166 
On Theories of Association 
normal coefficient, the tetrachoric r t , to pigmentation tables without a line of 
reference to a paper, which must have been perfectly well known to him, for it is 
entitled " A Reply to Certain Criticisms of Mr G. U. Yule." This may be strategic, 
but at the same time illustrates the peculiar character of Mr Yule's controversial 
methods. 
(3) On the Boas-Yulean "Theoretical Value" of the Correlation {Pearsons <}>). 
We now come to a very nice point indeed, namely Mr Yule's "theoretical 
value of the correlation coefficient," the method introduced into his Textbook 
of Statistics (p. 212) without a word of warning as to when it should be applied. 
Mr Yule now states that : " It would have been thought that anyone reasonably 
acquainted with the theoretical work of the last decade, and especially Professor 
Pearson and his collaborators, would have found no difficulty in the passage in 
question*." Now what exactly have Pearson and his collaborators done? They 
have applied the product-moment correlation to the presence of 0, 1, or 2 
protogenic units in theoretical Mendelian investigations. They have assumed that 
when a character goes by units, you may apply the usual product-moment methods. 
But they have objected in toto to the application of such a method to material 
where there was reasonable evidence of continuous variation. Does Mr Yule look 
upon ' death ' as the addition of one unit to ' recovery' ? Does Mr Yule look upon 
vaccination as the addition of one unit to 'absence of vaccination'? Does Mr Yule 
look upon 'mental defect' as the addition of one unit to normal mentality? In the 
three Mendelian types (RR), (BR), and (DD) there is a progression of one unit at 
each stage in the number of D's, but what are the units in the cases we have 
mentioned ? Or, does Mr Yule suggest that his " theoretical value of the correla- 
tion " is to be confined to those actual true unit additions to which Pearson and his 
collaborators have always confined them ? There is not a hint of this in his Text- 
book nor in his present paper. He has indeed carefully refrained hitherto from 
saying what are the characters of the attributes to which it is to be applied. He 
has suggested that " the ordinary theory of correlation, once that theory had 
been freed from any necessary relation to the theory of normal correlation, was 
applicable in its entirety to the 2 x 2-fold table f." Mr Yule says that this should 
be " a very obvious matter." Indeed ! — Then apparently 'vaccination' and ' mental 
defect' are a quantitative unit more than non-vaccination and normal mentality! — 
But how does this fit with Mr Yule's other assertion that these are discrete attri- 
butes and suitable for the application of his coefficient of association ? Let us 
see how Dr Boas investigates Pearson's r^k = 4>t- 
" Correlations of phenomena that cannot be measured but only counted may 
be treated in the following manner : If two events that have the probabilities 
p r and jh are correlated, we may say that those cases in which the event 1 occurs 
* Journal of R. S. S. Vol. xlv. p. 609. 
t Journal of R. S. S. Vol. xlv. p. 606. 
+ Science, May 1, 1909, p. 824. 
