lin'eech Slidisek7hin
CHAPTER 3
Charles Spearman: Behavioral and Statistical Scientist
Overview : This chapter is an outline of the significant achievements of the British
Psychologist, Charles Spearman. He was the originator of three important behavioral and
statistical domains: t I) Factor analysis; (2) A two-factor theory of intellieence, and (3)
The classical theory of mental tests. Spearman utilized both mathematical methods and
empirical psychological studies to pioneer research efforts in these three areas. It will be
demonstrated that there are strong links between the three.
Educational Experience
The British psychologist, Charles Edward Spearman, was born the 10th of September
of 1863 in London and died in the University College Hospital., London on September 17,
1.945 He spent a quarter of his long life of 82 years in the British Army, and as a result
didn't complete his Ph D. until. he was 41 years of age. Wilhelm Wundt, who initiated the
first laboratory in Experimental Psychology in Leipzig, Germany, directed his doctoral
study_ But Spearman was strongly influenced by the work of Francis Gallon, who had
made a strong case for intelligence testing Two well-known psychologists who studied
under Spearman were Raymond Cattell and David Wechsler, and five others who were
influenced by him were Anne Anastasi., J.P. Guilford, Philip Vernon, Cyril Burt, and
Arthur Jensen. Charles Spearman occupied chaired professorships at University College,
London from 1907-1931, initially assuming a position in Experimental Psychology and
then moving to the area of Mind and Logic. Although they were at the same college for
many years, Karl Pearson was not impressed by Spearrnan's research
The discussion that follows describes Spearman's contributions to factor analysi s,
theories of intelligence, and mental test theory, and to provide evidence that he was an
innovative force and pioneer in all three areas. Additionally, logical links will he made
available to connect the three areas. In other words, the three contributi ons are not
independent of one another.
The Classical Theory of Mental Tests
Most psychologists, and many other behavioral and social scientists, view Harold
Gulliksen's classic book, Theory vt Mental Tests (1950) as the definitive treatment of the
classical theory of mental tests Of course, when it was published in 1950 it was the only
comprehensive, mathematical development of psychological test scores. Since then there
have been a number of test theory models. See Crocker and Algina's Introduclian to
ClaSSItal and Mockrn Test .111cory (1986) for a more up to date discussion of the state of
psychometrics
In the classical theory a starling point is X = T--E, where X is observed score, T is true
score, and E is error score_ The investigator would like to kno w the true measurement,
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hut, unfortunately, errors of measurement obscure the true values. Therefore, one
objective of measurement procedures is to minimize errors of measurement. One way to
do this is to change test item format from true false to multiple choice to diminish the
probability of a successfUl guess Another is to lengthen the test. The Spearman-Brown
formula shows how test length influences reliability
Two important concepts in test theory are reliability and validity. Rel iability has to do
with consistency of measurement whereas validity focuses on whether the test or
measuring device actually measures what it purports to measure. Incidentally, the term
reliability - was introduced by Spearman. In mental test theory these two features of
measuring devices are treated mathematically. For example, the variances of the
components of test scores are additive; Var [XI - Var [TI Var [E]. Then reliability
is defined as the ratio of true score variance to observed score variance. o r, Var
-
[X]. Similarly, the proportion of observed variance which is due to error is Var [.E]/Var
[X) (Williams, Zimmerman, Zumho, & R.oss, 2003 ;, Lord & Novick,. 1968).
On page 1 of Harold Gul.liksen's text (1950), he says that most of the equations of th e
classical theory of mental tests are attributable to Charles Spearman (1904a, 1904b). One
such formula is Spearman's correction for attenuation. Errors of measurement diminish
or attenuate the correlation so this formula gives an estimate of what the correlation
would be in the absence of measurement errors. Then there is the Spearman -Brown
Prophecy Formula, which was mentioned above. It is even useful to plot curves showing
reliability coefficients against test length, prior to constructing a test.
In investigating test validity, the test scores to he validated are called X and they are
validated against a criterion called Y. For example, X might he Verbal. Scores on the
Scholastic Aptitude Test and Y could be grade point average in freshman year in colle ge.
Then the product moment correlation between X and Y is a validity coefficient.
Spearman derived another equation. that displays the effect of altering the length of both
the test and the criterion on validity Then the validity of a perfect test for pre dicting a
perfect criterion is given by Spearman's a generalized form of the correction for
attenuation. It yields an estimate of the true score correlations. For further information on
Spearman's contribution.s to test theory see "Spearman's Test Score Mo del - (Ng, 1974)
and "Charles .Spearman's Contributions to Test Theory." (Levy, 1995). It should be
mentioned that a number of the equations of the classical theory of mental tests depend,
in their mathematical derivation, on error scores being uncorrelated . (Zimmerman, &
Williams.,1977)
Factor Analysis
As a beginning to factor analysis, one obtains a set of n measurements on a set of
subjects or examinees. An n x n .correlanon matrix is then formed, employing productmoment correlations Some factor analytic technique is then applied to the matrix to find
a smaller more basic set of variables that can account for the variance in the original set.
This new set of basic variables is called "factors." The factor matrix is then geometrically
rotated to improve interpretation Several statisticians and behavioral scientists have
attributed the pioneering efforts of the development of factor analysis to Charles
Spearman. Lovie and Lovie (1993) have carefully analyzed the correspondence between
Charles Spearman and Cyril Burt, as both have been presumed to be responsible for the
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development of factor analysis, and concluded that Spearman is the true pioneer of this
methodology See also Cattell (1968)., Harman (1976), and Bartholomew (1995). All
three attest to Spearman's primacy.
Factor analysis and other multivariate statistical procedures are difficult to conduct by
hand so the eventual presence of pres,vritten computer packages such as the Statistical
Package for the Social Sciences (SPSS) and the Statistical Analysis System (SAS) was
welcome
As mentioned above., factor analysis is applied to matrices of intercorrelations among
Observed score variables. But since it is of greater interest to describe relations among the
underlying or latent variables, why not apply Spearman's correction to the observed score
correlations thereby generating a matrix of estimated true score correlations? This would
be done prior to the factor analysis proper. It would seem promising because the
estimated true score correlations would be larger and hence would account for a greater
proportion of the variance. One reason why this isn't done routinely is because there is
some controversy associated with Spearman's correction. For example, it sometimes
yields values greater than 1.100. (Zimmerman Se Williams. l997) Notice that here we have
provided a link between mental test theory and factor analysis. Some psychologists, such
as J.P Guilford and Raymond Cattell and Philip E Vernon, and Arthur Jensen have
employed factor analysis to assist in the creation of theories of intelligence , or personality.
Spearman's paper "General Intelligence Objectively Determined and Measured.,"
(1904a) was the pioneering contribution to factor analysis. The culmination and highlight
of Charles Spearman's factor analytic research is The Abilities of Alan: Their satin re art!
Measurement, (1927). His final work in factor analysis, which was completed by Wynn
Jones after Spearman's death, was Human Abilnies (Spearman & Jones, 1951;.
Bartholomew, I q).5)
A Two-Factor Theory of Intelligence
Charles Spearman created the first psychometric theory of intelligence, a two -factor
theory, and published it in the A enerfean Journal of Psychology (1904a), It was based on
mathematical notions together with empirical data collected from psychological studies
and philosophical concepts. His work in factor analysis paved the way for the application
of statistical methods to the study of human intelligence
In analyzing a table of mtercorrelations among test scores, Spearman noticed that they
could be transformed to produce a hierarchical pattern which revealed that all the tests
measured one common factor or, as he termed it, the "general factor (g)." But the various
tests exhibited differential correlations with the general factor Furthermore, each test
measured its own specific factor Hence, he had posited a "two factor theory of
intelligence " (Carroll, 1982)
It should be noted that the nature of a theory of intelligence can strongly dictate the way
in which intelligence tests are constructed. Some of the well-known intelligence tests are
specialty designed to measure Spearman's 'g " It is said that Raven's Progressive
Matrices, which is a non-verbal, cross-cultural test of intelligence, is one of the best
measures of Spearman's "g " John Raven worked closely with Charles Spearman in
constructing this test. Raven was a student of Speamian's.
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Tivehv Ilnoth SlittiNOctilm
Ii is of interest to mention that Charles Spearman developed a rank order correlation
coefficient (p,;), It is widely used to find the monotonic relation between two sets of
ranks, whereas the usual Pearson product-moment correlation coefficient is used to
analyze linear relations between two variables The latter coefficient is employed to
connection with factor analytic studies The Spearman correlation calculates a Pearson
correlation between two sets of observations that consist of integer ranks. One could say
that the Pearson formula is general whereas the Spearman formula is special—that is, the
general formula applied to a special case_ Two other coefficients, phi and point hiserial,
are also special cases of the Pearson r Maurice Kendall's correlation (T) yields relations
between two ordinal variables but it is considered to be a measure o f disarray The
Spearman and Kendall statistics are usually described in books on Nonparametric
Statistics such as Siegel and Castellan's .Vo.nparamerrue Statistics fi_a. the Behavioral
Sciences (1988)
Karl Pearson published an anonymous review of Spearman's The
of Man in
the journal Nature (1927) titled "The Mathematics of Intelligence" in which he attacked
Spearman for faulty mathematics and statistics In this and other publications Pearson
criticized the rank correlation and the correction for attenuation, inviting Spearman to
produce a paper containing algebraic proofs of his equations to support the unproven
assertions.. Pearson also pointed put that the correction for attenuation formula sometimes
yielded estimates {correlations) greater than one. This problem still exists. Zimmerman
and Williams (1997) conducted a simulation study of Spearman's correction and
concluded that the formula should he used only when the reliabilities of both tests are
relatively high and sample size is relatively large.
Charles Spearman countered Karl Spearman.'s criticisms by saying that the Galton Pearson model could no longer he used as a model. On the other hand he criticized
psychologists for not becoming familiar with the commendable work being carried out by
the Galton-Pearson school (Levy, 1995, p. 223)
Dr Jan Strelau, the Associate Editor of the European Psychologist, was interviewed in
the year 2000 and was asked to name the most significant developments in Psychology in
the 20th century. He named Charles Speann.an's work in intelligence theory, pointing out
that it was catalyst for the development for most subsequent theories of intelligence either
in a positive or a negative sense, He also pointed to the significance of Spearman's -g," as
the underlying concept in the construction of a number of intelligence tests. (Tele intery iews, 2000)
Charles Spearman utilized the method of tetrad differences in analyzing intelligence
test data to support his two-factor theory of intelligence. A tetrad difference is actually
the determinant of a 2x2 matrix. If it is zero the data are consistent with the notion of a
single general factor. The psychologist L.E. Thurstone was violently opposed to
Spearman's theory and generalized the British psychologist's criterion of tetrad
differences to examination of higher order determinants as a criterion for more than one
common factor. Thurstone's theory of intelligence postulates seven primary mental
abilities and he and his coworkers constructed seven tests to correspond to these factors.
Others developed hierarchical theories that retained "g - in the structure. One of these was
the. Canadian psychologist Philip Vernon and another was the American psychologist
Raymond Cattell, In Cattell's theory "g" is at the top of the hierarchy and it bec omes
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partitioned into fluid and crystallized intelligence, with Spearman - s specific ability
factors at the bottom of the hierarchy (Cattell, 1968)
Matrix formulations with the advent of high-speed computers have greatl y facilitated
the development of factor analysis and other multivariate statistical methods. It is now
common to distinguish between exploratory and confirmatory' factor analysis with the
latter more theoretical and hence used to test hypotheses
Concluding Comments
Applications linking all three of the domains of research investigated by Charles
Spearman are readily conceivable His intelligence theory could he used to guide
construction of a set of ability tests and then the test scores factor analyzed. !dental test
theory would he employed to investigate the psychometric properties of the ability tests.
Factor analysis itself is a way of studying validity, construct validity, and another
approach to validation is to correlate the test scores with an external criteri on, such as
rank order in class A method favored by Spearman to estimate reliability was to correlate
scores on parallel forms of a test.
References
Bartholomew, D.J. (1995). Spearman and the origin and development of test thecny.
British Journal pAlathematical and Statistical Psychology, 48, 211-220.
Burt, C (1909). Experimental tests of general intelligence. British -rimmed of
Psychology, 3, 94- 177.
Carroll, J.B. (198.2) The measurement of intelligence. In R J. Sternberg (Ed.),
Handbook of Human Intelligence (pp, 294120). Cambridge: Cambridge University Press.
Cana, R. B (1968) Spearman, C E. In D.E. Sills (Ed ). International Eneyelopedia of
the Social Sciences, (pp. 108.111) New York • Macmillan
Crocker, L
Algina, J. (198(x) triaroducOon to classical ft- modern test theory New
York: Harcourt Brace Javanovich.
Gulliksen, H .(1 950:). 950). Theory pi mental tests. New York
Harman, 11.11 (1976) Modern ,,factor analysis (30 ed rev.). Chicago: University
Chicago Press.
Joresk
, & Sorbom., D. (1984) .1.,ISREI. 17, Analysis of linear ,structural
relationships by maximum likelihood, instrumental variables, and least squares methods
Mooresville., IN: Scientific Software, Inc.
Levy, P. (1995). Charles Spearman's contributions to test theory, British Journal of .
Mathematical and Statistical Psychology, 48, 221-235,
Lord, F.M., & Novick, M.R. (1968). Statistical theories of mental test scores. Reading,
MA: Addison Wesley
Ng, K.T. (1974). Spearman's test theory model: A restatement Educatioin l and
Psychological Measurement, 34, 487-49S.
Siegel, S., & Castellani N.J., Jr, (1988). Nonparametric statistics fin. the behavioral
(2 " ed.), New York: McGraw-Hill.
SL'ilenCeS
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IdnuthSuitisucituis
Spearman, C..E. (1904a). "General intelligence's objectively determined and measured.
American Journai qif.Psyvho/ogy, 5, 201-293.
Spearman,. C. E (1904b). Proof and measurement of association between two things.
American journal qfp,sychology, 15, 72-101
Spearman, C E. (1927). The abilities rnan, their nature and nieasurement. New York:
Macmillan
Spearman., C E., & Jones, L W. (1951) Human fibilities. London.. Macmillan.
Tele interviews (2000). Interview of Jan Strelau European Psychohigist, 5, 1.4.
R.H., Zimmerman, D,W., Zumbo, B.D
Ross, D. (2003). Charles
Spearman: British behavioral scientist. Human Nature Review, 3, 114- I t 8
Zimmerman, D.W., &
R.H. (1977). The theory of test validity and correlated
errors of measurement Journal of Mathentattca1 PAychology, 16, 135,152
Zimmerman, DAV „ & Williams, R. H_ (l WI). Properties of the Spearman correction for
attenuation for normal and realistic non.nonnal distributions,Apphed P.sychologieal
Measurement, 21, 253;270
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