GENERAL- AND GROUP-FACTOR METHODS      301
The saturation coefficients can then be calculated either by the
summation method or by that of least squares : in this instance
both methods lead to the same figures, given for the pooled tests in
the first column of Table IIlA.
If the reader will partial out the first factor in the usual way by
subtracting the expected correlations from the observed, the
residuals will be found to yield a symmetric ^c compartite matrix,^31
having positive figures in the squares along the leading diagonal and
zero elsewhere. Each of the square sub-matrices (technically
termed * parts ') can now be treated as a separate hierarchy, and
factorized. If the entire matrix has 4 hierarchical^c parts,' its rank
will be 4 ([26], p. 7) ; the completion of the analysis will therefore
yield 4 * group-factors ' in addition to the first * general factor.'
To save space, instead of tabulating all the values for the
separate tests, I print tables of the pooled values only.^2
The complete factorial matrix as obtained by this method
will be the 4X5 matrix formed by the five columns of
Table IIlA (F^. The whole has the appearance of a
vertical column-matrix to which a diagonal matrix has been
subjoined, each element representing a sub-matrix of one or
more short columns.
Matrices having the form shown in Table IIlA are by no means
unusual in the resolution of compartite matrices and in the theory
of canonical reductions generally (cf. [26], II, p. 12 f.). By the
psychologist, however, both the compartite scheme and the pattern
into which it is resolved will be recognized as repeatedly occurring
in analyses of correlation-tables where the investigator has first
removed a general factor and then extracted a series of positive
non-overlapping group-factors. Examples are to be found in my
early analyses of emotional traits and educational abilities ([30],
P- 695; [33], p. 37; [35], Tables XX, XXIII), and later in Alexander's
analyses of correlations between intelligence tests of various types
([82], Tables on p. 148 and XXVI-XXIX), and most recently of all
^1	A ^c compartite matrix,' when rearranged in standard form, is a com-
pound matrix all of whose sub-matrices vanish except those lying along the
leading diagonal.    These non-vanishing sub-matrices are called its * parts/
In general, they will not necessarily be square; but they must be square if the
matrix is symmetrical ([26], II, p. 7).
^2	Incidentally this will serve to illustrate the treatment of * specific factors *
by the general-factor method.   To obtain the saturation coefficients for the
separate * tests' the figures in the factor matrices must, of course, here be
halved throughout.   For a more detailed example, see pp. 477 f.