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 positively
 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 argument
 may be tested by still further limiting the field of observation
 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