HIERARCHICAL CRITERIA 339

It is best, therefore, wherever the issue is crucial, to indicate the
probabilities for and against the chance hypothesis. The most
obvious procedure is to take the ratio of each residual correlation to
the standard error of the observed correlation from which it has been
computed, and then to compare the actual distribution of these ratios
(or its standard deviation) with the theoretical distribution that
would be expected if their fluctuations were due to chance (the
standard deviation of this distribution will, of course, be unity).
The probability that divergences as great as this, or greater, would
arise from random sampling can then be estimated in the usual
way. Better still, we may calculate y* = S(z — z0)2 (N—3),
where z0 = the expected value of z—e.g. tanh ~l fo/yg) if
only a single general factor, g, has been extracted—and enter
Fisher's P-table ([50], p. 110) with f(| n (n — i) — sn + %s (s — i)}
degrees of freedom, where n = number of tests and s = numbers
of factors so far extracted (cf. p. 463).1

errors of random sampling. For example, in my 1917 Report ([35], p. 59) and
elsewhere I found evidence for a special * verbal? or ' linguistic ' factor. In
The Abilities of Man (p. 237), however, Prof. Spearman examines the evidence
for or against' the assumed special ability for verbal-abstract operations * and
cites the ' decisive5 work of Davey [54]. If I understand his figures
rightly, the tetrad difference is apparently only 1-4 times its probable error.
Here, therefore, the evidence for a verbal factor certainly fails to reach the
level required for statistical significance : at the same time the odds are still
against rather than in favour of the appearance of an additional factor being
the mere effect of chance. (And the later work of Kelley and Stephenson now
seems unquestionably to confirm the hypothesis of a verbal factor.)

Because " there is no significant tetrad difference " we cannot infer that
" there is therefore no specific correlation or group-factor," particularly
when the odds are actually against the tetrad difference having arisen from
chance. We can only infer that the specific correlation, though probably
genuine, is not fully conclusive. To prove that a piece of evidence is not
significantly positive is not to prove that it is significantly negative. Never-
theless, this fallacious argument still constantly crops up. Hence it is worth
reminding the student that there are, not two opposed alternatives, but three,
with the lines but vaguely drawn between them': (i) the size of this figure
is consistent with its having probably arisen by chance; (ii) the size of this
figure is inconsistent with its having probably arisen by chance, but is con-
sistent with its being due to a special factor; (iii) the size of this figure is
consistent with either hypothesis, i.e. the sample is too small to enable us to
reach a decision.

1 The ^-transformation is scarcely needed with coefficients below '35.
These methods were employed in a succession of investigations by research
students working at the London Day Training College, and seemed to give
satisfactory results. Strictly speaking, the residuals obtained from one and
the same correlation table cannot be regarded as entirely independent,