44 THEORY OF STATISTICS. 
4. The associations observed between the attributes 4 and B 
in the universe of C's and the universe of y's may be termed 
partial associations, to distinguish them from the total associations 
observed between A and B in the universe at large. In terms of 
the definition of § 5 of Chap. IIL, 4 and B will be said to be posi- 
tively associated in the universe of Cs (cf. § 4 of Chap. II.) when 
40) (BC) 
Anas HERD ow a 
(ABO (1) 
and negatively associated in the converse case. 
As in the simpler case, the association is most simply tested by 
a comparison of percentages or proportions (§ 9, Chap. IIL), 
although for some purposes a coefficient of association” of 
some kind may be useful. Confining our attention to the more 
fundamental method, if 4 and B are positively associated within 
the universe of Cs, we must have, to quote only the four most 
convenient comparisons, 
(4BC) _ (40) (4BC)_ (BC) b 
me” ow OER enw Os 
(480) _ (450) © (450) _ (aBO) @ 
(BC) = (BC) (40) ~ (20) 
These inequalities may easily be rewritten for any other case by 
making the proper substitutions in the symbols; thus to obtain 
the inequalities for testing the association between 4 and C' in 
the universe of B’s, B must be written for C, 8 for vy, and wice 
versd, throughout; it being remembered that the order of the 
letters in the class symbol is immaterial. The remarks of § 10, 
Chap. IIL, as to the choice of the comparison to be used, apply of 
course equally to the present case. 
5. Though we shall confine ourselves in the present work to 
the detailed discussion, of the case of three attributes, it should be 
noticed that precisely similar conceptions and formule to the 
above apply in the general case where more than three attributes 
have been noted, or where the relations of more than three have 
to be taken into account. If, when it is observed that 4 and B 
are still associated within the universe of C's, it is argued that 
this is due to the association of both 4 and B with D, the argu- 
ment may be tested by still further limiting the field of observa- 
tion to the universe CD. If 
(ABCD)> (LCI aeD CONC, 
A and B are positively associated within the universe of CDs, 
and the association cannot be wholly ascribed to the presence and