DERIVATION OF CHIEF THEORIES 163

a ' multiple group-factor theor7'; for what are c multiple'
are the factors themselves, not the kinds of factors.

the ever-increasing growth of factor theories, much ambiguity is bound to
arise if each is merely labelled according to the number of factors (or kind of
factors) it postulates ; and accordingly it would be better to adopt more
informative titles. To American workers, who have entered the field of
factor-analysis comparatively late, the names often convey quite different
associations from those they possess for older writers in this country ; and to
younger students the nomenclature must be highly confusing. For this
reason, too, it seems wiser to rechristen what I formerly called the * Multiple-
factor Theorem ' (regarded as a supplement to the * Single-factor Theorem ')
the f Theorem of Added (or Superposed) Hierarchies} (see p. 164).

A word is needed on the relation between the multiple-factor hypothesis
and what was called above the dual-factor theory. With psychological data,
though analysis nearly always reveals a multiplicity of factors, it also produces
(with certain important exceptions) a well-marked duality of kinds. What-
ever mode of calculation is employed, the most striking distinction is in the
contrast between the first or dominant factor, with all its saturations positive
throughout, and the subsequent secondary or supplementary factors—
whether group-, bipolar, or specific—which show saturations that are partly
positive, and partly negative or zero. Essentially, the first or positive factor
represents an average; the other factors, deviations about an average.

This distinction, as we shall see, becomes of special importance when we
come to discuss correlations between persons. Yet from one point of view it
is artificial rather than real. In certain cases, we shall discover, the positive
factor may disappear ; and, if we are familiar only with tables of correla-
tions between traits, we shall be tempted to suppose that its disappearance is
exceptional. Actually, I believe, its presence is exceptional—due to the fact
that our collection of variables is exceptional. If there are no negative or zero
saturations in the * universal' or first * general' factor, that simply results
from the fact that, in selecting a group of traits or a group of individuals to
form our initial ' universe * or genus, we have excluded all instances that do
not belong to that genus. Supposing that we had taken a random, un-
selected group : then those that do not belong to the genus would have been
represented as well as those that do; and our general factor would have been
turned into a group- or a bipolar factor. If, for example, in testing cognitive
ability, we include, not only measures of intelligence, but also measures of
stupidity, then Spearman's g would show negative saturations for the latter.
Or again, if in correlating persons we include animals as well as men, then the
factor of general humanity would appear to be bipolar: it would define
humanity by stating what was not human, as well as what was. In practical
work, of course, we are obliged to start with somt more or less well-defined
group. But in theoretical work this limitation sometimes introduces a need-
less complication; and often, we shall find, it is more helpful to regard the
purely positive correlation matrix, so constantly found with psychological
data, as in theory a positive north-west quadrant cut out from a larger
bipolar matrix (cf. [93], p. 287).