IV.—PARTIAL ASSOCIATION. 
The two data give— 
a (AC)(BC : 
(4BC) = Ms | 
| ) 
(45) — AVE) Id) = (CB) ~ (BOY) 
(¥) ¥) 
Adding them together we have— 
(45)= ri | MACY BC) - (A)(C)BO) ~ (BXCNAC) +(A)BXO) } 
Write, as in § 11 of Chap. ITI. (p. 35)— 
(4)(B) A)(c (B)(C) 
(48), = NB) (40), NO) (py, BNO) 
subtract (45), from both sides of the above equation, simplify, 
and we have 
N 
(4B) ~ (4B)y = ((5[(AC) ~ (4C)IBO) - (BOY) (4) 
This proves the theorem; for the right-hand side will not be 
zero unless either (AC) =(4C), or (BC) = (£C),. 
7. The result indicates that, while no degree of heterogeneity 
in the universe can influence the association between 4 and B 
if all other attributes are independent of either 4 or 7B or both, 
an illusory or misleading association may arise in any case where 
there exists in the given universe a third attribute C' with which 
both 4 and B are associated (positively or negatively). If both 
associations are of the same sign, the resulting illusory association 
between 4 and B will be positive ; if of opposite sign, negative. 
The three illustrations of § 2 are all of the first kind. In (1) it 
is argued that the positive associations between vaccination and 
hygienic conditions, exemption from attack and hygienic conditions, 
give rise to an illusory positive association between vaccination 
and exemption from attack. In (2) it is argued that the positive 
associations between conservative and winning, conservative and 
spending more, give rise to an illusory positive association between 
winning and spending more. In (3) the question is raised whether 
the positive association between grandparent and grandchild may 
not be due solely to the positive associations between grandparent 
and parent, parent and child. 
Misleading associations of this kind may easily arise through 
a4 
49 
(3