IV.—PARTIAL ASSOCIATION. { 
complete independence of 4, B, and C in the sense that the 
equation 
(4B) _(4) (B) 
NTN NN 
is a criterion for the complete independence of 4 and B. If we 
are given JV, (4), and (B), and the last relation quoted holds 
good, we know that similar relations must hold for (48), (aB), 
and (a3). If &, (4), (8B), and (C) be given, however, and the 
equation (8) hold good, we can draw no conclusion without 
further information ; the data are insufficient. There are eight 
algebraically independent class-frequencies in the case of three 
attributes, while , (4), (B), (C) are only four: the equation (8) 
must therefore be shown to hold good for four frequencies of the 
third order before the conclusion can be drawn that it holds good 
for the remainder, 7.e. that a state of complete independence 
subsists. The direct verification of this result is left for the 
student. 
Quite generally, if &, (4), (B), (C), . . .. be given, the relation 
{4BC ..., J _ (4) (8B) (©) (9) 
A = 5% Fa : 
must be shown to hold good for 2" —n +1 of the nth order classes 
before it may be assumed to hold good for the remainder. It is 
only because 
2" —n+1=1 
when n= 2 that the relation 
4B) (4) (B) 
¥ 5A: 
may be treated as a criterion for the independence of 4 and B. 
If all the n (n>2) attributes are completely independent, the 
relation (9) holds good ; but it does not follow that if the relation 
(9) hold good they are all independent. 
REFERENCES. 
(1) Youre, G. U., “On the Association of Attributes in Statistics,” Phil. 
Trans. Roy. Soc., Series A, vol. cxciv., 1900, p- 257. (Deals fully 
with the theory of partial as well as of total association, with numerous 
illustrations : a notation suggested for the partial coefficients.) 
(2) YuLe, G. U., ‘““Notes on the Theory of Association of Attributes in 
Statistics,” Biometrika, vol, ii., 1903, p. 121. (Cf. especially §§ 4 and 
5, on the theory of complete independence, and the fallacies due to 
mixing of records.) 
LY