Yet the major effect of 11 +, in terms of human lives and hopes, surely lay with its primary numerical result—80 percent branded as unfit for higher education by reason of low innate intellectual ability. Two incidents come to mind, memories of two years spent in Britain during the regime of 11 + : children, already labeled sufficiently by the location of their school, daily walking through the streets of Leeds in their academic uniforms, readily identified by all as the ones who hadn’t qualified; a friend who had failed 11 + but reached the university anyway because she had learned Latin on her own, when her secondary modern school did not teach it and universities still required it for entrance into certain courses (how many other working-class teenagers would have had the means or motivation, whatever their talents and desires?).
Burt was committed to his eugenic vision of saving Britain by finding and educating its few people of eminent talent. For the rest, I assume that he wished them well and hoped to match their education with their ability as he perceived it. But the 80 percent were not included in his plan for the preservation of British greatness. Of them, he wrote (1959, p. 123):
It should be an essential part of the child’s education to teach him how to face a possible beating on the 11+ (or any other examination), just as he should learn to take a beating in a half-mile race, or in a bout with boxing gloves, or a football match with a rival school.
Could Burt feel the pain of hopes dashed by biological proclamation if he was willing seriously to compare a permanent brand of intellectual inferiority with the loss of a single footrace?
L. L. Thurstone and the vectors of mind
Thurstone’s critique and reconstruction
L. L. Thurstone was born (1887) and bred in Chicago (Ph.D., University of Chicago, 1917, professor of psychology at his alma mater from 1924 to his death in 1955). Perhaps it is not surprising that a man who wrote his major work from the heart of America during the Great Depression should have been the exterminating angel of Spearman’s g. One could easily construct a moral fable in the heroic mold: Thurstone, free from the blinding dogmas of class bias, sees through the error of reification and hereditarian assumptions to unmask g as logically fallacious, scientifically worthless, and morally ambiguous. But our complex world grants validity to few such tales, and this one is as false and empty as most in its genre. Thurstone did undo g for some of the reasons cited above, but not because he acknowledged the deeper conceptual errors that had engendered it. In fact, Thurstone disliked g because he felt that it was not real enough!
Thurstone did not doubt that factor analysis should seek, as its primary objective, to identify real aspects of mind that could be linked to definite causes. Cyril Burt named his major book The Factors of the Mind, Thurstone, who invented the geometrical depiction of tests and factors as vectors (Figs. 6.6, 6.7), called his major work (1935) The Vectors of Mind. “The object of factor analysis,” Thurstone wrote (1935, p. 53), “is to discover the mental faculties.”
Thurstone argued that Spearman and Burt’s method of principal components had failed to identify true vectors of mind because it placed factor axes in the wrong geometrical positions. He objected strenuously both to the first principal component (which produced Spearman’s g) and to the subsequent components (which identified “group factors” in clusters of positive and negative projections of tests).
The first principal component, Spearman’s g, is a grand average of all tests in matrices of positive correlation coefficients, where all vectors must point in the same general direction (Fig. 6.4). What psychological meaning can such an axis have, Thurstone asked, if its position depends upon the tests included, and shifts drastically from one battery of tests to another?
Consider Fig. 6.10 taken from Thurstone’s expansion (1947) of the Vectors of Mind. The curved lines form a spherical triangle on the surface of a sphere. Each vector radiates from the center of the sphere (not shown) and intersects the sphere’s surface at a point represented by one of the twelve small circles. Thurstone assumes that the twelve vectors represent tests for three “real” faculties of mind, A, B, and C (call them verbal, numerical, and spatial, if you will). The left set of twelve tests includes eight that primarily measure spatial ability and fall near C; two tests measure verbal ability and lie near A, while two reflect numerical skill. But there is nothing sacrosanct about either the number or distribution of tests in a battery. Such decisions are arbitrary; in fact, a tester usually can’t impose a decision at all because he doesn’t know, in advance, which tests measure what underlying faculty. Another battery of tests (right side of Fig. 6.10) may happen to include eight for verbal skills and only two each for numerical and spatial ability.
The three faculties, Thurstone believes, are real and invariant in position no matter how many tests measure them in any battery. But look what happens to Spearman’s g. It is simply the average of all tests, and its position—the x in Fig. 6.10—shifts markedly for the arbitrary reason that one battery includes more spatial tests (forcing g near spatial pole C) and the other more verbal tests (moving g near verbal pole A). What possible psychological meaning can g have if it is only an average, buffeted about by changes in the number of tests for different abilities? Thurstone wrote of g (1940, p. 208):
Such a factor can always be found routinely for any set of positively correlated tests, and it means nothing more or less than the average of all the abilities called for by the battery as a whole. Consequently, it varies from one battery to another and has no fundamental psychological significance beyond the arbitrary collection of tests that anyone happens to put together.… We cannot be interested in a general factor which is only the average of any random collection of tests.
Burt had identified group factors by looking for clusters of positive and negative projections on the second and subsequent principal components. Thurstone objected strenuously to this method, not on mathematical grounds, but because he felt that tests could not have negative projections upon real “things.” If a factor represented a true vector of mind, then an individual test might either measure that entity in part, and have a positive projection upon the factor, or it might not measure it at all, and have a zero projection. But a test could not have a negative projection upon a real vector of mind:
A negative entry … would have to be interpreted to mean that the possession of an ability has a detrimental effect on the test performance. One can readily understand how the possession of a certain ability can aid in a test performance, and one can imagine that an ability has no effect on a test performance, but it is difficult to think of abilities that are as often detrimental as helpful in the test performances. Surely, the correct factor matrix for cognitive tests does not have many negative entries, and preferably it should have none at all (1940, pp. 193–194).
6.10 Thurstone’s illustration of how the position of the first principal component (the x in both figures) is affected by the types of tests included in a battery.
Thurstone therefore set out to find the “correct factor matrix” by eliminating negative projections of tests upon axes and making all projections either positive or zero. The principal component axes of Spearman and Burt could not accomplish this because they, perforce, contained all positive projections on the first axis (g) and combinations of negative and positive groups on the subsequent “bipolars.”
Thurstone’s solution was ingenious and represents the most strikingly original, yet simple, idea in the history of factor analysis. Instead of making the first axis a grand average of all vectors and letting the others encompass a steadily decreasing amount of remaining information in the vectors, why not try to place all axes near clusters of vectors. The clusters may reflect real “vectors of mind,” imperfectly measured by several tests. A factor axis placed near such a cluster will have high positive projections for tests measuring that primary ability* and very low zero projections for all tests measuring other primary abilities—as long as the primary abilities are independent and uncorrected.
&
nbsp; But how, mathematically, can factor axes be placed near clusters? Here, Thurstone had his great insight. The principal component axes of Burt and Spearman (Fig. 6.6) do not lie in the only position that factor axes can assume. They represent one possible solution, dictated by Spearman’s a priori conviction that a single general intelligence exists. They are, in other words, theory-bound, not mathematically necessary—and the theory may be wrong. Thurstone decided to keep one feature of the Spearman-Burt scheme: his factor axes would remain mutually perpendicular, and therefore mathematically uncorrelated. The real vectors of mind, Thurstone reasoned, must represent independent primary abilities. Thurstone therefore calculated the Spearman-Burt principal components and then rotated them to different positions until they lay as close as they could (while still remaining perpendicular) to actual dusters of vectors. In this rotated position, each factor axis would receive high positive projections for the few vectors clustered near it, and zero or near zero projections for all other vectors. When each vector has a high projection on one factor axis and zero or near zero projections on all others, Thurstone referred to the result as a. simple structure. He redefined the factor problem as a search for simple structure by rotating factor axes from their principal components orientation to positions maximally close to clusters of vectors.
Figs. 6.6 and 6.7 show this process geometrically. The vectors are arranged in two clusters representing verbal and mathematical tests. In Fig. 6.6 the first principal component (g) is an average of all vectors, while the second is a bipolar, with verbal tests projecting negatively and arithmetic tests positively. But the verbal and arithmetic clusters are not well defined on this bipolar factor because most of their information has already been projected upon g, and little remains for distinction on the second axis. But if the axes are rotated to Thurstone’s simple structure (Fig. 6.7), then both clusters are well defined because each is near a factor axis. The arithmetic tests project high on the first simple structure axis and low on the second; the verbal tests project high on the second and low on the first.
The factor problem is not solved pictorially, but by calculation. Thurstone used several mathematical criteria for discovering simple structure. One, still in common use, is called “varimax,” or the search for maximum variance upon each rotated factor axis. The “variance” of an axis is measured by the spread of test projections upon it. Variance is low on the first principal component because all tests have about the same positive projection, and the spread is limited. But variance is high on rotated axes placed near clusters, because such axes have a few very high projections and other zero or near zero projections, thus maximizing the spread.*
The principal component and simple structure solutions are mathematically equivalent; neither is “better.” Information is neither gained nor lost by rotating axes; it is merely redistributed. Preferences depend upon the meaning assigned to factor axes. The first principal component demonstrably exists. For Spearman, it is to be cherished as a measure of innate general intelligence. For Thurstone, it is a meaningless average of an arbitrary battery of tests, devoid of psychological significance, and calculated only as an intermediary step in rotation to simple structure.
Not all sets of vectors have a definable “simple structure.” A random array without clusters cannot be fit by a set of factors, each with a few high projections and a larger number of near zero projections. The discovery of a simple structure implies that vectors are grouped into clusters, and that clusters are relatively independent of each other. Thurstone continually found simple structure among vectors of mental tests and therefore proclaimed that the tests measure a small number of independent “primary mental abilities,” or vectors of mind—a return, in a sense, to an older “faculty psychology” that viewed the mind as a congeries of independent abilities.
Now it happens, over and over again, that when a factor matrix is found with a very large number of zero entries, the negative entries disappear at the same time. It does not seem as if all this could happen by chance. The reason is probably to be found in the underlying distinct mental processes that are involved in the different tasks.… These are what I have called primary mental abilities (1940, p. 194).
Thurstone believed that he had discovered real mental entities with fixed geometric positions. The primary mental abilities (or PMA’s as he called them) do not shift their position or change their number in different batteries of tests. The verbal PMA exists in its designated spot whether it is measured by just three tests in one battery, or by twenty-five different tests in another.
The factorial methods have for their object to isolate the primary abilities by objective experimental procedures so that it may be a question of fact how many abilities are represented in a set of tasks (1938, p. 1).
Thurstone reified his simple structure axes as primary mental abilities and sought to specify their number. His opinion shifted as he found new PMA’s or condensed others, but his basic model included seven PMA’s—V for verbal comprehension, W for word fluency, N for number (computational), S for spatial visualization, M for associative memory, P for perceptual speed, and R for reasoning.*
But what had happened to g—Spearman’s ineluctable, innate, general intelligence—amidst all this rotation of axes? It had simply disappeared. It had been rotated away; it was not there anymore (Fig. 6.7). Thurstone studied the same data used by Spearman and Burt to discover g. But now, instead of a hierarchy with a dominant, innate, general intelligence and some subsidiary, trainable group factors, the same data had yielded a set of independent and equally important PMA’s, with no hierarchy and no dominant general factor. What psychological meaning could g claim if it represented but one possible rendering of information subject to radically different, but mathematically equivalent, interpretations? Thurstone wrote of his most famous empirical study (1938, p. vii):
So far in our work we have not found the general factor of Spearman.… As far as we can determine at present, the tests that have been supposed to be saturated with the general common factor divide their variance among primary factors that are not present in all the tests. We cannot report any general common factor in the battery of 56 tests that have been analyzed in the present study.
The egalitarian interpretation of PMA’s
Group factors for specialized abilities have had an interesting odyssey in the history of factor analysis. In Spearman’s system they were called “disturbers” of the tetrad equation, and were often purposely eliminated by tossing out all but one test in a cluster—a remarkable way of rendering a hypothesis impervious to disproof. In a famous study, done specifically to discover whether or not group factors existed, Brown and Stephenson (1933) gave twenty-two cognitive tests to three hundred ten-year-old boys. They calculated some disturbingly high tetrads and dropped two tests “because 20 is a sufficiently large number for our present purpose.” They then eliminated another for the large tetrads that it generated, excusing themselves by stating: “at worst it is no sin to omit one test from a battery of so many.” More high values prompted the further excision of all tetrads including the correlation between two of the nineteen remaining tests, since “the mean of all tetrads involving this correlation is more than 5 times the probable error.” Finally, with about one-fourth of the tetrads gone, the remaining eleven thousand formed a distribution close enough to normal. Spearman’s “theory of two factors,” they proclaimed, “satisfactorily passes the test of experience.” “There is in the proof the foundation and development of a scientific experimental psychology; and, although we would be modest, to that extent it constitutes a ‘Copernican revolution’”(Brown and Stephenson, 1933, p. 353.
For Cyril Burt, the group factors, although real and important in vocational guidance, were subsidiary to a dominant and innate g.
For Thurstone, the old group factors became primary mental abilities. They were the irreducible mental entities; g was a delusion.
Copernicus’s heliocentric theory can be viewed as a purely
mathematical hypothesis, offering a simpler representation for the same astronomical data that Ptolemy had explained by putting the earth at the center of things. Indeed, Copernicus’s cautious and practical supporters, including the author of the preface to De Revolutionibus, urged just such a pragmatic course in a world populated with inquisitions and indices of forbidden books. But Copernicus’s theory eventually produced a furor when its supporters, led by Galileo, insisted upon viewing it as a statement about the real organization of the heavens, not merely as a simpler numerical representation of planetary motion.
So it was with the Spearman-Burt vs. the Thurstone school of factor analysis. Their mathematical representations were equivalent and equally worthy of support. The debate reached a fury of intensity because the two mathematical schools advanced radically different views about the real nature of intelligence—and the acceptance of one or the other entailed a set of fundamental consequences for the practice of education.
With Spearman’s g, each child can be ranked on a single scale of innate intelligence; all else is subsidiary. General ability can be measured early in life and children can be sorted according to their intellectual promise (as in the 11+ examination).
With Thurstone’s PMA’s, there is no general ability to measure. Some children are good at some things, others excel in different and independent qualities of mind. Moreover, once the hegemony of g was broken, PMA’s could bloom like the flowers in spring. Thurstone recognized only a few, but other influential schemes advocated 120 (Guilford, 1956) or perhaps more (Guilford, 1959, p. 477). (Guilford’s 120 factors are not induced empirically, but predicted from a theoretical model—represented as a cube of dimensions 6 × 5 × 4 = 120—designating factors for empirical studies to find).
The Mismeasure of Man Page 35