THE REAL ERROR OF CYRIL BURT
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 nega-
tive projections of tests).
The first principal component, Spearman's g, is a grand aver-
age 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 mea-
sure spatial ability and fall near C; two tests measure verbal ability
and lie near A, while two reflect numerical skill. But there is noth-
ing 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 mean-
ing cang- have if it is only an average, buffeted about by changes in
the number of tests for different abilities? Thurstone wrote of g
(l94o. 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