Appendix to 'Personality, Biology and Society':
Description and discussion of
FACTOR ANALYSIS
LITTLE EXPERIMENTATION IS PERMISSIBLE ON HUMAN
PERSONALITY; SO IT IS IMPORTANT TO MAKE THE BEST OF CORRELATIONAL
DATA. FACTOR ANALYSIS ALLOWS LARGE CORRELATION MATRICES TO BE
SUMMARIZED AND REDESCRIBED IN MORE SIMPLE TERMS; AND IN THE COURSE
OF THIS IT WILL SOMETIMES INDICATE POSSIBLE CAUSAL STORIES AS
TO THE MAIN INFLUENCES THAT MIGHT HAVE BEEN AT WORK TO GENERATE
THE CORRELATIONAL ARRAY. HOWEVER, THE METHOD HAS HAD DETRACTORS
AS WELL AS SUPPORTERSAS THE FOLLOWING INTRODUCTORY QUOTATIONS
INDICATE.
"In my view the primary object
of factorial methods is neither causal interpretation, nor statistical
prediction, but exact and systematic description. And I suspect
that most of the confusion has arisen because factors, like the
correlation coefficients on which they are based, have been invoked
to fulfil these three very different purposes, and so have made
their appearance at three very different levels of thoughtlike
the famous legal firm of Arkles, Arkles & Arkles, which, 'more
to its own satisfaction than that of its clients, canvassed three
different lines of business in three small offices on three different
floors."
C.BURT, 1940, The Factors of the Mind. London University
Press.
"[Factorial analysis is] a brilliant but misguided departure
from the central path of empirical psychology."
O.L.ZANGWILL, 1950, An Introduction to Modern Psychology.
London : Methuen.
"....factor analysis provides no way to discover or explicate
the processes that in combination constitute intelligence."
R.J.STERNBERG, 1977.
"....despite certain limitations, factor analysis can
aid in the identification and characterization of cognitive
processes."
J.B.CARROLL, 1988.
"[The Cambridge psychologist, Sir Frederick Bartlett] regarded
factor analysis as a technical culdesac of psychology."
Paul KLINE, 1993, A Handbook of Psychological Testing.
London : Routledge.
"One issue on which [John Carroll (1994, Human Cognitive
Abilities, CUP)] is adamant is that, contrary to the clamor
of critics, the track record of factor analysis provides a sound
assessment for understanding the nature of cognitive abilities."
Kevin LAMB, 1994, Mankind Quarterly 24.
"The history of the development of factor analysis is not
edifying. Pearson and Spearman, who together had the skills to
develop it rapidly, quarrelled and feuded. ....Thomson....has
been neglected. Garnett....was ignored. The new wave of American
factorists in the 1930's lumped all the British workers in the
field together, called them Spearman's school, and set out to
rubbish them. Thurstone in particular turned the clock back....
The squabbles of the 1930's have continued with scarcely a break....
[Burt's] real influence is felt through the work of those of his
students, such as R.B.Cattell and H.J.Eysenck, who went on to
build factorial theories that have attracted widespread attention,
and through the continuing line of thinking which is prepared
to allow both g and more particular kinds of ability....
....one thinks of the work of Philip Vernon, and more recently
of course Jensen has laid emphasis on the value of Burt's contributions."
S.F.BLINKHORN, 1995, 'Burt and the early history of factor analysis.'
In N.J.Mackintosh, Cyril Burt: Fraud or Framed? Oxford
University Press.
"I inspected the subject index of some wellknown texts in
experimental cognitive psychology and found that the term factor
analysis never appears in the subject index."
D.DORFMAN, 1995, Contemporary Psychology 40.
The procedures that are involved in factor analysis (FA)
as used by psychologists today have several features in common
with the procedures for administering Rorschach inkblots. In both
procedures, data are first gathered objectively and in quantity;
subsequently, the data are analysed according to rational
criteria that are timehonoured while not fully understood
by most users; finally, in case it had not been enough to have
a choice between various analytic schemes, users are allowed
to interpret the 'results' in their own ways. It is probably
the combination of these three attractive facilities that defies
prophets of doom and accounts for the enduring popularity of FA
in psychology (especially when experimentation is impossible).
FA allows psychologists {and other users}:
(i) to handle the almost lifelike complexity of multiple measures
while reducing that complexity to slightly more manageable proportions;
(ii) to demonstrate semblances of methodological sophistication,
numeracy, computer wizardry and respect for their tribal elders;
and
(iii) to enjoy the subjectivity and originality of the artist
within the pretty secure framework provided by wellrehearsed
routines of reason and science.
These three achievements are loosely associated with the three
broad stages of FA:
(i) the collection of multivariate data and its
preparation for analysis;
(ii) the transformation (normally reduction) of
the correlation (r) matrix to a factor matrix; and
(iii) the final rotation and interpretation of the
factors.
Users who take each stage seriously can feel themselves embarked
on a voyage of discovery the very conservatism of which makes
for outcomes that are at once original and reasonably truthful.
At the same time, both the 'conservative' and the 'creative' aspects
of FA can be abusedespecially because of the large number of
particular steps that are involved. A common accusation of undue
'conservatism' would be that the authorized procedures are too
often applied with unreasoning faith in the wisdom
of the ages. It is the unthinking, mechanical use of FA that attracts
the criticism "Garbage ingarbage out!" On the other
hand, the 'creative' options within FA sometimes seem to allow
such a wide discretion to the researcher as to make FA "more
of an art than a science." Whereas Popper advised scientists
to weed out false hypotheses so that truth may flourish, FA can
sometimes seems to have more hypotheses at its end than at its
beginning (e.g. by allowing rotation of the axes to theoretically
favoured positions). Not the least galling to the critic is the
discovery that factorists actually rejoice in such liberties.
A casual user may want FA to provide the solutionand
FA will oblige; but with greater sophistication, a user
will end not just with factors but with ideas for
what factors could be expected in different, improved data sets.
The aims and limitations of Factor Analysis (FA): an analogy.
{OR: A View from the Motorway Bridge....}
The central aim of FA is to reduce the covariance
in a correlation matrix to more manageable proportions.
[The covariance is the 'amount of correlation',
or, precisely, 'the sum of all the squared correlations
between the variables in a table of correlation coefficients (usually
Pearson r's)'. (Squaring correlations eliminates minus
signs and highlights stronger correlations.)] The idea is to explain
the covariance of a relatively large number of variables in terms
of a rather smaller number of factors. [Such explanation
is not a discovery of causation except by happy accident.
What is being carried out is an accounting exercise in
which scores on variables are finally 'accounted for' statistically,
and rendered predictable from, scores on hypothetical factors.]
It could be said that factors are parsimonious packages, each
involving bits often quite large bitsof the covariance found
amongst the larger number of original variables. If the analysis
has gone smoothlywithout too many departures from conventional
assumptions and expectationsthe resulting factors will be promulgated,
instead of the original variables themselves, as being capable
of serving as the source of the original observed covariance.
That is, the factors could be causal but certainly enable
prediction with economy.
Each resulting factor is thus a hypothetical source of the covariance
in the studyand being a mere hypothesis is a headstart over not
emerging as a factor at all! Yet a factor's relation to variables
is no different from the relations that may be envisaged to exist
among variables themselves. When variables X (say, 'students'
knowledge of St Augustine') and Y ('knowledge of St Thomas Aquinas')
both correlate better with Z ('knowledge of Aristotle') than they
do with each other, the principle of parsimony in scientific explanation
inclines us to look with optimism at the possibility that Z actually
influences both X and Y; or at least that Z taps, or reflects
especially well an important underlying factor, say Zf, that itself
influences both X and Y. [For example, a fourth variable or factor,
'number of philosophy courses passed by students', might
explain, or account for quite a lot of the variation
('differences') between people in their knowledge of all three
philosophers; and the 'Aristotle' item might have tapped that
knowledge especially well because Aristotle is the best known
of the three philosophersmaking it especially unlikely that a
person could know a lot about philosophy without knowing quite
a bit about Aristotle. (By contrast, had the correlations with
'Aristotle' been weak, the covariance in the matrix would
be chiefly in knowledge of both Augustine and Aquinas. Why might
such differences have arisen, independently of knowledge of Aristotle?
Since both Augustine and Aquinas are Saints, individual differences
in knowledge of them might have reflected differences in degree
of religious interest and exposure.)] A factor is just
like a variable that correlates especially well with several other
variables.
Suppose we wish to learn from scratch about motor carsstarting
from only the most elementary knowledge that differently shaped
and titled bodies of metal can be seen moving around the roads.
That is to say, suppose that we are able to see some of the differences
between different makes of motor car, yet we cannot drive them
ourselves, read about them, experiment with them or even measure
them in any sophisticated way. (In fact, such restrictions are
pretty similar to those under which psychologists labour when
trying to find out anything very interesting about people!)
Frustrating as our situation may be, we can at least make frequency
counts of the appearance of the different makes of car in various
locations; and we can do a lot of counting quite easily if we
stand on a motorway bridge and count the frequency with which
the makes appear in the various lanes underneath. (In such a study,
the lanes would be analogues of the psychologist's tests,
or variables; the makes of car would be equivalent to individual
testees, or 'subjects'; and the correlations between
appearancefrequenciesperunit time (between the makes' scores
for frequencyofappearanceinLane1, frequencyinLane2, Lane3,
etc.) would provide the contents of the correlation matrix
to be submitted to FA.)
Now, suppose all the laneappearancefrequencies (LAF1,
LAF2, etc.) are positively correlated (i.e. the
same makes appear more often in all the lanes).
We can now talk of a factor of 'general frequency'.
We will conclude that, no matter which measure we take,
one 'source' of each variable's variance is that for whatever
reasonthere are simply more of some makes of car than of others.
(Which make is more frequent might still differ by locatione.g.
in shopping precinct car parks, or on motorways in another
country. But the overall positivity of our correlations, the 'positive
manifold', offers some assurance as to the generality, or at least
the notcompletespecificity of frequency differences.)
This is our first discovery. [By contrast, had the average of
the interLAF correlations (r's) been
zero, we would have falsified the hypothesis that there
are, quite generally, more of some cars than of others.]
However, even amidst such general positivity of intercorrelation,
further factors might be detectably at work. For example, if the
r's between LAF1 and LAF6
(between, say, frequencies on the extreme 'inside' and
'outside' lanes of the Londonbound carriageway) were significantly
lower, we would begin to think that the makes of car also differed
in some other respect than just that of 'general frequency'.
If we knew the significance of cars' being in such lanes, we might
conclude that makes of car differed in the typical speeds at which
they are drivenwhether from basic capacity differences between
them or because they attracted purchasers having different requirements
or propensities (such as levels of risktaking). We might be able
to check this hypothesis, perhaps by recording 'speed' as an additional
variable, or by making some motorway journeys and counting the
appearancefrequencies of various makes on the 'hard shoulder'.
For example, if LAF1 correlated with hardshoulderfrequency
more than LAFs generally correlated with each other,
we might conclude that there were indeed capacity differences
between makes; and we might interpret a factor of 'speediness
vs breakdown frequency' as being one of underlying differences
in 'roadworthiness'.
Yet further factors might be recognized, even within the constraints
of such a hamstrung study. Suppose the LAF r's
are generally positiveyielding the first, 'general frequency'
factor; and lower (or even negative) between the extreme inside
and outside lanesyielding a second, 'speed/roadworthiness' factor.
But now suppose the extreme inside lane, LAF1, correlates
significantly less with the other inside lanes (LAF2
and LAF3) than they do with each other. In this
case we might note a potentially specific, third source of variance,
and perhaps hypothesize that LAF1 is specially influenced
by drivers' preparations for turning off the motorwaytowards,
say, some leafy suburb in which makeownership was unrepresentative
of general motorway prevalence, and not in ways reflecting
choice of makes having low capacity for speed. Thus we would resolve
that our next 'motorway FA' would include a frequency count from
an actual exit lane so as to be able to confirm the existence
of a 'leafy suburb' factor and precisify its nature. [The inclusion
of such an 'exit lane' variable might have proved particularly
valuable if the first FA had thrown up not the two quite intelligible
factors of 'general frequency' and 'speed/roadworthiness' but
two equally large and independent factors of 'inside lanes frequency'
and 'outside lanes frequency'. For, by distinguishing (cf. rotating
to) a special 'leafy suburb' factorwhich would have 'pulled
in' some of the variance on LFA1, and thus some
of the negative covariance between the inside lane variables'we
might have strengthened the 'inside lanes factor' and given it
a positive r with the 'outside lane factor'. Thus,
putting 'exit lane' variance to one sideor extracting that
factormight actually have allowed the more interpretable
factors of 'general frequency' and 'speed/roadworthiness' to emerge
in their own right.]
[For an actual example of FA on physical objects, see Laumann
& House, 1970, Sociol. Res. Here the analysis is of
the contents of 800 living rooms. It yielded two factors of 'social
class' and 'tradition vs modernity'. Richardson
(1978, Neuropsychologia) provides a readily intelligible
FA of handedness: he finds writing, hammering and throwing to
provide the best items for measure the general factor of right
vs lefthandedness.]
Such are the main possibilities of discovery and discourse that
factor analysis enables. The provision of a 'summary' is certainly
important in much of the work in human individual differences,
where the measures that are available can so seldom be described
as being established 'tests' of anything that is well understood
in its untested form. At the same time, much more interest attaches
to summaries of sense than to summaries of nonsense: careful and
wideranging consideration of which variables to use and of which
factors to aim for (at the last stage of factor rotation)
will show FA at its best.
References for psychologists
The best introduction for psychology students is probably:
CHILD, D. (1971). Essentials of Factor Analysis.
Shorter (severalpage) summaries of FA are provided by:
CRONBACH, L.J. (1961 et seq.) Essentials of Psychological
Testing.
CATTELL, R.B. (1971) Abilities.
LEMKE & WIERSMA (1976). Principles of Psychological Measurement,
Chap. 8.
CATTELL,R.B. & KLINE,P.(1977) The Scientific Analysis of
Personality and Motivation.
KLINE, P. (1981) in F.Fransella, Personality.
RUST, J. & GOLOMBOK, S. (1989). Modern Psychometrics,
Chapter 9.
KLINE, P. (1993) Handbook of Psychological Testing.
The classic, comprehensive account is:
HARMAN, H.H. (1976) Modern Factor Analysis, 3rd edn. Chicago
University Press.
Readers wishing helpful accounts and demonstrations that bring
out the psychological relevance [or, some may cruelly claim,
irrelevance!] of factoranalytic procedures and arguments
might consult:
THURSTONE, L.L. (1934) Psychological Review 41.
GUILFORD, J.P. & GUILFORD, T. (1936) J. Psychology.
CATTELL & DICKMAN (1962). Psychol. Bull. (re
reality of 'oblique factors')
FRANSELLA, F. & ADAMS, B. (1966) Brit. J. social &
clinical Psychol.
CATTELL et al. (1970). Brit. J. Psychol.
EYSENCK, H.J. (1971) Brit. J. social & clinical Psychol.
BRAND, C.R. (1972) Brit. J. social & clinical Psychol.
NEISSER, U. (1979) Bulletin of the Brit. Psychol. Socy.
GOULD, S.J. (1981) The Mismeasure of Man.
JENSEN, A.R. (1981) Bias in Mental Testing.
BLAHA & WALLBROWN (1982) J. consulting & clinical Psychol.
(re 'hierarchical FA')
CARROLL, J.B. (1982) in R.J.Sternberg, Handbook of Human
Intelligence.
KLINE, P. & BARRATT, P. (1983). Adv. in Behav. Res. &
Ther. 5. (re 'obliquity')
EYSENCK, H.J. & EYSENCK, M.W. (1985) Personality and Individual
Differences:
A Natural Science Approach.
JENSEN, A.R.(1985). Behavioral & Brain Sciences. (re
'hierarchical FA')
McCRAE, R.R. & COSTA, P.T. (1985) J. Personality.
PARKES, K. (1985). Person. & Indiv. Differences 6 (re
'number of factors to extract')
OBRIGG & CHEEK (1986). J. Personality. (re 'confirmatory
FA')
EMLER, N. (1987) J. Child Psychol. & Psychiatry.
SCHONEMANN, P. (1987) in S. & C. Modgil, Arthur Jensen:
Consensus and Controversy.
MEYER et al. (1988) Personality & Individual Differences
9.
BRAND, C.R. & EGAN, V. (1989) Personality & Individual
Differences 10.
CARROLL, J.B. (1993) Human Cognitive Abilities.
Factor Analysis: its three main stages.
(i) The main issues and disputes that arise regarding the
selection and preparation of data for FA are as follows.
(a) There should be some varianceor, quite simply, some
differences between testees [or between whatever units have been
assessed]. More precisely, there should be an appropriate range
of scores on most variables: so, usually, standard deviations
should approximate those normally found for the variables
to be analysed. This may seem an obvious point; but variancefailures
can easily arise in two ways.
(1) The subjects who have been chosen for the study may not differ
markedlyby population standardsbecause of some kind of preselection:
e.g. IQ's will hardly range below 105 in studies
of British university students; and the attitudescores of students
of psychology will seldom range sufficiently widely to allow researchers
to explore the further reaches of 'authoritarianism', 'toughmindedness',
'realism' or 'social conservatism'.
(2) When new, 'experimental' items are used in a questionnaire
survey, some of them will have less variance than the researcher
had envisaged: e.g. sympathy for 'civil servants',
'teachers' or 'gypsies' may be much higher than the test constructor
believed from reading the newspapers.
[A useful rule of thumb might be to arrange visual display of
frequency distributions and/or scattergrams before proceeding
to FA. An FA may proceed using nonnormal variance, but
its conclusions will need to be appropriately qualified.]
(b) Flooding the datamatrix with related variables (e.g.
with different facets of extraversion, such as liveliness,
jocularity, adventurousness, sociability and activity or even
with sensationseeking, risktaking and impulsivity) will produce
a large extraversion factor, accounting for an impressive
percentage of covariation. Conversely, even the most familiar
dimensions of differential psychology will have a struggle to
'emerge' if no marker variables have been put in to define them.
[Ideally, the FA researcher seeking 'the dimensions of personality
/ attitudes / abilities / etc.' should be able to say of what
population of variables the study's selected variables
provide a sample. (A specified sample might involve 'all the tests
mentioned by particular authors' or '....mentioned in a particular
journal over a period of time.')]
(c) The inclusion of scores that depart extremely from sample
means can bias resulting r's and thus factors. Often,
such outliers are the result of clerical errors in scoring
or at the stage of data input at the computer keyboard. Sometimes
they are just that one subject is markedly older, and more traditionally
'religious' than the rest: such subjects should make researchers
question 'Of what population are my subjects intended
to provide a sample?' [Particularly in small sample (when N
< 100), there is a case for setting aside subjects whose scores
are more than three standard deviations high or low on any variablewhile
prominently documenting such exclusion when reporting the
study.]
(d) Whether to analyse items or item parcels was
long a matter of dispute between the mighty psychometricianpsychologists,
Hans Eysenck and Raymond Cattell. Individual items seldom have
any substantial percentage of their variance determined by the
factor that they are intended to 'measure'. Even IQ items seldom
correlate at more than .20 with final IQ; and personality itemssuch
as for extraversiontypically correlate at rather less than
.10 with the trait that gleams in the expectant researcher's eye.
Analysing individual items can thus lead to peculiar results in
small studies and to 'bloated specifics' (e.g. involving
two or three items mentioning 'punctuality') in larger studies.
Cattell therefore favoured using parcels of four or five items
(all of some preestablished broad type) as the unit of analysis.
[This enabled Cattell to develop measures of a rather larger number
of personality factors and dimensions than Eysenck indeed, Cattell's
usual six, independent, 'secondorder' dimensions of personality
are the forerunners of the 'Big Six' dimensions of modern personality
psychometry (e.g. Ormerod & Billig, 1981, Person. &
Indiv. Diffs 2.; Deary & Matthews, 1993, The Psychologist;;
Matthews & Oddy, 1993, Person. & Indiv. Diffs 14;
Brand, 1994, Psychologica Belgica). However, Eysenck (e.g.
1992, Person. & Indiv. Diffs.) still tends to maintain
that his 'Gigantic Three' personality dimensions are more substantial
and scientifically wellspecified in terms of established underlying
mechanisms and processes. At present, it can only be suggested
that item parcels will be particularly suitable when there appears
to be some special, 'parceladded' meaning to high or low scores
on all the items. For example, individual 'honesty' items
('Have you ever cheated / shouted / lied / etc. ?') may
reflect real variance in behaviour; however, when a subject denies
virtually any human failing, i.e. 'lies' on quite a number
of items, hypocrisy can properly be suspected. Thus a 'parcel'
of items may pick up real variance in 'hypocrisy / faking good'
that is poorly captured by individual items.]
(e) In order to enable relatively clear isolation of meaningful
factors from sheer error variance, it is helpful to include in
the FA several variables that have nothing to do with any of the
meaningful factors that are expected. Some such mock variables
might even be composed of mere random numbers. Provision of such
hyperplane stuff was long a feature of Cattell's work;
and it will serve to prevent overenthusiasm about factors that
emerge and overimaginative rotations of factor axes {see (iii)
below}.
(f) FA serves none but an illustrative purpose unless the number
of subjects is at least four times the number of variables.
Correlations themselves are less reliable when N's
are low and when the subjects' scores derive from untried tests;
and factor calculation involves multiplying variables' correlations
and is thus particularly affected by unreliability.
(g) FA only searches for linear relations. To check for Ushaped
relations between variables, deviation scores
[testees' scores minus the mean score on each variable,
disregarding the sign] may be included.
(h) When dichotomous or other noncontinuous variables
are included in FA (as often happens when test items
are among the variables), their correlations should be expressed
as phi coefficients or as point biserial coefficients.
It is quite in order to combine these measures of association
with Pearson r's in the final correlation matrix.
(ii) At the second stage of FA the analysis of the
correlation matrixthe main question is that of which FA package
to use.
(a) The first choice is whether to analyse only the covariance
in the r matrix, or whether to analyse the total
variance and covariance in the data. Normally the latter seems
more reasonable in psychology: one hardly wants to forget the
unshared variance  which may well be shared with other variables
that did not happen to be included in the study.
(b) Making the 'total variance' choice {as above} requires a communality
assumption about how much each variable has in common with
itselfi.e. about the reliability of each variable.
(1)Perhaps the simplest assumption to make is that, by definition,
each variable correlates at unity (1.00) with itself. This is
what is assumed in quite the most popular type of FA today, known
as Principal Components Analysis. The assumption
introduces the maximum possible unshared variance and so requires
more computational effortthus it only became popular as computers
became available to perform all the necessary donkey work.
(2)Alternatively, an attempt can be made to provide a realistic
estimate of the actual reliability of each variablerather
than flood the FA with specific (and often 'error') variance which
then has to be distinguished and set aside in special factors
during the course of the analysis. One possibility is to insert
for rXX the known reliability of Test Xe.g. as
given in the test handbook. However, such knowledge may not be
available or securely based; and, even if it is, it will be for
a population that probably differs in some significant ways from
the researcher's present sample. Thus the more usual procedure,
used in Centroid Factor Analysis (and in Principal Factor
Analysis, after a previous Principal Component Analysis has
been undertaken) is to estimate each variable's reliability initially
as being as high as the highest correlation which the variable
enjoys with any other variable in the matrix. [This is reasonable
enough in principle: variables cannot truly correlate with other
variables at a higher level than their own reliabilities allow.
Later in the factor analysis {see below} this estimate is corrected
if it seems to have been incorrect: this process is called 'reiterating
the communalities.']
(c) Here is how centroid analysis proceedswith the help of
a worked example. (The centroid method was the original method
that was first widely used in psychology of the 1920's. Though
other methods improve on it in particulars, no factor analysis
will produce results with most psychological data that are importantly
different as far as the general user is concerned.)
Here is a specimen r matrix to be analysed  the
Original Matrix 1 
VIQ PIQ SE BE
Verbal Intelligence (VIQ) .60 .00 .00
Performance Intelligence (PIQ) .00 .00
Social Extraversion (SE) .40
Behavioural Extraversion (BE)
1.For each variable, its r's with all other variables
and with itself (in so far as its variance is covariancethe 'communality
assumption') are added together. (The communality assumption normally
inserts the variable's highest r with any other
variable in the leading diagonal of the r matrix.)
Thus is created a matrix with a communality assumption the Original
Matrix 2.
VIQ PIQ SE BE ¦ Sum of r's
VIQ (.60) .60 .00 .00 1.20
PIQ (.60) .00 .00 1.20
SE (.40) .40 .80
BE (.40) .80

4.00 = Sum of sums
2.00 = (Sum of sums)
2.Each such sum of its correlations, for each variable,
is then expressed as a proportion of the square root of the sum
of all such sums, in order to express the loading of the variable
on the major centroid factor (i.e. to express the correlation
of each variable with the centroid factor).
**Thus: Sum of r's / (Sum of sums)
= Loading on
first centroid
factor
VIQ 1.20 / 2.00 = .60
PIQ 1.20 / 2.00 = .60
SE .80 / 2.00 = .40
BE .80 / 2.00 = .40
The first factor has now been extracted. It consists simply
of the variance that it shares with the variables that 'load'
it, as above.
All there is to 'see' of a factor is its correlation with the
variables that yield it. A factor might be said to have 'absorbed'
a certain amount of the variance of the variables that load it.
{Note: The major centroid factor here is hardly very 'helpful'.
It has no obvious interpretation; and implies that the variables
had this mysterious factor 'in common' when the original correlation
matrix was largely composed of zeros. In fact the picture will
change as the FA proceeds; but this first centroid factor can
still serve as a caution against taking loadings of .70 too seriously
when trying to understand the 'nature' of a factor.}
3. Next it is asked what the original correlations would have
been if only the first centroid factor had been at work
to generate individual differences in the data. These new, hypothetical
r's are, in each case, the product of the variables'
loadings on the first centroid factor.
These hypothetical r's make up the Product of Loadings
Matrix.
(E.g. the expected r or VIQ with itself, if only the first
centroid had been at work would be the product of VIQ's loadings
on the centroid, i.e. .60 x .60 = .36.)
VIQ PIQ SE BE
VIQ (.36) .36 .24 .24
PIQ (.36) .24 .24
SE (.16) .16
BE (.16)
4.Each r in the original r matrix
(using Original Matrix 2) is then reduced by the above hypothetical
values in the Product of Loadings Matrix. These operations effectively
partial out i.e. set asidethe variance from the first
centroid that had led to the original r's being
as high and as positive as they were.
The result of this operation is the Residual r Matrix:
VIQ PIQ SE BE
VIQ (.24) .24 .24 .24
PIQ (.24) .24 .24
SE (.24) .24
BE (.24)
5. The process involved in the above four steps is now repeated
upon the Residual Matrix. To speed up analysis, the number of
negative r's in the matrix is minimized by the process
of reflection: the direction of scoring of a variable
can be reversed, and its name changed (e.g. from Extraversion
to Introversion). This allows larger 'sums of sums', and thus
factors accounting for more variance. (It avoids wasting timean
important consideration in the days of handcranked calculation!)
For example, reflection of SE and BEwhich thus become Social
Introversion and Behavioural Introversionwill yield the following
Reflected Residual Matrix:
VIQ PIQ SE BE
VIQ (.24) .24 .24 .24
PIQ (.24).24 .24
SE (.24) .24
BE (.24)
6. Since the above residual values are all the same, analysis
of this particular Reflected Residual Matrix will yield the same
loading for all four variables on the new, second centroid.
Loading on 2nd Centroid
VIQ PIQ SE BE Sum of r's [= Sum / (Sum of sums)]
VIQ (.24) .24 .24 .24 .96 .49
PIQ (.24).24 .24 .96 .49
SE (.24) .24 .96 .49
BE (.24) .96 .49
3.84 = Sum of sums
1.96 = (Sum of sums)
7. The above 2nd Centroid will not seem any more helpful or interesting
to a psychologist than was the first. These two factors will certainly
require the extra attention provided by the rotational procedures
of Stage (iii). However, an important landmark has in fact
been reached. For the next Product of Loadings Matrix will have
.49² , i.e. .24 in every single cell, just like the Reflected
Residual Matrix above. Hence, subtraction of the new Product of
Loadings Matrix from the Reflected Residual Matrix will yield
a new, second Residual Matrix having zero in each of its cells.
At this point the search for factors can and must haltfor no
factors are required to account for a matrix involving no correlations!
[Normally, of course, the conclusion will not be so 'neat'so
the decision to search for no more factors will depend on whether
the latest residual matrix is deemed to contain any statistically
significant correlations: analysis will thus proceed until
there is nothing significant left to analyse.]
The result of this particular example of factor analysis is thus
that two centroid factors (each loaded by all the variables to
some degree) account statistically for the total variance in the
original matrix. This solution is ideal in its neatnessbut not
in its meaningfulness, as will be considered below.
(d) The chief technical problem that arises in factor analysis
(when unities are not inserted in the leading diagonal, as they
are in Principal Components) is that of how to stabilize the
communalitiesi.e. of how to equate the estimated
(as above, (b) and (c1)) and the obtained communalities.
The obtained communalities for each variable are given by the
sum of the squares of its loadings on all the obtained factors.
Thus, above, Social Extraversion has an obtained communality (normally
written h²) of
.40² + .49² = .16 + .24 = .40
In this case (because the analysis came out so neatly) the obtained
communality is just what had originally been estimated. If that
had not been so, however, the correct procedure would have been
to repeat the whole analysis again  using the new communality
estimate (presumably superior for having been 'obtained' empirically)
until stability (or insignificant instability) was achieved.
[Even though an initial guess has been made as to what communality
values to place in the leading diagonal, new and better estimates
can be made as the analysis proceeds. This is because the obtained
communality of a variable with a number of independent factors
is equal to the sum of the squares of its loadings on each of
them. Thus, at any stage in the analysis, the initial estimate
of the communality whether it was made as in the centroid method,
or whether it was the value of unity upon which Principal Component
Analysis proceeds until Principal Factor Analysis takes overmay
be revised and the whole analysis restarted. Although most psychological
researches avoid the issue by settling for Principal Components,
it was not uncommonly a matter of concern to Cattell and other
factorial specialists that numerous such reiterations
should be made to 'stabilize the communalities'.]
(e)There is always a choice for the factorist as to when to
stop extracting factorsfor reallife matrices hardly ever
resolve entirely cleanly. On this matter, disputes between Cattell
and Eysenck used to be particularly heated, for Cattell felt that
Eysenck systematically underestimated the number of factors worth
extracting from 'personality' data. In fact, the criterion that
'none of the r's in the last residual matrix should
be significant' became generally displaced by the KaiserGuttman
Criterion.. This specifies that it is not worth extracting
factors whose variance (= the sum of the squares of the factor
loadings that are made by the variables, also called 'latent root'
or 'eigenvalue') does not reach unity: The criterion effectively
ensures that each recognized factor will account for as much variance
as any single variable in the study; and it is standardly used
in Principal Components Analysis. Another criterion is that each
recognized factor should account for at least 10% of the
total variance in the study: such a percentage is calculated as:
(Latent Root x 100) / (Number of Variables). This provides a stricter
criterion when the number of variables exceeds tenfor it recognizes
fewer, and not more than ten factors. Cattell, however, always
preferred a third criterion, that of the scree test.
To apply this test, numerous factors are first extracted and the
percentage of the total variance for which each accounts is ascertained
as above. It is then demanded that the least substantial of
the factors to be recognized should itself account for significantly
more variance than the largest factor that is to be rejected
as too insubstantial. For example, the factors emerging from
a study might be arranged as below.
In such a bargraph, the last and least substantial factor to
be accepted will account for a percentage of variance that puts
it clearly above the line or 'scree' made by the many further
factors having the gradually diminishing '% variance' that would
be expected by chance. In this case, the third factor would be
the last to be accepted. The burden of the scree test is that
the recognized factors should be clearly distinguishable from
the many 'garbage' factors in terms of the percentage of variance
for which even the least of the recognized factors accounts. A
typical empirical 'scree' is shown by Howarth & Browne (1972,
Brit. J. soc. & clin. Psychol. though these authors
themselves used the lenient KaiserGuttman Criterion to draw their
own conclusion that the Eysenck Personality Inventory reflected
many more factors than Eysenck himself typically envisaged. A
fourth possibility is to extract a number of factors that has
been determined in advance. This feat may be particularly impressive
if the hypothesis is that 'there will be a certain number of specified
factors' and if subsequent inspection confirms that the main factors
are indeed of the particular nature expected. However, it is a
trick that will rarely work in everyday empirical psychologyexcept
with a few tests that have already been well refined by factor
analysis. {In any case, there is no journal in mainstream psychology
that even encourageslet alone requiresauthors to submit predictions
prior to datacollection.}
(f) The last stage of the factor analysis proper consists of the
factor matrix. This shows the factors extractedconstituted
as factors are by their collection of loadings from
each of the original variables. These loadings are essentially
correlation coefficients. Although a factor accounting for a decent
proportion of overall variance is a thing of beauty and an everpresent
cause for 'further research'...., strings of factorloadings of
less than .50 provideof themselves very little direct definition
of a factor. Rather, they are a reminder that not a single variable
could be found that shared even a quarter of its variance with
the putatively meaningful factor  even under favourable conditions
in which the variables themselves had every chance of contributing
to the very nature of that factor. Again, particular attention
should be paid to the final communality (h²) values
of variables before hastening to interpret a factor in terms of
the variables that load it: for it can easily turn out that a
variable that has the highest loading on one of the extracted
factors has still more variance that is unaccounted
for by any factor at all than variance which it shares with the
factor in question. If this is not noticed, a factor can all too
easily be 'understood' in terms of those very components of item
and testvariance that it was not in fact reflecting. (Lower
factor loadings are, however, of interest when the unreliability
of the original variables is high  as with the notinfrequent
exploratory items of personality and social psychology that take
researchers little effort to think up and testees little time
to complete.) Lastly, although a factor is sometimes said, impressively
enough, to account for 20% or even 60% of the variance (of an
original r matrix, or of the matrix of extracted factors),
there is a snag. It must never be forgotten that 'the variance'
in question is 'the covariance amongst these variables in the
present subjects'with or without the assumption of Principal
Components that each variable has its own true and unique variance.
FA will not go out of its way to remind users of reallife variance
that was not included in the analysissuch as 'error variance',
testretest variability, variance from other variables, and variance
occurring amongst types of testee not included in the present
sample.
(iii) The third and last broad stage of FA is that of rotation
of the axes.
This serves two purposes. It allows 'psychological' errors that
have occurred at the previous stages to be corrected; and it offers
a new set of criteria for deciding 'just what the factors really
are'. That it should do the latter may seem strange in
view of what has already been said. However, the simple point
is that, when any two independent factors are plotted orthogonally
[at a right angle] to each other  intersecting at the point at
which loadings on each of them are zero  it may turn out that
most of the variables, when their two loadings are plotted at
one point against the independent axes, fall not on either of
the axes but towards the middle of the four quadrants.
******************************
Thus, the two centroid factors in the example can be plotted at
right angles because they are, by their manner of extraction,
independent of each other. The two 'IQ' variables will both fall
at the same position in the twodimensional space, being loaded
positively and indeed having identical loadings on both factors.
Centroid 1


.60  IQ(V & P)

.40 

________________ __________________ Centroid 2
 .49






To plot the 'Extraversion' variables similarly requires 'reflection'
once moreof two of the loadings of +.49 on Centroid 2. These
positive loadings were made by variables (originally Social and
Behavioural Extraversion) that by then had been reflected and
renamed 'Introversion'. After rereflection, the loadings of the
'E' variablesboth Social and Behaviouralcan be represented as
follows.
Centroid 1


.60  IQ(V & P)

E(S&B) .40 

________________ __________________ Centroid 2
.49  .49 (after reflecting
 SI and BI)





Here all the variables (VIQ, PIQ, SE and BE) have their
loadings represented by points in twodimensional space that are
well into the quadrants rather than close to the line made by
either factor. This is because no single variable had a nearzero
loading on either factor.
Moreover, in the example, it can hardly be suggested that either
of the factors looks at all 'helpful' or intelligible. Centroid
1 is defined positively by all the variables. And Centroid
2 (after reflection of BI and SI) suggests there is a complementary
opposition amongst all the variables  now with intelligence
being opposed to extraversion. What can such factors mean?
 Especially when one centroid factor seems but a contrast to
the other  taking away, as it were, what the other centroid factor
put in place?
*****************************
Clearly, what has happened is that many or all of the original
variables are loading many or all of the factors to some degree.
For sure, the analysis has yielded a 2dimensional factor space
defined by NorthSouth and EastWest axes; and these axes have
been selected according to a rational criterion  that the first
one of them account for 'the most variance' while the other be
independent of the first. (In the particular example used, we
also know these two factors are exhaustive: they account between
them for 100% of the variance.) Yet these may not seem particularly
compelling reasons for continuing to describe the 2dimensional
space in terms of such axes. The situation is far from that which
makes 'NorthSouth' and 'EastWest' singularly useful (and widely
understood) ways of describing differences between reallife geographical
locations. Particularly when most variables cluster elsewhere
in 2D space than around such axes, it may seem preferable to
rotate the axes to some position (of simple structure)
such that some variables load primarily on one of the new
rotated factors while the remaining variables chiefly define the
other.
**********************************
In the example, what is evidently required is an approximate 45
rotation of the axes so that one will line up with VIQ and PIQ
while the other will line up with SE and BE.
The precise loadings of the variables can be read off from their
positions along each of the rotated centroids. One rotated centroid
will be loaded at about .75 by the IQ variables while the Extraversion
variables will project on to it at around zero. The other centroid
will now be principally defined by Extraversionwith both BE and
SE loading it at about .60.
*************************************
Such a rotation of the axes may seem even more appealing when
(as in the example) we are used to calling our variables by broad
titles such as 'verbal ability' and 'spatial ability' and when
we do not generally expect some variables (like IQ and Extraversion)
to have much to do with each other. {Even if one large and broad
factor brought many mental abilities together as loading on it,
some psychologists would still feel tempted to tease them apart
rather than struggle to understand a second factor (independent
of this broad g factor) that would, by its mixture
of positive and negative loadings (as on the second centroid in
the example), imply that there was somethingsome factorthat
tended to facilitate some particular abilities while actually
impairing others.} However, setting theoretical (psychological)
prejudices aside, the 'objective' way to resolve the question
of how to rotate is the criterion of simple structure.
Here the search is for that rotation (or set of rotations when
more than two factors are involved) that maximizes the number
of nearzero, nonsignificant factor loadings in the factor matrix
as a whole. Put another way, the idea is that a majority of variables
should end up identified principally with just one factor;
and, by the same token, that each factor should be defined
in terms of a relatively small number of variables. [The appeal
of searching for simple structure will thus be greater as and
when the researcher feels sure of what most of the variables were
measuring in the first place. By contrast, Arthur Jensen has considered
it folly to attempt to break up clear g factors
by an obsessional search for simple structure. Parsimony certainly
dictates that strong, general first factors should be taken pretty
seriously unless simple structure rotation provides clear theoretical
improvements.]
The possibility of rotating to welldefined clusters of variables
in factor space (like the IQ' and 'E' clusters in the example)
will increase enormously if we loosen the shackles of orthogonality.
For, if we allow ourselves to rotate axes to positions in which
the angles they make with each other depart from 90, we are free
to identify whatever factors we like. Of course, such factors
will themselves be correlated: so our final psychological theory
will have to explain not only the factors we have created by oblique
rotation but also the correlations that obtain between
them. [The correlation between two oblique factors is equivalent
to the cosine of the angle between them.] It is to allow this
freedom while imposing at least a certain restraint upon it that
criteria are developed allowing 'oblique' rotations with their
closer approximation to most nakedeye criteria of simple structure.
[The most widely used set of such criteria is that of the Promax
programme that can substitute for the orthogonalitymaintaining
rotations of Varimax. Both rotational programmes endeavour to
approximate to ideal simple structure, but Promax necessarily
achieves this goal more nearly.] For psychologists, there may
often be a certain relief in being able to identify (oblique)
factors that are singularly well defined by favourite and putatively
'pure' tests. On the other hand, major arguments frequently ensue
as to whether beautiful psychometric positioning of the axes is
any substitute for the experimental and biological criteria that
might also seem relevant to defining the 'true' dimensions of
personality and individual differences. [For example, it was because
of his mistrust of simplestructure criteria for rotation that
Eysenck (1971, op.cit.)  as he explains  rejected
Ferguson's rotational positioning of the dimensions of social
attitudes. Ferguson, Thurstone, Adorno, Altemeyer and others have
preferred to work with two independent attitude dimensions of
AUTHORITARIANISM vs HUMANITARIANISM
and RELIGIONISM / MORALISM vs HEDONISM / LIBERTARIANISM.
By contrast, Eysenck urged the superiority of a large, general
and seemingly familiar factor of SOCIAL CONSERVATISM vs
LIBERALISM
together with a less well defined factor of
TOUGHMINDEDNESS/REALISM vs TENDERMINDEDNESS/IDEALISM
as a superior rotation by which to describe and 'explain' the
same 2dimensional attitude space that has been so generally found.
{See Quotes XXV.}]
However, even if there is a risk of being driven up a psychometrician's
blind alley by oblique rotations that initially seem so convincing,
the possibility of scaling a tree and surveying the wider picture
is still with us. For, since oblique factors are themselves intercorrelated,
these correlations can in turn be analysed by FA to yield what
are called secondorder factors that account for them.
Indeed, it often turns outat least in the Eysencks' workthat
secondorder factors that emerge in this way are not too different
from those that emerge when the search for simple structure is
made within the constraints of orthogonality in the first place.
[By contrast, Cattell always preferred to suppose that what he
expressly called his sixteen primary (often oblique)
personality factors did possess a psychological reality that no
reduction of them to secondorder factorsand no demonstration
of their unreliability (Eysenck, 1972, Brit.J.soc.clin.Psychol.)could
dispel.] Most recently, Carroll's (op.cit.) work on human
mental abilities has documented the nearuniversal emergence of
general intelligence at the second or thirdorder
level of analysis even if simple structure criteria has
initially broken g up amongst 'primaries'.
Today, it is easier for FA users to experiment with different
rotational options for themselvesthough the ready availability
of Principal Components is quite enough to meet the needs of many
users. The main hope of course is to be able to arrive at factorial
solutions that seem theoretically convincing, though the failure
of many 'educational' and 'cognitive' psychologists to acknowledge
even the widespread phenomenon of g shows that this
is easier said than done. {See Quotes III and Quotes VIII for
discussion of 'the dimension(s)' of human personality and abilities.}
Lastly, it should be mentioned that, if obliquity has been adopted,
communalities for variables and eigenvalues for factors cannot
any longer be calculated because the assumptions on which they
depend no longer apply.
Concluding counsels
Thus it is that, after however many false starts, detours
and meanderings through seemingly endless reiterative feedback
loops, the three broad stages of the action in FA draw acolytes
finally towards the goal. The last 'outlier' has been removed,
the last 'communality' stabilized, and the last factor rotated
by hand to its final artistic position. Yet prudent players of
this exquisite game will not lose their heads at this point. Apart
from recalling the ways in which their analyses have already failed
to live up to the best possible practice, factorists will still
show a lively appreciation of several remaining issues as they
interpret their factors and dimensions. (The term 'dimensions'
tends to be usedrather than 'factors'when factors are
singularly large, relatively independent, and somewhat
more likely to emerge at a higher order of analysis.)
(a) Factorists will at least consider Scott Armstrong & Soelberg's
(1968, Psychol. Bull.) suggestions as to how they
might demonstrate the reliability of their results. Most
dutifully, in particular, will they prepare themselves to explain
why they did not divide their samples in half, complete
their FA on each half separately, and then steel themselves to
assess the similarity of the factors thus obtained. [Factor similarity
may be estimated by treating the loadings of factors by variables
just like the scores on tests by individuals. That is, one correlates
the loadings of factors by variables just like one correlates
the scores on tests by people. If two factors emerge as strongly
correlated, this is because they are both relatively highly loaded
by the same variables.] By conventional standards in FA, Eysenck
and his coworkers have been exemplary in their frequent presentations
of coefficients of factor similarity (CFS's). However,
Vagg & Hammond (1976, Brit.J.soc.clin.Psychol.) long
ago pointed out that reliability is far from being the only issue.
The CFS "simply accounts for the direction of the factors
and the shared items: it does not include the relative magnitude
of the factors or the item loadings which are shared." Vagg
& Hammond's preference was to calculate factor scores
[using loadings as weightings for variables] for each subject
on the factors that emerge in both the sample of which the subject
was a member and in another sample of which the subject was not
a member. They then correlated such factor scores, just as if
their subjects had taken two tests. As a further sophisticationto
guard against individual items getting 'misplaced' in the course
of FAthe scores on the original items can be correlated with
the factor scores that the subjects obtain on the similar factors
that emerge in different samples. Such itemscale r's can
then be used to make up scales representing the confactor.
Such scales are composed of items that load relatively highly
on both of two factors that have been 's found to be similar by
the criterion of r's between factor scores.  Needless
to say, little modern practice in FA exemplifies such high advices,
chiefly because of the limitations imposed by the usual small
samples with which psychologists work. [Not that nonfactorial
psychology is any better! Nonfactorial psychologists typically
neglect the most elementary examination of the reliability of
their findings. They are often content to have come up with a
measure that merely because of its unreliability in the
restricted samples useddoes not make for political heresy by
correlating with g.]
(b) At the same time, consideration should be given to the question
of how easily a result might have emerged that the factorist would
not have liked from a theoretical point of view. [Brand
(1972, op.cit.) provides an explicit example of the comparison
of alternative rotational solutions related to different theories.]
(c) As Guilford often stressed with regard to r's, there
is seldom any serious substitute for a scattergam: so scatterplots
of factor scores against each other should likewise be made
to enable Ushaped relations to emerge if they exist and to check
for outliers.
(d) Last, although differential psychology has doubtless benefited
in some ways from its isolation from the transient schools of
thought that have preoccupied experimental (lately 'cognitive')
psychology, there are other criteria of psychological validity
for tests and factors than those of a purely psychometric nature.
Those who find themselves either tired of or overenthusiastic
about FA should therefore try to consult volumes that urge the
merits of combining factoranalytic with experimental and psychogenetic
understandings of human psychological differences, such as Eysenck
& Eysenck (1985, op.cit.) or other volumes from the
Reference List [above, pp.5,6.].
Acknowledgments: I am grateful to Boris Semeonoff and Vincent
Egan for their
improving comments on an earlier version of these notes.
FINIS
(Compiled by Chris Brand, Department of Psychology,
University of Edinburgh.)
For coverage of dimensions of intelligence
and personality,
and of the essentials of factoranalytic method, see:
BRAND, C.R. (1996) The g Factor.
Chichester : Wiley DePublisher.
"The nature and measurement of intelligence is a political
hot potato.
But Brand in this extremely readable, wideranging and uptodate
book is not afraid to slaughter the shibboleths of modern "educationalists".
This short book provides a great deal for thought
and debate."
Professor Adrian Furnham, University College London.
The book was first issued, in February, but then withdrawn, in
April, by the 'publisher' because it was deemed to have infringed
modern canons of
'political correctness.'
It received a perfectly favourable review in Nature (May
2, 1996, p. 33).
For a Summary of the book, Newsletters
concerning the
depublication affair, details of how to see the book for scholarly
purposes, and others' comments and reviews,
see the Internet URL sites:
http://laboratory.psy.ed.ac.uk/DOCS/crb/internet.html
http://www.webcom.com/zurcher/thegfactor/index.html
For Chris Brand's 'Get Real About Race!'his
popular exposition of his views on race and education in the Black
hiphop music magazine 'downlow' (Autumn, 1996)see:
http://www.bhs.mq.edu.au/~tbates/intelligence/Brand_downlow.html
A reminder of what is available in other Sections of P,
B, & S.
Summary Index
for PERSONALITY, BIOLOGY
& SOCIETY
(This resource manual of quotations about
individual and group differences, compiled by
Mr C. R. Brand, is kept on the Internet and in Edinburgh University
Psychology Department Library.)
Pages of Introduction
3  11 Full Index, indicating key
questions in each Section.
12  14 Preface.  Why quotations?  Explanations and apologies.
15  51 Introduction: Questions, Arguments and Agreements in
the study of Personality.
Some history, and a discussion of 'realism vs 'idealism.'
52  57 Introductory Quotes about the study of personality.
Sections
General problems
1 'Situational' vs 'personological' approaches to human
variation.
2 'Nomothetic' vs 'idiographic', 'subjective' and relativistic
approaches.
3 Personality dimensions  by factor analysis and otherwise.
4 'Superstructure' and 'infrastructure'  the 'mind/body problem'.
5 Nature vs Nurture?  Or Nature via Nurture?
6 The role of consciousness in personality and 'multiple personality'.
7 The 'folk psychology' of personality components.
Intelligence
8 The measurement of intelligence.  Does g exist?
9 The bases of intelligence.  What is the psychology
of g?
10 The developmental origins of g differences.  The nature
and nurture of g.
11 The importance of intelligence.  The psychotelics of
g.
12 Piagetianism: Kant's last stand?
13 Cognitivism: 'The Emperor's New Mind?'
Propensities
14 Neurosis, emotion and Neuroticism.
15 Psychosis, psychopathy and Psychoticism.
16 Crime and criminality.
17 Genius and creativity.
Popular proposals  psychoanalytic, phrenological and prophylactic
18 Psychoanalysis: 'Decline and Fall of the Freudian Empire'?
19 Hemispherology: a twentiethcentury phrenology?
20 Psychosocial Engineering: therapy, training or transformation?
Group differences
21 Age and ageing  especially, the role of g in 'lifespan
development'.
22 Psychological sex differences.  Do they exist? Must they exist?
23 Social class.  Does it matter any longer?
24 Racial and ethnic differences.  Their role in 'lifestyles'
and cultural attainments.
Ideological issues
25 The psychology of politics and ideological extremism.
26 The politics of psychologists and allied coworkers.
27 Equality and Community: the 'utopian' package of political
aims.
28 Freedom and Responsibility: the 'legitimist' package of political
aims.
Pragmatic questions
29 Carry on differentializing?
30 Carry on psychotesting?
Appendix: Factor analysis.  'Garbage in, garbage out'?
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