466 APPENDIX I

second-factor hierarchy, subtracting it from the residuals left by the
first factor (with their original signs), testing the significance of the
new residuals, rearranging them in bipolar order so as to obtain an
approximate hierarchy of positive and negative quarters, estimating
figures for the diagonal, adding the columns, and dividing by the
square root of the grand total—as just described. In theory this
procedure should be continued with one set of residuals after
another, until the final residuals are virtually zero for the number of
decimal places to which we are working : in practice, it is un-
necessary to calculate detailed figures for non-significant residuals,
i.e. as a rule, for more than three or four factors.

19. As a final check, square all the saturation coefficients and
find the sums for each test, if necessary adding in a small estimated
fraction for any test that still shows an appreciable residual which has
not been explicitly factorized. The sum of the squares for each test
should be approximately identical with its variance ('communality'
or < self-correlation 3) as estimated at the very outset.

With the present set of correlations the results obtained by the
foregoing procedure are shown in Table VII, A, L They fit the
intercorrelations reasonably well to two decimal places. Strictly,
however, they must be regarded as first approximations only* To
obtain more accurate figures, we should take the sum of the squares
of the saturations for each test (710, -614, . . ., ^389) and insert
it (or a figure still further increased or reduced) in place of the
c estimated self-correlation ' (70, -60, . . ., -40) originally used for
obtaining the * completed totals' in Table IV; with these new
estimates we should then have to repeat the whole process.

After two or three such repetitions I reach the following figures
for the variances or self-correlations :

7216, -6351, -3794, -3994, -6266, -3924

The saturations ultimately obtained are given below in Table
VII, A, ii. The residuals remaining after extracting these three
factors are all less than -ooi.1 It is clear that, if we worked to
further decimal places, we could approximate as closely as we wished
to ihd given intercorrelations with three factors only. In practice, of
course, accuracy to four decimal places is seldom required: here,

1 This, of course, could be predicted at the outset (p. 461), With 6 tests we have
15 intercorrelations. If the 6 self-correlations or variances are themselves to be
determined from these 15 intercorrelations, then there are only 15 degrees of freedom
for determining the saturations. To determine the 6 saturations for the first factor,
6 degrees of freedom will be required: to determine the 6 saturations for the second
factor, only 5 (having determined 5 saturations the 6th follows automatically, because
the total is by hypothesis zero) j to determine the 6 saturations for the third factor,
pnly 4 will be required. But 6 + 5-h4*ai5» Hence no more factors can be
determined unless the variances include some arbitrary quantities,