3 i^6 THE MISMEASURK O F M A Na general factor common to all tests and a specific factor peculiar
to that test alone. He denied that clusters of tests showed any sig-
nificant tendency to form "group factors" between his two levels—
that is, he found no evidence for the "faculties" of an older psy-
chology, no clusters representing verbal, spatial, or arithmetic abil-
ity, for example. In his 1909 paper, Burt did note a "discernible,
but small" tendency for grouping in allied tests. But he proclaimed
it weak enough to ignore ("vanishingly minute" in his words), and
argued that his results "confirm and extend" Spearman's theory.
But Burt, unlike Spearman, was a practitioner of testing
(responsible for all of London's schools). Further studies in factor
analysis continued to distinguish group factors, though they were
always subsidiary tog. As a practical matter for guidance of pupils,
Burt realized that he could not ignore the group factors. With a
purely Spearmanian approach, what could a pupil be told except
that he was generally smart or dumb? Pupils had to be guided
toward professions by identifying strengths and weaknesses in
more specific areas.
By the time Burt did his major work in factor analysis, Spear-
man's cumbersome method of tetrad differences had been
replaced by the principal components approach outlined on pp.
275—280. Burt identified group factors by studying the projection
of individual tests upon the second and subsequent principal com-
ponents. Consider Fig. 6.6: In a matrix of positive correlation coef-
ficients, vectors representing individual tests are all clustered
together. The first principal component, Spearman's g runs
through the middle of the cluster and resolves more information
than any other axis could. Burt recognized that no consistent pat-
terns would be found on subsequent axes if Spearman's two-factor
theory held—for the vectors would not form subclusters if their
only common variance had already been accounted for by g. But if
the vectors form subclusters representing more specialized abili-
ties, then the first principal component must run between the sub-
clusters if it is to be the best average fit to all vectors. Since the
second principal component is perpendicular to the first, some
subclusters must project positively upon it and others negatively (as
Fig. 6.6 shows with its negative projections for verbal tests and pos-
itive projections for arithmetic tests). Burt called these axes bipolar
factors, because they included clusters of positive and negative pro-