344 THE FACTORS OF THE MIND

With, a covariance matrix, or with a correlation matrix whose
self-correlations are known, we have merely to calculate and
standardize the product-sums for the rows or columns taken in pairs.
To reduce the \n (n — i) intercolumnar correlations to a single
figure, their average has commonly been taken. Let us glance first,
therefore, at the criterion which this direct calculation appears to
supply.

If in factorizing the original table of observed correlations we
follow the method of weighted summation, there is a very simple
relation between the factor variances of the correlation matrix and
its square or higher powers : we have, in fact (with the usual
notation), Rm = LFmL'. With a perfect hierarchy this means that
the (unadjusted) product-sum on which the intercolumnar correlation
for any two tests is based is simply vl times the observed correlation
between those tests. Thus, on summing all the product-sums to
obtain an average, we have ZZfij = Zr^ (Zr^ = v^SSr^.
But now, it may be asked, why correlate a second time ? Why not
be content with ' first moments * ? For, on dividing both sides by
vv we apparently obtain a very simple test which has indeed been
actually employed,1 viz. ZZr^ = (Zrig) 2: i.e. the sum of the
correlations should be equal to the square of the sum of saturations
for the one and only ' general' factor. This requirement, it will be
noted, is very similar to that which we reached on the basis of the
tetrad-difference equation, except that the correlations on the one
side of the equation, and the saturation coefficients on the other
side, are not squared before they are summed.

To this proposal, however, there are two obvious objections.
First, with a bipolar hierarchy, such as that which might be found
in a table of residuals obtained by the centroid method, both
ZSfij and Łns = o ; hence, if we attempt to sum the residuals (as
Thurstone does for his ' first moments' criterion) we are bound to
adopt some arbitrary convention for changing negative coefficients
to positive. This difficulty, it will be observed, is obviated by the
squaring just referred to. Secondly, with the centroid method of
calculating ris, the equation is true of all tables ; this particular
criterion therefore turns on the identity of Tig as determined by the
methods of simple summation and weighted summation respectively.
But the simplicity of the reduction thus obtained with the method
of weighted summation prompts a practical suggestion : if after all
only the sum of the intercolumnar correlations is required, is there
not a much speedier way of reaching, or at least of estimating, the
final figure ?

1 By J. E. Watson, in an unpublished thesis. , : -