WORKING METHODS FOR COMPUTERS 463

4. Add each estimated figure to the total of the corresponding
column : (2-19 + "7° = 2*89, etc.).

5. Add these completed totals : (2-89 -f 2-08 + . . . = 12-48).

6. Find the square root of this grand total: (Vi2'48 = 3-533).

7. Divide the augmented total of each column by this square
root : (2-89 -r- 3-533 = -818, etc.).

The quotients (last line of Table IV) should yield approximate
values for the saturation coefficients for the first factor. It is no
longer possible to check them straight away by comparing their
squares with the estimated self-correlations, since the latter are
intended to include the contributions, not of the first factor only,
but of all the factors; but the differences should seldom be large.

(ii) Second Factor

Formula : Residual correlation, r'ab = r^ — rag r^g

Saturation coefficient, /as =

V j i

8. Multiply each of the saturation coefficients by each of the
others, as before, thus obtaining a perfect hierarchy of theoretical
correlations, attributable to the first factor, as shown in Table V :
(-818 X -818 = -669, -818 x -589 = -482, etc.).

9. Enter the observed figures above them, and subtract the
theoretical figure from the observed, as shown: (-70 — -669 =
•031, etc.). The remainders will form a table of residuals, with both
positive and negative signs. With the general-factor method, we
have now to f actorize this entire table of residuals as a single matrix.1
If we add the residuals as they stand, we shall find that the total of
each column, like the total of the deviations about an average (or
rather an average gradient), comes exactly to zero. We must
therefore adopt the procedure described above for factorizing a
bipolar table : (ci, Table III, there is no need to print a fresh table
to illustrate the working).

to. We begin as usual by rearranging the residual inter correla-
tions so as to bring out the general pattern. We may assume that
this is approximately hierarchical, and quartered into positive and

1 With an actual table the residuals should first be tested for statistical significance.
Here N « 120 ; and 4 out of the 15 residuals are significant, x* (see p. 339) = 37*5;
hence P < o-oooi ([no], p. 540). The s.d. of the ratios (residual —• s.e, of corre-
lation) is i'60—well over the theoretical value of i-oo (this test gives a slightly
higher value for P). As we have seen (p. 368) with simple summation, the use of x* is
not strictly valid; but here the least-squares procedure leads to virtually the same
results (p. 474)*