CHAPTER XIII

SIMPLE SUMMATION METHODS AND WEIGHTED
SUMMATION METHODS

(a) Differences in Procedure.—General-factor methods
may all be reduced to two main groups, according as they
rely on c first moments' or ' higher moments ' in their
attempts to fit a theoretical hierarchy to the empirical
table. For purposes of practical calculation, this means
that the less elaborate methods are content to use a simple
summation of the correlations, while the more ambitious
use a weighted summation. Spearman's well-known formula
for saturation coefficients, Thurstone's centroid formula,
and my own earlier summation formula, all treat the
saturation coefficients as proportional to the unweighted
sum or average of the corresponding columns of the corre-
lation table. On the other hand, Hotelling's method of

* principal components,' Kelley's c trigonometrical method,'
and the c method of least squares' in its various forms, are
tantamount to requiring each row of the correlation table
to be appropriately weighted before the columns are
summed, the weights in every case being proportional to the
saturation coefficients themselves.

Let us first consider what this last requirement implies (i) from a
theoretical standpoint and (ii) from a practical standpoint.

(i) If we express the essential requirement algebraically > we are at
once led to the fundamental equation which we have already
encountered, namely, R = LVU or RL = LV ; and this, it will be
remembered, is the equation for the * latent roots' (7} and the

* latent vectors* (i) of the correlation matrix R. Thus, if we put F'
(the matrix of saturation coefficients) = P*L'9 and use its elements
as weights, the required condition is immediately fulfilled : for, on
multiplying the rows of correlations by the corresponding satura-
tions, we have F'R = 7*Lr .LFL' = 7. 7*L' = W. If, on
the other hand, we conceive the problem geometrically, we may

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