356 THE FACTORS OF THE MIND

itself, no matter how those factors have been obtained. Conse-
quently, they may be used to compare different factorial analyses of
one and the same correlation matrix, and in particular to examine
the effect of rotation or other linear transformations.

Summary.—Let me briefly recapitulate this somewhat
prolix discussion, and indicate what I take to be the most
practicable methods at present available. For testing signi-
ficance the simplest method is to calculate x2 from the
squared residuals. This is much speedier than calculating
all possible tetrad differences. For estimating hierarchical
tendency the best criteria would seem to be those based on
first or second moments. If we are interested more
particularly in the relative contributions of the single
factors (e.g. in the importance of the first factor—c the
general factor ') then the simplest procedure is to take the

(unadjusted) variance of its saturations, 73j2 = --, except

where the number of tests is much greater than the number
of significant factors : in that case it will be wiser to
examine the contribution to the total significant variance
instead of the contribution to the total standardised, variance,

i.e. to take Ot\£ = -=^- instead of $\£ = —. If we desire

a single index for all factors, then, if n remains unaltered
in the tables to be compared (as in comparing factors from
the same correlation table before and after rotation), the
simplest procedure will be to take the adjusted standard
deviation of the two sets of factor-variances; In com-
paring results from different correlation tables, where n
may be different, it will be better to take the tetrad-differ-
ence ratio, 7j22. As before (except where the number of
tests is much greater than the number of significant
factors) we may assume Sty = n, and the formula is the
equivalent to examining the (unadjusted) variance of all
the saturations. The c condensation-ratio ? (/52) requires
slightly more elaborate calculations; but would seem to
have both the best theoretical basis and the most intelligible
meaning, and so to be most appropriate for formal com-
parisons on a systematic scale.