SIMPLE   AND WEIGHTED  SUMMATION METHODS   327
of all. It will be seen that the latter, too, form only an
imperfect approximation ; they are certainly much closer
than those which are got by Thurstone's method (as cal-
culated by Kelley), but tend towards over-compensation.
Whether our data permit us to discover one significant factor only
(r = i, as in Spearman's deduction of g) or whether we decide to
seek for a larger number (r > i), the method of analysis which I
have here described will always yield the hypothetical correlation
matrix of rank f which best Jits the actual correlation matrix (total or
residual) empirically given. Thus, the essential feature of the
6 simple summation' method as compared with that of the * least
squares ' is that the former is content with a matrix of rank/that
yields a slightly poorer fit—in fact, as we have seen, with the first
set of values, instead of the last, emerging in one and the same
sequence of successive approximations. Moreover, with the least-
squares method the factors are fully independent in that, not the
factor-measurements only> but also the factor-loadings are uncorrelated
(Thurstone's factor-loadings nearly always show a low correlation).
Hence, where further transformations are required, the algebra and
arithmetic become much simpler. Finally, if we construct an arti-
ficial correlation table from a given set of saturation coefficients
obeying these conditions, and then factorize it, we shall with this
method get back to the original saturation coefficients : this result
is not obtainable with Thurstone's procedure or Spearman's, nor
(to judge by his own example) with Kelley's.
Two defects are often attributed to the method of least squares.
First, it is said that, unlike the factors reached by simple summation,
those deduced by the method of least squares cannot be rotated :
for, if they are rotated, they lose their essential quality of being
* principal components,' i.e. of being principal axes of the frequency
ellipsoids. But the latter property as an end in itself is useful only
in special cases (e.g. empirical prediction). And, as must now
be evident, since one set of factors is merely an approximation to
the other, it must be just as legitimate to rotate the one as the other.
What is more, with the least-squares factors, owing to the non-
correlation of the factor-saturations, such rotations can be much
more easily performed. Secondly, the followers of the two-factor
theory have objected that with the least-squares method the factors
must change whenever a new test is added to the battery. Actually
this is also true with a two-factor analysis ; but when the original
battery involves only one significant common factor, it is easier to
select the additional test so that the change is negligible. The