HIERARCHICAL CRITERIA 343

either reverse the signs of half the columns in the table before
correlating or (better still) square the correlations after correlating.
By an * unadjusted correlation' I mean a product-moment
coefficient based on the absolute deviations, instead of on the devia-
tions about the means.1 Spearman's form of the criterion uses
ordinary or ' adjusted' correlations.2 He himself has drawn
attention to two defects attending its use.3 But in the simpler
form in which I have proposed it the criterion would seem to escape
both of these objections. In one guise or another, the inter-
columnar correlation has been freely employed in the past ; and the
principle involved has consequently become familiar to students
who are unacquainted with the conception of matrix rank. Hence
it will perhaps make the simplest starting-point for our discussion.4

gives the correct statement : " if the correlational matrix is of rank one then
the correlation between any pair of columns is + I or — i." ([84], p. 135 :
the converse is not necessarily true unless the unadjusted correlation is used.

1 [102], p. 179 f. Note that (as was there pointed out) "adjusting the
absolute product-moment so as to obtain a product-moment about the
mean " (though nearly always carried out by those who have used the
c intercolumnar correlation * as a criterion) " spoils the test of proportion-
ality," if by a hierarchy we are to understand a matrix of rank one. Thus,
Dr. Carey's example of a perfect hierarchy (loc. cit. sup., p. 2), where the rows
of correlations form an arithmetical progression, would not be a perfect
hierarchy by my criterion, though it would be a perfect hierarchy if the
ordinary adjusted correlation between columns was employed.

To bring out the analogy with more familiar devices and also to avoid
confusing the beginner by introducing more precise technical terms, I shall
continue to call the standardized absolute product-moment a ' correlation,'
just as I sometimes speak of the absolute root-mean-square as an* (unadjusted)
standard deviation.5 With a bipolar table, such as a table of residuals
obtained with the simple summation method, the mean of each row or
column is zero : hence the unadjusted and adjusted correlations and standard
deviations are identical. As I have pointed out elsewhere, there are grounds
for regarding all correlation tables obtained in psychology as essentially
bipolar tables, the ordinary initial table of positive coefficients constituting
simply the north-west quarter of a doubly symmetrical bipolar table.

2 It was first systematically employed by Spearman and Hart in their joint
paper on * General Ability, its Existence and Nature * [24].

3 Abilities of Man, p. ix.

4 I am much indebted to T. L. Barlow for making a comparative study
of the following formulae (and several others) at my suggestion. In place of
my own somewhat elaborate algebraic deduction, based on the f canonical
expansion of the correlation matrix,' I have mainly followed his simplified
method of exposition, as being more easily intelligible to the ordinary student.
Space compels me to omit the concrete arithmetical examples, which
illustrated and gave point to his analogies.