CHAPTER XIV

TESTS OF SIGNIFICANCE AND OF HIERARCHICAL
TENDENCY

BEFORE accepting a set of mathematical factors we require
to establish the statistical significance of the variances and
saturation coefficients that specify them. This involves a
number of somewhat neglected and difficult questions.
To attempt a formal and technical inquiry into the whole
problem would be out of place in a discussion intended for
the general student. I shall content myself with comparing
some of the more obvious and more familiar proposals, and
shall endeavour to indicate in the broadest way how they
are related to each other and to the methods adopted in
general statistics.1 For the most part I shall assume that
the factors under discussion are the factors immediately
resulting from an analysis by weighted summation—i.e.
are the factors specified by the latent roots and vectors of
the correlation matrix: for with this method the factors
and their saturations are independent of each other, and
so lend themselves to the simplest form of demonstration.
But, as we have already seen, many writers prefer to con-
vert these factors forthwith into another set having differ-
ent values and possessing, as they believe, a psychological
meaning which the first set cannot claim. First of all,
therefore, it may be well to consider what general changes
such a conversion is likely to entail.

1 My review does not profess to be complete. Kelley's interesting dis-
cussion of the problem ([85], pp. 10-17) relates primarily to the steps in his
own rotation-method of factorization, but could be brought into line with
what follows. Similarly, Hotelling ([79], p. 437) gives a method for testing
the * reality' of his components (which are obtained from corrected correla-
tions) based on the reliability coefficients: but here I have confined myself
throughout to correlations obtained from a single application of the tests.
Thurstone also suggests 4 or 5 supplementary criteria ([122], pp. 65-6) in
addition to the two I have discussed below.

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