HIERARCHICAL CRITERIA	355
perfectly hierarchical or a perfectly random arrangement, all the
intercolumnar correlations should become numerically equal. By
its mode of calculation each correlation is the ratio between a
product-sum and the geometrical mean of two square-sums. But if
all these ratios are numerically equal, their squares will be equal ;
and, in order to find their average, instead of calculating the sum of
the squared ratios, we can first square the numerators and denomina-
tors and then take the ratio of the sums. We thus obtain :
where Vf and v_h as before, denote the factor-variances and the test-
variances.^1
This expression is obviously related to the last of our trace-ratios ;
and, if the preliminary calculations required by the trace-ratio have
already been carried out, that formula may be used instead. With
either version, however, the arithmetic is somewhat laborious ; and
from the few applications that have been made I am inclined to
endorse Mr. Barlow's verdict : " the fourth (moment) criterion is of
theoretical interest, but on the whole too elaborate and delicate for
ordinary practical use." It would seem to be most useful when the
first two latent roots of the initial correlation matrix are nearly
equal.
The hierarchical criteria that I have attempted to deduce are
descriptive in the first instance of the correlation matrix itself.
Calculated as trace-ratios, they are independent of any form of
analysis. In expressing them in terms of the factor-variances,
however, I have assumed that the factors in question would be those
obtained by weighted summation (or some closely equivalent method),
i.e. that their variances would be identical with the latent roots
of the correlation matrix, or at any rate with a set of figures giving a
close approximation to the dominant roots. Thus converted they
become descriptive of the derived factorial matrix rather than of the
initial correlation matrix. It is but a natural extension of the under-
lying principle to assume that, when computed as ratios of the
variances or standard deviations of the factor-variances, the same
formulae will supply comparable criteria for the factorial pattern
^1 Students who have used the same summation device to calculate
Spearman saturation coefficients will be aware that, with the appreciable
deviations that occur in non-hierarchical tables, the result of the abridged
method of averaging often departs widely from the average obtained in the
regular fashion. That occurs here : but my object is, not to deduce the
criterion from the intercolumnar correlation, but rather to show the relation
between the two principles.