THE PSYCHOLOGICAL REVIEW - University of …psych.colorado.edu/.../PSYC5112/Readings/VectorsOfMind_Thurstone… · THE PSYCHOLOGICAL REVIEW THE VECTORS OF MIND1 BY L. L. THURSTONE

VOL. 41, No. 1 JANUARY, 1934

T H E PSYCHOLOGICAL REVIEW

THE VECTORS OF MIND1

BY L. L. THURSTONEThe University of Chicago

Under the title of this address, 'The Vectors of Mind/I shall discuss one of the oldest of psychological problems withthe aid of some new analytical methods. I am referring to theold problem of classifying the temperaments and personalitytypes and the more recent problem of isolating the differentmental abilities.

Until very recently the only attempt to solve this problemin a quantitative way seems to have been the work of Pro-fessor Spearman and his students. Spearman has formulatedmethods for dealing with the simplest case, in which all of thevariables that enter into a particular study can be regarded ashaving only one factor in common The factor theory that Ishall describe starts without this limitation, in that I shallmake no restriction as to the number of factors that are in-volved in any particular problem. The resulting factor theo-rems are quite different in form and in their underlying as-sumptions, but it is of interest to discover that they are con-sistent with Spearman's factor theory, which turns out to be aspecial case of the present general factor theory.

In this paper I shall first review the single-factor theory ofSpearman. Then I shall describe a general factor theory.Those who have only a casual interest in the theoretical as-pects of this problem will be more interested perhaps in theapplications of the new factor theory to a number of psycho-

1 Address of the president before the American Psychological Association, Chicagomeeting, September, 1933.

1 I

2 L. L. THURSTONE

logical problems. These psychological applications will con-stitute the major part of this paper.

It is thirty years ago that Spearman introduced his single-factor method and the hypothesis that intelligence is a centraland general factor among the mental abilities. The literatureon this subject of factor analysis has tended temporarily toobscure his contribution, because the controversies about ithave frequently been staged about rather trivial or evenirrelevant matters. Professor Spearman deserves much creditfor initiating the factor problem and for his significant con-tribution toward its solution, even though his formulation isinadequate for the multi-dimensionality of the mental abilities.

Spearman's theory has been called a two-factor method ortheory. The two factors involved in it are, first, a generalfactor common to all of the tests or variables, and second, afactor that is specific for each test or variable. It is lessambiguous to refer to this method as a single-factor method,because it deals with only one common or general factor. Ifthere are five tests with a single common factor and a specificfor each test, then the method involves the assumption of onecommon and five specific factors, or six factors in all. Weshall refer to his method less ambiguously as a single-factormethod.

We must distinguish between Spearman's method ofanalyzing the intercorrelations of a set of variables for a singlecommon factor and his theory that intelligence is such acommon factor which he calls 'g.' If we start with a giventable of intercorrelations it is possible by Spearman's method,and also by other methods, to investigate whether the givencoefficients can be described in terms of a single common factorplus specifics and chance errors. If the answer is in theaffirmative, then we can describe the correlations as the effectof (i) a common factor, (2) a factor specific to each test, and(3) chance errors. In factor theory, the last two are combinedbecause they are both unique to each test. Hence the analysisyields a summation of a common factor and a factor uniqueto each test. About this aspect of the single-factor methodthere should be no debate, because it is straight and simplelogic.

THE VECTORS OF MIND 3

But there can be debate as to whether we should describethe tests by a single factor even though one factor is sufficient.It is in a sense an epistomological issue. Even though a set ofintercorrelations can be described in terms of a single factor,it is possible, if you like, to describe the same correlations interms of two or three or ten or any number of factors.

The situation is analogous to a similar problem in physicalscience. If a particle moves, we designate the movement byan arrow-head, a vector, in the direction of motion, but if itsuits our convenience we put two arrowheads or more so thatthe observed motion may be expressed in terms that we havealready been thinking about, such as the x, y, and z axes.Whether an observed acceleration is to be described in termsof one force, or two forces, or three forces, that are parallel tothe x, y, and z axes, is entirely a matter of convenience for us.In exactly the same manner we may postulate two or morefactors in a correlation problem instead of one, even when onefactor would be sufficient. To ask whether there 'really' areseveral factors when one is sufficient, is as indeterminate as toask how many accelerations there 'really' are that cause aparticle to move. If the situation is such that one factor is notadequate while two factors would be adequate, then we maythink of two factors, but we may state the problem in termsof more than two factors if our habits or the immediatecontext makes that more convenient.

Spearman believes that intelligence can be thought of as afactor that is common to all the activities that are usuallycalled intelligent. The best evidence for a conspicuous andcentral intellective factor is that if you make a list of stunts, asvaried as you please, which all satisfy the common sensecriterion that the subjects must be smart, clever, intelligent,to do the stunts well, and that good performance does notdepend primarily upon muscular strength or skill or upon othernon-intellectual powers, then the inter-stunt correlations willall be positive. It is quite difficult to find a pair of stunts,both of which call for what would be called intelligence, asjudged by common sense, which have a negative correlation.This is really all that is necessary to prove that what is

4 L. L. THURSTONE

generally called intelligence can be regarded as a factor that isconspicuously common to a very wide variety of activities.Spearman's hypothesis, that it is some sort of energy, is notcrucial to the hypothesis that it is a common factor in intel-lectual activities.

There is a frequently discussed difficulty about which morehas been written than necessary. It has been customary topostulate a single common factor (Spearman's 'g') and tomake the additional but unnecessary assumption that theremust be nothing else that is common to any pair of tests.Then the tetrad criterion is applied and it usually happensthat a pair of tests in the battery has something else in com-mon besides the most conspicuous single common factor.For example, two of the tests may have in common the abilityto write fast, facility with geometrical figures, or a largevocabulary. Then the tetrad criterion is not satisfied and theconclusion is usually one of two kinds, depending on which sideof the fence the investigator is on. If the investigator is outto prove 'g,' then he concludes that the tests are bad becauseit is supposed to be bad to have tests that measure more thanone factor! If the investigator is out to disprove 'g', then heshows that the tetrads do not vanish and that therefore thereis no 'g.' Neither conclusion is correct. The correct con-clusion is that more than one general factor must be postulatedin order to account for the intercorrelations, and that one ofthese general factors may still be what we should call in-telligence. But a technique for multiple factor analysis hasnot been available and consequently we have been stumblingaround with 'group factors' as the trouble-making factorshave been called. A group factor is one that is common totwo or more of the tests but not to all of them. I use the termcommon factor for all factors that extend to two or more ofthe variables. We see therefore that Spearman's criterion,limited as it is to a single common factor, is not adequate forproving or disproving his own hypothesis that there is aconspicuous factor that is common to all intelligence tests. Ifhis criterion gives a negative answer it simply means that thecorrelations require more than one common factor. We do


not need any factor methods at all to prove that a commonfactor of intelligence is a legitimate postulate. It is provedby the fact that all intelligence tests are positively correlated.

There is only one limited problem for which Spearman'smethod is adequate, namely, the question whether a singlefactor is sufficient to account for the intercorrelations of a setof tests. The usual answer is negative. His criterion thatthe tetrads shall vanish is rarely satisfied in practice. Onemight wonder then why it is that numerous examples havebeen compiled in which the tetrad criterion is satisfied. Thereason is simply this—that in order to satisfy the criterion, thetests must be carefully selected so as to have only one thing incommon. Another way by which the criterion is satisfied isto throw out of the battery those tests which do not agree withthe criterion. The remaining set will then satisfy it. Thereason for these difficulties is that Spearman's tetrad differencecriterion demands more than his own hypothesis requires.His hypothesis does not state that there shall be only onecommon factor or ability. He himself deals with manyfactors. But his tetrad difference criterion requires that thereshall be only one common factor.

Now it happens that one can readily put together severalbatteries of tests such that within each battery the criterion issatisfied and therefore we have a common factor ' g' in eachbattery. But if we take a few tests from each of thesebatteries and put together a new composite, then the criterionis not satisfied and we then require more than one factor. Weare then faced with the ambiguity that we have severalbatteries of tests each with its own single common factor.Which of these common factors shall we call the general one?Which is it that we should call 'g'i Spearman's answer isthat we should use a set of perceptual tests as a reference andthat if we are dealing with that particular common factor, thenwe should call it general but that if we are dealing with one ofthe other common factors, then we should call it a specialability, a group factor, a sub-factor of some sort. It would bemore logical to assign a letter or a name to each of thesemental abilities and to treat them as related dependent abili-

6 L. L. THURSTONE

ties. If we choose one of them, such as facility in dealing withperceptual relations, as an axis of reference (Spearman's *g'),then it should be frankly acknowledged that the choice isstatistically arbitrary, for we could equally well start withverbal ability or with arithmetical ability as an axis of refer-ence. The choice of the perceptual axis of reference might bemade from purely psychological considerations, but it wouldnot rest on statistical evidence.

The multi-dimensionality of mind must be recognizedbefore we can make progress toward the isolation and de-scription of separate abilities. It remains a fact, however,that since all mental tests are positively correlated, it ispossible to describe the intercorrelations in terms of severalfactors in such a manner that one of the factors will be con-spicuous in comparison with the others. But the exactdefinition of this factor varies from one set of tests to another.If it is this factor that Spearman implies in his theory of in-telligence, then his criterion is entirely inadequate to define it,because the tetrad criterion merely tells us whether or not anygiven set of intercorrelations can be described in terms of oneand only one common factor.

Let us start with the assumption that there may be severalindependent or dependent mental abilities, and let it be aquestion of fact for each study how many factors are needed toaccount for the observed intercorrelations.2 We also makethe assumption that the contributions of several independentfactors are summative in the individual's performance on eachone of the psychological tests. If we do not make this as-sumption, the solution seems well-nigh hopeless, and it is anassumption that is either explicit or implicit in all attempts todeal with this problem.

We may start our analysis of the generalized factor prob-lem by considering the many hundreds of adjectives that arein current use for describing personalities and temperaments.We have made such a list. Even after removing the syno-nyms we still had several hundred adjectives. It is obvious at

* This point of view is represented by the work of T. L. Kelley as distinguishedfrom the work of Spearman and his students.


the start that all these traits are not independent. For ex-ample, people who are said to be congenial are also quite likelyto be called friendly, or courteous, or generous, even thoughwe do not admit that these words are exactly synonymous.It looks as though we were dealing with a large number ofdependent traits.

The traditional methods of dealing with these psychologicalcomplexities have been speculative, bibliographical, or merelyliterary in character. The problem has been to find a fewcategories, called personality types or temperaments, in termsof which a longer list of traits might be described. Psy-chological inquiry has not yet succeeded in arriving at a list offundamental categories for the description of personality.We are still arguing whether extraversion and introversion arescientific entities or simply artifacts, and whether it is legiti-mate even to look for any personality types at all.

In the generalized factor method we have one of thepossible ways in which a set of categories for the scientificstudy of personality and temperament may be established onexperimental grounds as distinguished from literary verbosityabout this subject. It is our belief that the problem can beapproached in several rational and quantitative ways and thatthey must agree eventually before we have a satisfactoryfoundation for the scientific description of personality.

The problem has geometrical analogies that we shall makeuse of. If we have a set of n points, defined by r coordinatesfor each point, we may discover that they are dependent inthat some of these points can be described linearly in terms ofthe coordinates of the rest of the points. If the adjectiveswere represented by these points, it is as though we were todescribe most of them in terms of a limited number of adjec-tives. But such a solution is not unique, because if we haveten points in space of three dimensions, then it is possible, ingeneral, to describe any seven of the points in terms of theremaining three. It is just so with the personality traits, inthat a unique solution is not given without additional criteria.

But before demanding this sort of reduction it would be ofgreat psychological interest to know how many temperaments

8 L. L. THURSTONE

or personality types we must postulate in order to account forthe differentiate traits that we use in describing people.Again the geometrical analogy is useful. If we have a tableof three coordinates for each one of ten points and if thatmatrix has the rank 2, then we know that all ten points lie in aplane and consequently they can all be described by two co-ordinates in that plane instead of by three. The applicationof this analogy would be the description of ten traits in termsof two independent traits which would then become psy-chological categories or fundamental types. They wouldconstitute the frame of reference in terms of which the othertraits would be described and in terms of which interrelationscould be stated.

But since we have no given frame of reference to beginwith, we do not have the coordinates of the points that mightrepresent the adjectives. We therefore let each adjectiverepresent its own coordinate axis, so that with n traits we shallhave as many axes. These coordinate axes will be oblique,since the traits are known to be at least not all independent.The projection of any one of these traits A on the oblique axisthrough another trait B is the cosine of their central angle, butthis is also the correlation between the two traits A and B.The correlations can be ascertained by experiment, and thenour problem becomes that of finding the smallest number oforthogonal coordinate axes in terms of which we can describeall of the traits whose intercorrelations are known.

The actual data that we must handle are subject to chanceerrors. It is therefore profitable to see how the geometricalmanner of thinking about this problem is affected by thechance errors. It can be shown that each test may be thoughtof as a vector in space of as many dimensions as there are in-dependent mental factors. If the test is perfect, then it isrepresented geometrically by a unit vector. If the test has areliability less than unity, then the length of the vector isreduced. In fact, the length of a test vector in the commonfactor space is the square root of its reliability. If the test haszero reliability, then it determines nothing and this fact has itsgeometrical correspondence in that the test vector is then of


zero length so that it determines no direction at all in the spaceof mental abilities.

The obtained correlation between two tests is the scalarproduct of the two test vectors. If the two tests are perfect,then their scalars are both unity so that the true correlationbetween the two tests is the cosine of the angular separationbetween the two vectors.

We shall consider next two of the fundamental theorems ina generalized theory of factors.3 Let us take a table of sevenvariables as an example In Fig. I we have shown such atable toward the right side of the diagram. We may call itthe correlational matrix. In a table of this kind we show thecorrelation of each test with every other test. For example,the correlation between the two tests A and B is indicated inthe customary cell.

In the diagonal cells of a table of intercorrelations we areaccustomed to record the reliabilities, but that is incorrect infactor theory unless the tests have been so chosen that theycontain no specific factors. It is necessary for us to make adistinction between that part of the variance of a test which isattributable to the common factors and that part of thevariance which is unique for each test. The part which isunique for each test may again be thought of as due to twodifferent sources, namely, the chance errors in the test and theability which is specific for the test. The reliability of a testis that part of the total variance which is due to the commonfactors as well as to the specific factor. It differs from unityonly by that part of the variance which is due to chance errors.We need in factor theory another term to indicate that part ofthe total variance which is attributable only to the commonfactors and which eliminates not only the variance of chanceerrors but also the specific variance. We have used the termcommunality to indicate that part of the total variance ofeach test which is attributable to the common factors. It isalways less than the reliability unless a specific factor is absent,

•The present generalized factor theory has been described in two lithographedpamphlets by the writer. These are: 'The theory of multiple factors,' The Universityof Chicago Bookstore, Chicago, HI., and 'A simplified multiple factor method,' TheUniversity of Chicago Bookstore.

IO L L. THURSTONE

in which case the communality becomes identical with thereliability. It is these communalities that should be recordedin the diagonal cells, but they are the unknowns to be dis-covered by the factorial analysis.

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THE VECTORS OF MIND II

ample we have assumed three factors, so that we have threecolumns and seven rows. Let us suppose that the first factorrepresents verbal ability and that the second factor representsarithmetical ability. These particular abilities are not inde-pendent, but we may ignore that for the moment. Then theentry at indicates the extent to which the test A calls forverbal ability and the entry a2 indicates the extent to which itcalls for arithmetical ability. Since we have assumed threefactors in this diagram we have three factor loadings for eachtest. A table in which the factor loadings are shown for eachtest we have called a. factorial matrix, while the square table con-taining the intercorrelations we have called a correlationalmatrix.

The experimental observations give us the correlationalmatrix, so that it may be regarded as known. The object of afactorial analysis is to find a factorial matrix which correspondsto the given intercorrelations. There is a rather simpletheoretical relation between the given correlational matrixand the factorial matrix. This relation constitutes the funda-mental theorem of the present factor theory. It is illustratedby an example under the correlation table and it can also bestated in very condensed form by matrix notation in that thefactorial matrix multiplied by its transpose reproduces the cor-relational matrix within the observational errors of the givencorrelation coefficients.

When we want to make an analysis of a table of inter-correlations, the first thing we want to know is how many in-dependent common factors we must postulate in order toaccount for the given correlation coefficients. One of ourfundamental theorems states that the smallest number of inde-pendent common factors that will account exactly for the givencorrelation coefficients is the rank of the correlational matrix.Although this theorem is of fundamental significance it is notpossible to apply it in its theoretical form, because of the factthat the given coefficients are subject to experimental errorsand the rank is therefore in general equal to the number oftests. It is always possible to account for a table of inter-correlations by postulating as many abilities as there are tests,

12 L. L. THURSTONE

but that is simply a matter of arithmetical drudgery andnothing is thereby accomplished. The only situation which isof scientific and psychological interest is that in which a tableof intercorrelations can be accounted for by a relatively smallnumber of factors compared with the number of tests.

There is no conflict between the present multiple factormethods and the tetrad difference method. When a singlefactor is sufficient to account for the given coefficients, thenthe rank of the correlational matrix must be I, but the neces-sary and sufficient condition for this is that all of the secondorder minors shall vanish. Now if you expand the secondorder minors in a table of correlation coefficients you find thatyou are in fact writing the tetrads. Hence the tetrad differ-ence method is a special case of the present multiple factortheorem.

It would be possible to extend the tetrad difference methodby writing the expansions of the minors of higher order and inthat manner to write formulae for any number of factorswhich correspond to the tetrads for the special case when onefactor is sufficient. Such a procedure is unnecessarily clumsy.In fact, there should be no excuse for ever computing any moretetrads. Several better methods are available which givemuch more information with only a fraction of the labor thatis required in the computation of tetrads.

We return now to the multiple factor analysis of person-ality. In Table I we have a list of sixty adjectives that are incommon use for describing people. These adjectives togetherwith their synonyms were given to each of 1300 raters. Eachrater was asked to think of a person whom he knew well and tounderline every adjective that he might use in a conversa-tional description of that person. Since it was not necessaryfor the rater to reveal the name of the person he was rating itis our belief that the ratings were relatively free from theinhibitions that are usually characteristic of such a task.With 1300 such schedules we determined the tetrachoriccorrelation coefficient for every possible pair of traits. Sincethere were sixty adjectives in the list we had to determine 1770tetrachoric coefficients. For this purpose we developed a set

THE VECTORS OF MIND

TABLE I

LIST OF ADJECTIVES USED FOR FACTOR STUDY

perseveringcraftyawkwardself-importantdeterminedfriendlypatientsarcasticcongenialhard-workingstubborncapabletolerantcalmpeevish

religiousimpetuousfickledomineeringfrankpessimisticspitefulquietdisagreeablereservedrefinedunnaturalbashfulself-reliantbroad-minded

haughtysubmissivesuspiciouscourageoussternheadstrongjealousgenerousdependablefaithfulreservedsolemnearnesttalentedfrivolous

eccentricingeniousaccommodatingtactfulcarelesstidyprecisesystematiccheerfulconscientiousgraspingsatisfiedcynicalcourteousunconventionalquick-tempered

of computing diagrams which enable one to ascertain thetetrachoric coefficients with correct sign by inspection.4 Eachcoefficient can be ascertained in a couple of minutes by thesecomputing diagrams.

The table of coefficients for the sixty personality traits wasthen analyzed by means of multiple factor methods5 and wefound that five factors are sufficient to account for the co-efficients. We reproduce in Figure 2 the distribution of dis-crepancies between the original tetrachoric coefficients andthe corresponding coefficients that were calculated by meansof the five factors. It has a standard deviation of .069. Theaverage standard error of thirty tetrachoric coefficients chosenat random in this table is .052.

It is of considerable psychological interest to know that thewhole list of sixty adjectives can be accounted for by postulat-ing only five independent common factors. It was of courseto be expected that all of the sixty adjectives would not beindependent, but we did not foresee that the list could beaccounted for by as few as five factors. This fact leads us to

* 'Computing diagrams for the tetrachoric correlation coefficient,' The Universityof Chicago Bookstore.

6 'A simplified multiple factor method,' Univ. Chicago Bookstore, 1933.

14 L. L. THURSTONE

s urmise that the scientific description of personality may notbe quite so hopelessly complex as it is sometimes thought to be.

Next comes the natural question as to just what these fivefactors are, in terms of which the intercorrelations of sixtypersonality traits may be described. Each of the adjectivescan be thought of as a point in space of five dimensions and thefive coordinates of each point represent the five factor com-ponents of each adjective.

jOO

-JO -ZOFifttt Factor Ptstduals

FIG. 2. Frequency distribution of fifth factor residuals in a study of sixty personalitytraits.

We shall consider a three-dimensional example in order toillustrate the nature of the indeterminacy that is here involved(Fig. 3). Let us suppose that three factors are sufficient toaccount for a list of traits. Then each trait can be thought ofas a point in space of three dimensions. In fact, each trait canbe represented as a point on the surface of a ball. If twotraits A and B tend to coexist, the two points will be closetogether on the surface of the ball. If they are mutually ex-clusive, so that when one is present the other is always absentand vice versa, then the two traits are represented by twopoints that are diametrically opposite on the surface of theball such as A and D. If the two traits are independent and


uncorrelated, such as A and C, then they will be displaced fromeach other in the same way as the north pole and a point onthe equator, namely by ninety degrees.

Now suppose that the traits have been allocated to pointson the surface of the sphere in such a manner that the correla-tion for each pair of traits agrees closely with the cosine of thecentral angle between the corresponding points on the surfaceof the sphere. Then we want to describe each of these traitsin terms of its coordinates, but we should first have to decide

FIG. 3. Correlation AB is high and positive. Correlation AC is zero. Corre-lation AD is — I. E and F represent constellations. Location of the orthogonalaxes is arbitrary.

where to locate at least two of the three coordinate axes.This is an arbitrary matter, because the internal relationsbetween the points, that is, the intercorrelations of the traits,remain exactly the same no matter where in the sphere welocate the coordinate axes. That is, when the points havebeen assigned on the surface of the ball, we have not therebylocated the respective coordinate axes. It is possible to de-termine uniquely how many independent common factors arerequired to account for the intercorrelations without therebydetermining just what the factors are. We may therefore useany arbitrarily chosen set of orthogonal axes.

It is psychologically more illuminating to investigateconstellations of the traits. In the factor analysis of theadjectives, constellations of traits reveal themselves in thatthe points which represent some of the traits lie close together

16 L. L. THURSTONE

in a cluster on the surface of a five-dimensional sphere. Oneor two examples will illustrate this type of analysis withrespect to the personality traits. We find, for example, thatthe following traits lie close together in a cluster, namely,friendly, congenial, broad-minded, generous, and cheerful. Itwould seem, therefore, that as far as the five basic factors areconcerned, whatever be their nature, these several traits arevery much alike as far as can be determined by the way inwhich we actually use these adjectives in describing people.

Another such cluster of adjectives which are used asthough they signified the same fundamental trait are theadjectives patient, calm, faithful, and earnest. They cling closetogether in the factorial analysis. Another small group isfound in the three traits persevering, hard-working, and sys-tematic, which lie close together. Still another one is thecluster of traits capable, frank, self-reliant, and courageous.

It is of psychological interest to note that the largestconstellation of traits consists of a list of derogatory adjectives.Such a cluster is the following: self-important, sarcastic,haughty, grasping, cynical, quick-tempered, and several otherderogatory traits that lie close by. It clearly indicates that ifyou describe a man by some derogatory adjective you arequite likely to call him by many other bad names as well.This lack of objectivity in the description of the people wedislike is not an altogether unknown characteristic of humannature.

The schedules used in this study contained 120 adjectives,since every one of the 60 adjectives in the principal list wasrepresented also by a synonym. This enabled us to ascertainthe correlation between each pair of synonyms and we used itas an index of the consistency with which the trait was judged.Let us consider this correlation to be an estimate of the relia-bility of the judgments of the trait. By factor analysis weknow the communality of each trait. It is that part of itstotal variance which it has in common with the other ad-jectives, namely, the five common factors. The differencebetween these two variances is the specificity. It shows themagnitude of the specific factor in each adjective.

THE VECTORS OF MIND

TABLE IISpecificity Adjective

.18 persevering4 0 crafty.46 awkward.10 8elf-important.16 determined.14 friendly.02 patient.16 sarcastic.16 congenial.34 hard-working.34 stubborn.16 capable.04 tolerant.27 calm.09 peevish.58 religious.36 impetuous.07 fickle.19 domineering.40 frank.22 pessimistic.04 spiteful.12 quiet.19 disagreeable.08 reserved.27 refined.18 unnatural48 bashful.14 self-reliant

Specificity Adjective.05 broad-minded.00 haughty.07 submissive.17 suspicious.27 courageous.01 stern.01 headstrong.25 jealous.16 generous.17 dependable.13 faithful.00 solemn.04. earnest.59 talented.39 eccentric.23 ingenious.30 accommodating.26 tactful•34 tidy.22 precise.22 systematic.03 cheerful.11 conscientious.30 grasping.68 satisfied.29 cynical.12 courteous.14 unconventional.41 quick-tempered

We have listed these differences for each adjective in TableII and they yield some psychologically interesting facts. Wefind, for example, one adjective which has a surprisingly highspecificity of nearly .60, and we want to know, of course, whatkind of trait it represents. We find that it is the adjectivetalented. It seems reasonable to guess that this adjectiverefers largely to the intellectual abilities which are not repre-sented by this list. When this study is repeated, we shallinclude several adjectives of this type, so that intelligence maybe investigated as a vector in relation to the personality traits.

Another item with a high specificity is the adjectiveawkward. This means that the trait awkward has somethingabout it which is unique in our list of sixty and which is not

2

18 L. L. THURSTONE

represented by the five common factors. This trait is prob-ably ease and facility in body movement, which is certainlynot represented in the rest of the sixty traits. When thisstudy is repeated we may or we may not include severaladjectives of this type, depending on whether we want to in-clude this additional factor in a study of personality. Anotheradjective with high specificity is religious and the explanationis undoubtedly along the same lines because this is the onlyadjective in the list that refers to any kind of religiousness.

Studies of this sort should be repeated until every im-portant trait is represented by several adjectives. The anal-ysis should yield as many independent factors as may berequired. When the factorial analysis is complete, the spe-cifics should all vanish or they should be relatively small.Then the communalities and the reliabilities will have nearlythe same value. The constellations to be found in such ananalysis will constitute the fundamental categories in terms ofwhich a scientific description of personality may be attained.

There is no necessary relation between the number offactors and the number of constellations. A system of testsmight be found which can be accounted for by several factorseven though it contains no constellations. Fortunately theconstellations can be isolated in a very simple manner whenthe coefficients have been corrected for attenuation.

I turn next to a factor study of the insanities. I have useda very elaborate set of data which Dr. Thomas Verner Mooreof Washington, D. C, collected and investigated by otherfactor methods. Dr. Moore worked with a list of forty-eightsymptoms, thirty-seven of which are listed in Table III. Herecorded the presence—absence, or a rating or test measure ofeach symptom for each of several hundred patients. Withthese records it was possible to ascertain to what extent anytwo symptoms tend to coexist in the same patient. For ex-ample, the extent to which the two symptoms excited and de-structive tend to coexist in the same patient is indicated by thetetrachoric correlation of + .71. The records were sufficientlycomplete so that we could prepare a table of intercorrelationsof the tetrachoric form for thirty-seven symptoms or 666 co-


TABLE IIIMULTIPLE FACTOR ANALYSIS OF PSYCHOTIC SYMPTOMS

References: (i) Moore, T. V., Amer. J. Psychiat., 1930, 9, 719-738. (2) Moore,T. V., Series of Research Publications of Ass'n for Research in Nervous and MentalDiseases, Vol. 9, 324-339; Baltimore, Williams and Wilkins.Code PsychoticNo. Symptom

1. Alcoholism of Parents2. Anxious3. Attacks, Number of Previous6. Delusions, Bizarre9. Depression

10. Destructive11. Disorientation in Space13. Euphoria14. Excited15. Fallacies, Autistic16. Fallacies, Logical17. Giggling18. Hallucinations, Auditory19. Hallucinations, Others23. Homicidal24. Insane Relatives25. Insight, Absence of26. Irritable27. Memory, Total Defect

Code PsychoticNo. Symptom28. Memory Ratio29. Mutism30. Negativism31. Neurasthenia32. Perception Defect33. Reasoning34. Refusal of Food35. Retardation36. Sensibilities, Loss of Finer37. Shut In38. Stereotypism of Actions39. Stereotypism of Any Kind40. Stereotypism of Attitudes41. Stereotypism of Words42. Suicidal43. Tantrums44. Tearful45. Voices, Speaking to

efficients. These computations were also made by the com-puting diagrams.

The multiple factor method was then applied to the tableof 666 coefficients and we found that five factors are sufficientto account for the correlations, with residuals small enough sothat they can be ignored. The communalities were thencomputed for each one of the symptoms and we found thatabout ten of the symptoms do not contain enough in commonwith the other symptoms to warrant their retention in a factorstudy. In other words, about ten of the symptoms are eitherso specific in character or so unreliable as to estimates tharf.tthey do not yield significant correlations with the several'other symptoms. This left twenty-six symptoms which aremore or less related and for which the factorial clusters ofsymptoms could be profitably investigated.

In Table IV we have listed the psychotic symptoms whichlie in each of several constellations. We find, for example,

20 L. L. THURSTONE

TABLE IVPSYCHOTIC SYMPTOMS

Cluster A, Catatonic Group Cluster B, Cognitive GroupMutism Logical fallaciesNegativism Total memory defectShut in Perception defectStereotypism of action ReasoningStereotypism of attitudes Disorientation in spaceStereotypism of wordsGigglingCluster C, Manic Group Cluster D, Hallucinatory Group

Destructive Bizarre delusionsExcited Auditory hallucinationsIrritable Other hallucinationsTantrums Speaking to voices

Cluster E, DepressionAnxiousDepressionTearfulRetardation in movement

that the following symptoms are functionally closely related,namely, mutism, negativism, being shut-in, stereotypism of action,stereotypism of attitudes, stereotypism of words, and giggling.These seven traits are evidently related in that they tend to befound in the same patients and we recognize the list as descrip-tive of the catatonic group. Another constellation consists inthe presence of logical fallacies, defect in memory, defect inperception, and defect in reasoning. This is a constellation ofsymptoms that indicates a derangement of the cognitive func-tions of the patient as contrasted with derangements of theaffective aspects of his mentality. Another group of threesymptoms that lie close together in the factorial analysisconsists of the traits destructive, irritable, and having tantrums.

0$. fourth cluster contains the symptoms delusions, auditorya.rhd other types of hallucinations, and speaking to voices. Afifth group contains the symptoms anxious, depressed, andtearful.

It is not likely that this analysis of the tetrachoric correla-tions between symptoms has given us anything like a de-pendable classification of the insanities, because the study can


be much improved when it is done the next time. But ourresults indicate that by the multiple factor methods it shouldbe possible to arrive at a rational classification of the insanitiesand of personality types.

Another application of the present factor methods is ananalysis to ascertain whether the vocational interests of collegestudents can be classified in constellations and whether theycan be described in terms of vocational interest types, small innumber compared with the list of available occupations.These should eventually be related to the temperamental andpersonality traits and to the constellations of mental abilities.

ActorAdvertiserArchitectArmy OfficerArt CriticArtistAstronomerAthletic DirectorAuctioneerAuto SalesmanBankerBiologistBotanistBuilding ContractorBusiness ManagerCattle RancherCertified Public AccountantChemistCivil EngineerClergymanClub SecretaryCollege ProfessorCongressmanDentistDiplomatic ServiceEconomistElectrical Engineer

TABLE VExplorerFactory ManagerFarmerFloristForeign CorrespondentForest RangerFruit GrowerGeologistHigh School TeacherHistorianInsurance SalesmanInventorJewelerJournalistJudgeLandscape GardenerLawyer, CriminalLawyer, CorporationLibrarianManufacturerMathematicianMechanical EngineerMusicianNewspaper ReporterNovelistOffice ManagerOrchestra Conductor

PharmacistPhilosopherPhotographerPhysicianPhysicistPoetPress AgentPrinterPrivate SecretaryProfessional AthletePsychologistPublic SpeakerRadio AnnouncerRailway ConductorReal Estate SalesmanRetail MerchantSales ManagerScientistSculptorSecret Service ManShip OfficerSociologistStockbrokerSurgeonTax ExpertVocational Counstia

We asked three thousand students in four universitiesto indicate their likes and dislikes on a list of eighty of thebetter known occupations (Table V) which are available for

22 L L. THURSTONE

college students. The tetrachoric correlations indicate theextent to which those who are interested in engineering, forexample, can also be expected to have some interest in physics,or in chemistry. Those who are interested in engineering tendto dislike law and journalism and salesmanship, so that thesecorrelations are negative. Our first question will be to ascer-tain the number of factors that are necessary to account forthe observed intercorrelations. Previous inquiry indicatesthat the number of types may not exceed six or eight.

The scoring of the individual schedules might be reduced toa number of scores equal to the number of factors. Thesescores would be the coordinates of a point which represent thesubject in a space of as many dimensions as there are factorsrequired by the intercorrelations. The occupational likes anddislikes of the subject might be estimated by the coordinatesof the point which represent his own interests. In thismanner it may be possible to make a limited list of occupationsthat are typical of the interests of each student.

The question might be raised whether a student should beadvised to enter that occupation for which his interests aretypical. It might well be argued that he has a better chanceof success if he enters a profession for which his interests areunusual. But that question refers, of course, to educationaland vocational guidance while our present problem concernsthe methodology of isolating types or constellations of traits.If we should find that vocational interests group themselvesinto a relatively small number of types it would have psy-chological interest in relation to the personality traits andmental abilities of the same subjects. It would be anotherquestion to decide how such data could or should be used inguidance.

An application of the factor methods has recently beenmade by Mrs. Thurstone. Her problem was to ascertainwhether radicalism is a common factor in people's attitudes onvarious disputed social issues. She gave eleven attitude scalesto about 380 students in several universities. Records of theintelligence examination of the American Council on Educa-tion were available for the majority of these students, so that


the study includes twelve variables. The factorial analysisreveals a conspicuous common factor that we recognize asradicalism. The second factor residuals were comparablewith the standard errors of the given coefficients. Thestandard error of the mean correlation coefficient was .047while the standard error of the second factor residuals was. 056.

Since the two factors were sufficient to account for the inter-correlations, we have plotted in Figure 4 the factor loadings asthe two coordinates for each of the twelve variables. Several

FIG. 4. Factor study of radicalism with attitude scales by Thelma Gwinn Thurstone.

psychological interpretations can be made from this diagram.The variables which are heavily loaded with radicalism are atti-tudes favorable to evolutionary doctrines, favorable to birthcontrol, favorable to easy divorce, favorable to communism, andit is of interest to note that intelligence is positively correlatedwith these radical or liberal attitudes. In the opposite direc-tion we find conservatism in attitudes favorable to the church,

24 L. L. THURSTONE

favorable to prohibition, the observance of Sunday, and beliefin a personal God. Inspection of the original coefficients aswell as the factorial analysis shows the more intelligent collegestudents tend to be radical or liberal on social questions andthat they tend to be atheistic or agnostic on religious matters.

In naming the common factor which is most prominent inthese attitude scales there may be some question as to whetherit should be called conservatism or religion. The most piouspeople are likely to be against evolutionary doctrines, againstbirth control, and against easy divorce. It may also be thatwhat we have called radicalism-conservatism is a factor whichis nearly the same as the religious factor. It is a psychologicalquestion of considerable interest to ascertain whether thisconspicuous common factor is to be attributed to tempera-mental and intellectual differences in people or merely to re-ligious training.

One or two observations may be made about the secondfactor. We note first that patriotism and communism arediametrically opposite in the factorial diagram and this is whatwe should expect. It is surprising that intelligence shouldcorrelate negatively with patriotism among college students;but when we inspect the original coefficients we find that thestrongest correlation with intelligence is negative .44 withpatriotism. In other words the most intelligent collegestudents tend to be lukewarm in their patriotism as judged bythe attitude scale for this trait. It is unfortunate that we didnot have a scale for measuring attitude toward pacifism,because it is a more disputed object than war. As is to beexpected, the second factor of nationalism is most heavilyrepresented in patriotism and in the glorification of war, whilethe opposite attitude, namely, international-mindedness, isrepresented by the attitudes of those Americans who arefriendly toward communism and toward Germany. Theseexamples will suffice to illustrate the possibilities of factorialanalysis in dealing with the affective and temperamentalattributes of people.

Until recently practically all of the studies that have beenmade by factor methods have been confined to the cognitive

THE VECTORS OF MIND

traits and especially to the mental abilities that are rep-resented by psychological tests. It is on this type ofmaterial that Professor Spearman and his students haveworked extensively with the tetrad difference method. Ishall describe the analysis of a set of nine psychological testswhich have been investigated by Mr. W. P. Alexander of theUniversity of Glasgow and I reproduce here with his per-

TABLE VICORRELATIONS OF NINE TESTS USED BY W. P. ALEXANDER

Westfield State Farm—71 CasesCorrelation Matrix

Test

I. . .2. . .3 . . .4 . . .s---6 . . .7 . . .8 . . .9 . . .

1

• 4Si• 373• 337• 717• 75O. 768. 786. 612

2

451

428470277393407S57616

3

373428—3622554 0 2415334415

4

337470362—1893°4257203528

s71727725s189—6957528 0 0335

6

75°3934 0 2304695—834850478

Description of Tests1. Stanford-Binet.2. Pintner-Paterson Scale.

7

768407415257752834—917433

3. Healy Puzzle 11—Picture Completion.4. Porteous Maze Test.5. Thorndike Reading Test6. Otis Group Test.7. Otis Self-Administering Test.8. Terman Group Test.9- Alexander Performance Scale.

8

7865573342038 0 0850917—450

9

6 1 261641552833547843345O

mission a section of his unpublished data (Table VI). Thenine tests are all well known except the new performance testwhich he has recently devised. When we apply the multiplefactor methods to this correlational matrix we find that mostof the variance of each test can be accounted for by postulatingonly two factors. These we have plotted in Fig. 5.

In this diagram the abscissae represent the first factor thatwas extracted and the ordinates represent the second factor.As is to be expected the centroid of the whole system is on the

26 L. L. THURSTONE

#-axis since this is implied in the approximation procedure thatwas used for this problem. Even in a first glance at this dia-gram we are struck by the fact that the nine tests dividethemselves into two groups or clusters. We turn immediatelyto the tests to see what psychological abilities may be found inthe tests that constitute each of these two constellations. Wethen find that all of the verbal tests fall in one of these con-stellations and that all of the performance tests fall in the other

i'o/*y ability

Verbal ability

FIG. 5. Two factor analysis of nine tests used by W. P. Alexander.

constellation. This suggests a rational method of establishingthe psychological categories that we should call mental abilities.There seems to be little doubt in naming one of these con-stellations verbal ability and we shall name the other onetentatively as manipulation, since that is common to all of theperformance tests. The verbal tests that cling together in thisanalysis are the Stanford-Binet, the Thorndike reading test,the Otis group test, the Otis self-administering test, and theTerman group test. The performance tests that group them-selves apart from the verbal tests are the Pintner-Patersonscale, the Healy Puzzle II (picture completion), the Porteusmaze test, and Alexander's new performance test.

It is to be noted that these two constellations are entirelyindependent of the location of the two orthogonal axes throughthis system of nine tests. A characteristic of the multiplefactor problem is that the location of the axes is arbitrary and


that hence the factorial components are to that extent arbi-trary and without fundamental psychological significance.This limitation is entirely obviated if we center our interest onthe constellations of mental traits that are revealed in thefactorial analysis. The relations between the constellationsare invariant under rotation of the orthogonal axes, and hencewe have here something more or less permanent in terms ofwhich we may define psychological categories and mentalabilities.

As illustrative of this method we have found the centroid ofeach of the two constellations in Alexander's nine tests.These centroids are indicated by the small black circles. Ifwe had a fairly large number of tests in each constellation wecould attach some confidence to the centroid of the constella-tion as a definition of a mental ability. I am here using theterm mental ability to identify a constellation of tests which liein the same cluster in the factorial analysis and which correlatenearly unity when they are corrected for attenuation.

In Alexander's data we find that the two constellations arenot independent. This is seen on the diagram by the factthat the central angle between the two constellations is not aright angle. Since all mental abilities are positively corre-lated, we should expect that all of the constellations of mentalabilities will be positively correlated also, and that is the casein the present data. Now if we are to use these constellationsof traits as the categories for psychological description and ifwe are to define mental abilities and personality types in termsof constellations, then it is immediately apparent that ourcategories will not all be independent. This indicates that wemust look eventually to still more fundamental categories interms of which to describe the mental abilities. In the mean-time it will be useful to describe them in terms of constellationsof known degrees of dependence and we shall then be using asystem of oblique coordinates instead of the orthogonalcoordinates of conventional mathematics. Ultimately weshall want to find that particular location of the orthogonalaxes which corresponds to independent genetic elements.

I shall now describe briefly the results of a multiple factor

28 L. L. THURSTONE

analysis of a set of twenty tests that were recently used by Dr.William Brown in support of the Spearman single-factorhypothesis of intelligence. We can sympathize with Dr.Stephenson who computed all of the fifteen thousand tetraddifferences for this table and we shall not even venture toguess how long it must have taken to make the computations.

The multiple factor method in its simplest approximationform was applied to this set of data, and we find that two

-16 -14 ~l£Second Factor

FIG. 6. Frequency distribution of second factor residuals for 20 tests used byDr. William Brown. 'A test of the theory of two factors,' Brit. J. Psychol., April,1933, Approximate solution by multiple factor methods.

factors are sufficient to account for the given coefficients.The distribution of second factor residuals is shown in Fig. 6.The standard deviation of the distribution of residuals is .039.In Fig. 7 we have plotted the two-factor coordinates for thesetwenty tests and we then see that they fall into two constella-tions. Reference to the tests shows that all of the verbal testsare in one group and that all of the perceptual tests are foundin the other group. We might regard these two groups asrepresenting two mental abilities, one of which we should callverbal ability, while the other one might be called visual formperception. The correlation between pure measures of thesetwo abilities is the cosine of the angular separation between thetwo centroids.


Brown's interpretation of these relations which he has in-vestigated with the tetrad difference method is that one ofthese two clusters represents Spearman's 'g ' and that what-ever is left over is attributable to verbal ability. It seems tous that it would be just as logical to call the verbal cluster thegeneral one and to attribute the residuals to a special per-ceptual factor. In that case the verbal constellation would becalled the general 'g.y This reasoning might be extended toeach one of the many constellations of abilities which are

FIG. 7. Two-factor analysis of the twenty tests used by Dr. William Brown in 'A testof the theory of two factors,' Brit. J. Psychol., April, 1933.

represented by groups of psychological tests. But the evi-dence is as yet entirely insufficient to demonstrate that anyone of the constellations of mental abilities is a principalintellective factor.

Since all mental tests are positively correlated, it followsthat the tests must lie in a limited quadrant or octant of thegeometrical representation of the mental abilities. Instead of

30 L. L. THURSTONE

adopting one of the constellations as a reference it might bemore logical to find a point in the octant of the mental abilitiessuch that the most diversified series of tests all correlatepositively with the imaginary test at this central point. It isnot inconceivable that a point might be located by this meanswith some degree of certainty. It would then represent thecentral intellective factor and those tests would be the besttests of intelligence which correlate highest with the imaginarypure test, which is represented by this central vector. Itwould not be the centroid of any particular set of tests unlessthese should happen to be evenly distributed over the space ofthe mental abilities and that is a condition which can not beguaranteed beforehand. It is not inconceivable that theconstellation of perceptual relations of Spearman will lie closeto this central vector, because we may expect those tests to lieclose to it which correlate best with judgments of intelligence,and that is the practical criterion by which intelligence tests havebeen constructed. If that should turn out to be the case, thenSpearman's 'g' would be a close representation of a principalintellective factor even though his tetrad criterion is am-biguous and inadequate. If we follow Spearman's procedurein carefully picking out a list of tests, as diversified as possible,which all satisfy his criterion, then we should be defining acommon factor which is close to the center of the space ofmental abilities.

But this procedure necessarily leads to at least some nega-tive factor loadings in all but the first factor, and it is difficultto make a psychological interpretation of negative abilities.It is much more likely that the opposite procedure will give asolution that makes sense for psychology and for genetics.This solution would be to find a set of orthogonal axes throughthe fringe of the space of mental abilities rather than throughthe middle of it. The geometrical representation of the solu-tion will probably be as follows. The mental abilities can berepresented as points within a cone. The axis of the cone willrepresent a fictitious central intellective factor. The funda-mental abilities which have genetic meaning will be repre-sented by a set of orthogonal elements of the cone in space of


as many dimensions as there are genetic factors. All mentaltests will then be described in terms of positive orthogonalcoordinates, corresponding to the independent genetic factors.Negative loadings will disappear. For rough and practicaldescriptive purposes the axis of the cone of mental abilitiesmay be used as an axis of reference which is central in the spaceof mental abilities. It will be a fictitious central intellectivefactor but it will probably have no fundamental psychologicalor genetic meaning.

In conclusion I want to suggest the course of investigationwhich is likely to lead to a scientific description and under-standing of mental abilities and personality traits and theiraberrations. Our first task is to establish the identity of theseveral mental abilities which reveal themselves as distinctconstellations in factorial analysis. Among these abilities itis quite likely that we shall find verbal ability, perceptualrelations, and arithmetical ability to be distinct, thoughpositively correlated.

In these studies it is probably best not to pivot on anysingle constellation as fundamental to all of the rest. It isbetter to use an analysis which allows as many factors toappear as are necessary to account for each new set of testsand to name the constellations when they appear. Thesecategories should be frankly regarded as temporary and sub-ject to redefinition in successive experiments. Eventually weshould be able to work with a rather limited number of mentalabilities and trait constellations. These categories will beeither more or less conventionalized, or else they will be soconsistent that some physical or even genetic significance maybe attached to them.

In extracting each constellation it is essential to make surethat the specific variance in each variable has vanished or thatit is small enough so that it can be ignored. If one of thevariables retains a considerable specific variance after thecommon factors have been extracted, it is necessary to repeatthe experiments with additional tests which are similar to theone which has shown a specific variance. Only when thereliability and the communality are nearly equal can we be sure

32 L. L. THURSTONE

that the common factors account for the total variance in eachtest. In order to favor this outcome it is best to assemble thetest batteries in such a way that there are several similar testsof each kind that are to be investigated. This is in a sense theopposite of the precautions which have been current in factorstudies where the experimenters have been careful to avoidsimilar tests simply because they disturb the tetrad criterion.Instead of concealing the specific variance with the chanceerrors, as is done in tetrad analysis, it is more illuminating toinvestigate the nature of the specific variance of each test byincluding several similar tests of each kind in the test battery.

When the mental abilities have been defined in terms of alarge number of elementary tests for each ability, it will be ofconsiderable interest to ascertain experimentally the extentto which training of one ability transfers to another ability andto relate such transfer effects to the known correlations be-tween the abilities.

It is my conviction that the isolation of the mental abilitieswill turn out to be essentially a problem in genetics. It willbe profitable to ascertain what part of the variance of eachability can be attributed to the parents. It is not unlikelythat this type of genetic research will constitute one of themeans for identifying the independent elements in the mentalabilities.

We hope that the development of factorial methods ofanalysis will give us the tools by which to reduce the com-plexities of social and psychological phenomena to a limitednumber of elements. These methods should be useful notonly for developing the theory of mental abilities and temper-amental traits but also in meeting the practical demands ofeducational and psychological guidance.[MS. received October 27, 1933]

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THE PSYCHOLOGICAL REVIEW - University of …psych.colorado.edu/.../PSYC5112/Readings/VectorsOfMind_Thurstone… · THE PSYCHOLOGICAL REVIEW THE VECTORS OF MIND1 BY L. L. THURSTONE