THE MISMEASURE OF MANrespects, their answers are already foreshadowed; and every-
where, they are rendered susceptible of eventual decisive solution."The method of tetrad differences
In his original work, Spearman did not use the method of prin-
cipal components described on pp. 275—278. Instead, he developed
a simpler, though tedious, procedure better suited for a precom-
puter age when all calculations had to be performed by hand.* He
computed the entire matrix of correlation coefficients between all
pairs of tests, took all possible groupings of four measures and
computed for each a number that he called the "tetrad difference."
Consider the following example as an attempt to define the tetrad
difference and to explain how Spearman used it to test whether
the common variance of his matrix could be reduced to a single
general factor, or only to several group factors.
Suppose that we wish to compute the tetrad difference for four
measures taken on a series of mice ranging in age from babies to
adults—leg length, leg width, tail length, and tail width. We com-
pute all correlation coefficients between pairs of variables and find,
unsurprisingly, that all are positive—as mice grow, their parts get
larger. But we would like to know whether the common variance
in the positive correlations all reflects a single general factor—
growth itself—or whether two separate components of growth
must be identified—in this case, a leg factor and a tail factor, or a
length factor and a width factor. Spearman gives the following for-
mula for the tetrad difference7" 13 r2A~r 23 ^1 4
where r is the correlation coefficient and the two subscripts rep-
resent the two measures being correlated (in this case, 1 is leg
length, 2 is leg width, 3 is tail length and 4 is tail width—so that ri3
is the correlation coefficient between the first and the third mea-
sure, or between leg length and tail length). In our example, the
tetrad difference is
(leg length and tail length) x (leg width and tail width) -
(leg width and tail length) x (leg length and tail width)
Theg calculated by the tetrad formula is conceptually equivalent and mathemati-
cally almost equivalent to the first principal component described on pp. 275-278
and used in modern factor analysis.