504 
Correlation between Vaccmatioti and Smallpox 
Summary. 
1. The correlation coefficient between severity of disease and previous 
vaccination in the late London epidemic was lower than in any of the previously 
investigated cases. 
2. There seem stronger reasons for assuming normal distribution for the whole 
population exposed to risk of infection, than for the population attacked by the 
disease. 
3. If the former assumption is correct the smallpox correlation tables of 
Pearson, Macdonell, and of this paper, correspond only to a portion of the normal 
correlation surface. 
4. Values of r calculated from such " curtailed " normal distributions are 
lower than the true value, whether product moments are used or fourfold formulae, 
or contingency*. As to the first method a complete investigation is given, as to 
the other two I have failed to find a mathematical proof, and base the statement 
upon calculated instances, unfortunately not numerous. 
5. Values of r have been calculated for each age group from the figures of the 
London epidemic. These appear to shew that the protection of vaccination lasts 
without perceptible diminution for about 30 years. This result is opposed to the 
opinion of most of those who have had exj^erience of smallpox. My hypothesis 
given under (2) would explain the discrepancy, for the real values of the correla- 
tion at early ages would be considerably higher than the apparent values drawn 
from smallpox cases only. 
6. The correlations between area of soars and severity, number of scars and 
severity, foveation of scars and severity in the London epidemic agree closely with 
those calculated by Macdonell from Glasgow statistics. They are all considerably 
lower than the correlation between vaccination and severity. The difference would 
be expected, if the normal distribution held for the total population admitted to 
hospital. For the tables from which the three former correlations have been 
calculated are " curtailed " by the omission of the unvaccinated group. 
7. The figures from the London epidemic do not enable us to decide whether 
the correlations of severity with number and foveation of scars are secondary to 
that between severity and area of scars ; though they may be so. 
8. The gross correlation between vaccination and severity is lower than the 
mean net correlation at various ages, owing to the very high correlation between 
age and vaccination among inmates of smallpox hospitals. 
In conclusion, I have to thank Prof. Pearson, of University College, and 
Dr Ricketts, Superintendent of the Smallpox Hospitals of the Metropolitan 
Asylums Boai'd, for much help in the preparation of this paper. 
* No examples of contingency formulae are given in the paper, as none of the tables were adapted to 
this method. 
