486 APPENDIX I

and from them we could obtain the figures shown in Table XIc,
approximately if we followed his * graphic' procedure,1 exactly
if we used a more adequate method and retained enough factors and
decimals. If we started with weighted summation, the rotation
could still be carried out, though Thurstone's criticism of such
methods seems to imply that it could not : indeed, the necessary
calculations would be much easier, owing to the zero correlations
between the rows.2 But in either case it is evident that we save a
vast amount of labour, and probably reach more precise figures, if,
when we want a c simple structure,' we seek it by the direct * group-
factor method' according to the procedure described above.

Of the three main methods described in this Appendix I recom-
mend simple summation for general purposes, weighted summation
for exact research, and the grouf-factor method for special cases of
discontinuous distribution. The student, however, who is new to
factorial work should practise all three. Let him begin with a
small fictitious table of unit figures only (cf. next Appendix) ; and
then try his hand with physical measurements before proceeding to
mental. Let him take, say, half a dozen body measurements for half
a dozen persons (e.g. sitting height, length of leg, and of arm, girth
of chest, waist, and hips for a homogeneous and for a dichotomous
group, i.e. mixed males and females, or pyknics and leptosomics),
and factorize covariances and correlations for both traits and persons
by every procedure available, including * simple averaging * (p. 396),
Physical measurements present much the same problems as mental;
and the interpretations are easier to visualize.3 The theoretical
advantages of weighted summation will quickly become obvious.
Like Thurstone's method, it yields a factor-pattern of minimal
rank; like Hotelling's, it yields the best possible fit. It would
thus seem to combine the merits of both.

1 Thurstone in his rotations would presumably begin by seeking the factors con-
taining zero saturations (which with him are usually overlapping group-factors) and
leave the general factor (i.e. the factor with no negative or zero saturations) to emerge,
if at all, only when the group-factors do not account for the correlation.

1 The method has already been described above (p, 304); and therefore need not
be repeated here.

* The idea that there is a ' contrast * between results from physical and mental
measurements (the latter alone yielding hierarchies) and that neither * bodily dimen-
sions ' nor * assessments of physical maturity * show signs of general factors (Spearman
[56], p. 142, Thomson [132], p. 279) is not borne out by my own data. Both with
adults and with children the factor-variances follow the same rough progression as is
obtained with psychological tests (Table VII, p. 358). For Spearman^ own chief
example ([56], p. 141) vijn = -49, "I2> *°9: (the two group-factors—covering the
3 head measurements and the 2 skull-breadth measurements'—were only to be ex-
pected). Of course, if we mix linear with non-linear measurements (e,g, weight)
and anatomical with physiological ([56], p. 144), the hierarchical tendency is bound
to be obscured.