HIERARCHICAL CRITERIA 347

(or partial1 correction) of rf so that it ranges from o to i.
The analogy with the analysis of variance is clear, but for
obvious reasons far from perfect.2 Nevertheless, if, after
finding the factor-variances, we base the calculation of vj2
on one or other of the trace-ratios enumerated above, we
shall find that several instructive results may be reached.

Economy Ratio. — First, however, let us glance at the most
simple substitution of all. If with Thurstone we could assume that
the complete correlation matrix has an exact, assignable rank r
(where r^n)9 we could take the total variance to be analysed,

namely ay5, as approximately equal to r<rw2, and write v}3 — - .

The above formula for p would then reduce to an easily intelligible
expression, depending essentially on the ratio of the number of tests
(n) to the number of common factors (r), or, in more technical
language, on the ratio of the order of the correlation matrix to

I / n \
its rank. We should have p = -— — ( - — i j = E (say), an

expression which I have called the * economy ratio.' When r = n,
E = o ; and the factorization leads to no economy in the number of
variables. When r = i, E = I ; the economy is perfect, and the
matrix hierarchical. And between these limits E might be regarded
as measuring the simplicity of the correlational pattern, and
therefore of the group of processes tested.

With an empirical correlation or covariance matrix, however, to

1 Strictly rf = , _ ^ _ ,. Thus, if the correlation in the

_ j

* infinite population * is zero, if tends to the value -— r= - = - when

n Jy — I n

the sample is large enough for us to write N for (N — i) after the fashion
of the older formulae. On the need for correcting if see Kelley [47], p. 240
and references.

2 In particular, here and subsequently, the usual indications for the degrees
of freedom will not hold. For more precise work, however, an entirely
different line of attack would seem desirable : e.g. it would be better to
deduce the expression for the total variance either from the distribution
of the original scores or from independent data showing the amount of
unreliability included in those scores. This, however, leads to complicated
expressions and laborious calculations, such as the inexact data of the psycho-
logist would hardly warrant.