496 Correlation between Vaccination and Smallimx 
where g =^ a + d, I = b + e, m = c +f, 
A' = « + 6 + c, t = d + e -\- f, 
N = s+t = (j + l+ m. 
This sixfold table can be turned into a fourfold table by adding the middle 
column to either the first or second. If the distribution of the two characters 
among the population is a normal one, the same value of r will be obtained by 
applying the " fourfold table " formulae to either of these. This, however, will not 
be the same as the value calculated by the same formulae from the fourfold 
table obtained by omission of the first column, thus : 
TABLE D. 
Recovered 
Died 
Totals 
Unvacciiiated ... 
b 
c 
u 
Vaccinated 
e 
f 
V 
Totals 
I 
m 
71 
where u = b + c, v = e + f, n = I + m = u + v. 
We have no accurate evidence on the distribution of immunity among the 
population at large, but the following reasons point to the assumption of normal 
distribution being more correct for the former sixfold than the latter fourfold 
table. 
(1) The same agent, vaccination, is known to protect against attack and also 
against death. On the former supposition this fact is intelligibly represented in 
the table, for if an array representing a group of persons all vaccinated to an 
equal degree were to be vaccinated further, the severity of disease in each 
individual case would be reduced. It follows that some who would have con- 
tracted a fatal attack would now recover, i.e. some cases would be transferred from 
column 3 to column 2. Similarly others would be transferred from column 2 to 
column 1, and would not suffer from smallpox after exposure. In the fourfold 
table the first transfer is similarly represented, but the second is not. Whatever 
unit may ultimately be used to measure severity, the hypothesis of normal distri- 
bution applied to the latter table cannot explain a reduction of severity to 0 or a 
negative quantity. 
(2) Among vaccinated persons the severe cases are much less frequent than 
the mild. The hypothesis of normal distribution amongst those affected with 
smallpox requires that in the total, and also in each array, the cases shall cluster 
round a certain mean severity and that cases of very great or very slight severity 
shall be extremely rare. If, however, normal distribution of immunity applies to 
the whole population, we can readily see how grades of slight severity can be the 
commonest. 
