THE REAL ERROR OF CYRIL BURT 289Spearman argued that tetrad differences of zero imply the exist-
ence of a single general factor while either positive or negative val-
ues indicate that group factors must be recognized. Suppose, for
example, that group factors for general body length and general
body width govern the growth of mice. In this case, we would get
a high positive value for the tetrad difference because the correla-
tion coefficients of a length with another length or a width with
another width would tend to be higher than correlation coefficients
of a width with a length. (Note that the left-hand side of the tetrad
equation includes only lengths with lengths or widths with widths,
while the right-hand side includes only lengths with widths.) But if
only a single, general growth factor regulates the size of mice, then
lengths with widths should show as high a correlation as lengths
with lengths or widths with widths—and the tetrad difference
should be zero. Fig. 6.8 shows a hypothetical correlation matrix for
the four measures that yields a tetrad difference of zero (values
taken from Spearman's example in another context, 1927, p. 74).
Fig. 6.8 also shows a different hypothetical matrix yielding a posi-
tive tetrad difference and a conclusion (if other tetrads show the
same pattern) that group factors for length and width must be rec-
ognized.
The top matrix of Fig. 6.8 illustrates another important point
that reverberates throughout the history of factor analysis in psy-
chology. Note that, although the tetrad difference is zero, the cor-
relation coefficients need not be (and almost invariably are not)
equal. In this case, leg width with leg length gives a correlation of
0.80, while tail width with tail length yields only 0.18. These differ-
ences reflect varying "saturations" with g, the single general factor
when the tetrad differences are zero. Leg measures have higher
saturations than tail measures—that is, they are closer to g, or
reflect it better (in modern terms, they lie closer to the first princi-
pal component in geometric representations like Fig. 6.6). Tail
measures do not load strongly ong* They contain little common
variance and must be explained primarily by their s—the informa-
tion unique to each measure. Moving now to mental tests: if g rep-
resents general intelligence, then mental tests most saturated with"The terms "saturation" and "loading" refer to the correlation between a test and
a factor axis. If a test "loads" strongly on a factor then most of its information is
explained by the factor.