214 
On Theories of Association 
(9) On the Limitation in Value of the Boas-Yulean (f>. 
Given two total variate frequencies, if we can assert nothing of the nature of 
the distribution, the maximum value of the uncorrected mean square contingency 
coefficient depends on the number of cells and cannot for a finite number of cells 
exceed a certain limit. Mr Yule has spoken of this fact as if it were a serious blot 
on the method of contingency. We do not agree with him, but it is singular that 
if he thinks so, he should not have rejected the use of <f>, the " theoretical value of 
the correlation." The fact that </> had a maximum limit was known to Mr Yule*, 
yet he never throughout his paper refers to it as detrimental to his own " theoretical 
value of the correlation." Consider any table : 
i h m 2 
iV 
Ml 
>!. 2 Wl, 
nonio , 
m 2 
N 
This is the most general form the fourfold can take for given n 2 , m lt m 2 . We 
then have 
= ocN I J n 1 n i > m 1 m i : 
This will be a maximum of a positive kind when x takes the largest possible 
value, i.e. when x is equal to the lesser of n^j/iVand n^mJN. It will be a maximum 
of negative kind when x is equal to the lesser of and — ^ . Thus 0 always 
lies between definite limits which may be most restricted. 
Consider the table 
2269 + .v 
97261 -x 
99530 
1 1 - X 
459 + .? 
470 
2280 
97720 
100000 
Here the limits are given by x= 11 and # = — 459 or we have the two tables 
2280 
97250 
99530 and 
1810 
97720 
99530 
0 
470 
470 
470 
0 
470 
2280 
97720 
100000 
2280 
97720 
100000 
These give </> = -0106 and <f> = - "4499. 
* Journal of B. S. Soc. Vol. lxxv. p. 604. 
