WORKING METHODS FOR COMPUTERS 459

(+ 3;°4 -T- 3'Bo = + -80 ; . . .; - 3.42 ~ 3*80 = — -90). The
quotients should be the saturation coefficients required.

With this and other bipolar tables it is instructive to continue
with the full method of weighted summation, using the saturation
coefficients as weights, and repeating the process by successive
approximation as described above (section B, pp. 451-3). In this
case, as in the preceding example, it will be found that the saturation
coefficients remain unchanged.

II. MULTIPLE-FACTOR ANALYSIS

Where a given table of correlations appears to be produced by
more than one factor, we have to split the matrix of observed
coefficients into the sum of two or more simple c hierarchies 9
(matrices of rank one). Each of these component hierarchies may
cover either (i) the entire series of tests (in which case the factor is
described as a * general factor '), or (2) a limited group of tests only
(in which case it is called a f group-factor ').

(i) GENERAL-FACTOR METHODS

The former procedure consists essentially in applying (i) the
ordinary single-factor analysis to the correlations actually observed
in order to determine the first factor, and then (ii) the modified
bipolar analysis to the successive tables of residual correlations in
order to determine the remaining factors.

The hypothesis of summed hierarchies now implies that the
variance for each test (its c self-correlation/ as it is sometimes
inaccurately termed) should be the sum of the squares of its factor-
saturations, just as the intercorrelation between any two tests is the
sum of the products of their factor-saturations.

With a table of covariances the calculations are perfectly straightforward.
If n is the number of tests and r the number of factors, we have (in general)
<|n (n + i) independent values in the covariance matrix, and nr — \r (r — i)
independent parameters to calculate in the factorial matrix.1 Hence, if the
variances are known, there will be r = n factors, of which, of course, only the

1 As will be seen from the procedure described below, there are n degrees of freedom
for the first factor; only (n — i) for the second (since the totals must be zero);
only (n — 2) for the third ; and only (n — p + i) for the pth factor, since in general
the totals must be zero, not only as they stand (i.e. when weighted by the signs of the
first factor which are all positive), but also when weighted by the signs of any of the
(P *» i) preceding factors. This latter point can easily be proved algebraically (see
Notes), and may be verified from the tables of saturations given below: it will also
be found to hold good approximately for the tables analysed by Thurstone's centroid
method (e.g. the five columns of saturations in his table 25, Vectors of the Mind,.
p» 1x7; the slight discrepancies are due to the substitution of fresh values for the
diagonal residuals in the case of the later factors).