THE REAL ERROR OF CYRIL BURT
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 pref-
erably it should have none at all (1940, pp. 193-194).
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 uncorrelated.
But how, mathematically, can factor axes be placed near clus-
ters? Here, Thurstone had his great insight. The principal com-
ponent 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 reified his factors, calling them "primary abilities," or "vectors of
mind." All these terms represent the same mathematical object in Thurstone's sys-
tem—factor axes placed near clusters of test vectors.