HIERARCHICAL CRITERIA 353

Here \/vs and \/vs + i are standard deviations about the principal
axes of the concentric frequency ellipses, and are consequently
proportional to the lengths of those axes. They thus measure the
tendency of the ellipses to broaden into circles or to c condense ? into
a single straight line.1 The above expression may accordingly be
described as the c condensation ratio ' for two factors, p can be
treated as a correlation between two variables in a population-
sample of N individuals ; and the usual tests of significance applied.
This method of determining whether one factor-variance is signifi-
cantly greater than another has, in fact, been proposed by Hotelling
([79], p. 434). It will serve to show when we have reached a pair of
residual factors whose difference is so small that it cannot safely be
attributed to anything but chance : it will not serve to show that
we have obtained a difference so large that the larger of the two
factors is definitely significant.

Systematically and completely carried out, however, this principle
would lead us to examine, not merely the (n — i) differences be-
tween successive factor-variances taken in pairs, but the \ n (n — i)
differences between all possible pairs. But that is precisely the
type of situation which the analysis of variance has been devised to
meet ; and once again we may approach it from the standpoint of
an intra-class correlation. Except for the final division by the
number of items added, the factor-variances represent the means
of the squares of their saturation coefficients ; and, just as we have
summarized all the differences between the factor-saturations by
their standard deviation (or its square), so we can summarize all the
differences between the factor-variances by their standard deviation
(or its square). Following the same lines as before, we shall be led
to a criterion based upon the unadjusted variance of the factor-

variances themselves, viz. - ZVA The observed value of this

n J

expression we can compare with its maximum value ; and we obtain
another correlation ratio v)22 = ._, - = f gA the second of the

three trace-ratios described above. If we compare this formula
with the equality reached for * principal tetrad-difference criterion *
(p. 335), we may describe the result as converting the tetrad-
difference criterion into a tŁtiad-ratio criterion.

If, as before, we put Zvf = Hvt = #, we can give this ratio a

--

different interpretation. We have ,vu2 = /_ ^N = — ~. Thus,
r A * a *

1 Cf. Marks of Examiners, p. 255, Yule and Kendall, p. 232, or figure
21 in Brown and Thomson's Essentials of Mental Measurement, p. 122.

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