460 APPENDIX I

first few will usually be statistically significant. With a table of inter-
correlations, where the variances or ' self-correlations' are unknown, we have
only \n(n— i) independent values in the correlation matrix. We have,
therefore either (i) to make some convenient but arbitrary assumption about
the nature of these variances, e.g. that they are all equal and standardized at
unity: in that case we should obtain n factors—most of which could have no
real'statistical significance and would be due solely to the additions made to
the true variances to bring them all up to unity. Or, alternatively, we can
(ii) follow the procedure described above, and insert an estimated * self-
correlation ' to fit the requirements of each successive hierarchy solely. But
now, if there are more than one factor, the total variance (i.e. the self-
correlation used in calculating the first factor) should be at least equal to the
sum of the self-correlations for all the factors, and generally the variances
required for calculating for the ^>th factor should each be the sum of con-
tributions of the (r — p + i) factors that remain to be calculated. This
involves readjustments in the estimated self-correlations, reduces the number
of ascertainable factors, and leads to a lengthy process of successive
approximation. It yields, in the end, not n factors, but the minimum
number required to account for the intercorrelations as given, or, in technical
language, a factorial matrix of minimum rank. Actually, however, the figures
given for the intercorrelations are themselves approximate abridgments for
irrational fractions. Hence, with large correlation tables, there is little
point in seeking a factorial matrix having exactly the minimum rank de-
ducible from the figures given; and it is even arguable that some small
allowance should be included for variance due to the * specific factor? or rather
to that part of the test that does not overlap with any other test in the table.
In these circumstances the working procedure I have suggested assumes
that the variances for the different processes tested will differ in the main
according to their complexity, and that (in default of other evidence) this
can be estimated from the maximum covariance for which each could be
responsible (see p. 286). The inserted figures will therefore be (i) not less than
the total variances that would be obtained if the successive approximations
were carried through to the end, but (ii) never so large as to make the total
variances exactly or even approximately equal.1

By way of illustration let us now take actual figures ; I choose a
set of correlations printed and analysed in my 1917 Report.* For
simplicity I shall here confine the analysis to six tests only, namely,
Composition, Reading, Spelling, Handwork, Writing, and Drawing,
and seek figures correct to two decimal places. A more detailed
analysis by the summation method, and a full discussion of the
inferences to be drawn, may be found in the Report itself.

1 The insertion of the reliability coefficients usually makes the proportionate
variances approximately equal, and tends unduly to diminish the variances of the
more complex tests, since the simplest tests usually have the highest reliability,

1 LQC. cit., Table XVIII, p. 52. I am indebted to Miss G. Bruce for checking the
calculations : a few of the figures originally printed have required correction, par-
ticularly those for Writing, where two tests have here been amalgamated into ome.