HIERARCHICAL CRITERIA 337

variances and the n factor-variances respectively) and
J£V/2, the hierarchical requirement is equivalent to demonstrating
V ' Evf* = £vf, which is obviously satisfied if all the factor-vari-
ances except the first, i.e. z>2, v^ . . ., ?WJ = o. The expression
itself suggests the possibility of a simple criterion in terms of the
standard, deviation of the factor-variances.

The True Hierarchical Issue. — But the whole issue, as it
seems to me, should now be formulated rather differently.
The parties to these controversies often write as though
there were only two extreme alternatives : either to explain
all the variation by the smallest conceivable number of factors
(namely, one, in Spearman's case) or to explain it by the
greatest conceivable number (which they take to be n, the
number of correlated tests). Rather, it would seem, the
alternatives lie between (i) seeking a factorial matrix that
will exhibit the variance as mainly concentrated in a few
dominant factors of varying importance and (ii) accepting a
factorial matrix that will exhibit the variance as distributed
almost equally among a large and indefinite number. And
so far as concerns the first or dominant factor — * the general
factor,' as it is commonly called — our task is not to demon-
strate that it will account for everything, but simply to
discover how much it will account for.

Save for exceptional cases in which the tests have been
specially selected, there must always be more non-specific
factors than one. Consequently, unless the sample tested
is so small that the issue cannot really be decided, the
answer to the tetrad-difference criterion must be the same in
every case : namely, no empirical correlation table forms a
perfect hierarchy. Thus, the crucial issue has been wrongly
conceived. The question to ask is not, is this empirical table
a hierarchy or is it not ? but, how strong is its tendency
towards the hierarchical pattern ? And in theory the same
inquiry should be made, not only about the initial matrix,
but about each of the residual matrices (so long as they
are significant) in turn. The most obvious mode of ap-
proach therefore will be this : having extracted a complete
series of factors in order of their maximal contribution to
the total variance, to inquire : first, how many of these factors

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