CHAPTER IV. 
PARTIAL ASSOCIATION, 
1-2. Uncertainty in interpretation of an observed association—38-5. Source of 
the ambiguity : partial associations—6-8. Illusory association due 
to the association of each of two attributes with a third—9. Estima- 
tion of the partial associations from the frequencies of the second 
order—10-12. The total number of associations for a given number 
of attributes—13-14. The case of complete independence. 
1. If we find that in any given case 
(4B)> or $3) 
all that is known is that there is a relation of some sort or kind 
between 4 and B. The result by itself cannot tell as whether 
the relation is direct, whether possibly it is only due to fluctuations 
of sampling” (cf. Chap. III. § 7-8), or whether it is of any other 
particular kind that we may happen to have in our minds at the 
moment. Any interpretation of the meaning of the association is 
necessarily hypothetical, and the number of possible alternative 
hypotheses is in general considerable. 
9. The commonest of all forms of alternative hypothesis is of 
this kind : it is argued that the relation between the two attributes 
A and B is not direct, but due, in some way, to the association of 
A with C and of B with C. An illustration or two will make the 
matter clearer :— 
(1) An association is observed between vaccination” and 
« exemption from attack by small-pox,” i.e. more of the vaccinated 
than of the unvaccinated are exempt from attack. It is argued 
that this does not imply a protective effect of vaccination, but is 
wholly due to the fact that most of the unvaccinated are drawn from 
the lowest classes, living in very unhygienic conditions. Denoting 
vaccination by A, exemption from attack by B, hygienic conditions by 
C, the argument is that the observed association between 4 and J 
is due to the associations of both with C 
Q°; 
il