448 APPENDIX I

would now be called) for indications of further " subordinate
factors." All through, however, the common underlying assump-
tion is that any empirical table of correlations or covariances can
be treated as the sum of a rapidly diminishing progression of
hierarchies, each attributable to an independent factor.1 At par-
ticular points and for particular methods certain supplementary
assumptions have to be made, for which the chief justification is
practical convenience. Where these diverge from the practice of
other writers, or where they have been questioned or criticized, I
shall take occasion to interpolate a note in their defence,

To determine the saturation coefficients ra(J, the easiest formula
for general use is that first given and illustrated in my 1917 Report
on the Distribution and Relations of Educational Abilities* I have

1 Cf. page 164.

» L.cfc. Report, No. 1868 (P. S. King & Son, is. 6&). Tables XVIII-XXIV. It
was more fully described in my 1915 paper on ' General and Specific Factors underly-
ing the Primary Emotions' (briefly reported in [30]). Thurstone, who has since
adopted the same formula, prefers to call it the ' centroid method ' (see [84], 1935,
chap, iii, eq. (13), p. 94) ' his working procedure, however, differs from mine (a) in
the mode of estimating the missing diagonal values ; (6) in the treatment of negative
residuals ; (c) in insisting on a subsequent rotation of the factorial axes. In discussing
my introduction of the essential formula, Thomson adds : " it is not quite clear how
he (Burl) would have filled in the blank diagonal cells " ([132],, 1939, p. 25). How
the missing value is filled in at the outset does not greatly matter, provided the figure
is checked and corrected by the results thus provisionally obtained. With a nearly
hierarchical table, the empty cells would have been filled by applying the proportion-
ality formula given in my earlier article (cf. below, p. 450, footnote i). With
more irregular tables a direct calculation is no longer feasible ; but, as I have else-
where indicated, " the difficulty can be easily overcome by successive approximation :
in the spaces for the self -correlations, trial values can be inserted by smoothing
the several columns ; these are then checked by computing fu =» »V, when tkg has
been found by equation xxy " (i.e. by the summation formula, [9^], p. 285 : cf.
' Methods of Factor Analysis with and without Successive Approximation/ [102],
p. 178). Where a multiplicity of factors is assumed, the check of course requires us
to calculate r jut = 2 n^4, where g now denotes all the general and group-factors entering
into test k : in either case the figure required is really the square of the multiple
correlation of each test with its essential common factors, i.e. with the infinity of
tests involving those same factors (p. 286). By * smoothing' I meant the rough
process that teachers and examiners so often adopt, and sometimes call by that name,
when, for example, they wish to estimate an average or total for a boy who has
been absent for one of a series of examination papers and so give him a rough
allowance—an allowance which could no doubt be calculated more exactly, were
such exactitude warranted (e.g. by a proportion based on the totals for all candidates
except that boy and all papers except the paper missed, as in fitting a contingency
table, p. 147) ; fortunately, as I pointed out, the figure suggested by the general trend
of the correlation pattern is, as a rule, sufficiently near the mark to render much
recalculation needless, unless a high degree of accuracy is required or the number of
correlations to be summed or averaged is very small. For these latter cases I
suggested a simple ' product formula ' for estimating ?%* directly from the inter-
correlations alone, viz. :

« -!)(«- 2)

As is demonstrated in my fuller notes, in theory, (i) the process of successive
approximation, if carried out mechanically and completely, with a matrix of rational
coefficients, progressively reduces the reinserted diagonal elements (i.e. the values
for the variances or communalities) to a set whose sum is the smallest possible that is
compatible with the assumed positive-definite character of the completed correlation