482

APPENDIX I

TABLE X

ONE SUB-MATRIX OF RESIDUALS AND CALCULATION OF SATURATION
COEFFICIENTS FOR 2ND GROUP-FACTOR

Residuals.
Totals.

[-09]
•12 •06
•12
[-16] •08
•06 •08 [•04]
•27 -36
•18

Total ....
•27
•36
•18
•8i«-90»

Sat. Coeff. .
•30
•40
•20
•90

Squares . .
•09
•16
•04
—

factor method thus leads back to the original saturations from which
the table was constructed.

Unless we are to regard Thurstone's requirements as excluding
a priori anything like a general factor, we may say that the factor-
pattern obtained fits his description of a * simple structure' : all
the coefficients are positive, and each test and each factor (except
the general) has a maximum number of zero saturations.

<?.—The above formula is not valid when the correlation matrix is
partitioned into more than 9 sub-matrices, i.e. when there are more than
three different rectangular blocks influenced by the first or general factor only.
In that case, instead of omitting the square diagonal blocks, R^9 R^t etc.,
we have first to estimate their expected totals. This is done by treating the
remaining totals URZV £R$V • • • £RZ& etc'? as foxing a hierarchy with
the entries in the leading diagonal unknown. Applying the simple-summa-
tion formula in its original form, we can deduce appropriate values for
£R1V £R2Z> • • • etc., and thence the grand total for the entire hierarchy,
SZRpq say. The formula for the saturation coefficients is then as follows:

pq

where / as before extends over all the tests correlated except those affected
by the same group-factor as a, Ria, . . , Rfa> denote the / sub-matrices in
which the ath test falls (including the square diagonal blocks), and ^ and #
extend over all the / sub-matrices without exception.