Karl Pearson and David Heron 
180 
age group of the vaccinated — surely a "natural" process — then we shall modify 
the coefficient of association obtained by dividing between vaccinated and un- 
vaccinated. If we equalised the totals of vaccinated and unvaccinated, then Q 
would have changed, if the division were taken at an interval of 25 years since 
vaccination. The whole process is quite arbitrary, and we believe wholly without 
validity. 
(6) On the Coefficient of Association and the Assumption of Discrete Variates. 
Mr Yule has asserted that when we free ourselves from any necessary relation 
to the theory of normal correlation, the ordinary theory of correlation is applicable 
in its entirety to the 2 x 2-fold table. Apparently he holds that the same is true 
for his coefficients of association and colligation although these different methods 
lead to diverse and often contradictory results, for the simple reason that selection 
vastly affects one and does not affect the other. He has not considered the 
surface of equal association, nor discussed the cases to which the " four-point " 
surface, involved in using </> as Boas and he propose, can be legitimately applied. 
It is accordingly of interest to see what happens when the coefficient of association 
is applied to various types of frequency surface. 
It is no valid reply to criticisms based upon such an investigation to say that 
such frequency surfaces do not occur in practice. Mr Yule has never entered 
into any discussion of the character of the distributions to which he applies his 
association ; it is sufficient for him that they give a fourfold table, and he makes 
no appeal, as we do, to experience as a basis for the adoption of any coefficient. 
It is therefore possible to test his association against the clear idea of correlation 
on any distribution whatever. 
If Mr Yule replies that this is not fair treatment because the coefficient of 
association applies to discrete quantities, we answer that he has never defined 
it an amusing task to equalise the total frequencies ; he will then discover that it takes an interesting 
form, that of an old friend of Mr Yule's, and the correlation will then be recognised as '50. 
TABLE VII. 
First Variate. 
a 
b 
C 
d 
e 
/ 
9 
h 
i 
j 
Totals 
a 
11 
20 
30 
35 
40 
45 
50 
40 
30 
2 
303 
b 
20 
4400 
600 
700 
800 
900 
1000 
800 
600 
40 
9860 
c 
30 
600 
9900 
1050 
1200 
1350 
1500 
1200 
900 
60 
17790 
d 
35 
700 
1050 
13475 
1400 
1575 
1750 
1400 
1050 
70 
22505 
e 
40 
800 
1200 
1400 
17600 
1800 
2000 
1600 
1200 
80 
27720 
f 
45 
900 
1350 
1575 
1800 
22275 
2250 
1800 
1350 
90 
33435 
g 
50 
1000 
1500 
1750 
2000 
2250 
27500 
2000 
1500 
100 
39650 
h 
40 
800 
1200 
1400 
1600 
1800 
2000 
17600 
1200 
80 
27720 
i 
30 
600 
900 
1050 
1200 
1350 
1500 
1200 
9900 
60 
17790 
j 
2 
40 
60 
70 
80 
90 
100 
80 
60 
44 
626 
Totals 
303 
9860 
17790 
22505 
27720 
33435 
39650 
27720 
17790 
626 
197399 
