VARIOUS USES OF FACTORS 41

matic attention should be paid to the appropriate selection of
tests and persons than is usually the case. The first prin-
ciple requires us to select samples of tests and persons that
shall be relatively homogeneous in the relevant characteris-
tics ; the second, to select samples homogeneous in the
irrelevant traits, introducing a relevant discontinuity into
the sub-groups; the third, to select samples appropriately
heterogeneous in the irrelevant characteristics—a point that
is far more frequently disregarded.

If the hypothesis to be tested is sufficiently definite, and
if the alternative factors are comparatively few, such pre-
cautions should certainly lead to a fairly simple pattern both
in the correlation table and in the factorial matrix. But
even so, neither the precautions nor the ensuing simplicity
will, as a rule, enable us to transcend the fact that our own
table is merely one specimen taken from the enormous
number of analogous tables that presumably await investiga-
tion. In general, therefore, to justify a factorial prediction,
or any other inductive generalization from the figures
obtained by factor-analysis, the first prerequisite must be
to base conclusions, not on a single sample or a single set,
but on a series of such sets and samples.

It is for this reason that I have elsewhere proposed criteria which
may serve to test the stability of factors from one investigation to
another. Of these the simplest is the ' symmetry criterion.' If
RI and J22 are two correlation or covariance matrices dependent
upon the same dominant factors, then R^^ ^ -^2^1? *>e- ^ product
of the two matrices should be approximately symmetrical.1

At the same time, let me add that there are grounds (which again
cannot here be set out in detail) for doubting whether the coefficient
of correlation is after all the best measure on which our inductive
predictions are to be based. Strong reasons can be adduced, to a
large extent following from the arguments just given, for preferring
covariance to correlation, wherever covariance is legitimately
calculable, and for basing predictions and statements of probability
upon the regression coefficients rather, than upon the coefficients of
correlation themselves. A correlation coefficient is descriptive
solely of the set of figures on which it is based : it cannot profess to

1 A worked example is given in Table IV [128], p. 68. Other instances are
given in the earlier theses by Williams and Davies (cf. [119] and [130]).