kept open for
some functional linkage. This compromise was criticised decades ago by Guilford and Zimmermann (1963, p. 289): “[This amounts to] a hollow and accidental victory for oblique methods of rotation”.
The pseudo-solution by oblique rotations is mentioned here to illustrate the consequences of a
frustrating search for SS and the attempts of researchers to get rid of the self-created evil by
questionable decisions.
Cattell and Radcliffe’s (1962) attempt to eliminate, by suppressing unwanted variance, actually
existing complexity that was not removed by SS rotation is just another symptom of misguided
SSM: “If we are right in assuming that behaviour which has a large personality factor variance will generally be
factorially complex, then the unwanted common factor variance will, as a rule, be far from negligible.” The authors intend developing “unifactor scales” and to this end they manufacture a process “… which
will reduce the contribution of unwanted factors by suppression” (p. 125).

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Chapter 1 – Critique of the simple structure doctrine

06. Where did we go wrong?
How did the erroneous idea of SSM come about? Human liability to cognitive
errors probably contributed much. Gestalt psychologists focusing on “Gestalt
laws” have repeatedly warned about detrimental side effects. “Thought misled by
Gestalt tendencies” (Witte 1974) and “‘Visual Prägnanz’ [conciseness] as an obstacle to
problem solving” (Kanizsa, 1975) seems to affect visualisations of data points in factor space, clusters of variables are irresistibly attracted by surrounding coordinates
due to the laws of proximity and grouping. Apparently, researchers adhered to
“intuitively compelling” perceptions that provided an anchoring of the variables.
It was easy to employ “curve fitting” by utilising freely rotatable coordinates. Gorsuch (1974, p. 164 f) talks of a “visual compellingness” that guided Thurstone in
his early visualisation of extracted factors resulting in his oft-quoted five criteria of
simple structure rotation29.
Arkes (1991) describes another reason for judgmental errors possibly underlying the argument of “easy interpretation” of SS factors: “Suppose a person adopts a
quick and dirty strategy to solve a problem. Because it is so quick, it is easy to execute. This is a
benefit. Because it is dirty, it results in more errors than a more meticulous strategy. This is a
cost. Although the choice of this strategy results in [more errors] …, this cost may be outweighed
by the time and effort saved” (p. 487). Arkes goes on to call this, not quite accurately, a
“strategy-based judgment error”. He should have called it a “cost-saving judgment
error”.
In a similar vein, Edmonds (2002) refers to a universal bias for simplicity. “…
the simpler theory is not more likely to be true and is not likely to be nearer the truth … For
human beings it is much easier to elaborate … [a failing] theory, or otherwise tinker with it,
than to undertake a more radical shift ( for example, by scrapping the theory and starting again).
This elaboration may take on many forms, including … complicating the model with extra
equations or rules … or using more complicated functions.” Edmonds summarises: “…
Model selection‚ for the sake of simplicity is either simply laziness … [or] due to pragmatic
reasons”. He advocates, as a matter of principle, foregoing “simplicity for fear of
innovation”. “… The elaboration of [an existing] theory in order to fit a known set of data
should be resisted … The lack of success of a theory should lead to a more thorough and deeper
analysis than we are usually inclined to perform.”
Without doubt, the “power of words” has boosted SSM’s immunity. Hardly
anyone dares oppose a term indicating the seductive attributes simple and structure.
29

1. Each row of the factor matrix should contain at least one zero.
2. If there are m common factors, each column of the factor matrix should have at least m zeros
3. For every pair of columns in the factor matrix, there should be several variables for which entries approach zero in the one column but not in the other
4. For every pair of columns in the factor matrix, a large proportion of the variables should have entries approaching zero in both columns when there are four or more factors
5. For every pair of columns in the factor matrix, there should be only a small number of variables with nonzero
entries in both columns.

Chapter 1 – Critique of the simple structure doctrine

33

Had Thurstone named his rotational principle, say, the Principle of Solitary Factor
Contribution, which would have been more modest and closer to the truth, its validity would have soon been doubted. Now, after a worldwide dispersion of the magic term “simple structure”, it will be hard to get rid of it.
Hard-core SSM has been sealed off by leading methodologists. They countered an apparent need for more complex modelling by more sophisticated procedures that showed flaws right from the outset. The cautious attitude of an exploratory researcher, who is prepared to be surprised by unexpected findings, has increasingly been replaced with an attitude of mere administrators of variables, who
determine the units of their domains arbitrarily and assign them functions as they
see fit.
The current epistemic climate in psychology places feasibility at the top of value rankings. Models that can be manufactured and imposed or inflicted on nature
garner more respect than models emerging from nature itself. The intricacy of
models constructed by amateur tinkerers easily conceals the fact that discoveries
of true value mostly occur by careful bottom-up observations.
Using Lakatos’ (1978) historical approach, the SS principle may be taken as resulting from a “hard core” attitude accompanying long-term developments of
conventional research. Core beliefs may turn hard core when a community of
researchers takes them for granted and no longer questions them30. Michell (2000)
considers blind clinging to fundamental premises a symptom of “scientific pathology”: “A hypothesis is accepted without a serious attempt being made to test it and this failure
of critical inquiry is ignored” (p. 648). Schönemann described such a symptom by
mentioning the outrage of “traditionalists” when Bargmann wanted to obtain
significance values for SS statistically: “[They] scorned it [Bargmann’s Test] as a sacrilege
of their cherished belief that simple structure is a law of nature” (Schönemann, 1994, p.
293).
Long before Lakatos, Fleck (1935) presented case studies to describe sociological excesses of “thought collectives”: “Once a fully developed, closed system of opinions
has been formed, it will consistently persist against all opposition … What does not fit the system
will not be seen or is concealed … or is declared as not opposing the system by employing huge
exertion.” (p. 35). “The tendency of opinion systems to persist proves that they should be seen as
… stylistic structures. As harmonic entities … they exhibit special stylistic features that determine every single cognitive function. … [They create a ‘harmony of deceptions’], which cannot
possibly be dissolved within the ambit of a certain way of thinking.” (p. 45).
Lakatos maintains that hard-core conditions in scientific research will lead, after a while, to empirical anomalies. As the number of anomalies grows, research
programmes move into a “degenerative” phase. “Protective belts” are constructed
and reinforced when danger looms. For the past two and a half decades or so,
one-sided orientated methodologists have been developing highly complicated
30

“It may be contended that we should cling at all costs to the conception of simple structure, because we have no
satisfactory alternative” (Reyburn & Raath, 1949, p. 127).

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Chapter 1 – Critique of the simple structure doctrine

procedures for analysing multivariate data. These methods are clearly protective
belts whose function is to immunise and shield the doctrine of SS from its disappearance (also refer to section 08).

07. Unheeded critical voices
Did no one ever take umbrage at the SSM principle? Yes, the limitations of this
principle were challenged occasionally, but its basic legitimacy was almost never
questioned.
W. Stephenson (1956) was one of the earliest doubters. He realised that attempts to make personality traits dependent on singular factors will clash with
empirical complexity: “The problem is to explain complex traits in terms of relatively few
primaries” (p. 7). He used more liberal rotation methods for what he rightly named
“compound traits”. But the usage of his method was laborious, it was much like
the Circumplex procedures of Hofstee et al. (1992) and de Raad et al. (1994)
(more in section 08) and their confusing results that were also based on SSM’s
rotation ideal.
Stephenson favoured the factorial Q design which aims at analysing “person
variables”. He obtained them by so-called Q-sort assessments that he himself had
developed. Person variables were intercorrelated and factorised unlike what is
required for factorial R designs that select behavioural variables (such as test results, ratings, not persons). Stephenson thereby circumvented the unsolved question: Which are the basic features that make complex experience, behaviour and
trait variables, theoretically understandable? The answer can only be found by
employing exploratory R studies, since the constructs needed to interpret Q results are not provided by the Q FA itself.
Another researcher highly dissatisfied with Thurstone’s SSM was J. P. Guilford
(1974, pp.498 f): “Thurstone’s principle of simple structure … is by no means sufficient if we
want logical psychological meaning … In numerous instances [in Thurstone's and his students’
studies] tests of very different character are often thrown together significantly on the same factors.
This does not make good psychological sense … Rarely were varimax factors clean-cut and easy
to interpret psychologically …”.
Guilford reacted to deficient results of FA of intelligence variables by proposing what he called a “structural model” of intelligence. He tried to improve the
factorial SSM results by “logical and psychological” means. Alas, he did not detect
inherent errors in Thurstone’s principle. He set out to rectify factorial deficiencies
by applying non-factorial conceptual tools.
A more recent critic providing similar arguments is Allen Yates. Better than
anybody before or after him he hit the nail on the head: “The factors that result from
cluster-oriented factor analysis [from blind application of simple structure] are simply an index of
success an investigator has had in putting together groups of collinear variables (clusters) – regardless of how complex these same variables might be in terms of their latent determinants. In other

Chapter 1 – Critique of the simple structure doctrine

35

words, manifest collinearity among variables is an indication only that they share the same pattern of causal determination; it does not in any way suggest that the shared pattern involves just
one latent causal factor.” (Yates, 1987, p. 39).
Yates’ monograph delivers the most concerted attack on the model of SS that
I have found. His last chapter gets started with: “Only a radical reorientation of current
perspectives will allow researchers to apply exploratory factor analysis in the manner envisioned by
its originators as a powerful technique for routine discovery of the underlying bases of observed
covariation” (Yates, 1987, p. 323).
But even Yates does not touch the core of the SS calamity. According to him,
Thurstone’s original approach that had generally been ignored was “more liberal”
(Thurstone even ignored it himself later). Yates developed new rotation algorithms. His alternatives (Direct Geomin and Direct Geoplane) are extremely complicated and researcher-dependent, much like the procedures of other researchers who
tried to escape the unwelcome side effects of SS analyses. The success of Yates’
innovation was meant to be a way out of the dilemma, but psychometricians did
not take notice of his approach.
SSM problems have also been discerned by Rozeboom (1991), who saw no
way out: “For diagnosing the causal grain of common-factor space, rotation to simple structure
is so disquietingly fallible that we would surely prefer another criterion were any plausible alternative at hand. (Churchill's aphorism on the inferiority of democracy comes to mind here, namely
that democracy is the worst form of government there is – except for all the others …Read ‘simple
structure’ for ‘democracy’ and ‘rotation criterion’ for ‘form of government’.)” (p. 587). Rozeboom’s HYBALL rotation, in which multiple coordinates (sub-spaces) instead of
individual coordinates were successively rotated (“somewhat more holistic than a simple
sequence of planar rotations”, p. 587), proved to be just as dubious a compromise as
the one Yates had invented.
Schönemann and Borg (1996) were among the critics categorically doubting
SSM. With much resignation they stated: “the simple structure criterion (formulated as
Varimax criterion) [is] routinely applied in factor analytical practice”. And, more critically,
they add: “Given many systems of variables in the first place, the important question why a
simple structure should be expected completely falls by the wayside” (Steiger, 1994, p. 204).
Basically, “the hypothesis of a simple structure was not very plausible …” It asserted that
test questions with non-zero loadings on all factors were “impossible” and, in so
doing, “puts the cart before the horse” (according to Guttman, 1992, p. 186).
A more general underlying flaw of researchers was diagnosed by Gigerenzer
(1978), albeit for another example of dimensional model generation. Uncritical
model users in psychology tend to ignore the fact that their models are, by presuppositions, connected to the psychological domain of observations. These presuppositions are not directly tested and Gigerenzer says that what he calls “proposition of implication” is generally ignored: “[This proposition] holds that every mathematical system (e.g. a method for dimensional analysis) implies some psychological theory about the

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Chapter 1 – Critique of the simple structure doctrine

respective domain of observations […]” (p. 110).31 The formalised system of relations
(FRS) is directly and inseparably entwined with the empirical relations system
(ERS) (Gigerenzer refers to supporting views of leading methodologists Suppes
and Zinnes (1963)). Should FRS and ERS diverge in essential aspects, divergence
artefacts ensue, leading to “theoretically worthless results” (p. 111).
Applied to the present context this means SS (an example of FRS) assumes,
without evidence, that the variables accessible at the ERS level are based on solitary sources of variance. Gigerenzer demands accordingly: “To prevent an investigative
result being interpreted as divergence artefact, the researcher has to … explicate the observed
psychological model that was implied by the mathematical model …” (p. 116).
This, though, is precisely what factor analysts have thus far neglected to do.
They believed that their analyses, applied with mathematical precision, would
more or less automatically deliver structural models of psychological reality. Utilising independent reasoning was deemed superfluous and even disreputable; this
carried the blemish of “subjective” intrusions into the discourse on “objective”
facts.

08. Can non-factorial procedures take us forward?
 What do Circumplex procedures achieve?
Methodologists have often reacted to unsatisfactory EFA results by inventing
additional procedures. Mathematical superstructures should cure the symptoms
(Revenstorf, 1980, p. 12). Elaborate calculations were conducted to cope with
multiple loadings of variables (e.g., questionnaire items) deviating from the SS
ideal. Hofstee et al. (1992) and their AB5C model (Abridged Big Five Dimensional
Circumplex) is an example. The authors tried to improve SS results of adjectival
personal descriptions by applying Wiggins’ (1979) Circumplex method. The Circumplex method allows two factorial dimensions to be associated with variables
simultaneously, and thus does away with SSM restrictions, but anyway only
partially.
The Circumplex model is just another hybrid compromise, the method “does
not deal adequately with those … variables that load highly on more than two … factors”.
Circumplex results are thus neither “definitive nor comprehensive” (Hofstee et al.,
p. 161). The SS principle is not shaken; the dimension problem is not stirred. The
authors merely “propose a partial liberalization of simple structure” (p. 147), while what is
required is the abolition of the SS principle for solving the complexity problem.
31

Gigerenzer’s implication proposition antedated Smith and Jones’ (1975) “central thesis” published in a paper on multi-dimensional scaling: “All data analysis and all scaling involve fundamental
assumptions about the psychological processes that lead to the data and the scaling solutions under consideration.
In particular, current scaling methods, in our view, should be regarded with deepest suspicion, precisely because
they are based on doubtful or untested psychological assumptions.” (p. 44).

Chapter 1 – Critique of the simple structure doctrine

37

Acton and Revelle (2004) surveyed psychometric criteria for an application of the
Circumplex (p. 26). They also found that this method fails when “all items are best
described by more than two factors”. The requirements for Circumplex application are incidentally very discerning – the authors name ten requirements –, making the Circumplex hardly commendable even if data sets can be described by only
two factors.
 Are confirmatory procedures an alternative?
Confirmatory factor analysis (CFA) and its more flexible sequel “structural equation modelling” (SEM), seem to have largely displaced, with sophisticated algorithms, exploratory (EFA) procedures. Instead of considering, as EFA researchers
generally do, unpredictable latent variables in their calculations, it has become
fashionable to invent latent variables willy-nilly and to use confirmatory techniques in order to find fits between the thought-up model and empirical reality.
To this end vast numbers of fitting attempts are conducted, mostly via blind trial
and error.
But structural equation models are still geared to the SS principle. Therefore
these models also do not uncover the complexity of sources of variance32. “The
more recent structural equation models – initially as ever euphorically celebrated … exacerbate
factor analysis’ problems” (Schönemann & Borg, 1996, p. 241). Nearly always they fall
short of underlying structures33. Critical remarks are sometimes recorded: “The
assumption of simple structure is probably a typical … simplification bias”, but unfortunately
“necessary” (Beauducel & Wittmann, 2005, p. 43). “… Simple structure models of
personality are unlikely to meet conventional or even fairly relaxed goodness-of-fit criteria …
Overemphasis on simple structure … may explain some of these problems” (p. 44)34. SEM
results are sparse (“poor results”, Beauducel & Wittmann, 2005, p. 42). In a critical review, MacCallum and Austin (2000) point at problems of the confirmatory
method and the “confirmation bias” of its users. Users tend to make do with goodness-of-fit values and arbitrarily chosen criteria, resulting in make-believe fits. Instead,
results should be evaluated using factual information and not primarily formal
32

33

34

Basilevsky (1994, p. 415) describes the trend to dissect singular factors from the examined
variables in CFA practice as follows: “… we may wish to impose zero restrictions on the loadings. Values
other than zeroes can also be used, but zeroes are most common in practice”.
“The LISREL manual discreetly conceals the fact that none of the latent causes have been positively defined”
(Schönemann & Borg, 1996, p. 250). The “indeterminability problem”, a purely mathematical
formal problem, “was actually compounded” in the LISREL case “because many more latent
variables were postulated there than in the multiple factor analysis model …” (p. 250).
A recent attempt to solve the acknowledged problems with SEM research has been undertaken
by Marsh et al. (2010) who try to combine the advantages of unbiased exploratory analyses with
confirmatory procedures (the approach is called exploratory structural equation modelling, ESEM).
Marsh et al. address symptoms of the malaise which they are eager to correct, but the underlying
source of the symptoms (SS bias) is not recognised. The Big Five factors are not questioned,
they are fully replicated, the ostensible advantage are improved statistical properties in detail.

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Chapter 1 – Critique of the simple structure doctrine

measures35. When models do not make the fit criterion, this is often ignored
(“working with imperfect models” is, the title of an article by MacCallum (2003)).
Or it remains unconsidered that the selected model’s fit could easily be surpassed
by other, non-tested models. Even Kaiser did not think much of CFA: “I cannot
resist saying that, for me at least, the earlier exploratory thrashing about was much more fun –
and perhaps even represented more progress – than the forthcoming confirmatory prettying-up”
(Kaiser, 1970, p. 406).
Sobering results from other CFA and conventional EFA comparisons can be
found in Church and Burke (1994) and Ferrando and Lorenzo-Seva (2000). Criticism by Cliff (1983) beats the same drum. A recent comprehensive CFA study
done with simulated data finds as follows: “… trait models [of personality] assuming
simple structure tend to be rejected with CFA …”. “… there will always be some small distortion of simple structure” (Beauducel & Wittmann, 2005, p. 72). When applying SSM to
data sets (especially personality data) that are subjected to CFA, “a gap [is created]
between the large body of results based on exploratory factor analysis and CFA in personality
psychology” (p. 73).
Conventional EFA is orientated toward SSM but, as a rule, for many variables
unwelcome secondary loadings are noticeable. These cannot even be manipulated
when factor loadings are blindly distributed beforehand (“there is no [prior] knowledge
of secondary loadings” (Beauducel & Wittmann, 2005, p. 43). Vittadini (1989) considers LISREL results indeterminate because latent model variables are made dependent on manifest variables: “one may actually be confirming the model because the
manifest variables are determined by other variables than those hypothesized, which happen to
have the same pattern of relationship to the manifest variables as given by one's hypothesis …
One can never regard structural hypothesis as true as opposed to ‘confirmed’” (p. 428).
Only where latent variables have already been verified by exploratory analyses,
may confirmatory procedures be applied. Velicer and Jackson’s (1990) reasoning
points that way: “Exploratory analytic approaches … should be preferred except for those
cases where a well-defined theory exists. Exploratory approaches avoid a confirmation bias, do
not force a theory-oriented approach prematurely, and represent a conservative strategy.” (p. 21).
But it does not make much sense to conduct CFA calculations for obtaining a
result that has already been discovered by means of EFA36 .

35

36

“The LISREL model’s practically boundless plasticity … not only undermines its claim to statistical inference
but also gets close to soliciting abuse, because if you have sufficient patience you are bound to discover some kind of
causal model that does not have to be declined for the currently available data.” (Schönemann & Borg, 1996,
p. 250).
Rost (2002): “The model specialist [specialist in modeling with structural equations, log-linear models, itemresponse models, etc.] … cannot use data at all if he is not told which variable is supposed to interact with which
other variable …, which latent variables should be there, … etc.” My comment: The present-day “model
specialist” can gain an approximate insight into the latent parameters of human thinking, feeling,
and behaviour right from the outset. The exploratory factor analyst from days gone by, however, was unable to gain this knowledge, not even with the benefit of hindsight, despite having

Chapter 1 – Critique of the simple structure doctrine

39

 How should mathematical tools of research be generally evaluated?
A growing tendency in current science is to trust mathematical techniques blindly
and to let statistical tool makers dominate research. Papers of some self-critical
methodologists contain warnings: “Those who firmly believe that rigorous science must
consist largely of mathematics and statistics have something to unlearn. Such a belief implies the
emasculation of the basic substantive nature of science. Mathematics is content-less, and hence not
– in itself – empirical science … rigorous treatment of content or subject matter is needed before
some mathematics can be thought of as a possibly useful (but limited) partner of empirical science” (Guttman, 1971, p. 42).37 Schönemann quotes the sceptical Guttman: “There
remains the danger of seeking data merely to fit axioms”, and comments: “In hindsight, these
warnings sound positively prophetic in anticipating the present malaise in mathematical psychology some 20 years before Cliff (1992) noticed it …”. Schönemann (1994, p. 294) also
mentions Narens and Luce (1993) as critics of the malaise.
As early as 1975, a pioneer of mathematical psychology, William K. Estes,
complained about the shortcomings of his field: “… it is clear that many investigators
in our field are not entirely happy with their current situation” (Estes, 1975, p. 263). He
laments the chasm between mathematical and content orientated psychology and,
to support his own review, quotes Leont’ev and Dzhafarov (1973, p. 20): “An
analysis of the present situation shows that contemporary psychology and contemporary mathematical instruments are still not compatible enough with one another to allow mathematization to
assume a central place in the development of psychological knowledge; the reason for this is not
only the low level of sophistication of the latter … What is required is a continual interaction
between mathematics and psychology, an interaction that … would lead to a revision of existing
mathematical methods into forms more amenable to the proposed mathematized conceptual systems.”
Access to psychological content is not primarily achieved through formal
models but through the totality of experience in psychological domains.
Knowledge and belief acquired by previous experience are only specified and
tested by research. In this process, accounts that can be communicated in everyday language may play a significant role. The fit of formal modelling of psychological data should be evaluated by taking into account non-formalised information.
Critical writers have repeatedly commented on this:

37

masses of data to analyse. Today it is often hypothesised that these parameters can be dreamt up
and that they only need to be shaken out by trial and error through model fitting.
This quote was taken from an article by Barrett (2003), who criticised conventional psychometrics because its measurement operations rely upon untested assumptions of quantitative structure for psychological attributes (intelligence, personality). He pleads, alternatively as it were, for
openness toward application issues. Barrett propagates so called “applied numeric” that would
liberate research from the constraints of questionable theoretical expectations and be pragmatically much more useful (a rebuff of theoretical claims).

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Chapter 1 – Critique of the simple structure doctrine

 “Neither algorithmic sophistication, nor axiomatic rigor alone are apt to advance our
knowledge much if they are cultivated in an empirical vacuum” (Schönemann, 1981,
p. 412).
 “May I … insist once again on the absurdity of divorcing the mathematical or statistical
evidence from evidence procured by other means? … The sole claim of mathematical analysis should be to verify, by appropriate calculation, the hypotheses commonly advanced on
the basis of much broader and more general lines of evidence.” (Burt, 1949, p. 107).
 “Let us try to be free of … a priori mathematical and statistical considerations and prescriptions – especially codes of permission. Instead, let us try to think substantively …
and focus directly on the specific universe of observations with which we wish to do our
business.” (Guttman, 1971, p. 346).
 “Factorists with more mathematical training than the rest of us have been addressing
themselves to problems … on a technical rather than upon a fundamental level… In
most cases [their models are] irrelevant … to the problems … of those for whom factor
analysis is a research tool…” (Butler, 1969, 252-3).
 “These techniques [“for rotating factors into ‘psychologically meaningful positions’”] …
became the stock in trade of practicing factor analysts … It is probable that future historians will be severely critical of them and of their users; critical of the techniques because of
… their users’ extravagant claims on their behalf.” (Maxwell, 1959, p. 228).
Given the growing formalistic alignment of psychological research, we should go
out of our way to nurse and nourish non-mathematical methods of knowledge
acquisition. These sources of information are often pushed aside as merely “phenomenological” or “hermeneutical”. But if they were allowed to operate “on a
fundamental level” (Butler) and if they were required to critically review the formalists’ “extravagant claims” (Maxwell), one should keep them alive. In the universe of our knowledge, mathematical resources can fulfil only part of our desires38. Precision in detail and ingenious operations with numbers are useful, but
not always essential. Numerical tools can be harmful if wrongly designed or excessively applied – as the SSM debacle shows – while more basal, comprehensive,
holistic, albeit possibly at times less focused methods for acquiring knowledge are
needlessly forced to stay outside39. Gawronski (2000) also pleads for a holistic
approach in research in order to examine whether methodically carved out observations are in fact reconcilable with our comprehensive background knowledge.

38
39

This is also the primary concern of dissident Sigmund Koch (1999) and other lone voices criticising science that is continually drifting off toward one-sided objectivism (Bridgman, 1959).
S. Jevons (1873): “I tend to grumble about mathematical writers because so often they cheer all the things they
can do without pointing out that what they do is just the very minutest part of what could actually be done. They
exhibit the general tendency of not even mentioning the existence of stubborn or intractable problems …” (quoted in Rescher, 1985, p. 124).

Chapter 1 – Critique of the simple structure doctrine

41

Discussion of chapter 1 and outlook
This chapter does not downgrade FA, but just its habitualised wrong application
of today. Gigerenzer and Strube (1978) emphasised that applying research methods in psychology should always go along with “critically scrutinizing their assumptions”. By adhering to this recommendation, Yates’s ideal could have been reached
earlier. Yates considered a paradigm change inevitable,40 because without fundamental changes the “pathology” of factor analytical research could not be cured
while the “morass” of its previous results would endure.
Ultimately, the ritual called Little Jiffy, decried by Gigerenzer and Strube
(p. 81), should come to an end. (Little Jiffy is a recipe-like application of FA culminating in a Varimax rotation). Also, the spirit of criticism that was revitalised at
the seminal “Munich Symposium” and withered away again needs to be revived.
Its proponents were Kallina (1967), Kalveram (1970), Gigerenzer and Strube
(1978), and Revenstorf (1980). By employing FA guided by CSM (see introduction
to Varimin in chapter 2) the Thurstone doctrine will certainly be shaken up. After
further refining this approach, perhaps debugging it, an optimal understanding of
variance sources underlying manifest observables may eventually be achieved.
Even Henry F. Kaiser (1927–1992), who created the preconditions for Little
Jiffy, might readily have supported a radical new orientation of factor transformation – unfortunately he passed away too soon. In closing a talk on “Second
generation Little Jiffy” to the Psychometric Society he explicitly offered, should
such a situation arise: “For the future, I can assure you of one thing: if any of you folk …
come up with some Big Breakthroughs I shall be waiting in the wings ready and eager to paste
them together to produce the next generation Little Jiffy” (Kaiser, 1970, p. 414). In his eulogy, Gene Glass (1991) characterises Kaiser as “disrespectful”, a trait that seemed
to suit Kaiser well, because, Glass continued: “Irreverence must be a necessary ingredient
in the recipe for creativity. Whoever worships received wisdom too ardently will never see beyond
it.” (p. 159-171).
Complex structure modelling might appear “disrespectful”, because it turns Kaiser’s Varimax criterion upside down. But Kaiser might have welcomed such insubordination, since he practised it himself and expected his students – and probably those of coming generations – to do likewise.

40

Some current formal modelling experts seem categorically to exclude any “paradigm change”.
Rost (2003), for instance, talks about “laws cast in iron that will survive fashion tides … and will
even emerge from them stronger”. Rost believes that psychology’s zeitgeist can affect the use of
methods only marginally, but would not lead to “the ... arsenal of research methods proving to be wrong
or unusable. Rather [fashion] will result in the method arsenal being expanded and broadened by important aspects.” To complete his generalising review, Rost would have to add that flawed methods contained in this arsenal can cause huge damage, and must be fundamentally redesigned.

Chapter 2
Finding complex structures

Introduction
Empirical observations never assert themselves more vigorously than when they
thwart our expectations and disabuse us. When I began contemplating Varimin
rotation as a possibly better alternative to Varimax, I was concerned that my expectations might be frustrated. I could merely hope that the new method would
surprise me by revealing complex structures (CS) of analysed variables and that
they would model examined domains more appropriately than conventional simple structure (SS) procedures.
My uncertainty was justified. Varimin structures of FA could not differ too
much from initial structures which are complex to start with. If complexity of
factorial structures were actually as veridical as I kept hoping it, why then had the
partial advantage of complex initial solutions not been recognised during the decades of practice with this method? I feared that Varimin rotation might increase a
model’s complexity to such an extent that valid models of psychological reality
might be missed. Without knowing exactly where complex structure modelling (CSM)
was headed, this could not be ruled out.
Are factors of complex solutions interpretable? Thurstone and his followers
maintain that an interpretation of extracted factors requires a simple structure (SS)
transformation. Apparently, they had not considered or were not aware of a sim-

44

Chapter 2 – Finding complex structures

ple method, useful for feature and componential interpretations, developed by
linguistics, by phonologists in the first place (1952), called “minimal pair comparison”. Two variables A and B may both be complex because each bears multiple
latent features. These cannot be captured in their entirety. But if (n-1) features of
A and B are equally pronounced and if only one feature has opposing characteristics (e.g., positive vs. negative factorial loadings), then it is possible, by comparing
the meanings of A and B, to attribute a conspicuous difference between A and B
to the contrasting feature only (cf. Table 2.01).
Table 2.01: Clarifying the minimal pair procedure: Variables A and B display contrasting values of feature d only. Similarly pronounced characteristics
of A and B (a, b, c, and e) need not be considered if the characteristic,
causing contrast (d), is to be identified.

A chair and a stool differ by the presence or absence of a backrest, a mare and a
stallion by their gender, a mountain and a hill by their height. It is more reliable to
identify an individual contrasting characteristic by minimal pair comparison, as
manifested by Varimin CSM, than to try to embrace communalities in a cluster of
unanalysed SSM variables. The issue of interpretability of factors will be examined
further under “Question II”.
In this chapter, the advantages of complexity-oriented factor analyses will be
discussed by using ten empirical test runs. For this purpose, data sets are preferably used whose characteristics are largely transparent and almost self-evident even
without factor analyses. Why did methodologists hardly ever make use of this
obvious testing strategy when the efficiency of their procedures needed proof?

Chapter 2 – Finding complex structures

45

Questions and Test Runs
Chapter 1 found fault with conventional EFA. It was argued that correlations
among manifest variables of an empirical domain were generally based on multiple
variance and covariance sources. Individual variables do not reveal simple structures; but almost exclusively complex structures. Thurstone’s SS principle presuming a preponderance of monofactorial conditions of manifest variables was discarded as misconception.

Figure 2.01: Simple structure (A) and complex structure (B) of relationships between sources of covariance
(possibly latent) and manifest variables.

Thurstone’s simple structure model (SSM) differs from the complex structure
model (CSM) as follows: SS rotation (Figure 2.01A) assigns only one extracted
factor to individual manifest variables, or as few factors as possible. On the other
hand, CSM rotation (Figure 2.01B) aims to link individual variables with as many
extracted factors as empirically possible. How is this goal achieved? By replacing
SSM rotation with its inverse. The most frequently used SSM procedure, Varimax
(see Figure 2.02), is replaced with what I call Varimin, a term denoting that what is
maximised by Varimax is minimised by Varimin. Varimax increases the variance of
the squared factor loadings per factor by pairwise rotation of the factor coordinates. This procedure is repeated until the sum of loading variances for all factors

46

Chapter 2 – Finding complex structures

cannot be increased any further. Criterion V (cf. equation 1), which is the Varimax
criterion, is maximised. Varimin coordinates are rotated with the aim of iteratively
reducing the variance until the sum of the squared loadings, i.e., criterion V, cannot be reduced any further.

b
V  n  jp

p 1 j 1  h j
m

h = factor communality
b = factor loading

n

4

m  n b2

jp


  2

p 1  j 1 h j






2

Equation 1
p = running index for factors 1 to m
j = running index for variables 1 to n

In transformations towards SS, orthogonal procedures are the method of choice,
as it also is for Kaiser’s Varimax-rotation (cf. Figure 2.01). But unlike SSM, CSM
does not also consider oblique rotation, oblique rotations had been introduced to
come closer to the SSM utopia which runs counter to a CSM realism.

Figure 2.02: Number of articles with reference to factor rotation aiming at simple structure (SS). Result of
an Internet search with keywords indicating rotation procedures (Science Direct, Elsevier,
2008).

Chapter 2 – Finding complex structures

47

Assuming an initial structure of Figure 2.03A is available, applying Varimax transforms this structure into structure 2.03C. In 2.03C, the coordinates intersect clusters of variables, and for individual factors the sum of their squared weights is
maximised. Applying Varimin rotation transforms 2.03A into 2.03B. The distance
between the clusters of variables and the coordinates is increased as much as possible. The sum of the squared weights for the factors is minimised.

Figure 2.03: Two-factorial initial solution with fictitious variables (A) after Varimin rotation (B) and
Varimax rotation (C).

Introducing Varimin as a procedure for factor transformation engenders new
questions. Six of the most urgent ones are dealt with in the following, including,
wherever possible and advisable, support by empirical checks.
Question I:
Why use complex structure rotation at all?
Are initial solutions not complex enough?
A general result obtained by numerous Varimin rotations allows the conclusion
that initial factor solutions tend to reliably announce the result obtained from
Varimin rotations, but only where not more than two factors are validly interpretable. With three and, above all, more factors the results of unrotated and Variminrotated factors may considerably differ. Three and more varimin-rotated factors of
an analysis are generally more interpretable compared with factors from initial
solutions. Since they are more interpretable in case of k>2 substantive extracted
factors, I recommend that Varimin rotations should always be applied to initial
factors disregarding that – with only two interpretable factors – numerical differences of loadings between initial and Varimin-rotated factors are generally negligible.
Why are solutions with three or more Varimin-interpretable factors less safely
interpretable on initial extraction levels or not at all interpretable? The reason is

48

Chapter 2 – Finding complex structures

that once a factor has been extracted, the variance contained in a correlation matrix is exploited for that factor more than optimally. After initial (PCA) extractions, factors are uncorrelated (they show zero intercorrelations or intercongruences). Factors of initial solutions therefore are exactly orthogonal to each other.
The mathematical constraint towards zero congruence among extracted factors
ignores empirical deviations from an imposed model of complete factorial independence. To some extent, Varimin considers not quite model-fitting underlying
correlations among factors. Therefore, after applying Varimin rotations some
small factor intercongruences are generally noticeable. For comparison, after applying Varimax, intercongruences are considerably larger. Thus, given orthogonal
rotation in both cases, Varimax does not achieve orthogonality as perfectly as
Varimin does and minor deviations from perfect independency of factors by
Varimin rotation is a reasonable compromise with empirical conditions.
 Question II:
How can Varimin-transformed factors
be interpreted?
SS rotation of factor solutions is traditionally deemed necessary because initial
factor solutions are complex and factors are regarded as not interpretable if manifest variables have more than one substantial factor loading (Guilford, 1952, p. 27,
Burroughs & Miller, 1961, p. 37, Überla, 1971, p. 175, Gorsuch, 1974, p. 162,
Comrey, 1978, p. 653 f., Reise et al., 2000, p. 292).
The argument that factorial complexity of variables impedes or even prevents
interpretation loses its weight if the concepts ‘distinctive features’ and ‘minimal
pair comparison’ are considered methodically. These concepts have been developed in phonology by Jacobson and Halle (1956). They have been widely
acknowledged as a significant methodological improvement in linguistics. In a
generalised form, the logic of this procedure can be transferred to and used for
other disciplines.
The following example provides details: Phonemes are distinguishable by “features”, provided they have a distinctive function within a specific language. By
individual observation and objective research, three categories of features were
found to be relevant for every phoneme specifying their articulation: Duration of
articulation (short vs. long)41, sonority (voiced, unvoiced), and the location of articula41

Other distinguishing features generally used in determining the articulation mode (plosive,
fricative, nasal, etc.) are more specific and cannot be determined with the phoneme sample chosen here. [m] and [f] are phonemes with longer articulation (as opposed to short articulation of
the plosive sounds [b], [p], [t], [k]). The articulation of [m] is categorised as a nasal sound,
whereas [f] is classified as a fricative sound. As all plosive phonemes are short, it would have
been possible to create a bipolar category of “plosive” vs. “non-plosive” phonemes and to expect these to emerge as a factor in this assessment.

Chapter 2 – Finding complex structures

49

tion (bilabial, labial-dental, etc. all the way to uvular). Here, the terminology of FA
may be applied, because it is safe to say that phonemes are manifest units. Underlying (“latent”) sources of variance, distinctive features, are conceivable as factors.
Every phoneme can be defined by three sources of variance. [b], for example, is a
phoneme defined by short duration of articulation (plosive), is voiced, and is bilabial (formed with both lips).
Now, how can the three features of phoneme [b] be detected? Minimal pair
comparison is required. To form a pair for [b], another similar unit from the same
domain is needed. In German, the phoneme [p] shares two features with [b] (same
length and location of articulation), [b] and [p] differ only regarding sonority. Using phonemes as variables including [b] and [p], their differences/similarities can
be rated, yielding a matrix of intercorrelations. A factor analysis should extract
three distinctive features as factors, and a subsequent Varimin rotation should
identify the three factors F1, F2, and F3.
Phonemes [b] and [p] should show similar loadings for, say, factors F1 and F2,
but then F3, should show contrasting loadings. A factor analyst would only need to
interpret the difference between [b] and [p] in F3: In this case, F3 would exhibit
the difference regarding sonority. This interpretation could be verified if more
minimal pairs were put together taken from this data set, for instance, the pairs [d]
vs. [t] as well as [g] vs. [k] which all differ regarding sonority. They would exhibit
the same F3 difference42.

Test run 1: Evaluating phoneme similarities
(Data new: unpublished)
The above considerations were used in an experiment with two German-speaking
students, one a psychology student with an obvious gift for languages and the
other a student of advanced linguistics with phonology as sub-discipline. Both
participants were asked to assess similarities among 10 German phonemes ([b],
[d], [f], [g], [k], [m], [n], [p], [t], [v] on bipolar seven-point Likert-scales. These 10
phonemes under investigation were used in pairs with all combinations (=45
combinations) in random succession.

42

The factor analytical research strategy favoured here can be directly tied to theoretical and empirical approaches, where concepts and other cognitively represented objects may be perceived
as bundles or as structures of more or less latent components (“component model” of objects
with common settings (Feger, 1979), “Feature Pattern Analysis” (Feger & Brehm, 2001). Also,
the strategy is in line with the efforts to explain the phenomena of “experiencing similarities”.
Shepard (1974) analysed them as “hidden structures” and Tversky (1977) as “collections of features”.

50

Chapter 2 – Finding complex structures

The phonemes had to be rated according to their similarities on scales such as
[d] 3---2---1---0---1---2---3 [m]
[b] 3---2---1---0---1---2---3 [d] etc.
and on 43 other scales. If a phoneme represented a scale polarity, e.g., if [d] was to
be rated on a scale like [d] 3—2—1—0—1—2—3 [m], the participants were instructed to mark level 3 on the left (maximal ‘resemblance’ or identity).
The ten resulting profiles comprising 45 judgments each were intercorrelated
for each participant and subjected to independent principal component analyses
(PCA). The first three extracted factors (=components) were Varimin rotated,
since three interpretable factors were expected (eigenvalues of the first five factors: 2.23, 1.58, 1.40, 1.12, 1.05 for participant 1, and 4.86, 1.68, 1.09, 1.03, 0.59
for participant 2).
The validity of Varimin rotations is estimated by how well the factorial loadings comply with the expected classifications. According to phonological classification, the following phonemes have short duration of articulation: [b], [d], [g], [k],
[p], [t] (called plosives). The following phonemes [f], [m], [n], [v] are articulated
longer with applying continuous air flow. Phonemes [b], [d], [g], [m], [n], [v] are
voiced, whereas [f], [k], [p], [t] are unvoiced.
While articulation duration and sonority result in bipolar classifications without
further differentiation, the location of articulation has four alternatives with linear
order. In the following examples, the location of articulation ranges from “fully in
front” to “far back”: (1) bilabial phonemes [b], [m], [p], (2) labiodental phonemes
[f], [v], (3) alveolar phonemes [t], [d], [n], and (4) velar phonemes [k], [g]. As a next
step, the loadings of the Varimin rotated factors, based on participant data, are
related by point-biserial correlations to length of articulation (short=1 and long=2)
and sonority (unvoiced=1 and voiced=2). The four ordinal levels of the location of
articulation (1=fully in front, to 4=far back) are product-moment correlated with
the obtained factor loadings.
This correlation is shown in Table 2.02. It can be seen that the correlations are
almost all larger than .90. The only exception is the psychology participant’s correlation for articulation location (r= .671). F1 represents articulation duration while
F2 represents sonority. Apparently, because of her training, the linguistics student
had acquired a finer ability to perceive locations of articulation (the Tucker F3
congruence for both participants amounts to only .574). But as the loadings for F3
are much larger for the linguistics student than for the psychology student, averaging both sets of data with Fishers Z transformation results in an F3 correlation of
r=.922 with articulation location. It is thus safe to display unified sets of data for
all three factors.

Chapter 2 – Finding complex structures

51

Table 2.02: Correlations of factor loadings with objective rankings and Tucker
congruencies of factors for two female participants.

Figure 2.04 shows the Varimin results of the two students’ combined data. Positive factor loadings are represented by dark circles, negative ones by lighter circles.
The different sizes of the circles represent absolute loadings. Zero loading would
be represented by a point without dimension. If these three factors were not already correlated with expert judgments, the minimal pairs [t] vs. [f] and [d] vs. [n]
might have been formed for F1, articulation duration as a distinctive feature. In
case of F2, the minimal pairs of “sonority yes” vs. “sonority no2 would have stood
out: [b] vs. [p], [d] vs. [t], [g] vs. [k] and [v] vs. [f]. In F3, the minimal pairs [p] vs.
[k] and [b] vs. [g] would have manifested the “front – back” contrast of articulation locations
Varimax results of the two students differ considerably. The linguistic student’s results are as follows: Varimax F1 with positive sign clusters long, voiced phonemes [m], [n], and [v], and with negative sign the short, unvoiced phonemes [t] and
[k],. The short, voiced phonemes [g] and [d] with positive sign are clustered by F2.
The long, voiced phoneme [f] has a negative F2 loading. Unipolar factor F3 clusters
the bilabial plosive phonemes [p] and [b]. It is clear that Varimax rotation clusters
similar phonemes. But these clusters have multiple phonetic features that do not
stand out as phonetic features. Phonetic features underlying similarities and differences among phonemes are not factorially discovered, rather they are disguised.
(The results of the psychology student would not change this conclusion). Thus,
CSM-transformed factors (by Varimin) can be interpreted as latent sources of
variance without difficulty, as long as minimal pairs are formed. The interpretation
of factors of SSM-oriented factor analyses is considerably more difficult, since
global similarities among factorially clustered variables, based on underlying multiple
features, are unsuitable in principle to identify these features. Seemingly paradoxically, the CSM result is simpler than the SSM result.

52

Chapter 2 – Finding complex structures

Figure 2.04: Varimin-transformed factor loadings of ten German phonemes (based on similarity judgments).

Chapter 2 – Finding complex structures

53

Differential results from Varimin and Varimax factor rotation are just as convincing in a study using similarity judgments of British coins.

Test run 2: Similarity judgments of British coins
(Novel data: unpublished)

British coins were chosen because similarity ratings of coins, even more than of
phonemes, are based on perceivable objective features. i.e., with British coins on
size, shape, and colour (see Figure 2.05). Coins of adjacent values differ in size
(e.g., 1 pence small, 2 pence large, 5 pence small, 10 pence large, etc.).
Natural pairs of coins are thus formed by their size, additional pairs or groups
suggest themselves with colour of the metal (e.g. 5 and 10 pence are silver, 1 and 2
pence are not silver) or with their shape (e.g. 5 and 10 pence are round, 20 and 50
pence are heptagonal). These features are expected to influence the similarity ratings
among these coins. The diameters and weights of the coins are given in Table
2.03.

54

Chapter 2 – Finding complex structures

Figure 2.05: British coins with the attributes of colour, size, and form.

Chapter 2 – Finding complex structures

55

The main experiment was conducted using a German student (TS) attending a
college in Cambridge, UK, She was asked to rate all eight current British currency
coins (1, 2, 5, 10, 20, and 50 pence and 1 and 2 pound) according to similarity. The
coins were stuck on cardboard and presented in pairs: Each coin was paired with
every other coin, beginning with the pairs 1 penny43 – 2 pence, 1 pence – 5 pence etc.
up to the pair 1 pound – 2 pound, with 28 pairs in all.
The student was asked to hold one of the eight coins and compare it with all
pairs of coins on the cardboard. On a seven-point bipolar Likert scale she had to
indicate whether the coin in her hand resembled the coin on the left or the coin
on the right on the cardboard. For example, she may think the 50 pence coin resembles the 5 pence coin better than the pound coin. In this case she should mark
the 5 pence – 1 pound scale at a point close to the 5 pence side. An all embracing, holistic judgment was requested. All features influencing similarity and difference
were to be considered concurrently, but the face value of the coins was to be ignored. The eight coins obtained 28 ratings each.
Table 2.03: Diameter and weight of British coins.

In this manner, an assessment profile for each coin, consisting of 28 individual
similarity ratings, was created. The profiles for the eight coins were intercorrelated,
the correlation matrix was subjected to Principal Component Analysis (PCA), and
the extracted factors were subjected to Varimin and Varimax rotations. I expected
that the features of the British coins would assert themselves factorially by a
Varimin rotation, but not by a Varimax rotation, and that the Varimin rotation
would show the features of coins better than the initial solution.
The results (Figure 2.06) met all expectations44. The most prominent factor
(F1), distinguishes with positive and negative signs markedly silver coins from
non-silver ones. The second factor reveals, with bipolar loadings, the larger and
smaller coins. The third factor distinguishes between the two shapes of coins45.
An interpretation by meticulous minimal pair comparisons is not necessary, given
the transparency of this “latent” set of conditions.
43
44

45

Colloquially, one pence is often preferred to one penny, which is the correct expression.
Circle diameters indicate loading levels. The lighter circles represent positive loadings, the dark
ones negative loadings. Numerical values are displayed in small print in the cells of the matrix.
Percentages of factor communality appear below the columns.
The slight variations in loading levels of F2 (especially the 2 pence coin) may possibly be attributed to the subjective feature evaluations of the coins. Actually, the 2 pence coin seems very
large considering its low value. Size seems to make a greater impression than shape.

56

Chapter 2 – Finding complex structures

Which information does the initial solution provide? (see Figure 2.06): Among the
initial factors, F2 may be interpreted as a manifestation of coin size. F1 seems to be
the colour factor, but the loadings of the 5 and 10 pence coins do not tie in with
colour. F3 cannot be interpreted as a shape attribute and remains a mystery in the
initial solution. The initial solution is thus less satisfactory as are many other initial
results of other such studies. This suggests one practical conclusion: PCA factors
should always be rotated with Varimin, even if the result does not differ significantly from an initial solution.
The Varimax solution: Only in half the coins is the ideal of a monofactorial
loading achieved. Both the 10 pence and the 20 pence coins deviate significantly
from a solitary ideal loading. While multiple factor loadings have always been
common in simple structure practice, they were tolerated as an exasperating nuisance. More importantly, Varimax does not reach descriptive simplicity at a contextual level. Neither colour, nor form, nor size is represented by Varimax factors
– they are not even hinted at.
To test whether the results, obtained with participant T.S., may be generalised,
the experiment was repeated with eight students who had no experience with
British coins. The results were averaged and analysed in the same manner as were
the results of T.S. The eigenvalues 2.07, 1.90, 1.30, 1.20 allowed an extraction of
four factors. In the case of T.S., only three factors were extractable (eigenvalues:
2.87, 1.92, 1.24, 0.97).

Chapter 2 – Finding complex structures

57

Figure 2.06: Varimin (A), Initial (B) and Varimax-transformed (C) solutions. Similarity judgments of
British coins.

The factorial congruences of the solutions to be compared, i.e. of T.S. and the
students, were significant for F1 (factor of colour) and for F2 (factor of size)46,
amounting to .981 and .973 respectively. However, there was not even a hint of
congruence between F3 of T.S. and either F3 or F4 of the students (.312 and .224,
respectively). However, the students’ data showed a high correlation between the
eight F3 loadings and the eight face values of the coins, while the largest correlation T.S. achieved with face values was only .14 (for F1).
It seems that the similarity judgments of the students were influenced not only
by the colour and size of the coins but also by their monetary value. They ignored
the shape of the coins which was clearly manifest for T.S. as F3. It had been men46

The interpretation of Varimin F2 as a factor of coin size is supported by correlating the F 2 loadings with the measured size of a coin: r = .81 (T. S.), r = .85 (students). The weight of the coins
in grams only marginally correlates with the factor loadings, although it correlates with their size
up to r = .80.

58

Chapter 2 – Finding complex structures

tioned in the written instructions to the students that the monetary value of the
coins should be disregarded, but this was less emphasised compared with the oral
instructions given to T.S.
 Question III:
Interpretations of factorial simple structure-solutions
must have been fairly satisfactory in the past,
why else would they have been constantly used?
The variables in the studies referred to so far were physical objects whose perceived features had been assessed and compared by test participants. Such data are
rarely subjected to SSM factor analysis. Similarity judgements and ratings of complex givens are unsuited for SSM analyses. Alternative multidimensional procedures are used for this purpose, e.g. multidimensional scaling (MDS).
In psychology, FA is extensively used for rating people, for self-and external
assessments, assessing personality traits, behavioural dispositions, attitudes etc.
Traits and behaviours are rated by verbal items without objective references. The
validity of factors extracted from semantic material cannot be appraised by objective means. Factors extracted from such material are prone to noncommittal interpretations with considerable subjective freedom.
I think the lack of rigorous methods for validating factors obtained from verbal material is one of the reasons why FA of psychological data has often also
been regarded as fairly satisfactory. The reason – I guess – is that the modelling
quality of SSM factors cannot be based on evident criteria, as is the case with
phonemes or coins. Factors based on semantics can easily somehow “make
sense”, they can almost always be interpreted in one way or another. Using verbal
variables, denoting mental or otherwise non-evident matter, deficiencies of SSM
cannot readily be identified. Although occasional reports exist about significant
correlations between factorial self- or external assessments and objective behavioural data, factor analysts have to construe meanings conceptually anyway for
determining latent sources of factorial variance.
In the following study, correlations between rating preferences are factoranalysed. At this stage, semantic problems are avoided because the variables used
are adverbs without content indicating mere degrees of Likert-scale judgments.
They are taken from a study by Carl (1968) about response-style behaviour. With
the help of expected Varimin results it can be shown, nevertheless, how meanings
of SS factors are construed.

Chapter 2 – Finding complex structures

59

Test run 3: Differentiation of response styles at responding to questionnaires

(Data: Carl, 1968).

Carl (1968) aimed to determine response sets of participants responding to Likert
scales with five response alternatives. He collected from N = 580 participants
Likert ratings for 580 items of the Minnesota Multiphasic Personality Inventory (MMPI)
from 100 persons. Across items he summed the ratings 1 to 5 separately for each
rating point and for each participant. The scale ranged from “strong approval” to
“strong disapproval”, intermediate steps of the scales were not verbalised). Carl
excluded artefact correlations by separating five subsamples of items, rendered
parallel with regard to content, one subsample of items served for one rating
point.
For the present purpose, a 5 x 5 intercorrelation matrix was picked from Carl’s
paper, each correlation was based on frequencies of usage of respective rating
points. For this matrix, the initial PCA factor structure was determined that was
rotated using Varimin and for comparison, also Varimax.
An expert in response set research would expect certain results: Acquiescence,
the tendency to respond affirmatively, is well known and should generate a factor
with positive loadings for the two affirmative response alternatives and negative
loadings for the two disapproving response alternatives. Equally well known is the
tendency to give extreme responses, and this should produce a factor in which the
two extreme alternatives, one for extreme affirmation and one for extreme disapproval, should have positive loadings and the intermediate alternatives between
the extremes negative loadings. At least these two factors should come to light by
Varimin rotation.

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Chapter 2 – Finding complex structures

Figure 2.07: Initial (A), Varimin- (B), and Varimax structure (C) of response set data
(source: Carl, 1968).

The Varimin result is shown in Figure 2.07B. The expected acquiescence factor is
F3 and the extreme response factor F1. However, a substantial unexpected F2 is
evident and demands interpretation. The matter can be followed up with
Herrmann (1965), who identified what he called “distinctiveness of judgment”
(Urteilsnuanciertheit) as another response tendency. Accordingly, in scales with response alternatives respondents are not only distinguished by approving and disapproving a statement, but also by the extent to which they differentiate between
“approve” and “greatly approve”, “disapprove” and “greatly disapprove”, as well
as between “approve /disapprove” on the one hand and “undecided” or “don’t
know” on the other. In pertinent literature, this response variance is rarely discussed, presumably because it does not distort results to the extent that the other
response tendencies do (Hinz et al., 2003). In my opinion, factor F2 of the present
Varimin analysis may safely by interpreted as “distinctiveness of judgment”.
Are the three response sets already recognisable in the initial factor structure
(cf. Figure 2.07A)? The initial factor structure resembles the Varimin factor structure. In both solutions, the F1 factors are virtually identical. Hence, the initial F1
factor may also be interpreted as indicating an extreme response set. The initial
solution, however, is not satisfactory with F3, i.e., because the middle rating alternative, used for refraining from judgment, features a considerable negative loading. This should not be the case, because judgment abstention is supposed to be
dependent on factors other than negative judgment. Furthermore, the Varimin

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solution for F2 is more distinct. In the initial solution “disapproval” and “strong
disapproval” in F2 are not distinguishable. Also, the numerical difference between
“approval” and “strong approval” in the initial solution is considerably weaker
than in the Varimin solution. The initial F2 factor can thus hardly represent distinctiveness of judgment.
Our main concern here is the Varimax result (Figure 2.07C). How is this to be
interpreted? Comparing Varimax and Varimin solutions (2.07C and 2.07B) sheds
light on this. The Varimax rotation led to a bipolar clustering of “strong approval”
(positive loading in F1) and “disapproval” (negative loading in F1). Why? It is noticeable that the “strong approval” and “disapproval” loading signs have opposite
signs of the three features as differentiated by Varimin. Contrasts in the Varimin
feature profiles are also found for Varimax factor F3, a bipolar factor as well. In
the Varimin result, there is no polar opposite for the “undecided” category of
judgment, hence “undecided” remains fairly isolated in Varimax F2.
These findings conform to the results of the Varimin-Varimax coin factor
comparison. In short, variables with similar Varimin profiles of features tend to be
clustered by Varimax transformations47. In the case of bipolar factors, Varimax
also clusters variables with opposite profiles (with their signs reversed). Varimax
does not analyse features. Varimax clusters variables, using Varimin features, feature differentiation is thus prevented48.
What might a conventional factor analyst publish after a Varimax analysis of
these response data? He might argue that acquiescence is not a monofactorial
construct, as has been traditionally assumed. Rather, he would go on, a distinction
should be made between acquiescence I (F1) associated with extreme response
tendency and acquiescence II (F3) without such tendency. Moreover, he might interpret F2 as an “undecided” factor and simply ignore that F2 shows considerable
negative loading even for “strong disapproval”. He might rely on his SSM data
processing and believe he had discovered three new psychological constructs (acquiescence I, acquiescence II and ‘undecided’. Since the terminology does not
appear senseless, nobody might notice that these factors represent a rather useless
collection of variables, as they are constantly put forth by SSM. The following analysis of verbal data from an MDS study further exposes this non-committal
practice of interpretation.
47

48

Zimmermann (1953) remarked that “a test which actually contains variance on two or more factors may
appear with all of that variance confined on a single factor.” This he calls the “composite factor”. The author thus sticks to the literal meaning of the initial factors (centroid factors), which he deems
composed in the rotated factor. “It is my feeling that the failure to give composite factors the attention they
merit must be considered either a serious oversight or a serious error or omission” (Zimmermann, 1953, p.
389).
Overall (1964) explicitly claims: “Rotation to simple structure can be understood as an elaborate approach to
cluster analysis. It identifies clusters of tests which measure the same things, but there is no assurance that these
‘same things’ are simple and primary dimensions.” (p. 271), and “there is no need to assume that simple structure factors will correspond to any particular set of fundamental dimensions of the objects …" (p. 276).

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Chapter 2 – Finding complex structures

Test run 4: Semantic features of kinship terms

(Data: Marx & Hejj, 1989)

The participants were asked to sort cards hierarchically with 16 kinship terms. The
results were used by Marx and Hejj (1989) to determine a matrix of similarities (cf.
Tables 2.03–2.5, p. 112 in the original). The authors fairly successfully applied an
NMDS process to obtain the semantic terms of kinship from the resulting stacks
of cards49. To this end, the original matrix of sort sequences serving as indicators
for distance or dissimilarity (DS) in the 16 terms was mirrored diagonally and
extended into a square matrix. Then, the DS-measures were transformed into
measures of similarity (S) by S = 1 – DS/100050. The highest similarity value of
each relevant column/row was inserted into the diagonal. The columns of the
similarity matrix were subsequently correlated amongst themselves and the intercorrelation matrix was subjected to PCA. Five substantial factors were extracted51
and transformed by Varimin and Varimax.
Figure 2.08A shows the Varimin solution. The variance of loadings of the general factor F1 is only minimal. Apparently, F1 is a methodical product and may
therefore be ignored52. Factors F2 up to F5 are bipolar. With contrasting signs and
by applying minimal pair comparison, the expected semantic features stand out:
Lineality (F2), nuclear family (F3), gender (F4), and age or generation (F5)53. For
example, the minimal pair of brother and sister shows differing loadings only with
factor F4. It should thus be interpreted as the “gender” factor (male vs. female).
Other minimal pairs are easily identifiable, e.g., father and son or mother and daughter,
whose loading directions contrast only with F5, the “generation” factor. The factor
structure does not show subtler differences, e.g., distinctions between the youngest, the middle and the oldest generations (as for son, father, grandfather). Also,
differences due to the first-person perspective cannot be recognised by factors,
e.g. the distinction between (my) brother and (my father’s) son – denoting the same
person. The result shows the main kinship characteristics only.
Figure 2.08B shows the result of a Varimax rotation of the same factors that
were rotated by Varimin for Figure 2.08A. Of the five rotated factors, F1 and F2
cannot be interpreted at all and F3 only by taking considerable liberties. F3 combines grandparents and grandchildren. That may make some sense, since the genera-

50
51
52
53

These linear transformations were made to help interpret the individual values in the table.
Eigenvalues: 4.31, 2.34, 1.54, 1.09, 1.05, 0.90, 0.79, 0.75 …
F1 seems to be a result of the hierarchical clustering procedure.
These interpretations of Varimin factors F2, F4 and F5 are also found in a similar MDS study by
Romney and d’Andrade (1964) (gender, generation, consanguinity). In F2 (lineality), the present
study also differentiates between “nuclear family” and non-nuclear family (by F3). Moreover, the
study by Romney and d’Andrade offers an excellent introduction to the terminological and
methodological principles for componential analysis of concepts. It also gives insight into the
conclusions based on the existence of discriminative stimuli (sememes) in search of definitions.

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63

tional extremes in the same lineage are here grouped. This cluster might be termed
“very young or very old in the dominant lineage”.
Regarding factors F2 and F4, a certain similarity is noticeable between the
Varimax and Varimin results Varimax F2 loads the kinship terms of the non-lineal
lineage (“extended family”). Unlike the Varimin solution, Varimax does not indicate lineal kinship (“close relatives”) with opposite signs ([+] “feature present” vs.
[-] “feature absent”). Instead, all Varimax factors are unipolar. In F4, the members
of the nuclear family are grouped, but non-members again remain without a sign
(no negative sign). Also, the F4 loadings for granddaughter, niece, and female cousin are
quite high, which, however, has no semantic reason.

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Figure 2.08: Varimin- (A) and Varimax-rotated factors (B) of similarities among kinship terms (data
source: Marx & Hejj, 1989).

Nonetheless, while Varimax factors are not entirely uninterpretable, semantic
overlaps in this solution render most interpretations unsatisfactory. This becomes
evident by using the semantics of kinship terms. The sources of semantic similarity and dissimilarity (generation, gender, etc.), that should be revealed by FA, remain hidden in the Varimax solution. Grandparents and grandchildren are clustered according to lineality and generation, but Varimax does not distinguish the
features.

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Table 2.04: Data of hierarchical sorting (source: Marx & Heij, 1989).

 Question IV:
Can any substantial information be gained
from the bipolarity of Varimin factors?
The bipolarity of Varimin factor loadings deserves special attention. Bipolarity is
quite common in Varimin solutions, but less common in Varimax or other simple
structure solutions, if it occurs there at all. The occurrence of negative Varimin
loadings may mean that these variables (as opposed to positively loaded variables)
might have a detrimental effect on the respective co-variance source. For example,
in the response set data of Carl (1968) it may be assumed that variables with negative factor loadings might have an inhibitive effect: A subject giving many extreme
yes-no answers would obviously not give many moderate answers, and vice versa.
This is due to his/her inclination to either avoid or favour moderate judgments54.
To bipolar factors of acquiescence or distinguishability, motivational and functional interpretation of sign differences in factor loadings are applicable.
Bipolarity in kinship data has to be interpreted differently. For example, in the
factor for gender, bipolarity cannot be regarded as functional. The male-feature is
present not because the female-feature is absent. Rather, it is an organismic condition, precluding (as a rule) the feature female if male is present55. Lineality, however,
only needs a yes- or no-answer about the lineage position of a relative in the genealogical tree. In such cases, a positive or negative sign indicates the presence or
absence, respectively, of a feature. This interpretation of plus-minus signs is more
common in fields such as linguistics than in psychology.

54
55

Although it is possible that participants submitting many negations equally tick “undecided” or
give moderate affirmations, this does not seem likely.
This is not meant to comment on the spiritual polarity in “Animus vs. Anima” by Carl Jung,
which maintains that male and female tendencies occur in one and the same biological gender.

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If a negative loading of a Varimin factor indicates the absence of an effect or function, it may sometimes be concluded that this factor is irrelevant for the respective
variables. If a positively loaded bipolar factor is present, this may also depend on
other variables of the sample56. I therefore suggest one should always interpret
negative signs initially to mean “feature not present” only then, – and with contextual knowledge – a functional interpretation (constraint or polarised characteristics) should be made, if appropriate57 58.
The following transparent example should aid a better understanding of the
determinants of bipolarity in factors.

Test run 5: Intellectual development in childhood
(Data: Humphreys & Davey, 1988)
In a longitudinal study, Humphreys and Davey (1988) used four age-adjusted intelligence tests on children aged three months to nine years. The authors wanted
to determine the consistency of the children’s intellectual development in this
period. I copied their resulting intercorrelation matrix for 14 test repetitions in the
longitudinal section and applied PCA and Varimin- and Varimax-rotation. Figure
2.09 shows the results (the initial solution is not listed; it is virtually identical with
the Varimin solution). Only two factors are substantial (eigenvalues: 5.705, 1.473,
0.827, 0.788 …).
The first Varimin factor (communality of 40.7%) represents the stable percentage of general intelligence. In the longitudinal test period, the individual averages
of factor loadings roughly remain the same. The second factor is bipolar and
shows 10.6% communality. It represents the time-dependent variance in intelligence development. This variance is due to specific favourable or unfavourable
56

57

58

That is why multivariate causal models which were introduced by confirmatory FA and structural equation models are questionable. Modelling sources of covariance – which is the main objective of multivariate analysis – does not necessarily reveal causal sources.
Because the rotation method allots positive or negative signs irrespective of the assigned content of the measurements, it additionally has to be decided whether the signs, resulting from the
statistical program, should be kept or be reversed to improve their comprehension. To facilitate
comprehension, variables with a factor indicating functionally restricting influence or where the
factorially expressed material has to be regarded as not present, may be given, by sign reversal, a
negative sign if it is not present originally. It has become apparent (a rule of thumb so far) that
absence of a feature or negative influence of a variance source within a factorial vector tends to
appear specifically in those variables that have (1) negative loadings or (2) predominantly either
plus or minus signs.
A neglected hint from the beginnings of centroid factor analysis which grants informational
value to unrotated bipolar factors can be found in a note by Zimmermann (1953): “It is wellknown in dealing with intellectual variables that the first centroid loadings are usually all positive and the second
centroid, as well as those that follow, divide positive and negative variables equally. What is apparently overlooked is the tendency for the second centroid to split the most obvious dichotomy, the third centroid to split the
next most obvious dichotomy and so on. For example, if the battery contains both linguistic and quantitative
tests, the second centroid will most likely separate these two major groups …” (p. 389).

Chapter 2 – Finding complex structures

67

individual influences exerted on the participants: education, varying psychological
or physical condition or illness etc.
The bipolarity of F2 and the monotonous succession of the factor loadings are
to be interpreted as follows: The amount of covariance not exhausted by F 1 is
spread evenly across all ages. Changes within a single test interval are smaller than
across two or more test intervals. They are largest between the first and the last
measuring point.
Accordingly, test results of adjacent test intervals correlate more closely than
those spaced further apart. Test results in the middle range of the longitudinal
section are equidistant regarding their difference from results at the beginning and
at the end, as indicated by correlation coefficients. In the Varimin model, the testing occasion located in the middle between the first and the last testing is regarded
as the zero point of the F2 loading-vector. Factor F2 thus represents the degree of
individual instability in intelligence test performance with respect to the average
value of its stable level.
This example is informative in as much as it shows that the F2 factor loadings
are related to those in F1. The particular meaning of this relationship has to be
specified with the help of appropriate contextual knowledge. In the present case,
negative F2 loadings do not indicate missing or contrary effects or logical exclusion, but instead differences in temporal change of individual intelligence test
performance which may have, on an individual level, sometimes a positive and
sometimes a negative direction. As the F2 factor loadings are polarised positively
or negatively, a scale emerges showing changes in the test results (degree of fluctuation) steadily increasing in the course of this longitudinal study.

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Chapter 2 – Finding complex structures

Figure 2.09: Varimin (A) and Varimax solution (B) of longitudinal data of intelligence development
(data source: Humphreys and Davy, 1988).

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69

The Varimax solution eliminates the general factor and thus ignores the fact that
interindividual variance of intelligence test performance remains fairly constant in
this longitudinal study. Instead the Varimax solution comes up with two factors
which might be wrongly interpreted as two independent kinds of intelligence. One
of them (F1) might be conceived as affecting early childhood and eventually being
superseded by the functioning of the second sort of intelligence (F2) – an absurd
notion that is unlikely to be entertained even by SSM factorists. Given repeated
longitudinal section data, these researchers might simply regard their SS method as
unsuitable and would probably not use it.
A different example is shown in the following study: Here, the negative characteristics of a factor not only reveal the absence of a feature, but also indicate the
presence of a scalometrically independent feature.

Test run 6: Body size and body shape in cattle
(Data: Rasch/Weber, 1962)
Rasch (1962) gathered twelve measures of the body bulk of 107 female cattle
(heifers). Height, width, and length were measured. Our usual factorial processing
according to E. Weber’s intercorrelation matrix produces the bifactorial solution
(eigenvalues: 7.69, 1.20, 0.74 …) in Figure 2.10. The initial structure is not shown,
it is virtually identical to the Varimin structure.

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Chapter 2 – Finding complex structures

Figure 2.10: Varimin (A) and Varimax (B) solution of body size and body shape measurements in
cattle (data source: Rasch, 1962).

The first factor (communality 64.0%) represents the body bulk (body size, or
mass) without specifying spatial dimensions. It shows that cows varying above or
below average in height also tend to vary above or below average in length and in
width. Here, F1 can be interpreted as the general factor of morphometric measurements. The bipolar second factor (communality 10.1%) represents the variance
in body shape. Among large and small cattle relatively delicate (slender) or bulky
(fat) animals exist. The polarity indicates that growth in height occurs ‘at the expense of width’, as it were, and growth in width takes place ‘at the expense of
height’.

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In the Varimax solution the general factor of body bulk is lost. The variables slenderness and fatness are evident as well, but they are represented as non-polar orthogonal dimensions. The interconnected and somewhat opposing manner of
expansion in height and width does not take effect in the SSM result. However,
when using Varimin, this manifests itself as an influential bipolar factor for body
shape, aside from the dominant volume or mass factor59.
Even in psychological data, interpreting negative factor loadings as bipolar
traits can seem obvious, as demonstrated in the following test run.

Test run 7: Intelligence tests and performance tests

(Data: Holzinger & Swineford/Jöreskog & Sörbom, 2003)

Holzinger and Swineford’s data are used occasionally in educational textbooks to
demonstrate model calculations, for instance by Jöreskog and Sörbom (2003). The
table of intercorrelations was taken from their website (Google search: <LISREL
8.52 Jöreskog>).
Holzinger and Swineford used three tests each for visual, verbal and for speed
performances (speed tests), respectively. Figure 2.11 contains the Varimin and
Varimax standard results. Unipolar Varimin F1 (Figure 2.11A) apparently is general
factor g, general intelligence, which is generally expectable from intelligence test
series. Bipolar factor F2 contrasts, by positive sign, three speed tests against six
other tests (negative signs), where the emphasis is clearly on “concentration” and
“power” rather than on speed.
This is a case of bipolarity in the domain of intelligence. Knowledge of psychological context allows negative loadings to be interpreted as due to functional
opposition. A heightened ability and inclination of a participant for speed should
have a positive impact on speed tests. In tests requiring concentration and deeper
problem solving, tendencies and abilities to the advantage of speed will probably
have some negative effect. A corresponding counter effect is possible regarding
concentration skills and aptitude. These tendencies may be counter-productive for
tests requiring skills and aptitude for speed. This differential speed effect seems to
have given rise to the bipolarity of F2.

59

Bipolarity of factorial constructs has often vanished when treated with SSM: By SSM rotation,
responses to emotion items with negative valence turn out to be statistically independent from
responses to emotion items with positive valence. Against all common sense, SSM psychologists
believe that positive and negative emotions are functionally unrelated (Diener & Emmons, 1983,
Watson & Clark, 1988, late correction after model comparisons by Crawford & Henry, 2004).
Optimism and pessimism appear similarly independent in factorial SSM results. According to
common experience, these two attitudes are functionally bipolar (opposite) expectations towards the future (Marshall et al., 1992). The polarity in the gender typology is also lost by SSM
methods. Due to factorial orthogonality, generated by SSM, the traits femininity and masculinity
are deemed unrelated (Bem, 1981).

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The bipolarity of bipolar Varimin F3 needs another comment. It is noteworthy that
the bipolarity of F3 is not limited to verbal vs. visual performance, both belonging
to the “power” category. The contrast here is found as verbal vs. nonverbal performance. It is possible – but mere speculation so far – that an inclination and
ability preferring verbal performance might be somewhat disadvantageous (inhibitive) for nonverbal intellectual performance or, vice versa, that nonverbal inclinations and abilities are somewhat disadvantageous for verbal performance. This,
however, cannot safely be concluded60.

Figure 2.11: Varimin (A) and Varimax solution (B) of intelligence tests (data: Holzinger and
Swineford)

60

These tests should also be run with two additional instructions, with a speed instruction to
encourage the participants to solve the tasks fast and with a power instruction to emphasise quiet concentration and deliberation. Depending on the test, a differential increase or decrease in
performance will probably result.

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The Varimax solution (Figure 2.11B) causes g to vanish, and differentiates the
three types of tests. In this way, no antagonism is implied between these tests and
their factorial conditions. Jöreskog and Sörbom’s LISREL solution draw the same
wrong conclusion as is drawn from the Varimax solution of 2.11B, however by
exerting considerably more statistical effort. The authors thus verified the SSM
model and assumed three latent sources of variance that they deem independent.
Jöreskog and Sörbom did not find the more straightforward and theoretically
more plausible solution, something that Varimin has revealed without mathematical extra investment.
Bipolar solutions do not always have variables with positive or negative loadings, sometimes they may have near-zero loadings. This may actually be informative as shown by the following example.

Test run 8: Psychophysiological activity indicators
(Data: Köhler & Troester, 1991).

Köhler and Troester (1991) tried to validate palmar sweat (PSI, palmar sweat index) as an indicator of psycho-physiological activation. They tested 50 subjects for
three states of rest and for one state of strain (Strain: Participants were to successively subtract 7 from the number 2007). On these four test occasions the authors
collected 16 psychophysiological values per person: PSI values taken at the middle
and at the index fingers (PSI-M, PSI-F), spontaneous fluctuations of sweat production (SF), skin conductance level (SCL), and heart rate (HR). These five values
were taken 16 times per person and they were correlated intra-individually. The
correlations were averaged across all 50 participants.

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Figure 2.12: Varimin (A) and Varimax solution (B) of human physical measures (data: Köhler and
Troester, 1991).

Figure 2.12 shows the results of a factorial standard analysis of the intercorrelations. Because of their loading pattern and the order of eigenvalues 3.50, 0.76,
0.33, 0.20 … two factors were deemed interpretable. The initial solution is not
shown, as it is very similar to the Varimin-rotated solution.
Results: Varimin F1 is a general factor showing that individual differences of the
five measurements apparently have the same indicator value, i.e. activation.
Varimin F2 represents additional variance from skin conductance and heart rate.
The sweat variables seem to have nothing to do with this source of variance. This
is indicated by their near-zero loadings. The F2 loadings with opposite signs in
SCL and in HR seem to indicate that these variables have an antagonistic relationship: There are participants whose skin conductance reacts more strongly to activation than their heart rate. In others, the heart rate reacts more strongly than skin
conductance61.
61

Troester, the first author, briefly expressed approval of my Varimin interpretation of his results
in an email correspondence.

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Although both variables generally indicate the same function of activation in F1, a
small residue of variance remains. This can be explained by slight preferences for
either SCL or HR. One might speak here of a “forked effect”: The greater an
effect takes a direction X, the smaller it is for direction Y, and vice versa. Sweat
production is not influenced by these coordinated preferential effects.
Varimax solution presents two insoluble puzzles. The activation effect is split
into two independent components – puzzle number one. Physiologically, two
independent sources of activation are hardly imaginable. The second conundrum
has SRL belonging to one of the activation branches and HR to the other. Thus
Varimax succeeds in fouling up the sources of variance beyond recognition.
The above examples show the following: FAs of variables allowing for hypothetical complexities of their sources of variance engender results that can be
more easily interpreted than results that have been forced into the straitjacket of
SS. The question of latent conditions will be further dealt with in detail below.
Question V
Can CSM-orientated factor analysis
capture method-dependent variance sources?
In 1959, Campbell and Fiske introduced a new methodological technique, multitrait-multimethod analysis (MTMM). The authors tackled a previously largely neglected phenomenon. Personality researchers are not only confronted with a multitude of latent traits, they also have to expect variance whenever they use different testing methods to assess people’s behaviour. Soon, other determinants of
variance such as by changing samples of informants, were included (selfassessment vs. external assessment). Situational factors, possibly influencing the
covariances of measures (test rerun effects etc.), were included. In subsequent
decades numerous MTMM data sets were analysed. As multiple sources of variance were successfully revealed by using this “complexity friendly” MTMM procedure, it seems reasonable to expect corresponding outputs from CSM factor
analyses of MTMM data.

Test run 9: Knowledge test with varying test methods
(Data: Campbell & Fiske, 1959).
Campbell and Fiske (1959) developed a somewhat subjective procedure for revealing from MTMM data methodical sources of variance. Effectively, this amounted
to systematic inspections of tables of intercorrelations. The authors selected correlation data from published papers to demonstrate how their procedure worked.
For test run 9, one of Campbell and Fiske’s correlational data sets is taken, the
data had been used before by Cronbach and Vernon for other purposes – without
indicating who had collected the data. Apparently, students as participants had

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Chapter 2 – Finding complex structures

been submitted to some physics knowledge test that included subject matters of
mechanics and electricity, the tasks were presented either by verbal questions or
with the aid of visual displays.

Figure 2.13: Varimin (A) and Varimax (B) solution of knowledge tests (source of data: Campbell &
Fiske, 1959).

Figure 2.13 shows the results from applying Varimin and Varimax analyses. Considering the order of eigenvalues (2.67, 0.62, 0.45, and 0.27), traditional interpretation rules would call for interpreting F1, at most F2 in addition. By inspection of
these correlations, without FA, Campbell and Fiske identified effects of two
sources of variance62.
Results: Varimin F1 represents the general factor associated with a high proportion of communality. Intelligence differences amongst the participants and differences in diligence for physics may have contributed to this, as did, albeit to a far
lesser extent, F2 as a method factor (speech vs. image presentation) and F3 as the
factor for subject matter (electricity vs. mechanics).
Apparently, some participants found it easier to deal with verbal questions
while others preferred to solve problems with the aid of pictures (F2). Some seem
62

The unreliability of the criteria of eigenvalue ≥1 or the bend of eigenvalues in a Scree Plot, has
been noted before (Fabrigar, 1999, p. 287). The “parallel test” procedure also suggested by Fabrigar cannot be conducted with correlation data only. My own experience with re-analysing factorial data shows that the percentages of the summed communalities held by initial factors can
be helpful. A factor less than 10% of summed h2 is usually not interpretable. In Campbell and
Fiske’s data, the percentages of communality of the four initial factors amount to 66.7, 15.4,
11.3, and 6.7. If a minimum of 10% communality is set as a criterion, F2 and F3 still have to be
taken into consideration.

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77

to have had more interest and training for mechanics, others for electricity (F3).
Again, bipolarity turns out to be an indicator for competing conditions, not only
for the presence or absence of one single condition. Opposing effects in verbal vs.
visual test material are lost in a Varimax solution. The factorial content for electricity is not represented in a Varimax solution either. Only mechanics is deemed
to have been considered by Varimax (F3).
The last test run tries to establish whether Varimin is suited to expose a source
of variance that emerges when samples of informants are swapped.

Test run 10: Self-assessment and external assessment of children
(Data: Matson & Nieminen, 1987)

In a questionnaire survey on behaviour disorders, depression, and fear in children,
Matson and Nieminen (1987) used six scales with items to be rated by children.
They were shown to the children themselves and to their teachers. Figure 2.14
gives the result of their ratings. The eigenvalues are 3.98, 2.14, 1.41, 0.99, 0.86 …;
the percentages of explained variance in the initial factors are 33.1, 17.8, 11.8,
8.2 …, meaning that three factors are probably substantial.
The result of the Varimin solution is given in Figure 2.14A: The communality
for the source of variance in children and teachers is evident in the general factor
F1. A tentative interpretation suggests that the children’s actual dysfunctional
symptoms and their inter-individual variance may be expressed by the children’s
and teachers’ judgments. Caution is required, however, as in most questionnaire
surveys; because acquiescence may give rise to or boost a first factor. The argument that in this survey the children’s and teachers’ judgment may be different is
not valid because a disposition for acquiescence (tendency to say yes) in questionnaires may be equally distributed among teachers and children. Additionally, F1
might have had influence by social desirability (SD) in children and in teachers equally. The SD component is not identifiable, since all scales include negatively rated
experiences and behaviours.
The bipolar factor F2 seems to reflect differences due to variance of informants. As such, this is not in doubt, but its origin is not clear. The children might
tend to judge themselves as less maladjusted than their teachers did, or the other
way around, or they may be more or less prone to acquiescence or SD than their
teachers. These influences may add up in F1, much in the same way partial influences do.

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Chapter 2 – Finding complex structures

Figure 2.14: Varimin (A) and Varimax (B) solution of dysfunctional statements (source of data:
Matson & Nieminen).

The bipolar factor F3 apparently differentiates dysfunctional symptoms by opposite signs, both for children and teachers alike. On one side (by variables with
positive signs), disorders with disinhibiting effects (acting out, conduct disorder)
are clustered, and on the other side (by positive signs) disorders with inhibitive
effects (withdrawal, depression, anxiety). Here too, an antagonistic relationship
seems to be present. Dysfunctional dispositions may come into effect either with
or without inhibition. Disinhibition can lead to an unrestrained “acting out” (e.g.,

Chapter 2 – Finding complex structures

79

aggressions). Psychoanalytically speaking, the manifestation of fear occurs under
dissipating conditions.
The Varimax solution (Figure 2.14B) brings forth three clusters of scales for
pupils and teachers. One cluster combines the six scales assessed by pupils, the
second cluster combines the three scales for disinhibition assessed by teachers,
and the third combines the two scales for inhibition, also assessed by the teachers.
However, the Varimax result is marred by a number of runaway values and
multi-factorial loadings. Conceptually, these clusters are useless. A re-analysis of
Matson and Nieminen’s data again makes it clear that the term “latent” (invisible,
hidden) is not always an appropriate term for factors that are causal for manifest
variables. Their possible influence must be revealed by an analysis designed to
indicate the existence or non-existence of sources of variance. Whether they are
known or unknown, manifest or latent, does not play any role.
In the present case, Varimin F2 (informants) is explained by transparent conditions. Interpretation of F3 (dysfunctional inhibition vs. dysfunctional disinhibition)
had to refer to some less transparent psychological abstractions.

Discussion of chapter 2
Five methodological questions raised by Varimin, were addressed. Ten empirical
tests of its usage were conducted to help answer them.
Should the tests have covered more psychological data?
In my view, the efficiency of new research procedures should initially be put to
the test with data promising positive results, provided the procedure works. If
reasonably expected results are not achieved, it stands to reason that the procedure is faulty. This testing strategy, I believe, is too rarely used by psychologists.
Psychological data, above all descriptive verbal material from this domain, is not
suitable for testing FA methods. Even though language provides an abundance of
words and expressions denoting psychological phenomena, the semantics of mental experience and human behaviour is much less transparent than the semantics
of, say, kinship terms.
Obviously, FA, once it has passed its methodological checks, should be employed for solving problems in our discipline. Based on available methodological
research, a significant conclusion may be predicted even now: FA with Variminrotation of manifest verbal variables referring to psychological content is prone to
reveal essential components. However, these are more difficult to identify and to

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Chapter 2 – Finding complex structures

separate from each other than features we find in the perceivable and tangible
world we live in63.
Thus for interpreting Varimin factors of a psychological domain, factor designations based on manifest verbal variables (e.g., questionnaire items, lexical units)
cannot suffice, while users of SSM procedures took this for granted. Terms of
everyday language, e.g., trait concepts like conscientious, agreeable, open, sociable, etc.,
are not appropriate because it is not these terms in the first place, that we are interested in, but their sources of variance. An optimal approach should resemble
what we did for analysing components of the kinship domain, where the descent
to a “latent” level was required. Manifest terms like mother, sister, aunt etc. had to be
ignored as “merely manifest”, because their distinctive features were sought, not these
terms themselves nor their Varimax clusters64.
The empirical examples of this chapter using relatively transparent domains,
proved that Varimin factor solutions are indeed interpretable and, what is more,
could be interpreted more satisfactorily than Varimax factor solutions. Minimal
pair comparisons, permissible for complex Varimin interpretations, reveal differences between paired variables of only one feature. This is easier and more reliable
than looking for similarities in unanalysed feature aggregates that Varimax usually
clusters.
To be fair, three limitations have to be mentioned.
Firstly, using minimal pair comparisons requires suitable pairs of variables in the
data set. Reliable interpretations of factors may presuppose the presence of multiple minimal pairs displaying not only opposite loadings of focal factors but also
near-identities of non-focal factors. This requirement cannot always be considered
beforehand when variables are selected; so some factors may at times not be reliably interpretable because minimal pairs in the sample of variables are missing.
Additional pairs may be found only in extended samples of data.
The second limitation is due to the fact that the sources of variance found
with Varimin are always more abstract than those factors obtained from SS rotations. This may be made clear with the kinship terms: The feature generation, revealed by Varimin, is more abstract than the youngest and the oldest in the same lineage,
which is required, as shown, in a Varimax analysis which had to factorially combine grandparents and grandchildren. It will be more difficult, in CS analyses of data
from the psychological domain, to identify abstract features than, in SS analyses,
63

64

“Characterizing concepts in terms of features works very well for certain types of words … [those] that are very
well structured by physical, social, or biological dimensions, which then serve as the basis of semantic features.
Other types of words … which do not come from well-structured domains, are more difficult to characterize as a
set of features. … No methodology directly reveals the meaning components of a word. Despite the difficulty in
confirming what is or is not a feature of a word, it is generally accepted that … words are comprehended by accessing various features or meaning components.” (Just & Carpenter, 1987, p. 63 f).
This will be expounded and exemplified in chapter 5.

Chapter 2 – Finding complex structures

81

to make sense with more tangible feature compounds. By the same token it is
harder to conceptualise furniture than wardrobe, armchair or cupboard.
The third limitation is due to the fact that minimal pairs can hardly be formed
for general factors. General-factor loadings of variables of almost any domain are
generally consistently positive and show little variance. On the other hand, a general factor normally represents, to a considerable extent, the dominant source of
variance in which researchers are usually most interested.
It needs to be considered, in addition, that general factors may represent more
than one source of variance. A general factor of intelligence tests may be based on
intelligence plus ambition. In questionnaires, a general factor may represent trait
dispositions as indicated by the items used plus an inclination toward acquiescence.
Information about relative proportions of such additional sources of variance of
factors cannot be retrieved from the data itself. Varimin will display the complexity
of latent conditions only to the extent that sources of variance affect the variance
of manifest variables with different effect contributions.
A number of examples in this chapter have shown that Varimin rotations facilitate interpretations of initial complex structures provided they contain more than
two substantial factors. While conducting numerous factorial re-analyses, I did not
come across a single case where the result of a Varimin transformation was more
difficult to interpret than the result of an initial solution65.
For certain data types, the interpretability of SS results left much to be desired.
In practice, SS-oriented factor analyses are not generally used for such data. Alternative multidimensional procedures, such as MDS (or NMDS), MTMM, or Circumplex are preferred. The significant advantage of CS modelling is that it may be
used to analyse a large variety of data sets, while SS modelling procedures exclude
applications of that sort.
FA with SS rotation may remain useful, however, for particular purposes, not
for discovering “dimensions” in domains being examined, as was believed thus
far. It may be useful for clustering domain variables if this is what is desired.
However, in that case a discovery of cluster-producing features will not occur.
Varimax and especially oblique rotation procedures would prove useful only for
discovering covarying variables as clusters without regarding the sources of their
clustering.
Varimax may also be useful if variables have to be analysed that are known to
be based on only one single source of variance. That would be the case if underlying features revealed by Varimin were used as manifest variables, e.g., as items in
questionnaires, and if a correlation matrix of these variables is then factor analysed.
Varimin analyses may run into complications if it is not sufficiently known
right from the start whether the selected variables are based, truly without excep65

The Varimin program was tested on more than 500 data sets.

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Chapter 2 – Finding complex structures

tion, on a larger number of variance sources. So far I have assumed that variables
are generally multifactorially determined. This does not exclude exceptions, and it
is unclear how many exceptions are tolerable in particular cases and how they can
be recognised as exceptions.
This uncertainty as to how to treat Varimin rotated factors was discussed earlier when the question was raised: Should negative factor loadings be interpreted, and
if so, how? The analysis itself does not answer the question whether there are
functionally antagonistic determinants present in bipolar structures tagged with
plus or minus signs, or whether the signs merely indicate the presence or absence
of some condition. These questions can only be answered with contextual
knowledge of the selected domain and by considering relevant assumptions.
As was indicated previously, Varimin research requires samples of variables to
be representative of the analysed domain. From the start, the variables should be
suspected to reveal multiple latent conditions. Variables providing only little information about sources of variance should not be included. For example, if agedependent variables were introduced into a personality questionnaire (“I am becoming more forgetful”, “my health is deteriorating”), they would possibly produce an additional factor (“signs of aging”). With positive loadings, they would
contribute to such variance – if present –, but they would also yield negative loadings for variables unrelated to age. With meaningless negative loadings, communality would be “wasted” by such variables66.
Re-analyses with Varimin rotations do not always yield satisfactory results. Unsatisfactory results are mostly attributable to an inadequate sampling of variables.
Applying measures of sampling adequacy (Kaiser, 1970) may help to determine
routinely whether data sets are adequate for exploratory factor analyses.
The agenda also requires us to attend to the question how the problem of invariance can be solved. Do Varimin results deliver stable solutions or do they considerably change if additions to or deletions from a sample of variables are made,
or when samples of participants are changed? An unpublished study with intelligence data showed significantly more stability for Varimin than for Varimax results when artificial test variables were added to a previously analysed sample of
naturalistic variables67.
The method of optimal factor extraction is another issue still to be looked at.
In the present project so far, principal component analysis (PCA) was used exclusively, as has become standard in FA research. It is possible that initial results
might be slightly distorted by entering – as PCA requires – the number 1 into the
66

67

Negative factor loadings of a source of variance, which, for most variables, denotes “effect not
present in this variable” are incorporated into the variable concerned, thus adding to the communality of such factors. The communality in such cases would not be informative.
A lack of invariance in SS solutions was already criticised by Butler (1969): “the simple structure
concept does not solve one of the most crucial and fundamental problems of factor analysis, the problem of the
likelihood of factorial invariance” (p.13). “Normal varimax factors cannot be regarded as factorially invariant
…” (p. 24).

Chapter 2 – Finding complex structures

83

diagonal fields of correlation matrices, especially if intercorrelations are predominantly low. Methodological research should also test and compare the modelling
quality of the principal-axes and the maximum-likelihood method. Further improvements might ensue if other extraction methods were to yield better models68.
The present study limited itself to the development of Varimin as a CSM procedure for exploratory factorial research. The method stood numerous tests with
varying sets of data, only some were related to psychological issues. A more systematic treatment of psychological issues is a requisite. Two Varimin studies on
intelligence will be reported in chapter 4 and one on personality in chapter 5.
Lastly, there may be objections to the general strategy of this approach that
uses issues of formal data processing (in this case, factor rotation) to reach conclusions by considering much non-formal knowledge. Are non-quantitative arguments admissible for evaluating mathematical-statistical tools?
Such an objection has been discussed in chapter 1. SS and CS principles are
formal principles. Yet these were introduced with non-formal motives because the
value of formal operations (rotations) was made dependent on results suggesting
non-formal interpretations69. Such interpretations require psychological understanding, i.e., knowledge exceeding formalistic techniques. This is the reason why
the present study up to now has preferred examples where conclusions required
predominantly non-formal knowledge. By entertaining the non-formalised notion
of “understanding”, Thurstonean formalists would move closer to researchers,
who feel committed to “understanding” without much restraint. Where the positions held by model formalists overlap with those of less narrowly-focused researchers, the latter should be allowed to demand a say. They should be allowed to
point out the mistakes of formalists, provided they understand their language.
They should be allowed to test their mistakes and to rethink them if they cannot
be explained away. If their skills permit it, they should also be allowed to help
correct their errors.

68

69

This can hardly be expected in the light of the comparisons of extraction methods available so
far: “The major conclusion of this article is that there is little basis to prefer component analysis or factor analysis. For practical purposes the choice of method is not a decision that will greatly affect empirical results or substantive conclusions.” (Velicer & Jackson, 1990, p. 19).
“… consistent psychological meaning is by far the most important criterion for the success of a factor analysis that
is designed to illuminate psychological phenomena. The simple structure criterion was designed only as a means to
that end.” (Guilford & Hoepfner, 1969, p. 6). “The arguments in favour of rotation are not mathematical;
and in each research the investigator has to decide its merits on non-mathematical grounds” (Burroughs & Miller, 1961, p. 35).

Chapter 3
Decathlon data under analysis

Introduction
The historical beginnings of FA raise a perplexing question: Why was the new
method predominantly applied to analyse psychological variables such as mental
abilities, attitudes and personality traits? Such variables make use of words and
sentences as this is required for depicting psychological functioning. But verbal
communication about mental experience and performance tends to be imprecise,
semantically fuzzy and often ambiguous. If data sets with unambiguous variables
had been submitted to FA at the outset, for instance variables of physical performance, a consensus about the efficacy of FA might have been attained more
readily.
Such deliberations inspired me to subject sports results to FA. For decathlon
sports, athletes deliver performances of ten events. Fortunately, valuable data sets
of Olympic decathlon performances are available, in print and downloadable via
internet. I do not understand why FA results of these data have almost never been
published. I suspect the reason for this omission is that the results of factor analysing decathlon data by procedures committed to SSM are rarely interpretable, if
at all. This is a challenge for CSM. An analysis of decathlon data (10 events) with
CS rotation of its factors (by Varimin) should generate an easily comprehensible
factor structure provided the Varimin procedure is valid. An interpretation of

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Chapter 3 – Decathlon data under analysis

decathlon factors (related to physical conditions) should be much easier than an
interpretation of intelligence factors (related to mental processes). Researchers in
sports psychology have generally complained about inefficiencies of FA in their
field (Bös, 1987, Teipel, 1988, p. 341 et sqq., Büsch et al. 2001)70.

Description of data
Table 3.01a: Intercorrelations of decathlon performance (sources: Linden (1977)
and Kunz (1980)).

Notes:
Upper triangle matrix: Source Linden (1977). Performance at Olympics 1948-1976
Lower triangle matrix: Source Kunz (1980). Performance of Swiss top athletes.
The signs of correlations between race disciplines (higher achievement is obtained with
less time measures) and all other disciplines (higher achievement is obtained with more
length or height) have been reversed. Original correlations were negative.

70

Discussing the factorial validity of motor activity tests in sports, presented in a textbook by Bös
(1987), he states (p. 141): “A detailed analysis of the quoted [factor analytical] findings shows an ‘alarming
non-commitment’ (Orlik, 1967, p. 87) of the factor analytical dimension analyses.” The results often caused
“contradictions”; they did not permit “definitive interpretations … the validity of the statements was overrated”. “Factor analysis (was) not practical.” There were “doubts about the viability of factor analytical findings … Factor structures could only rarely be replicated.” (p. 461).

Chapter 3 – Decathlon data under analysis

87

Table 3.01b: Intercorrelations of decathlon performance (sources: website and
Zarnowski).

Notes:
Upper triangle matrix: Source website: Austrian top athletes (N = 2,400).
Lower triangle matrix: Source Zarnowski (Olympics 1989); Performance at Olympics
1948-1988.
Signs of correlations were partially reversed, see note in Table 3.01a

Linden (1977), who has so far conducted the only FA of decathlon variables (as
described by Basilevsky (1994)), obtained four factors. This author transformed
them using Varimax and interpreted them as follows: F1: short-distance run, F2: explosive arm power, F3: running endurance, F4: explosive leg power. Linden’s Varimax solution
yields no g factor that would indicate general athletic competence. The specificity
of his labels suggests that he did not reach the level of components characterising
the events. Why should arm power and leg power, for example, manifest themselves separately and only “explosively”? Why should the ability to run fast become evident only in short-distance races? Linden does not mention pole vault in
his categories; apparently it does not fit into his factorial system. I used three further decathlon data sets to back up the results: An intercorrelation matrix by Kunz
(1980), original data on individual athletes by Zarnowski (1989), and original data
from an internet source (2004). Tables 3.01a and 3.01b show mean intercorrelations of performances in 10 sports disciplines, obtained from these four sources.
1. Linden (1977)71
Linden’s correlation matrix is based on Olympic decathlon performances of n =
106 athletes at eight Olympic Games (1948–1976). Linden subjected this data to
FA. In the present study Linden’s data were used (as reported by Basilevsky
(1994)).
71

Because of their transparency, Linden’s data are occasionally used for exercises in statistics
courses. On the internet, they are currently available at http://math.usask.ca/~miket/f-03.pdf

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Chapter 3 – Decathlon data under analysis

2. Kunz (1980)
Kunz’s correlation matrix (p. 161) is based on decathlon performances of n = 27
Swiss top decathletes (the database contained results of 90 competitions, many
athletes participated in more than one event). While Kunz interprets n(n-1)/2
intercorrelations by inspection, he does not use FA. In an appendix (p. 212/13),
Kunz provides the original individual data.
3. Zarnowski (1998)
This source supplied the raw data of 233 decathletes whose scores were obtained
at eleven Olympic Games (1948–1988). The performances of 75 athletes who did
not participate in all 10 disciplines were ignored. Linden’s data are included in
Zarnowski’s data.
4. http://www.10k.at/site/zk_oesterreich/_home.html
From this Austrian website, raw data of N = 108 Austrian all-time decathlon athletes
were taken, including N = 64 all-time decathlon juniors (aged < 20).
These data can be found at werthner.casc.at/bin/results/alltime.php.
Table 3.02: Eigenvalues 1 to 6 for four principal component analyses with data
from Linden, Kunz, Zarnowski, and website.

Results: Factor analyses
Table 3.02 shows the six first eigenvalues of factors extracted from the four data
sets (by PCA). Applying the Kaiser-Guttman factor-extraction criterion, three
factors should be extracted for rotation in the datasets except for Zarnowski’s. In
this latter case, the criterion suggested just two factors. But Zarnowski’s data
would, with interpretability as a criterion, also suggest a three-factor solution.
The four initial PCA solutions were first transformed by Varimin and then by
Varimax for comparison. Firstly, similarities of the four factorial solutions were
obtained separately for Varimin and Varimax results. Table 3.03 shows congruence
coefficients of Tucker-Wrigley and Neuhaus (cf. Harman, 1968, p. 270).

Chapter 3 – Decathlon data under analysis

89

Table 3.03: Congruences of Varimin and Varimax-rotated factors F1, F2, and F3,
separately for four data sources. Low congruences in columns F1, F2,
and F3 are bold.

In Varimax solutions, congruences of factors are somewhat smaller for factors (F1,
F2 and F3) than for Varimin solutions indicating that Varimin solutions are more
invariant when data sources are changed. (cf. Table 3.03).
As congruences in Varimin rotated factors of different data sets are large, it
suffices to use the results of only one data set for factor interpretation.
Zarnowski’s dataset seems to be the most suitable source as it has the largest N of
decathletes from 49 nations. Zarnowski includes data from the less comprehensive Linden source. Zarnowski’s initial data are original performances, 10 per athlete per Olympiad. By way of example, the seven best Olympic performances are
shown in Table 3.04.
Table 3.04: Seven top decathlon records of Olympic participants based on official
record counting rules (source: Zarnowski).

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Chapter 3 – Decathlon data under analysis

Some athletes participated in more than one Olympiad. Table 3.05 shows the
Varimin factor loadings (data by Zarnowski) including Varimax factor loadings
and initial solution.
Table 3.05: Varimin, Varimax, and unrotated factor loadings for ten decathlon
disciplines.

Note:
Results of four race disciplines are printed bold to improve clarity and comparability.

1. Interpreting Varimin factors
F1 is the expected general factor g of decathlon events. Individual F1 factor scores
of the 233 athletes correlate highly with their overall decathlon performance (=
.95). Individual total scores, as officially calculated according to the rules of the
decathlon sports association, are weighted sums of the performances in the 10
track and field events72. To avoid redundancy, the official points total, provided by
Zarnowski, was not included as a variable in the sample for FA. Based on their
high correlation with the total score, F1 factor loadings are thus to be considered
as independent indicators of overall achievement in the 10 decathlon events.
Some events show higher F1 loadings than others (e.g., pole vault) and, accordingly, somewhat lower F2 and/or F3 loadings than events with lower F1 loadings
(e.g., 100 m race). Kunz (1980) already noticed conspicuous correlations between
pole vault and various more specialised events, such as race, jumping and throwing. He concludes that pole vault is an “exceptionally many-sided event” (Kunz, 1980,
p. 166). Many-sidedness implies dependence on multiple athletic abilities whose
joint effects may be revealed by g factor loadings. Another interpretation suggests
that high g loadings indicate particular contributions through training and that
lesser g loadings indicate more contributions through congenital advantage (for
example, bones and muscles in the case of the 100 m race). Varimin F2 is a bipolar
72

The decathlon scoring system was determined in 1985 by an IAAF committee. No mathematical or statistical explanation of the individual rating was published.

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Chapter 3 – Decathlon data under analysis

factor, with its highest loading, negative in sign, for the 1500 m race (r = -.65). No
other event shows a negative F2 loading. For interpreting Varimin factors, minimal
pair comparisons are often indispensable (cf. chapter 2). We find a minimal pair
with the 1500 m and 100 m races, as shown in Table 3.06.
Table 3.06
A minimal pair of variables.
Large difference of loadings on one factor only
1500 m race
100 m race

F1
.65
.49

F2
-65
.61

F3
.23
.42

The loadings of F1 and F3 for the two race events are similar, but loadings of F2
have opposite directions, thus giving rise to an optimal minimal pair. Endurance is
the descriptive term (or demand of endurance) characterising the 1500 m race and
speed or demand of speed as a descriptive term for the 100 m. The 1500 m race requires continuous power expenditure with longer duration while the 100 m race
requires an explosive expenditure of power for a short time period. Plausibly,
other race events also require additional constrained power, e.g., the 110 m hurdles (F2: .47), while the 400 m race, likewise plausibly, is positioned between the
400 m and the 1500 m race with an F2 score of .06 on the bipolar F2 scale.
Distinctions between more extended expenditure of effort vs. concentrated
expenditure can also be found among throw events. Shot put requires an “explosion” of strength (F2 = .51), as does the discus throw (F2 = .43), while for the
javelin throw energy expenditure (.11) is less tight. Kunz called the javelin throw a
“many-sided event”, “probably extremely demanding” (Kunz, 1980, p. 167). The
javelin throw differs from typical power disciplines, i.e., shot put and discus,
among the throwing events.
The proposed F2 interpretation (speed) also applies to differences among
jumping events. It makes sense to expect concentrated effort expenditure for the
long jump (F2 = .47). The high jump (F2 = .18) requires more skilful and coordinated body movements, not merely peaks of energy expenditure. The same holds
true for pole vault (F2 = .16). In sum, F2 seems to indicate the speed demands of
energy expenditure73.

73

For power-endurance (F2) in athletic performances there is an analogous polarity (speed-power)
in mental performances, which has been debated and empirically examined (Jensen, 1993, pp.
492-509) in intelligence research.

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Chapter 3 – Decathlon data under analysis

F3 is another bipolar factor which, by signs of loadings, distinguishes events requiring predominant energy expenditure either of the upper or lower extremities.
Muscle power of the arms is required for throwing events (discus throw, F3 = -.42,
javelin throw, F3 = -.35, and shot put, F3 = -.37). Muscle power of the legs is required for the race events of 400 m (F3 = .46) and 100m (F3 = .42) and the 110 m
hurdles (F3 = .30). The 1500 m race does not seem to require particular leg power
(F3 = .23), endurance of energy expenditure (F2) seems to be more important.
In sum, an interpretation of Varimin factors of decathlon scores is straightforward. Each event is characterised by g, (1) a general disposition, by (2) the predominant source of demanded muscle power (upper vs. lower body parts) and (3)
by the temporal pacing of energy expenditure (speedy vs. enduring).
The knowledge of conditions obtained from analysing the correlations of decathlon events appears to be applicable also for research in sports physiology.
Final conclusions can be attempted for generalising concomitant effects of training. Does discipline B benefit from training discipline A? At present it suffices to
state that an immediately comprehensible result has been achieved, as soon as the
way was paved for analysing co-functioning conditions.
For example, the 100 m race is not viewed as dependent on a single condition,
as Linden would have it (his one-factor label: short distance run). Instead, the manifestations of three factors are incorporated. A superior performance in 100 m
races requires a general physical potential (congenital physical condition plus generalised benefits by training). In addition, leg power is required as well as a concerted input of physical energy (power). The other decathlon disciplines can also
be characterised by variable, but conjunct contributions of the same factors,
whose functioning depends on linked aspects of the same process74.

2. Attempt at an interpretation of Varimax factors
The three decathlon factors are unipolar, after Varimin rotation they have only
positive loadings just like factors obtained from intelligence tests. No general factor shows up. Factor scores of the 233 Olympic athletes correlate with official
total scores, i.e. with external criteria for a general factor, as follows: r = .60 (for
F1), r = -.45 (for F2) and r = -.64 (for F3). For none of the three factors does a
conspicuous correlation between factor scores and Olympic total record appear.
This is to be expected since Varimax distributes communality of a general factor,
present to a large extent in an initial F1 factor, among additional factors F2, F3

74

It needs to be remembered that when evaluating differences in athletes’ achievements, only the
performance-related parameters responsible for these differences can be identified. The latent
parameters vary considerably between individuals. They remain latent and can only be researched with other methodological procedures.

Chapter 3 – Decathlon data under analysis

93

etc.75. The Varimax result is flawed even by its own criteria, because single loadings are found for only three of ten sports variables. According to SS, seven disciplines are thus unwelcome hybrids.
Are Varimax factors useful, though?
Varimax F1:
F1 shows highest loadings for shot put (.86), discus (.85), and javelin throw (.74).
The F1 cluster of variables might be termed “throw events”. However, pole vault
is not a throw event despite its high F1 loading (.51). Also, long jump and high
jump with considerable F1 loadings do not have anything in common with throw
events.
Varimax F2:
Varimax F2, with its highest loading for the 1500 m race (.94), appears to indicate
endurance. The 400 m race (F2 = .54) still fits this interpretation. But how to explain high F2 loadings for pole vault (.44) and high jump (.40)? One would be hard
pressed to claim similarities between the 1500 m race and two high jump disciplines.
Varimax F3:
It is difficult to make sense out of Varimax F3. Varimax F3 shows highest loadings
in the 100 m race (.86), in long jump (.77), in 100 m hurdles (.76), in the 400 m
race, and in high jump (.59). In F3, race and jumping are shuffled as if they were
related, which can hardly be explained76.
Conclusion: Varimax factors are not useful.
75

76

Negative signs present with two loadings need not be taken as unexpected since signs of factor
loadings are generally arbitrary. Highmore and Taylor (1954) also criticise the falsifying effect of
an SSM rotation in sports test data: “… the ‘basic’ factor, representing general athletic ability (in which we
are primarily interested), necessarily disappears, and the group factors [of simple structure rotation] show little relation to the classification indicated by the [initial] bipolar matrix” (p. 4).
After completing this manuscript, I discovered a more comprehensive Austrian data source
containing 4586 individual scores of 2674 decathletes: www.werther.at/zehnkampf in “AlltimeListe Wien, 1993-2004”, a non-Olympic selection. The above-reported Olympic results were
taken from the source “Wien 1993-2004”: For F1 (general factor) the congruence between the
Olympic and non-Olympic selection was .99, for F2 (the limb factor) .95 and for F3 (pacing of
energy factor) .94. Such high congruence in factor structures despite widely differing proportions of variance in the factors is interesting. The proportions of variance for the Viennese
(non-Olympic) athletes were 63.8%, 11.1%, and 10.1% for F1 to F3 respectively (total 85%), and
for the Olympic athletes 47.5%, 17.5%, and 11.2% (total 76.2%). The scores of Viennese athletes are certainly larger than those of the Olympic athletes due to different selection yardsticks.
This is indicated by a larger g factor proportion (F1) for the Viennese athletes. The sources of
variance for F2 and F3, however, are not much different in the two samples. This seems to indicate the fact that anatomical and physiological performance conditions represented by F2 and F3
are as valid for top athletes as for less successful ones.

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Chapter 3 – Decathlon data under analysis

3. Attempt at an interpretation of initial factors
It may be suspected that Varimin factors do not differ, or hardly differ, from initial
factors so that differing initial solutions may compete with Varimin solutions. The
case at hand contradicts this assumption. Even though there is a very high congruence (Tucker (1951), Wrigley & Neuhaus (1955)) between Varimin F1 and Initial F1 (cf. Table 3.07b), for Varimin F2 and initial F2, the congruence shrinks to .62
and for F3 to .69. Thus factors F2 and F3 for initial and Varimin solutions are not
congruent with one another.
Regarding an interpretation of initial F2, Table 3.04 also shows an extremely
high positive loading (.76) in the 1500 m race. Interpreting the score as an indication of stamina seems appropriate here, as shot put and discus throw also have
positive loadings (.46 and .45) as they did in the Varimin solution, representing the
opposite of stamina, i.e., power. But the initial F2 lacks a high (power) loading for
the 100 m race. Because of their sustained or concentrated energy input, the 1500
m and 100 m race should facilitate an ideal minimal pair comparison. But these
two running disciplines do not represent an ideal pair in the initial solution. Only
after Varimin rotation do they become useable. The initial F2 likewise lacks evidence of concerted race performances in the 110 m hurdle while Varimin can
extract this from the correlations.
Initial F3 would not even be remotely interpretable as a factor differentiating
between arm power and leg power. The javelin throw (F3 = -.45) and the 1500 m
race (-.40) should have lower arm power loading in the initial F3 than shot put (.22), as would be expected if F3 was interpreted to mean polarity of arm power
and leg power. By the same token, an interpretation of the initial F3 cannot (as
might have been hoped) be interpreted by adding other traits. The initial factors
F2 and F3 are thus difficult to interpret and much less plausible than the Varimin
factors F2 and F3.
Conclusion: Initial factor F1 is an approximation to Varimin F1 and may thus
be considered as usable. Initial factors F2 and F3, however, lack clarity and consistency, compared with Varimin-rotated factors. Initial factorial solutions, except
for F1, need not and should not be considered when final conclusions about FA
results of any data category are drawn.

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95

Table 3.07: Congruences among Varimin, Varimax and initial factors from Zarnowski’s data set (Olympic performances, N = 233). Note: Congruence values >
.70 are bold.

Table 3.07 shows congruence values of the factorial solutions that elicit the following commentary:
1. Table 3.07a: The first Varimin factor F1 shows relatively high congruences with
the three Varimax factors (F1, F2, F3). This is apparently caused by SS transformations distributing the loadings of initial factor F1 evenly amongst the
three Varimax factors. With Varimin, initial F1 is maintained or even optimised
(cf. Table 3.07b).
2. Table 3.07a: Varimin factor F2 seems to be related to Varimax F1 (.68) and to
Varimax F3 (.73), as if underlying parameters, revealed by Varimin F2 in its bipolarity, were distributed amongst Varimax F1 and F3.
3. Table 3.07b: Apart from a close resemblance of Varimin F1 and initial F1, there
is no significant configural congruence between the Varimin and initial solution. This has already been noted when attempting to interpret initial factors.
4. Table 3.07c: By comparing Varimax and initial factors it is noted that Varimax
F3 is surprisingly close (.93) to initial F1. It is hard to explain this similarity, but
this does not seem worth worrying about.

4. Expert rankings for validating Varimin factors
Twelve decathletes were asked via an internet survey to rank the 10 track and field
events of their sports. The instruction was:
1st ranking: Give rank 1 to the event requiring most arm activity and least leg activity. Give
rank 10 to the event requiring most leg activity and least arm activity. Ranks 2 to 9 should be
distributed according to relative arm-leg activity between the extremes.
2nd ranking: Give rank 1 to the event requiring an exertion of bodily energy with highest
concentration within a short time period and give rank 10 to the event requiring an exertion of
energy spreading over a longer time period (stamina). Ranks 2 to 9 should be distributed between
the extremes according to relative contributions of concentration and stamina.

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Chapter 3 – Decathlon data under analysis

The aim of this survey was to find out if the experience of decathletes corresponds with our interpretations of Varimin factors F2 and F3. In Figure 3.01, the Yaxis shows averages of the first rankings. On the X-axis, the Varimin factor loadings are indicated. The correlation is r = .80 (p = .003).
It stands to reason that in pole vault the athletes probably slightly overrated
the use of their arms, and the use of their legs was probably overrated for the
1500 m race. According to Kunz (1980, p. 166), as noted earlier, for pole vault the
functions of the entire body are required conjointly (see above, “versatility of pole
vault”).
For the 1500 m race, a continued optimal dosage of energy exertion seems to
be more important than leg power. This is shown plausibly in Figure 3.02 which
depicts the results of the second ranking (F2). The correlation between rankings
and Varimin F2 is lower (r = .70, p = .01) probably because the athletes overrated
the stamina for the 400 m race and underrated the stamina needed for the javelin
throw. It is also possible, though less probable, that Varimin factor loadings, and
not the rankings of athletes, slightly distorted the empirical givens. At this point it
suffices to state as a result that the majority of rankings of the athletes confirmed
the interpretation of Varimin factors.

Figure 3.01: Ten decathlon disciplines plotted for Varimin factor loadings (X-axis) and mean ranks
obtained on a scale (Y-axis) on which decathlon athletes rated relative amounts of arm and
leg energy required for good results in these disciplines.

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97

Figure 3.02: Ten decathlon disciplines plotted for Varimin factor loadings (X-axis) and mean ranks
obtained on a scale (Y-axis) on which decathlon athletes rated speed of energy expenditure
relative to endurance of energy expenditure required to obtain good results in these disciplines.

Discussion of chapter 3
Factorial analyses of athletic performance show that applying the conventional
EFA paradigm based on SS structure modelling (SSM) leaves much to be desired.
An interpretation of Varimax factors is either very difficult or even impossible.
Complex structure modelling (CSM), on the other hand, yields transparent structures that are easily interpretable. In addition, it is easier to interpret Varimin factors than initial factors which may also escape comprehension. It has also been
noted that a Varimin solution is more invariant than a Varimax solution against
changes of participating samples. Moreover, an interpretation of Varimin factors
of decathlon events was supported by self-assessments provided by experienced
decathletes.
Varimin results of this study correspond with the results of other sports psychological studies, i.e. with the results of Szopa et al. (1998), who reported, without using FA, multifunctional relationships among various sport activities. The
researchers used 42 tests of motor performance; the participants were 143 men
and 91 women. The authors summarised their results by distinguishing five main
motor abilities; the first three match perfectly with Varimin factors. Szopa et al.’s
“ability to develop global strength” manifests itself in Varimin F1, an “ability to
develop local strength (of lower or upper extremities)” manifests itself in Varimin

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Chapter 3 – Decathlon data under analysis

F3, and an “ability of muscular endurance” matches one polar characteristic of the
bipolar Varimin F2 factor.
The bipolarity of energy pacing (explosive vs. enduring), as indicated by
Varimin F2, is also related to physiological observations: “The energy at muscular
activity can be supplied either a) anaerobically, as it is during short bursts of activity of high
intensity with an accumulation of lactic acid as a result, or b) aerobically, as during more prolonged work, when oxygen intake balances the oxygen demand … In aerobic work, respiration
and circulation will play a dominating role …” (Åstrand, 1956, p. 307). Both physiological processes (a) and (b) may be used relatively independently despite bi-polar
amounts of contributions: “The development of the anaerobic and aerobic processes are not
parallel. It cannot, therefore, be expected that a test procedure where the capacity of the aerobic
processes is determined will give accurate information about the capacity of the man for aerobic
work” (Åstrand, 1956, p. 307). According to Milhorn (1982), the basic physiological requirement for physical stamina is cardiovascular fitness.
The demands of physical activity on muscles for different decathlon events are
rarely equal. This explains Varimin factor F2, showing that priority is given either
to the upper or the lower extremities, depending on the activity.
Experts will hardly be surprised by this result. Athletes competing in various
different disciplines “take up the challenge of competing across the whole range of athletic
disciplines … [but] it is not possible for a driving mechanism to exist which is equally optimal or
which can provide an identical maximum neural activation for all disciplines” (Tidow, 2000,
p. 245). Certain “movement affinities” occur, different activities require the use of
different muscle sub-systems.
The results of Varimax transformations are not supported by physiological
findings. SSM transformations of factors of decathlon variables were disappointing. Manning et al. (1988) gets right to the point in his recap of poor factor analytical results gained in anaerobic tests by conventional methods: “Results showed no
single factor emerged and that unrelated aspects existed among these tests and that they were not
measuring similar qualities. It is suggested that anaerobic tests that are used to evaluate anaerobic power be performed as specifically as the skill being tested.” – in other words: Conventional FA is not suited for analysing athletic activities. It should not be used, the
authors conclude. This begs the question: Why should one refrain from FA (with
SS-orientated rotation) of human physical performance, but not from FA of intellectual and other mental performance? The answer seems to be that in the realm
of ambiguous intellectual and mental processing expectancies are vague, if there
are expectancies at all, and hence deviations from expectation cannot be recognised and their underlying causes not identified.
The goal of earlier sports psychologists (Guilford, 1958, Pöhlmann et al., 1979,
Bös & Mechling, 1984 etc.) is being approached. Thurstone’s obstructive SS principle should not be used when an investigation of components of physical fitness
is required (Ismail et al., 1965). The harsh critique by these authors depicts the
present situation of conventional EFA research that Ismail et al. call totally “hope

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99

less” (see also Lykken, 1991, Michell, 1997, Koch, 1999, Breiman, 2001, Gigerenzer, 2004, Barrett, 2005). But this does not have to be the case, provided there is a
willingness to model complexity 77

77

An example of serious criticism of the failure of conventional multivariate research is found in
Barrett (2005, p. 45): “The pressure for change is building - and it looks like a paradigm change – for example, not merely a transition from say Classical Test Theory to IRT – but the entire loss … of psychometric test
theory altogether over time.” Borsboom’s “attack of the psychometricians”, to which Barrett refers, is considered an attempt to bridge the “home-grown rift between psychometrics and psychology”.
The modelling of psychometrics, which is deemed modelling without substance, has to be replaced by modelling with substance (Borsboom, 2006).

Chapter 4
Intelligence data under analysis

A complex structure analysis of IST data
(Intelligence Structure Test)
Point of departure and objectives
Chapter 4 is devoted to a factor analysis (FA) of intelligence data. Intelligence was
the first domain on which the pioneers of FA, Spearman, and his early followers,
Cattell, Thurstone, Vernon, and Burt, applied the new method. Textbooks today
distinguish between two main intelligence factors that were introduced by Cattell
and Horn (1963), namely “fluid” intelligence (Gf) (supposedly based on congenital
conditions) and “crystallised” intelligence (Gc) (based on long-term learning).
This distinction was the result of analyses that considered, without further ado,
SS structure modelling (SSM) to be right and safe. Today the distinction between
(Gf) and (Gc) is regarded worldwide as an established fact. In the present study, 18
samples of participants in the Intelligenz-Struktur-Test (IST), which is very popular in Germany, will be re-analysed in order to find out whether the alleged subdivision of intelligence is replicable if data formerly analysed by Varimax are reanalysed using Varimin, the Complex Structure Modelling (CSM) device. It seems
likely that CSM will engender different factors.

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Chapter 4 – Intelligence data under analysis

In the following, intelligence factors are subjected to Varimin and, for comparison,
to Varimax rotations. The interpretation of intelligence factors rotated to SS and
CS, provided in Study I, will be extended by collecting external validation data in
Study II.

Study I: Varimin analysis of IST factors
Objective
The first question we have to answer is whether Varimin rotation yields meaningful IST factors.

Data
In order to increase the stability of results, 18 IST intercorrelation tables, found in
psychology journals, were factorised. Factor loadings of eight subtest variables
were averaged across these 18 studies. In Table 4.01 the sources are provided,
with additional information. Test versions IST-55 and IST-70 were considered,
version IST-2000 did not provide sufficient correlation tables.

Expectation
Amthauer et al. (1999) obtained a two-dimensional factorial structure using the
IST-2000 test version. Following an SSM-orientated rotation, these authors interpreted their findings as manifestations of fluid and crystallised intelligence. I expect that after Varimin rotation IST factors must be interpreted differently and
that Varimin interpretations will be more satisfactory than Varimax interpretations.

Chapter 4 – Intelligence data under analysis

Table 4.01: Sources of intelligence test data. IST-55 was applied in test studies
1–8, IST-70 was applied in test studies 9–18.

103

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Chapter 4 – Intelligence data under analysis

Table 4.02: Mean factor loadings (averaged across 18 analyses) of two-factor and
three-factor solutions after Varimin and Varimax rotations of individual initial solutions.

Data analysis
A memory test (ME) was eliminated from the nine subtests of IST, leaving eight
for analysis. Reviewers of IST-70 (Schmidt-Atzert & Hommers, 1996, SchmidtAtzert, 1997, Brocke et al., 1998) likewise did not regard memory as sufficiently
represented by ME alone. The use of ME might also interfere with the factorial
structure of the whole test. When non-IST variables (for instance, on personality)
had been used in addition, the correlations of these variables were also discarded.
The 18 correlation matrices of eight remaining subtest variables were PCAfactorised in each database. For each factor matrix a two-factor and a three-factor
solution was rotated, using Varimin as well as Varimax. The obtained factor loadings were averaged across the 18 analyses using Fisher’s Z transformation. Before
determining averages, the sequences of factors for the 18 analyses were synchronised, because identical or almost identical factors did not always share the same
position within a sequence of extractions.
The signs of loadings varying between analyses were also coordinated in order
to equalise structures and to align them. The loading patterns in the various analyses were adjusted by visual comparison. Ambiguity occurred in just two or three
cases for three-factor solutions. Uncertainty appeared occasionally when two fac-

Chapter 4 – Intelligence data under analysis

105

tors with very similar loading profiles had to be coordinated. Unnoticed incorrect
decisions at such matchings are minimal and have been neglected.

Results with comments
Table 4.02 depicts aggregates of factor structure of Varimin and Varimax rotations.
1. Comparing two-factor and three-factor solutions
To begin with, we shall look at differences between two-factor and three-factor
solutions. In 18 original IST analyses, following SSM rotations, the authors often
kept a third extracted factor for rotation if the factor seemed to account for nonnegligible and interpretable variance. But the Varimin aggregates for F3 explained
only 2% variance. Obviously, Varimin F3 is irrelevant and is therefore discarded;
only Varimin F1 and F2 are apparently substantial78.
2. Comparing communalities
Compared with the Varimin-based factor aggregate, the Varimax aggregate shows
entirely different communalities. The two-factor solution has 54% (F1) vs. 46%
(F2) communality for Varimax, as opposed to 79% (F1) vs. 21% (F2) for Varimin.
The three-factor solution yields loadings of 38%, 34%, and 28% for Varimax factors F1, F2, and F3, respectively, as opposed to 79%, 19%, and 2% in the case of
Varimin factors.
The contrast regarding communality for F3 loading percentages is striking:
Varimax F3 explains 28%, Varimin F3 explains 2%. The relatively high percentage
for Varimax F3 would be a notable result if F3 were interpretable. But it is not,
because the variance of F3 factor loadings across those eight variables is very small.
Accordingly, they provide no information about conditions of performance.
Varimax F3 in Table 4.02B is thus an artefact and will be ignored.

78

My experience with Varimin rotations, still to be specified, holds that to be recognised as making a substantial contribution to the total variance, a factor should explain >10% of variance.

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Chapter 4 – Intelligence data under analysis

3. Interpreting Varimax F1 and F2 (Table 4.02B)
F1 and F2 profiles of earlier IST FAs, based on Varimax, are obviously similar to
the Varimax profiles at hand. Amthauer et al. (2001) used Oblimin-rotated factors.
Oblimin rotation is another SS procedure, and the loadings of Oblimin are generally similar to Varimax loadings. Varimax F1 of the present study is conventionally
interpreted as fluid intelligence (Gf or gf)79; Varimax F2 as crystallised intelligence
(Gc or gc). Fluid intelligence is ostensibly based on congenital conditions. Cattell
(1963) and Horn (1976) found that what they called “crystallised intelligence”
indicated intelligence transformed by educational and cultural learning. This idea
resulted from second-order FAs and SSM rotations.
The idea is plausible. Performance in intelligence tests depends, to some extent, on antecedent educational and cultural learning. Even some researchers who
did not find crystallised intelligence in their data when they employed conventional methods (Johnson & Bouchard, 2005) assumed that learning will modify intelligence test results (“learning processes preceding the test could cause a noticeable variance of
performance”) (p. 410). A FA of IST data should therefore disclose such additional
sources of variance of intellectual performance.
SSM, however, can hardly fulfil this expectation in the first place. By Varimax
rotation the main source of variance, general intelligence, loses its unity. Much of
F1 variance is passed on to F2 so as to give rise to two allegedly different types of
intelligence. An influence of learning on performance that is actually small is thus
blown up and falsified by scrambling F2 with g. General intelligence thus disappears by SS rotation and is recovered in a cumbersome way by second-order FA
procedures.
4. Jensen (1998) describes the result of simple structure rotation: “… So if you ask
where g went, the answer is that it has been divided up and lies ‘hidden’ among all of the tests’
smaller loadings on all of the orthogonally rotated factors. Its variance has not disappeared, it has
simply been obscured by being dispersed throughout the whole factor matrix.” (Jensen, 1998, p.
66).80 Interpreting Varimin F1: Basic intelligence (g).
Obviously Varimin F1 represents g, the base of intellectual performance. The
term basic intelligence emphasises the importance of Varimin F1 as a fundamental
condition for intellectual performance. This interpretation will be tested in Study
II by examining correlations of Varimin F1 with IST-70 performance on the one
hand and with culture-free tests on the other (see below). The abbreviation g will
henceforth be used to denote general intelligence factors obtained by applying
Varimin in order to facilitate the distinction from general intelligence g obtained
through Varimax and second-order procedures.
79
80

Alternative notations have been used for fluid and crystallised intelligence: Gf vs.Gc, gf vs. gc, gF
vs. gC.
The correlations of Varimax F1 and F2 with the IST-70 original totals are .74 and .63, respectively.

Chapter 4 – Intelligence data under analysis

107

5. Interpreting Varimin F2: Learning assets (L)
Varimin F2 is a bipolar factor. Traditionally, negative intelligence factor loadings
were not tolerated, SS rotation largely removed negative signs81. In chapter 1,
detailed reasons were given for interpreting bipolarity straightforwardly. Varimin
F2 has significant positive loadings on subtests requiring verbal operations (SC,
WS AN, CO). F2 loadings of figural tasks FS and CT bear negative signs. This
seems to indicate that F2 emphasises verbal operations playing a predominant role
in education. Only rarely do schools require students to master pictorial tasks and
to perform language-free formal operations. A relative deficit of learning advantage for FS and CT performance might result. Predominant educational practice requires linguistic operations. Number tasks, showing near-zero Varimin F2
loadings, may require less school training, and almost no educational practice
seems to be required for pure figural operations.
“Learning assets”, the label for Varimin F2, may be taken as a metaphor.
Learning requires storage. For many intelligent operations stored knowledge gives
helpful returns, much like interest on capital accumulation82. The distinctions
made here correspond to Weinert’s “triangle” of aptitude, knowledge and learning
(Weinert, 1996). In the first instance, aptitude or talent, according to Weinert, requires basic intelligence. Knowledge may be seen as accumulated learning assets. In
our present analysis, learning processes are presupposed, not directly investigated.
Learning generates an increase of knowledge and abilities during an individual’s
lifetime and education (Waldmann et al., 2003). Research clarifying the interpretation of Varimin F2 is taken up in Study II of this chapter.
Interpreting Varimin F2 as learning assets need not be the final word. It might
be argued that a bipolar factor F2 might indicate a preference for one of two polar
cognitive styles, either for holistic operations as are required for languagedemanding tasks (F2 positive), or for more analytical and detailed operations required for solving figural tasks (F2 negative). Numerical tests might require the two
operations in balanced proportions. Likewise, educational “learning assets” are
possibly more easily acquired by students who prefer holistic cognitive operation.
Differences of cognitive style may be based, just like basic intelligence, on genetic
endowment. A final decision about this issue is neither possible here nor required.

81

82

Berneyer (1957) comments along these lines: “The different methods of [factor] analysis [of mental
aptitudes] yield factors which have negative loadings … Such factors, Thurstone contends, must be devoid of ‘scientific meaning’ They do not permit us to ‘interpret’ the various tests as functions of the mental aptitudes which
those tests elicit.” (p. 23).
Cattell (1971) uses a similar metaphor. With his “investment theory” he attempts to connect
fluid and crystallised intelligence: Fluid intelligence is the “investment” made by learning
throughout a person’s life, says Cattell (Holling et al., 2004, p. 21). But in his SS-based analyses
his learning “investments” are confounded with intelligence: the capital and resulting interest are
mixed.

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Chapter 4 – Intelligence data under analysis

Summary of Study I
Mean factor loadings of Varimin F3 for IST subtests in three-factor solutions are
small while Varimax F3 subtests attract significant loadings. However, since F3
loadings are unstable across 18 studies, F3 appears to be an artefact and is furthermore ignored.
IST’s conventional F1 and F2 interpretations of Varimax F1 as crystallised and
Varimax F2 as fluid intelligence cannot also apply to Varimin solutions. Varimin F1
apparently represents basic intelligence, g,. Varimin F2 is best understood as an
additional source of performance, i.e., as learning assets (L) (deemed independent
of congenital basic intelligence). Participants of IST draw more or less advantage
from L depending on the amount of learning assets they have acquired. This
might hinge on antecedent availability and practice with tasks demanding cognitive
processes akin to those that verbal IST subtests require.

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109

Study II: Validations of Varimin-rotated IST factors
Objective
In what follows, the validity of Varimin factors F1 and F2, which have been interpreted as g (basic intelligence) and L (learning assets), shall be tested. Also, comparisons of correlations by Varimin factors g and L with correlations by Varimax
factors Gf and Gc will be made. Apart from IST data, the following data will be
considered: Firstly, school performances collected by Höger (1964) and
Cronemeyer (1983); secondly, spelling and numerical performances plus results
from a culture-free intelligence test by Schmidt-Atzert et al. (1995); thirdly, data of
two culture-free intelligence tests collected by Brocke et al. (1998).

Validation 1: School performance
Data 1: Höger and Cronemeyer
Höger (1964, p. 435) and Cronemeyer (1983, p. 172) independently collected IST
test data and the participants’ school grades (cf. Table 4.03). Höger’s school grades
were obtained from 519 gymnasium school pupils (age 10–13), Cronemeyer’s
grades were those of 656 high school graduates.
Table 4.03: Correlations between F2 (Varimin) of IST subtests (row 1) and school
grades. Sources: Höger (row 2) and Cronemeyer (row 3).

Notes:
SC sentence completions, WS word selections, AN analogies, CO communalities,
NU numeracy, NS number series, FS figure selection, DI dice

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Chapter 4 – Intelligence data under analysis

Expectation 1
If learning assets is the correct interpretation for Varimin F2, then this should be
revealed by differential F2 loadings of IST subtests and by differential correlations
between F2 loadings and school grades.
Assessment 1 and comments
Varimin factor loadings of the eight IST subtests, obtained from a factor aggregate
(study I data), and correlations of the eight subtests with school grades (data provided by Höger and Cronemeyer), need to be focused.
Clarifying Table 4.03 entries. In Höger’s study, the correlation of pupils’ SC subtest
scores (sentence completion) with their school grade average is .09 (line 2). Correlations in line 2 and line 3 and their means in line 4 (lines 2 and 3 correlate with
each other with r = .77) and were subsequently correlated with Varimin F2 factor
loadings of line 1 (cf. Table 4.02). The three correlations in the second to last
column of Table 4.03 (.62, .63, .67) are significant. They indicate that school learning is related to F2 loadings of IST subtests. The interpretation of F2 in terms of
learning assets has found support.
Varimin F2 (learning assets) is factorially independent of Varimin F1 (basic intelligence). School performance of pupils is expected to be influenced, in the first
place, by basic intelligence. But applying Varimin F1 (basic intelligence) in conjunction with F2 (learning assets) as independent variables for multiple regression,
while using school performance as dependent variable, a correlation of r = .67
between F1 and school performance alone is increased, by adding F2 as predictor,
to r = .80.
The original student data used by Höger and Cronemeyer are not available, so
the correlation of school achievement with F1 and F2 factor scores cannot be calculated on an individual level. This would have been preferable.

Validation 2: Culture-free IQ Test CFT, spelling
and numeracy test
Data 2
A database provided by Lothar Schmidt-Atzert (and partially utilised in Study I
(see Table 4.01, nos. 15–17) with anonymised test results of 908 participants included original IST-70 data plus individual IQ values from the language-free and
culture-free intelligence test CFT IQ. Results of a spelling test (dictation) and a
typical school numeracy test were also available. The sample was made up of 397
gymnasium students, 394 junior-high school students, and 196 secondary school
students. The sample of pupils has missing data for some variables and thus has a
somewhat lower N in the table.

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111

Expectation 2.1 (regarding basic intelligence):
The data provide an opportunity for validating Varimin F1 as a measure of basic
intelligence and for validating Varimin F2 as a measure of learning assets.
For each participant, two measures of general intelligence g, i.e. factor scores from
IST Varimin F1 and IQs from CFT, are available. Also available, for each participant, is one factorial measure of L (learning assets, factor scores of Varimin F2
from IST) as well as two derived measures of learning assets (Ld and Lp) based on
relative performance with IST subtests with positive F2 loadings and performance
with IST subtests with negative F2 loadings (see note to Table 4.04). It is expected
that the two g measures are highly correlated and that g indicators do not correlate
appreciably with the two L indicators83. The aim was to find out whether original
raw scores of IST would validate the construct L without using factorial F2 loadings.
Table 4.04: Intercorrelations among indicators of basic intelligence and
learning assets.

Note:
* LP = (SE+GE)/ SE+GE+FA+WU)*100
* Ld = (SE+GE)-(FA+WU)

83

Two measures are introduced, LP and Ld, as estimates of learning assets. IST subtests SC and CO
represent the positive pole of Varimin F2, best indicators of learning assets. Subtests FS and CT
represent the negative pole, best indicators of basic intelligence (see Table 4.03, first row). All
subtest variables had been transformed into standard values (Schmidt-Atzert had done so, getting
an average of 100). Next, two indicators for a learning component were set up, Lp and Ld, by
relating SC+CO and FS+CT:
Lp =(SC+CO)/(SE+CO+FS+CT)*100 (idealised mean = 50)
Ld =(SC+CO)-(FS+CT) (idealised mean = 0)
(L = learning indicator, p = proportion, d = difference)

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Chapter 4 – Intelligence data under analysis

Assessment 2.1 and comments
It can be seen that
 The three intercorrelations among measures of learning assets are large, as
they should be.
 Correlations between measures or estimates of learning assets and measures
of general intelligence (Varimin F1 and CFT-IQ) are low, as they should be.
 Varimin F1 (obtained from IST data) proves to be a measure of general intelligence since its correlation with a culture-free intelligence test (CFT-IQ)
is considerable (r = .73).
Expectation 2.2 (regarding learning assets):
If Varimin F2 obtained from IST data is a valid indicator of learning assets then
Varimin factor scores F2 should correlate significantly with typical school tests like
spelling and numeracy.
Assessment 2.2 and comments
Figure 4.01 visualises the correlation between learning assets L, obtained from F2
of IST, and one of the two variables of external validation (spelling performance).
Spelling performance should correlate significantly with L because good spelling is
possible only after sufficient learning and practice. In Figure 4.01 spelling data of
945 participants in the Schmidt-Atzert sample (Y-scale) are spread across the Lp
scale (i.e., the X scale). The correlation between Lp indicating learning assets and
spelling requiring much of those assets is not large, but highly significant (r = .30).
Without considering basic intelligence (indicated by F1 of IST), not shown in this
graph), it may be assumed that learning assets contributes significantly to spelling
test results.
Another meaningful result from Figure 4.01 is worth mentioning, but details
must be renounced Basic intelligence of participants with best spelling results
(those in the upper horizontal section D5) is not evenly distributed across sections
L1 to L5, instead L1 participants in D5 turn out to be more intelligent, L2 less intelligent and L5 participants least intelligent (intelligence measured by CFT IQ). Correlations with intelligence vary from D5 participants down to D1 participants, but
the tendency of change is progressively reverse: Participants whose spelling scores
are low and who possess scant learning assets tend to be more intelligent than
those whose spelling scores are low and who dispose of more learning assets. In
other words, good spelling scores can be obtained by participants with aboveaverage learning assets without extraordinary intelligence, or with above-average
intelligence without extraordinary learning assets.

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Figure 4.01: Distribution of spelling performance (Y-axis) across learning assets (X-axis) for five subsections for Y (spelling D1–D5) and five subsections for X (learning assets L1–L5) facilitate the
visual impression of the correlation.

More information about the Varimin and Varimax results can be gained from
Table 4.05.
Varimin results (Table 4.05)
Table 4.05 section A:
CFT IQ values indicating basic intelligence correlate .73 with Varimin F1 (basic
intelligence), but only marginally with Varimin F2 (.11). Why is the latter correlation so marginal? Apparently because F2, based primarily on IST subtests for
which learning assets are beneficial, does not contribute to scores of tests like
CFT, for which basic intelligence is mainly responsible.
Table 4.05 section B:
Spelling performance that is largely dependent on verbal practice in school correlates more highly with Varimin F2 factor scores (learning assets), r = .31. Spelling
performance is thus also correlated with basic intelligence (Varimin F1 factor
scores), which is not surprising.

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Table 4.05 section C:
The numeracy results meet expectations only partially. The correlation between
numeracy and Varimin F1, factor scores of basic intelligence, is high as expected (r
= .64). An expected additional, although somewhat lower, correlation with
Varimin F2 factor scores (learning assets) did not show up. The correlation was r =
.07, only barely significant.
Varimax results (Table 4.05)
Correlations between numeracy and Varimax factors are not elucidating As expected, the correlation of numeracy is somewhat higher for fluid intelligence (with
factor scores of F2, r = .59) than for crystallised intelligence (with factor scores of
F1, r = .44).
As expected, spelling and numeracy performances correlate better with crystallised
intelligence than with fluid intelligence (numeracy .64 with crystallised intelligence
vs. .21 with fluid intelligence, numeracy .50 with crystallised intelligence vs. .40
with fluid intelligence). Comparisons of Varimin vs. Varimax rotational validities
thus are favourable for Varimin once again.
Table 4.05: Pearson correlations between Varimin and Varimax factor scores F1
and F2 obtained from IST and CFT IQ, a dictation, and a test of numeracy.

Notes:
g = General or basic intelligence
L = Learning capital
f = Fluid c = Crystallized intelligence
“Gymnasium” (a 9-year selective high school for gifted students, with an academic focus).
“Realschule” (a 6-year selective high school for middle-tier students, providing an
allrounded education).
“Hauptschule” (a 5-year high school open to all students, focused on developing handson, practical skills).

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Validation 3: Culture-free IQ test FRT
Data 3
Brocke et al. (1998) published an intercorrelation table that has already been referred to in Study I (cf. Table 4.01, line 18). Apart from using the eight IST subtests, N = 279 for IST), they also applied correlations with the Figure Reasoning
Test (FRT) (N = 241 for FRT).The authors hoped “to come up with evidence for IST70’s internal validity” (p. 94), above all by correlations of “fluid” IST-70 subtests
with FRT. FRT’s correlation with Raven’s figural SPM, widely used for fluid intelligence, was r = .93.
Expectation 3
FA of intercorrelations of FRT variables should show a high loading on Varimin
factor F1, provided Varimin F1 is correctly interpreted as indicating general intelligence. FRT should show a low loading of Varimin factor F2, provided Varimin F2
is correctly interpreted to be the learning assets factor. Varimax factors F1 and F2,
allegedly indicating fluid and crystallised intelligence, should divide intelligence
into two sub-branches, without indicating which factor, F1 or F2, would represent
fluid and which crystallised intelligence.
Assessment 3 and comments
Table 4.06 shows the results of Varimin and Varimax analyses of the eight IST
variables (in rows 1 to 8, ME subtest excluded). The FRT variable was added in
row 9. It turned out:
Varimin result: FRT reveals the expected high Varimin F1 loading (.70, basic intelligence) as well as an expected low F2 loading (-.15, learning assets). The reason that
FRT is unrelated to F2 is that advantages by schooling and cultural experience are
not supposed to help solving FRT tasks just as such experience is not helpful at
solving IST Figure selection FS (-.12) or, even more so (-.59), for solving IST task
dice (DI).
The main outcome is that the factorial distinction between “fluid” and “crystallised” intelligence (via Varimax) is much less marked than the distinction between “basic intelligence” g and “learning assets” (via Varimin), L. The empirical
and conceptual separation of “intelligence” and “learning assets” is thus considerably more marked when these two constructs, intermingled by Varimax rotation,
are untangled by Varimin rotation.
The authors unfortunately did not apply FA to IST subtests together with
FRT subtests or with FRT total. They merely provided an alternative way of relating test results from the two sources. They employed multiple regressions using
IST variables as independent variables (IVs) and the FRT total as the dependent

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variable (DV). The calculations were complicated and the result was, in their own
view, unconvincing.
Table 4.06: Varimin and Varimax rotated factor loadings of FRT data (row 9) and
loadings of IST 70 subtest variables (rows 1 – 8).

Summary of Study II
After validating Varimin and Varimax factors, measured by tests that facilitate
their validation, Varimin comes out on top. Performances in two culture-free tests
(CFT and FRT) are better predicted by Varimin F1, interpretable as basic or general intelligence, than by Varimax F2, which since Cattell and Horn has been considered as indicating “fluid”, i.e., general intelligence. In other words, Varimaxbased general intelligence is outdone by Varimin-based general intelligence which
– in order to account for this property – I prefer to call “basic intelligence”.
The performances with two tests as external criteria (spelling and numeracy),
supposed to benefit from preceding cultural learning, are better predicted by
Varimin F2, the learning assets factor, than by Varimax F1 interpreted as crystallised intelligence, which allegedly manifests itself by effects of school and cultural

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learning. Varimax F1 should have shown its contribution to scores in spelling and
numeracy. But it did so less convincingly than Varimin F2.

Discussion of chapter 4
Studies I and II presented in this chapter were devoted to the question whether
CSM rotation (by Varimin) of intelligence test data would be more valid than conventional SSM rotation by Varimax. The answer was in the affirmative, the results
are convincing. Varimin separated the two main sources of variance of IST performance; intelligence and learning. The proposed interpretation of the two factors proved valid.
Proposing new research strategies with claims that they outdo conventional
measuring tools should go along with exposing weaknesses in past research. I
believe and repeat that the main weakness of past research lies in Thurstone’s SS
model, on which past research was based. SSM ignores what Jäger terms a “core
assumption” that belongs at the beginning of all intelligence research: “Each intelligence performance involves (along with other conditions) all intellectual abilities, albeit with significantly differing weights. Every performance's variance can be dismantled into its respective components.” (Jäger, 1997, p. 4).84 But Jäger’s own ways to attain this goal did not question the SS principle.
Varimin analyses of intelligence data showed that, to attain good test scores,
increased learning efforts in schools and beyond can compensate, to a certain
degree, for lack of intelligence, and vice versa: Educational psychology (Weinert,
1996) has been aware of this for quite some time, as has actually anybody with
common sense. What appears to be new, though, is that the concept of learning
influence has been freed, methodically and conceptually, from the concept of intelligence. The two concepts had been and still are generally bonded together, without
justification, by an SSM artefact called “crystallised” intelligence. CSM separates
the two conditions, nature and nurture, facilitating their comparison and theoretical evaluation.
Factor rotation aiming at SS, which assigns only one factor (one source of variance) to test variables, destroys a general factor g which is preformed by initial
factor solutions, instead of improving its representation. SSM spreads variance of
the main factor’s contribution to subsequently extracted factors attributing to
them an unjustified premium of communality.
To make sense out of Varimin F2, no new “ability” (such as “crystallised intelligence”) needs to be invented. F2 can be understood as “earned interest”,
84

In his data analyses, Jäger did not break away from the SS principle. Instead, he had to employ
alternative methods to prove empirically the concurrence of latent functional contributions to
intelligence test performance. This he tried to achieve in his BIS model. (Berliner IntelligenzStrukturmodell).

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summed by performance throughout life-long learning, from which test participants benefit differently, according to varying amounts of preceding practice.
Varimin F2 is a fertiliser, while F1 is a measure of an innate potential for growth of
its output.
A practical consequence of these results would have us recommending that the
IST authors revise the utilisation of their test. They should focus the user’s attention on basic intelligence g and put it centre stage. Thus far, the IST manual does
not even make mention of basic intelligence although the test can capably provide
information about basic g.
Information about learning assets accumulated by educational and school influence might also be obtained from IST. One might take raw test results from
IST, transform them by employing Varimin F2 factor loadings and by using suitable conversion tables. Two separate scores might be obtained to compare
summed F1 and F2 test values (e.g., by using Ld or Lp). The two scores can be
related, e.g., after standardisation. This strategy would allow for an assessment of
relative contributions of past learning and practice to levels of achievement. One
might eventually even be able to distinguish between “diligent” and “lazy” test
candidates, by relating individual effort expenditure to individual giftedness.

Chapter 5
Varimin factors from Big Five personality data

Point of departure
In this chapter I apply Varimin rotation to factorial personality data. Varimin rotation is expected to suspend the Big Five personality factors of extraversionintroversion, stability-neuroticism, openness, conscientiousness, and agreeableness
that presently make up the core of prominent personality dimensions. Theoretically, the loss of these dimensions will hardly be harmful since in fact there is no
theory requiring the Big Five as necessary building blocks. An analysis aiming at
complexity might lead to first steps towards theoretical re-orientation. Details
cannot be predicted, the course a complexity analysis of personality descriptions
will take is open to bottom-up surprises. The issues examined in chapter 5 have
been presented in my German-language publication, Basiskomponenten der
Persönlichkeit.85 Non-German readers are invited to contact me regarding any matters that have been dealt with insufficiently in the present monograph86.

85
86

Ertel (2011b).
Contact:: sertel@uni-goettingen.de

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Material and analysis
The data used here (N = 11,724) was obtained from the standardised Germanlanguage self-assessment form, NEO-PI-R (Ostendorf & Angleitner, 2004)87.
Intercorrelations of six facet variables for each of five factors were available, i.e., a
total of 30 variables (cf. Ostendorf & Angleitner, 2004, Table 1). A PCA of the
correlations and a subsequent Varimin rotation yielded the factor structure shown
in Table 5.01 and in Figure 5.01. The factorial results obtained by Varimax rotations of such data are well-known. In Figure 5.01 they are indicated by assigning
them the letters N, E, O, A, C as labels. The Varimax factors will be referred to by
using these labels. .

87

The correlation matrix was kindly provided by the first author, Fritz Ostendorf.

Chapter 5 – Varimin factors from Big Five personality data

Table 5.01: Varimin-rotated factors of the 30 NEO-PI-R facet variables
Data source: Ostendorf & Angleitner (2004). N = 11,724.

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Figure 5.01: Varimin-rotated factors of the 30 NEO-PI-R facet variables
Data source: Ostendorf & Angleitner (2004) N = 11,724.

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Towards an interpretation of Varimin factors of personality
The distribution of Varimin factor loadings engenders two preliminary comments
(cf. Figure 5.01):
1. Varimin factors, represented by dark and light circles (for positive or negative
loadings, respectively) is not entirely unrelated to the Varimax Big Five clusters
(N, E, O, A, C). Facet variables belonging to Varimax clusters have Varimin loading patterns that are similar to each other. This is not surprising, because the
Varimax clustering of the Big Five facets reveals similarity among these facets in
the first place. Any orthogonal transformation of variables preserves similarities
among them. The question remaining is what causes facet variables to have profiles of Varimin factor loadings that are similar to each other (more about this
issue below).
2. If one compares Varimin factor loadings column by column, it is clear that
Varimin factor loadings with the same loading signs are not located exactly within
the limits of Varimax dimensions, especially in the case of F2, F4, and F5 (not quite
as pronounced in the case of F3). This indicates that variables of the five Varimax
dimensions, which have thus far been conceived as being independent of each
other, may indeed have something in common. Variables of different Varimax
dimensions may have loadings on equal Varimin factors, albeit possibly of different amount and with different loading signs.

More preliminaries
Interpreting Varimin personality factors is more elaborate and more involved than
the general practice of ad-hoc labelling of factors. Findings must be embedded
semantically into contextual knowledge. It turned out, as a surprise, that Varimin
factors are conceptually interrelated. I shall pay attention to their relationships
with delineations at an unusually abstract level. The naming of Varimin factors can
hardly make use of familiar trait vocabularies (where one might find, e.g., openness,
conscientiousness, agreeableness or habitual technical terms such as extraversion, neuroticism).
In chapter 2, minimal pair comparisons of variables were recommended for
carving out the meanings of Varimin factors. In view of the larger number of variables of the present data set the description of detailed pair comparisons would
require a great deal of space. Pairs of facets will only occasionally be contrasted.
Since Varimin factor loadings generally have bipolar directions, a short-cut method
similar to contrasting minimal pairs has been applied in Table 5.02. Five variables
with extreme positive loadings on one factor have been grouped and contrasted to
five variables with extreme negative loadings on that factor. This procedure does
not produce distinct trait opposites such as assertive vs. yielding, altruistic vs. egoistic,

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warm vs. cold. In Table 5.02 Varimin factor terms are chosen to denote the respective polarities, not their positive or negative manifestations. This procedure requires sensitivity to semantic nuances. The size of loadings of a Varimin factor on
variables does not determine their meaning in the first place, as it has always been
taken for granted for interpreting Varimax factors.
In addition, the importance of factors is not determined by the amount of statistical variance they explain, but by their contribution to an emerging nomological
network. In the case at hand, for example, factors extracted later (with less communality) appeared to be semantically more basic than factors extracted earlier.
Factor F5, which was extracted last (level of activation), contributed most to understanding factors F2 to F4. According to Cronbach and Meehl (1955), one should
always aim for nomological networks when conducting factor-analytical research88. In this regard, Fischer (1967, p. 125) also demanded: “Actually, all factors of
any structure should be interpreted simultaneously, because they depend on each other to a significant degree.”

88

“A necessary condition for a construct to be scientifically admissible is that it occurs in a nomological net…”
(Cronbach & Meehl, 1955, p. 290). “To validate a claim that a test measures a construct, a nomological net
surrounding the concept must exist” (p. 291). “As research proceeds, the construct sends out roots in many directions, which attach it to more and more facts or other constructs” (p. 291). Constructs can enter a nomological net of theoretical rank, if they do justice to the nature of what is being described. They
should also be suited to being successfully applied in other areas of psychological research belonging to the same system. Fiske (1976, p. 877) provides support by saying: “Construct validation
requires the investigation of construct-operation units in an explicit conceptual framework”.

Chapter 5 – Varimin factors from Big Five personality data

Table 5.02: Facet variables with extreme positive and negative loadings
of five Varimin factors from NEO-PI-R data.

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Chapter 5 – Varimin factors from Big Five personality data

Varimin factor interpretations
1. Level of activation (Varimin F5)
When comparing the five most positively loaded F5 facets with the five most negatively loaded F5 facets (cf. Table 5.02), it is hard to avoid the interpretation that
the difference is basically engendered by amounts of psychophysical energy (activation) required for manifesting the respective behaviours. Initially we can disregard the fact that releasing psychophysical energy might either be beneficial or
inhibitory for the system as a whole (cf. in this context, State of functionality (factor
F1) as an extra feature).
Activation as a term was coined by general psychological research (Lindsley,
1951, Duffy, 1962)89. Factor F5 indicates the level of psychophysical energy associated
with behavioural and experiential phenomena. The facets of Deliberation (C6) vs.
Dutifulness (C3) constitute a minimal pair (cf. Table 5.02) regarding the expenditure
of energy in the Conscientiousness cluster. Dutifulness (positive F5 loading) normally
requires more energy-demanding actions than Sobriety and Thoughtfulness (negative F5 loading). The data reveal more minimal pairs of variables for F5 with contrasting levels of apparent energy expenditure (cf. Table 5.01). F5 shows higher
values across greatly differing Varimax facets: Activity (E4), Hostility (N2), Achievement striving (C4), and lower values for different A and O facets: A4 Compliance, A2
Straightforwardness, A5 Modesty, and O5 Ideas (open to)90.

2. Trend of activation (Varimin F4)
Trend of activation refers to a tendency to change the amount of energy expenditure.
While energy is being expended a person may want to generate even more energy
(ascending or positive trend of activation). On the other hand, too much energy might
be used up (depending on conditions of homeostasis), which is generally associated with the need to lower its level (descending or negative trend). Even in cases of low
energy expenditure an upward or downward trend of energy expenditure may
follow depending on discrepancies between actual and optimal activation (cf. Eysenck, 1973, on the differential phenomena of activation).
F4 has its strongest positive loading with Excitement seeking (E6), a prototype for
striving to heightened activation. Extraversion (E2 Gregariousness and E3 Assertiveness,) are also associated with upward trends of psychophysical energy release (effort).
89

90

“Characteristic individual differences in activation, or responsiveness, are suggested as the basis from which certain
other differences in behaviour may be derived” (Duffy, 1962, 322): Duffy alternatively uses “energy level”, “energy mobilization”, and “degree of excitation”.
“Active or controlled processes are energy or resource dependent, while passive or automatic processes are not…”
(Sanders, 1983, p. 74).

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The downward trend of activation is strongest with the C facets of Deliberation (C6)
and Dutifulness (C3). People who like to deliberate do not act hastily, they want to
consider issues in peace, to remain thoughtful, composed, and serene. The dutiful
person acts in predetermined ways and avoids breaking the rules, is obedient,
meticulous, loyal. Self-consciousness (N4) and Anxiety (N1) facets that load the Varimax factor N (Neuroticism) also have negative Varimin F4 loadings. With these
facets optimal mental energy expenditure (F5) is exceeded, a need to reduce energy
consumption is prevalent. This is consistent with clinical research: As a rule, clients suffering from mental instability are in want of a reduction of tension. This is
one of the main goals of all variants of psychotherapy. The extraversion facets are
likewise associated with an increased level of energy production (cf. F5), as was
shown. Extraverted and neurotic persons are thus similar in this respect. But extraversion is associated with some heightened need more activation (F4), a desire
for increased energy release, while neuroticism is associated with a need for diminished energy release.

3. Source of regulation (Varimin F3)
The Varimin factor F3 has pronounced positive loadings on the facets of Conscientiousness [Self-discipline (C5), Order (C2), Dutifulness (C3), Achievement striving (C4), and
Competence (C1)]. An inner source of regulation, an ego or will centre, plays a determinant role. This feature denotes a system-internal causation of behaviour and
may be called endodynamic regulation or endoregulation.
The opposite polarity is called exodynamic regulation, or exoregulation, which denotes regulation by non-ego determinants, i.e., by environmental stimulation, excitation, enticement, temptation, commands, etc. One should also distinguish between positively and negatively evaluated endo- and exoregulation. Forms of behaviour triggered by exoregulation, if positively valued, are frequent among facets
of Tender-mindedness (A6) and Altruism (A3). These forms of behaviour are responsive, not agentive, generally initiated by others without strong obligations. A modest
(A5) and compliant (A4) person avoids imposing his will on other people, so as to
maintain for them unobtrusive conditions of endoregulation.
Exodynamic regulation is predominantly found with facets of neuroticism. This
makes sense because a self-conscious (N4), vulnerable (N6), depressive (N3) and fearful
(anxiety) (N1) person feels helpless and delivered to his own mental and emotional
urges. Given such lack of will power, his troubled endoregulation is turned into exoregulation. The person is overwhelmed by his own emotions. Such lines of thought
were advanced by de Charms (1968), Heider (1958), and Deci and Ryan (1985).
Boekaerts, Pintrich, and Zeidner (2000), and Baumeister and Vohs (2004) provide
an overview of research focusing on self-regulation (unfortunately without sufficiently considering exoregulation). In Ertel (2011b) more theoretical support for
this view is provided, including references to Sigmund Freud’s ego-id polarity, Hen-

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ry Murray’s (1938) concept of press, and James Gibson’s (1963) endo-exo perspectives in his revolutionary theory of perception.

4. Mode of representation (Varimin F2)
The endo-exo distinction is abstract, but vital. A bipolar endo-exo concept is also
suggested to underlie major distinctions of mental representations. Representation
means that behaviour, regulated by internal or external determinants, is embedded
in and surrounded by a myriad of mental units bearing endo or exo characteristics.
Features of an inner world may be described as vague, unarticulated, holistic, feeling-like and subjective. These characteristics shall henceforth be called endomodal.
Features of mental representation belonging to an outer world are articulated, distinct, delimited, marked-off, detailed, and often called “objective”. These characteristics will henceforth be called exomodal. This is suggested by the negative polarity of factor F2 which is termed mode of representation.
With positive F2 loadings (indicating endomodal quality) facets of openness and neuroticism appear comparable. This finding may initially come as a surprise because
openness and neuroticism appear to be quite different. But within the context construed here, they are plausibly comparable. Hostile (N2), impulsive (N5), and depressive
(N3) people with a much disordered ego are out of touch with reality, and extremely subjective because of concomitant emotional traits. The frequently found
correlation between neuroticism and introversion, which thus far was hardly understood, suddenly seems plausible. By the same token it becomes comprehensible
that endomodal experiences are predominant in humans who are fantasy prone (O1)
and open to feeling (O3), showing much subjective supplements when dealing with
their surroundings, like those with an interest in aesthetics (O2), or people entertaining ideas (O4) and who are open to values (O6). In these cases subjective information
processing (endomodal qualities) develops spontaneously, perhaps by hereditary
disposition, while for people with an acquired neurotic dysfunctioning internal
tensions engender an inflation of endomodal ways of dealing with self and others.
An overview of pertinent research supporting the factor interpretation presented
here is contained in a meta-analysis by Mor and Winquist (2002) about selffocused attention and negative affect.
The concept of mode of representation cannot easily be built into further contextual knowledge. Many earlier attempts to grasp variance and distinctions in this
field are referred to in my more-extended study (Ertel, 2011b). They include references to William Stern (1930) (polarity of subjective vs. factual matters); to Carl Jung
(1930) (feeling vs. thinking); to P. Lewicki (2005) (internal vs. external encoding); to B.
Shanon (1993) (presentational vs. representational); to S. Epstein (2003) (experientialintuitive vs. rational); to P. S. Holzman and G. S. Klein (1954) (levelling vs. sharpening);
and to H. A. Witkin (1959) (field-dependence vs. field independence).

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5. State of functionality (F1)
Factor F1 manifests the personal system’s actual or enduring state of functionality.
The relationship of system components with each other and vis-à-vis their surroundings is more or less positive, balanced and undisturbed or negative, unbalanced and disturbed – with intermediate degrees separating the extremes. A positive state of functionality displays eufunctional system processes that are appropriate
and undisturbed, even if they may be tense and strained. Excessive or prolonged
stress and an unexpected trauma will cause dysfunctional symptoms indicating that
the system is unstable.
All neuroticism facets are negatively loaded with Varimin F1, which was to be expected. Of the remaining Varimax facets Modesty (A5) has slightly negative F1 loading (cf. Table 5.02), probably because this facet contains numerous items of an
implied ego weakness (Self-consciousness). Other variables have positive F1 loadings,
expressing eufunctional conditions in behaviour and experience, the strongest
being Competence (G1), Positive emotions (E6), and Open to actions (O5). The positive
signs that have been affected by the sign reversal of F1 and F5 loadings are informative and just as fit as the negative ones, because an absence of dysfunctionality and the presence of mental health are preconditions for approaching the
world and life with curiosity, competence and pleasure. Dysfunctionality requires
expenditure of energy to eliminating disturbance in the system, thus energy will be
lacking which must be freely mobilised for the person to become and remain open
to their environment.

Concluding remarks
Having characterised the five Varimin factors as aspects of a functional whole,
they shall henceforth be called basic components of personality functioning91. Personality
is considered as a system of constituents cofunctioning and cooperating with each
other with greater or lesser degrees of involvement for the system as a whole
(Cronbach & Meehl, 1955). The task of differential psychology is to identify general and individual parameters of these components.

91

The term facets would not, unlike components or constituents, indicate the contributory
function that basic traits have in shaping the whole. Endoregulation, for instance, is not only a
facet of personality, but also a component of its manifestation. An addition of basic as in basic
traits is recommended in order to indicate the latent level of its functioning.

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The interpretations of Varimin factors on probation
In order to test whether my interpretation of Varimin factors remains stable across
different judges, 26 students were presented with descriptions of the 30 Varimax
facets. The description of each individual facet was written on a card. The descriptions used are exemplified for gregariousness and aesthetics.
Gregariousness: “I need to have people around me, I dislike being alone, I think
parties and get-togethers are stimulating, I do not want to work alone in my profession.” (a facet of extraversion).
Aesthetics: “I have a keen sense for beauty in nature and in art and can be totally
absorbed by music. Ballet, dance, poetry fascinate me.” (a facet of openness).
Participants were asked to rank order the 30 facet descriptions five times, once for
every one of the five Varimin factor descriptions. The appendix contains Varimin
factor descriptions for F1 to F4. The description for activation (F5) e.g., was as
follows:
Increased energy expenditure. Please give top ranks (1, 2, 3 etc.) to facets associated with highest
energy expenditure and give further ranks to facets requiring less energy expenditure.
High energy expenditure means: The individual uses a great deal of energy. He/she may
want to achieve ambitious goals that require much energy consumption. It could also be that
his/her energy is absorbed spontaneously by strong impulses, urges, and emotional experience. Or
a great deal of energy is expended because of internal or external impediments that the individual
has to deal with. Energy does not only manifest itself as unhindered power, but may be consumed
by continued tension and blockage.
Reduced energy expenditure. Please rank lowest (30, 29, 28 etc.) facets with lowest apparent
energy expenditure.
Reduced energy expenditure means: The individual uses only small amounts of energy.
He/she may be pursuing less ambitious goals that can be reached with little effort. Maybe, by
nature, he/she does not seek motivational or emotional stimulation. Possibly, he/she only wants
to maintain his/her energy reserves. He/she may also need to overcome fewer internal and external impediments which, if present, would call for an increased energy input. At any rate, increased effort expenditure or continued tension and blockages occur less often in his/her case.
The participants could, for instance, have assigned ranks 1 and 2 to the facets
Activity and Vulnerability (these facets are highly positively F5 loaded), and the facets Compliance and Open to values might have obtained the lowest ranks 29 and 30
(these facets are highly negatively F5 loaded). Facets not yet ranked for increased

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131

or decreased energy expenditure were eventually to be given intermediate ranks by
the participants.
For each Varimin factor, ranks of the 30 facets were averaged across the 26
participants (possible range of means 1–30). For Varimin factor interpretations (F1
to F5), five mean rankings ensued. They were then ordered alphabetically by facet
descriptions. By the same token, the facets’ Varimin factor loadings were also ordered alphabetically by facet descriptions, eventually providing five vectors of
mean rankings and five vectors of factor loadings that were intercorrelated.
Figure 5.02 shows the result. As expected, the highest positive correlations of
mean ranks with Varimin factor loadings are mainly found in the diagonal fields.
Only the source of regulation factor deviates by showing a smaller correlation in the
associated diagonal field. Post hoc the suspicion arises that the participants were
not properly instructed about possible sources of regulation and that it was not
made sufficiently clear that exodynamic regulation could indicate either weak will
power (lack of ego regulation) or heightened susceptibility to external impulses. In
four out of five cases, however, the factor interpretations proved to be replicable
by independent judges.

Figure 5.02: Correlations between Varimin factor loadings of 30 facet variables with average ranks of
judged meanings.

132

Chapter 5 – Varimin factors from Big Five personality data

Re-interpreting Varimax factors by profiles of
Varimin factors
Varimax factors can be conceived as clusters of Varimin components. This can be
exemplified by reinterpreting neuroticism and extraversion in terms of Varimin factors
(for the remainder, refer to Ertel, 2011b).
1. Neuroticism: The most striking characteristic of people contributing to the
neuroticism cluster is disturbed functionality. Dysfunctionality goes along with an increased level of activated energy and with the desire to tone down energy expenditure. The energy expenditure of neurotically disturbed persons is dependent on
environmental impediments and on uncontrolled endodynamic impulses (quasiexoregulated). The readiness for and/or ability to endoregulation is missing. Excessive exoregulation is often called ego weakness by psychoanalysts. A disordered functionality also leads to unbalanced modes of representation. Information is chiefly processed endomodally (subjectively with excessive feeling quality). Increased selfconsciousness is accompanied by reduced assessments of reality. An increased energy
expenditure in disordered systems makes sense, because solutions are sought for
reducing painful activation (tension), often with support requested from experts
(therapists). Activated energy can be employed differently, depending on whether
the system is balanced or imbalanced. Many authors have noted this. Thayer
(1985) uses bipolar attributes to indicate undisturbed vs. disturbed energy expenditure within person systems: energetic arousal (in the case of extraversion, for
instance) and tense arousal (in the case of neuroticism). This terminological distinction
helps to compare the use of mental energy in undisturbed and disturbed systems.
2. Extraversion-Introversion: Most pronounced in individuals with high extraversion in Big Five questionnaires is an increased level of activation associated with an
upward trend92. Representations of internal and external experiences are preferably
processed in an exomodal manner, i.e., focused on external objectives. Endomodal concomitants are often neglected. The regulation of energy for extraverts is,
however, not uniform. While most extraversion facets are associated with exoregulation (behaviour is guided by external stimulation, social incitements are prominent), facets clustered as extravert do not totally lack endoregulation. Extraverts
manage their actions, they know what they want. The functional status of extraverts is balanced and has positive signs.

92

SS-orientated factor analyses make an activation/energy trait disappear. Zuckerman (1994)
comments: “…Energy Level is considered a major trait … and regarded as a basic aspect of temperament in
neo-Pavlovian models …, but it tends to get lost or subsumed under Extraversion in the Big Three and the
standard Big-Five” (p. 66).

Chapter 5 – Varimin factors from Big Five personality data

133

For the opposite pole, for introverts, the statements characterising extroverts must
be reversed. The behaviour of introverts is based on a sub-average energy level.
Moreover, introverts tend to lower their activation levels. Introverts tend to have
endomodal representations even when other people generally have exo-modal
experiences. Introverts also prefer endoregulated activities. They do not like to be
guided by other people. Their functionality status often indicates some lack of
adjustment.
Concluding from these two examples: neuroticism, extraversion-introversion
and the other Varimax “dimensions” are not elementary building blocks of some
personality theory, instead they are clusters based on definable Varimin profiles
(cf. Table 5.07). Watson and Clark (1997, p. 780) likewise emphasise that positive
affectivity is a “central feature of the construct [extraversion]” which combines a number
of sub-constructs (“positive emotional experience forms the core of the higher order construct”).
The question remains open why Varimin components of activation, regulation,
representation etc. combine as profiles called extra- and introversion. But this
question is of subordinate importance here.

Discussion of chapter 5
Objections to the present interpretation of Varimin factors are fairly predictable. It
might be argued that the suggested interpretations are not compelling, different
researchers might suggest different factorial meanings. It might likewise be objected that these interpretations are primarily based on semantics, not on objective
data. However:
1. Writers of questionnaire items and subjects responding to them operate with
the semantics of verbal units. Any production and comprehension of language is
dependent, in the first place, on interpretations, not on quantitative units.
2. With the aid of words and terms references are established to trans-linguistic
reality. The components of kinship terms (e.g. generation, gender etc., cf. chapter
2) are undoubtedly anchored in non-linguistic realms. In the semantic field of
personality, components obtained by the same methodology should no less be
considered as referring to some essentially trans-linguistic reality.
3. Doubts are merely legitimate as to whether terminological decisions are optimally appropriate and useful for more comprehensive contexts demanding an
overall understanding.
4. Terminological decisions made in this study seem to take directions towards a
developing conceptual network. Varimin factors emerging here reveal relationships

134

Chapter 5 – Varimin factors from Big Five personality data

to more encompassing themes and the theories of scholars ranging from Pavlov
to Freud to Murray to Shanon etc.
5. One of the most urgent theoretical desiderata in psychology is that a bridge be
built joining differential psychology with general psychology. SSM work in differential psychology has not yet built and apparently cannot build connections with
general psychology93. Surprisingly, CSM factors obtained from interindividual
covariances, using a bottom-up strategy, proved to be useful for describing processes of general psychological functioning.
6. It is possible that the componential concepts obtained from Varimin analyses of
personality data are liable to be used with bio-psychological and neuropsychological aims (with the help of constructs like energy, activation, regulation,
representation, functionality).
7. Objections to terminological decisions: I should nevertheless always consider
them if objections are embedded in theoretical contexts replacing and significantly
improving the beginnings that have been developed here.

93

“It would be advantageous, if not magnificent, if a between-subjects five-factor model would imply …
exchangeability [with a within-subjects model]… However, the required equivalence has not been shown, and [we]
expect that it will not, in general, be a tenable assumption.” (Borsboom, 2013, p. 213). Apparently, the
basic features gathered from inter-individual correlations of the NEO-PI-R facets by using
Varimin rotation (activation, regulation, representation, etc.), meets the demands formulated by
the authors: “If it is shown that a given set of … processes [within individuals] leads to a particular latent
variable structure [of personalities], we could therefore say that this set of processes realizes the latent [personality]
variables in question” (p. 215 f).

Insight and Outlook

My dispute with alleged misconceptions of factor analytical research may appear
“rebellious”. Could I have prevented giving this impression? When I went into the
lateral thinking mode and during subsequent years of empirical testing - not all
results have been presented here - did I perhaps not abide by the rules of proper
scientific conduct? I used verifiable data trying to prove the theory that SS modelling was an error. At the same time, I supplied evidence for the benefits of the CS
modelling strategy. Even at the risk of exaggeration I dare forecast that adjusting
psychological research to CSM might open new doors to latent domains of mental
processing. Componential analyses might diminish the chaos of unanalysed constructs that has long inhibited psychological progress. New perspectives might
become visible and initiate prolific developments.
Or should I state in mitigation that Varimin is just another rotation method to
be added, with pluralistic tolerance, to the long list of earlier such techniques aiming to solve the continuing problem of indeterminacy of extracted factors? Should
I maintain that conventional SSM data analyses, including SEM, might have the
promising future that most methodologists of today, endorsing these approaches,
are expecting.
What could prove me wrong claiming that simple structure as advocated by
Thurstone and his followers had gravely detrimental consequences to this day
(2013) and that SSM was and sadly remains the main source of the misery afflict-

136

Insight and Outlook

ing multivariate analyses? What could stop me believing that by taking a 180degree turn the past mistakes are rectified? Unrecognized logical errors in my
reasoning? Errors in my empirical work? I have braced myself for criticism and
would enjoy being confronted with surprising new facts. I would be happy if sceptical conventionalists were up in arms, if they would deal with the challenge by
undertaking empirical attempts to defend their approach against my claim that
they have failed. For me it is inconceivable that the majority of my results can be
brushed away. But I can wait and see whether this book is just an instance of an
aberrant scientist’s need for adventure or the beginning of a turn in multivariate
methodology towards ensuing conceptual and empirical innovations.

Review of chapters 1 to 5

Chapter 1: Critique of the simple structure doctrine
In chapter 1 the aim of the book is outlined: A methodological doctrine is revealed and critiqued that has prejudiced FA since its beginning. Simple structure
(SS), the guiding principle for factor rotation (Thurstone, 1935/1947) is unveiled
as questionable because it generally distorts latent sources of variance of manifest
empirical variables instead of revealing them. The critique is based on theoretical
considerations and is supported by many verbatim quotations from critical authors. The present calamity of factorial research is deemed to be due to these
methodological flaws. One-sided mathematical formalisation in the discipline has
lost its objectives by unjustifiably ignoring ordinary sources of gaining knowledge,
including common sense. The problem of SS cannot be solved by circumplex and
structural equation modelling (SEM) which suffer no less from SS errors. An alternative factor transformation leading to complex structures is demanded. A
paradigm change is overdue.

Chapter 2: Finding complex structures
This chapter resumes the preceding criticism. The rotation procedure Varimax
which is commonly used to generate SS is replaced with Varimin which aims to
manifest latent complex structures (CS). Varimin optimises the model of complexity which – being already announced by initial unrotated structures – still needs

138

Review of chapters 1 to 5

improvement. The new method raises various questions, of which five are discussed. How can Varimin factors be interpreted? Do latent sources of covariance
not already appear sufficiently complex with initial solutions? Are SS solutions not
fairly interpretable, how else could they have been routinely used? How to interpret the commonly encountered bipolarity of Varimin factor loadings? Is FA with
complex structure transformation applicable to data affected by method factors?
Ten empirical applications of Varimin transformation serve as exemplary tests.
Particular features of transformation to CS, revealing latent sources of covariance
(by Varimin), are elucidated by comparing pertinent results with those obtained
from transformations to SS (by Varimax). Varimax will remain useful for merely
clustering objectives. Attention is also drawn to limitations of the methodical innovation.

Chapter 3: Decathlon data under analysis
The results of a factorial study are reported using transparent sports data: Decathlon record scores covering 10 sporting events. The aim is to compare Varimin and
Varimax results regarding factorial stability and interpretability. It is shown that
Varimin factors reveal latent components of sports activity in interaction, while
Varimax factors yield obscure clusters of features. In addition, factor structures
obtained by Varimin rotation are more robust to changing data sources than those
obtained by Varimax rotation. The results of this study are consistent with pertinent non-factorial results of sports physiology.

Chapter 4: Intelligence data under analysis
Eighteen matrices of intercorrelations of eight subtest variables of the intelligence
test IST are subjected to principal component analysis, the resulting factors are
rotated by Varimin to model complex structure (CS). The 18 Varimin solutions are
aggregated, two factors result: Varimin-F1 represents a general factor g (“basic
intelligence”), Varimin-F2 represent a performance-modifying factor, apparently
based on previous educational training and learning effects (to be termed “learning assets”, l). The validity of Varimin-F1, basic intelligence, is ascertained by high
correlations between g and test scores of general intelligence, operationalised by
culture-free CFT and FRT. The interpretation of Varimin-F2 as learning assets
finds support by significant correlations with school grades and scores in orthography and arithmetic. The 18 PCA-factors are also transformed by Varimax to SS.
This transformation causes the splitting up of initial g into two seemingly separate
factors, called “fluid” and “crystallised” intelligence by convention. In addition,
differences between Varimax F1 and F2 of correlations with external criteria of
general intelligence versus school grades and training scores in orthography and
arithmetic that should emerge are missing. Apparently, SS modelling of intelligence test data amalgamates general intelligence with learning effects. Rotation of

Review of chapters 1 to 5

139

intelligence data to SS does not reveal independent contributions of latent functional components, but hinders their detection.

Chapter 5: Varimin factors from Big Five personality data
Varimin rotation is applied to five PCA factors obtained from 30 facet variables of
NEO-PI-R (Ostendorf & Angleitner, 2004). As expected, Varimin-rotated factors
do not replicate the Big Five (neuroticism, extraversion, etc.), but instead reveal five
distinctive bipolar factorial components: level of activation (high-low), trend of activation (ascending-descending), source of regulation (endodynamic-exodynamic), mode of
presentation (endomodal-exomodal), and status of functionality (eufunctional vs. dysfunctional). The well-known Big Five factors turn out to be clusters of Varimin
components rather than unique dimensions. The validity of the five features obtained by Varimin has largely been confirmed by independent rankings of the 30
NEO-PI-R facets using Varimin features as ranking criteria. Replacing SS analysis
on a broader scale through CS procedures might lead to building blocks for conceptualising personality and individual differences as future theoretical goals.

Appendix

Bipolar meanings of Varimin factors provided to participants as criteria for
rank ordering the 30 NEO-PI-R facets.

For factor F5, level of activation, refer to section Varimin factor interpretations p.126.
The interpretation of Varimin factors on probation for F4, F3, F2, and F1 are as follows:
F4: Trend of activation
Ascending trend
The individual often feels a need to expend more or greater energy than he or she
currently expends. Even though an energy release may be high, the person may
want to increase it. Energy expenditure can also be too low and thus cause discomfort. The person feels insufficiently stimulated and challenged and therefore
strives to make use of more energy. The manners in which persons attempt to
increase energy expenditure may differ – myriad forms of stimulation may be
sought to increase the intensity level of experience. The individual may also feel he
or she has an increased influence on his/her environment and other people. None
of this is crucial, though. Likewise it is not important whether the individual actually manages to increase the desired energy expenditure. It is only essential that a
need exists for increased activation of the individual’s energy reserves.

142

Appendix

Descending trend
The individual feels a need to reduce energy expenditure. The current level of
energy being expended is too high. The individual may seek to reduce an unpleasant excess of stress and tenseness. Possibly, and although the person may be expending little energy anyway, he/she may want to reach an even better state of
rest that would further improve the energetic condition. It is not critical in which
way the individual attempts to achieve a reduction of energy expenditure and
whether these attempts are met by success. The only aspect of importance is the
presence of a desire to reduce energy manifestations.
F3: Sources of regulation
Endodynamic regulation (self or internal regulation)
The behaviour of an individual follows a volitional programme. The person makes
decisions and seeks to reach various goals, wanting to influence and change external conditions. It is not important which activities the person initiates and whether
they are focused reflexively on the person him- or herself or on external objectives. The actions should merely be initiated and executed by the person’s ego and
not by others or circumstances. The acting individual should also not be urged by
their own uncontrolled emotional drives and should remain free from internal
constraints and compulsions.
Exodynamic regulation (external or uncontrollable internal regulation)
The individual’s behaviour is initiated by other people or environmental factors.
The person reacts to external stimulations, triggers, or seductions. Self-determined
decisions are rare. The person is unwilling or unable to manage his or her environment. Instead, his or her behaviour predominantly consists of reactions to
external challenges or to inner impulses and constraints. Possibly, the individual
cannot control him- or herself well despite being eager to do so. At any rate, the
individual is considerably dependent on external or internal will-restricting conditions.
F2: Mode of representation
Endomodal representation
The individual tends to prefer subjective views. Thinking about things and experiencing them is more important than the things themselves, feelings are more effective than what is transmitted by sensory channels. This kind of subjectivity may
have positive or negative effects. An openness to feeling qualities might result as
well as the neglect of objective and sensually discernible realities.
Exomodal representation

Appendix

143

The individual tends to prefer objective views. Experiencing things is less important than things as they are, feelings are less important than what is transmitted
via sensory channels. This kind of objectivity may have positive or negative effects. A precaution against subjective errors might result as well as a neglect of
beneficial subjective contributions to reality as a whole. The individual’s inclination to cling to what is factual might be inflated at the cost of fruitful subjectivity.
F1: Status of functionality
Eufunctional or balanced status
The individual is consistent with him- or herself; varying directions of thinking,
experience, and aspirations are in harmony. Energy expenditures, however strong
they may be, are aligned with one another having balanced relationships. Tensions
may be present, but they do not strain the overall system. Should major conflicts
occur, they do not survive for long and are soon resolved.
Dysfunctional
The individual is at odds with him- or herself. The person’s thinking, experience,
and behaviour are insufficiently aligned and not well adjusted; they are not in
harmony and interfere with each other. The relationship of activated energies is
unbalanced. Ambivalences may cause tensions and conflicts which tend to endure
and are only overcome slowly, if at all.

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E

xploratory factor analysis (EFA) is a statistical tool for digging out hidden factors which
give rise to the diversity of manifest objectives in psychology, medicine and other sciences. EFA had its heyday as psychologist Leon Thurstone (1935 and 1948) based EFA on
what he called the “principle of simple structure” (SS). This principle, however, was erroneous from the beginning what remained unrecognized despite subsequent inventions of
more sophisticated statistical tools such as confirmatory analysis and structural equation
modeling. These methods are highly recommended today as tolerable routes to model
complexities of observation. But they did not remove the harmful errors that SS had left
behind. Five chapters in this book demonstrate and explain the trouble. In chapter 2 the
ailment of SS is healed by introducing an unconventional factor rotation, called Varimin.
Varimin gives variables of an analysis an optimal opportunity to manifest functional interrelations underlying correlational observations. Ten applications of Varimin (in chpter 2)
show that its results are superior to results obtained by the conventional Varimax procedure. Further applications are presented for sports achievements (chapter 3), intelligence
(chapter 4), and personality (chapter 5). If Varimin keeps on standing the tests new theoretical building blocks will arise together with conceptual networks promoting a better
understanding of the domains under study. Readers may check this prognosis by themselves using the statistical tool (Varimin) which is provided by open access in the internet.

Suitbert Ertel

Factor Analysis

Suitbert Ertel

Factor Analysis

Healing an Ailing Model

ISBN: 978-3-86395-133-7

Universitätsverlag Göttingen

Universitätsverlag Göttingen