GENERAL- AND GROUP-FACTOR METHODS 299

holds good of measurements in terms of * group-factors' and of
' general factors ' respectively.

(b) Relations between the Saturation Coefficients.—Corre-
lation tables that lend themselves most obviously to the
6 group-factor ' mode of analysis usually arise when the
tests that have been correlated (or the persons, if we begin
by correlating persons) form a discontinuous selection :
(e.g. tests representing different mental levels or functions
[16], [82], examinations in different school subjects [35],
persons of different types of imagery [116], etc.). In such
cases the coefficients can be reasserted so that exception-
ally high correlations that break the hierarchy cluster in
blocks along the principal diagonal, while the remaining
rectangles fit more appropriately into the general hier-
archical order (cf. [30], Table I; [35], Tables XVIII, XX,
XXII, XXIII). Thus arranged, the whole array of co-
efficients can be partitioned into square and oblong sub-
matrices, grouping together those tests or traits which
belong to much the same categories and are therefore
attributable to the same ' group-factor.'

To illustrate the resulting scheme at its simplest, let us
suppose that both the variances for the different factors and
the saturation coefficients for the different tests are every-
where equal (say -5 throughout). With 4 group-factors
each found in two tests, we should then obtain an 8 X 8
correlation table such as the following (Table II), which
may be partitioned as shown and regarded as a ' compound
matrix ' ([26], p. xii; [73], pp. 5-8). The rank of the
matrix is evidently only 4; but the 4 conspicuous clusters,
and the positive correlations outside the clusters, suggest
an analysis into 5 factors. The non-mathematical student
will most easily grasp the relations between the alternative
modes of factorization if we now apply them to a simple
arithmetical example, such as this.

(a) To determine the factors by the group-factor
method we may follow the procedure I have outlined else-
where (' method a ' [93], p. 306 ; [116], p. 339).

On applying this procedure to an empirical table, the first step,
as we have seen, is to group the correlations so that the enlarged