308 
On Theories of Association 
We have seen that if the sampled population be really one of zero association, 
the sample may have in quadrant d in 87 °/ c of cases zero frequency, but in 12°/ 0 
of cases unit frequency. This unit must — working to unit individuals — be with- 
drawn from a, b and c. The most probable case is the addition of a unit to both 
a and d and their withdrawal from b and c, or again we may draw units respectively 
from a, b or c. We may consider only these four cases : 
(«) 
972 
22 
994 
m 
971 
22 
993 
5 
1 
6 
6 
1 
7 
977 
23 
1000 
977 
23 
1000 
(7) 
971 
23 
994 
(«) 
970 
23 
993 
5 
1 
6 
6 
1 
7 
976 
24 
1000 
976 
24 
1000 
These lead to 
a 
0 
7 
8 
Value of tetrachoric r t ... 
p. e. of r t = 0 for same marginal totals... 
Value of Association Q ... 
p. e. of Q = 0 for same marginal totals 
•39+ -15 
•00 ±'27 
•80+ -14 
■00 ±-23 
•36+ -16 
•00 ±-25 
■76+ -16 
•00 ±-24 
•38+ -15 
•00 ±-27 
•78+ 13 
■00 +-23 
•35+ -16 
•00 ±-25 
•75±-16 
■00 ±-25 
It will be seen that r t is always less than 2 - 5 and Q always greater than 
4"5 times the probable error. The value of r t obtained is always less than 
1*5 times the probable error of r t = 0 for the same marginal totals, while Q is 
3 times the probable error of Q = 0 for the same marginal totals. Thus the 
tetrachoric method warns us that the association is probably zero, while the 
association coefficient emphasises a high value of the association. 
To sum up : The correct process in these cases with zero in one quadrant 
is not to assert that the association is perfect with Mr Yule, but to apply first the 
probability test and determine whether the material may not rather be a random 
sample from an original population of zero association. The failure of the tetra- 
choric r t in these cases is rendered evident in the working, we reach non-convergent 
series and are thrown back on a limiting case ; if we place in the zero quadrant a 
small frequency less than 0"5, which would correspond to zero in the actual table, 
we find a finite value of r t , but one non-significant having reference to its probable 
error, unless we approach close to the limit of d = 0, in which case the probable 
error of r t is so far undetermined, because the ordinary process fails to be valid. 
Mr Yule's view that association is perfect when there is zero in any quadrant 
ignores the fact that he can only deal with a sample of the true population, and 
