250 
On Theories of Association 
unchanged by the artificial selection in which Mr Yule sees* "a most important 
property and one of special importance in such cases as those I have chosen for 
illustrations," i.e. vaccination and recovery. On the other hand the <j), or pseudo- 
rank correlation, is for a fourfold table immensely influenced, as we have shown, by 
selection of this artificial kind. If this <£, however, be artificially selected, so that 
it deviates widely from the <j> of the data actually provided, then this artificial 
value of <f> — for a table, say, in which the number of vaccinated has been made 
equal to the number of unvaccinated and the number of deaths equal to the 
number of recoveries — is Mr Yule's coefficient of colligation. From our stand- 
point it is hard to conceive a stronger argument against this coefficient of 
colligation than the fact that it is unequal to <j), unless you have artificially 
Diagram X. Kegression in Eye-colour. Mother and Son. Based upon means obtained 
by Gaussian assumptions. 
8 
~5&6 
5 2 
• 
Mean 
^ 
e 
• - 
«■ 
{Light, 
3 4 
Mother. 
5+6 
5?6 
(DarJc. 
doctored (f> to bring the two into agreement. What is the probable error uf 
0 thus doctored or how is it related to the probable error of <f> found from 
the undoctoied material ? Mr Yule, however, writes : " We have therefore the 
important theorem briefly mentioned without proof in p. 17 — the coefficient of 
colligation a> for any table is the product-sum correlation rf for the equivalent 
symmetrical table. These two coefficients r and a> form, accordingly, a natural 
pair, the first giving the actual correlation in the given table, the second the 
correlation in a derived table of standard form, thus enabling ns to compare the 
* Loc. cit. p. 587. 
t Mr Yule here as elsewhere terms <p the "product-sum correlation" and uses for it the letter ?•. 
This is wholly unjustifiable, it is merely a correlation of pseudo-ranks and not of true variates 
at all. 
