V

454 APPENDIX I

step 6. If it is not, each saturation coefficient must be divided by
um of squares of revised saturation coefficients
Total factor-variance as obtained in step 6.

With a perfect hierarchy, as, for example, in the present case, the
methods of weighted and unweighted summation give identical
results. If the trial values taken for the saturation coefficients
(and therefore for the self-correlations which are assumed to be the
squares of the saturation coefficients) had been inaccurate, we
should have taken the calculated values as giving better weights
and better communalities, and started all over again. The reiterated
calculations would ultimately lead to the figures given above. It
will be noticed that these final figures imply values for the self-
correlations which reduce the correlation table to a perfect hierarchy,
that is, to a matrix of rank one, or, in other words, of the lowest
possible rank which the observed inter correlations permit. Thus, with
the foregoing procedure of successive approximation, the methods
of unweighted and weighted summation both yield an analysis
entailing the smallest possible number of factors.

C. BIPOLAR HIERARCHIES

In earlier researches, where tests of intellectual abilities or educa-
tional attainments were applied to random samples of the popula-
tion, the correlations were usually positive throughout; but when
such tests are applied to more homogeneous samples (e.g. school
classes instead of complete age-groups), negative correlations are
often found, systematically distributed within the table. The
negative values are still more conspicuous when we correlate assess-
ments for emotional or temperamental traits instead of tests of
cognition. The elimination of a general factor has much the same
effect as selecting a homogeneous group ; and, when the summation
method has been used, the residual correlations yield a special kind
of pattern which may be called ' bipolar.'

In later work on factor-analysis, therefore, modified methods
proved essential in order to deal with negative correlations as well as
positive ([30], [3 3]). In principle either simple summation or weighted
summation may still be employed. The requisite modifications
may best be illustrated by taking once again an artificial example,

Table III shows a perfect bipolar hierarchy, constructed from the
saturation coefficients set out along the top and left-hand margins.
The pattern shows certain new peculiarities. In the upper half1
(or * chief/ if I may borrow a convenient term from heraldry) the

1 ' Half * as judged by the totals, not by the number of rows or columns, The
' dexter * half will be on the left of the spectator or reader.