286 THE FACTORS OF THE MIND

that the study of what (in the analysis of variance) is called * inter-
action * would indicate how mental processes modify one another
when they are combined. Moreover, almost from the start, increas-
ing evidence for supplementary factors of a more specific kind in-
dicated that a slightly better estimate for the test-variance would
be obtained by adding a small and varying amount for so-called
group or specific factors. This means that, in factorizing a covari-
ance or correlation table, we ought rather to identify the test-
variance with what I have called the ' complete communality,'
i.e. the sum of the squares of the saturations for all factors. If
we increase the number of our tests indefinitely, without appre-
ciably increasing the number of factors, this figure is identical
with the limit of the multiple covariance (i.e. of the square of the
multiple correlation of the given test with all the rest). However,
as we shall see later on, except with small tables, the contribution
to the variance made by non-general factors is relatively slight.

Of the better-known methods of factor-analysis, Kelley's
[85] is the only one that is expressed primarily in terms of
covariances.1 At the same time, however, he appears to
assume that the factor loadings or saturation coefficients so
derived will be precisely the same as those that would be
obtained from correlations. This assumption, as I have
endeavoured to prove, is not strictly true ([102], p, 193).

The relations between the results of the two procedures may be
indicated briefly as follows. As we have observed, in its initial
steps the problem of factor-analysis can be most generally conceived
as a problem of linear dependence or rank, rather than of statistical
dependence or correlation, which is only a special case of the former
([101], p. 69) ; the principle underlying the correlationist's procedure

i

1 Hotelling refers to the " replacement of correlations by covariances,"
but decides " not to treat of this obvious generalization except to discuss a
suitable criterion for weighting " ([79], p. 422). The special assumptions
that he males for the purpose of this criterion would, as he points out,
define * a natural unit of measure for each variate * (p. 510). Though ex-
pressed in different terms, the equation that he then reaches for the test-
variances (eq. 51) would seem to be identical with my own (eq. 12, [101],
p. 75). If so, my * natural units' would be the same as his. The * metric J
adopted in the hody of his paper, however, is " based on the assumption that
the essential quantity to be analysed is the unweighted sum of the variances,
where the total variance of each test is taken as unity," which would be
(as he points out) incompatible with his equation 51 (p. 421).