THE LOGICAL STATUS OF MENTAL FACTORS 107

same objection has been advanced against my own methods of
factor-analysis by Prof. Reyburn and Dr. J. G. Taylor.1 The
criticism overlooks the fact that, even when the factors do not differ
from the traits in number, nevertheless they differ from them
greatly in their mutual relations and in their comprehensiveness :
first of all, whereas the traits are mutually correlated, the factors
are (or should be) mutually independent; secondly, whereas each
trait accounts for only a small fraction of the total variance, and
that much the same fraction, the factors are so determined that the
first or £ highest common factor '2 shall account for as much of the

factors in the absolute sense may, like the error-factors, be conveniently
excluded from the final factorial classification (though my reasons are not
the same as Thurstone's). But Thurstone's postulate, while treating specific
factors as negligible, makes them at the same time responsible for the greatest
amount of variance that they can possibly contain.

Where we are dealing with a table of correlations, as distinct from a table
of covariances, my * summation ' method (simple or weighted), if carried
out automatically, does, as a matter of fact, yield factors whose number is
identical with the number specifying the lowest rank obtainable for the
table of intercorrelations. But that is not because I hold that a postulate
of economy requires the smallest number of factors, but because that
procedure happens to give definite and plausible figures for the variances.
In point of fact, Thurstone's own methods would seem to involve the
determination of far more common factors than I should ever venture to
extract. Thus, in his illustrative analysis (Primary Mental Abilities) he
extracts as many as 12 factors, in spite of the high probable errors, where
I should have thought barely one-quarter of that number were statistically
significant.

On page 92 [84] he declares himself ready to accept any method of factor-
izing the matrix of intercorrelations " provided the minimum rank of R is
not altered " ; at first sight this seems to mean that with 15 tests he would
extract 10 factors (cf. Table 2, p. 77). Yet, although here and elsewhere
he insists so strongly on the principle of minimal rank, he does not
show how his own method will secure precisely this number of factors;
and his working procedure, where " the diagonal entry may be given any
value between zero and unity " (p. 108) and the computer is advised in
practice to give it the value of the highest correlation, seems a double
contradiction of the demand for minimum rank, since this is equivalent to
demanding a minimum sum for the communalities. Indeed, the device of
fitting the leading diagonal with the largest correlation from each column
must almost inevitably increase the number of factors and raise it artificially
well above the number indicated by the minimal rank.

1 ( Factorial Analysis and School Subjects: A Criticism,' by H. A. Rey-
burn and J, G. Taylor, Brit, J. Educ. Psychol (in the press).

2 The phrase ' highest common factor ' was used in my earlier articles (e.g.
Brit. J. Psycb., Ill, 1909, p, 166) to denote the factor which would account