476 APPENDIX I

example, we might have started with the method of weighted summation,
without any preliminary determination of the self-correlations by simple
summation. We could then have begun by seeking the best-fitting factorial
matrix of rank one ; then of rank two ; and so on : i.e. the first fit would
contain in each diagonal cell the square of one saturation only ; the second
fit would contain the sum of the squares of two ; and so on. This would
again lead to a factorial matrix of minimum rank, with every column un-
correlated with every other. Such an approach might be of theoretical
interest, but is hardly a practical procedure, since, as a rule, it leads to pre-
cisely the same figures by what in the end is not a shorter but a more cumber-
some procedure.

Again, if we assume that the results of correlating persons and correlating
tests should be equivalent (as they should under certain conditions, e.g.
when the selection of both persons and tests is * random '), we should expect
each bipolar column of saturation coefficients to add up to zero both when un-
weighted and when weighted by one of the other columns. A process of
successive approximation with weighted and unweighted summation used
alternatively will usually yield a close approach to this result. But whatever
self-correlations we assume, and whatever rank we accept, there is an
obvious gain in having a factorial matrix whose columns are uncorrelated :
any further linear transformations into which the correlation matrix enters—
e.g. rotating axes, calculating regressions, comparing results with those of
correlating persons—are greatly simplified.

If, however, the self-correlations or variances are known from the outset,
then, as we have seen, successive approximation may be entirely avoided by
* table-by-table' multiplication—a procedure that is sometimes convenient
with a very small table of covariances (for illustrations, see [102], p. 185).
Or again a triangular matrix of positive saturations may be obtained by the
earlier procedure due to Lagrange (described, e.g., in B6cher's Higher Algebra^
1907, p. 131), which will be found to fit certain correlation problems quite
well.

Partial Correlations.—It should be noted that the residual correlations
obtained as above by simple subtraction do not give the correlations that
would be found in a sample population selected so as to be homogeneous for
the factors subtracted. For example, we may desire to know what degree
of correlation would exist between Reading, say, and Spelling (or Arithmetic)
in a class or form where the children had been so selected as to be upon
practically the same level of general ability. For this purpose the full
formula for partial correlation must be employed. This is equivalent to
dividing the residual correlation by the mean residual variance of the two
correlated tests1—a process which greatly enlarges the figures. However, in

1 [93], !>• 306, and L.C.C. Report [35], p. 57- In my earliest article [16] this
further adjustment was not applied, because the problem was to determine the re*
lative importance of the first or general factor and the remainder. And in most forms
of multiple-factor analysis it is, as a rule, omitted. That, however, should not obscure
the fact that multiple-factor analysis (and, indeed, single-factor analysis according to
Spearman's approach) is essentially a development of partial correlation : cf. Am. /*
PsychoL,XV, 1904, p. 256 (where Yule's formula for partial correlation is cited), and
Yule's determmantal solution for the partial variances ([no], pp. 267 et seq ).