HIERARCHICAL CRITERIA 351

evaluate the significance of the differences between these saturations
along the lines followed in analysing variance. If we start with a
table of residuals or with correlations obtained with a homogeneous
population, the first factor will be bipolar, the means of the satura-
tions will be approximately zero, and the adjusted variances
approximately equal to the unadjusted. If we start with a table
of initial correlations that are all positive, we can calculate the
adjusted variance of the first factor-saturations in the usual way, or
assume that the initial table is really part of a symmetrical bipolar
table. For brevity I shall here assume that bipolar conditions may
be presupposed throughout. We can then take the variance of

saturations as proportional to E ft\ = v± ; and the total variance

i

as proportional to EE frf = «rx + v2 + ... + vn. This latter
j i

expression also indicates the maximum value that v± can take ; it
is at the same time identical with the sum of the variances of all the
tests — the total variance in fact which our factorization analyses.
Let us therefore put

trR

where vr denotes the residual variance. We thus reach a more
concrete interpretation of the first of the three ' trace-ratios 7
suggested above. If we regard v^ as an ordinary correlation ratio,
we may test its significance by applying one of the simpler standard
formulae.1

We may note that, with this interpretation of 7]2, (a) for a perfectly
hierarchical distribution of the correlations, vl = Evf, Tjj2 = I, and
E = I as before ; (b) for a perfectly random distribution of the

correlations, v^ = v2 = ... = #w, 7]^ = -, and E = o as before.

With intermediate cases, however, we have no data for directly
estimating the total variance, Zvf. If, as is here assumed, we are
adopting a summation method with minimal rank, the figure for
Evf is merely a theoretical lower limit : the upper limit is n.*

1 Yule and McKendall, he. cit., pp. 409, 453-4. Fisher, loc. cit.y p. 245.
But see the comments in the following paragraph, which show that this is
only a rough and practical test.

2 The alternatives are similar to those described by Thurstone, when he
distinguishes between working with a * reduced correlation matrix * RQ
(in which the * communalities ' are substituted for the f complete test-
variance ') instead of with the i complete correlation matrix ' R± (in which
the * diagonal entries are unity ') ([15], p. 66). Where it is necessary
to distinguish between the alternative formulae, I shall affix the same sub-