IV.—PARTIAL ASSOCIATION. . 
(2) It is observed, at a general election, that a greater 
proportion of the candidates who spent more money than their 
opponents won their elections than of those who spent less. It 
is argued that this does not mean an influence of expenditure on 
the result of elections, but is due to the fact that Conservative 
principles generally carried the day, and that the Conservatives 
generally spent more than the Liberals. Denoting winning by 4, 
spending more than the opponent by B, and Conservative by C, the 
argument is the same as the above (¢f. Question 9 at the end of 
the chapter). 
(3) An association is observed between the presence of some 
attribute in the father and its presence in the son ; and also 
between the presence of the attribute in the grandfather and its 
prescuce in the grandson. Denoting the presence of the attribute 
in son, father, and grandfather by 4, B, and C, the question arises 
whether the association between 4 and C may not be due solely 
to the associations between 4 and B, B and C, respectively. 
3. The ambiguity in such cases evidently arises from the fact 
that the universe of observation, in each case, contains not 
merely objects possessing the third attribute alone, or objects 
not possessing it, but both. 
If the universe were restricted to either class alone the given 
ambiguity would not arise, though of course others might remain. 
Thus, in the first illustration, if the statistics of vaccination 
and attack were drawn from one narrow section of the population 
living under approximately the same hygienic conditions, and an 
association were still observed between vaccination and exemption 
from attack, the supposed argument would be refuted. The fact 
would prove that the association between vaccination and 
exemption could not be wholly due to the association of both with 
hygienic conditions. 
Again, in the second illustration, if we confine our attention to 
the “universe ” of Conservatives (instead of dealing with candidates 
of both parties together), and compare the percentages of Conserva- 
tives winning elections when they spend more than their opponents 
and when they spend less, we shall avoid the possible fallacy. If 
the percentage is greater in the former case than in the latter, it 
cannot be for the reasons suggested in § 2. 
The biological case of the third illustration should be similarly 
treated. If the association between 4 and ¢ be observed for 
those cases in which all the parents, say, possess the attribute, or 
else all do not, and it is still sensible, then the association first 
observed between 4 and C' for the whole universe cannot have 
yg due solely to the observed associations between 4 and B, B 
and CO. 
435