F. M. Turner 
503 
TABLE F. 
Diseased 
Not Diseased 
Totals 
Recoveries 
Deaths 
Unvacciiiated . . . 
39 
1525 
753 
2317 
Vacciuated 
927 
6505 
876 
8308 
Totals 
966 
8030 
1629 
10625 
?•= --4492. 
I think the above figures afford some explanation of the apparent small 
diminution of vaccination protection as age advances. If my contention be correct 
that a normal distribution in immunity may more correctly be assumed for the 
whole population exposed to attack, than for that part only which actually contracts 
the disease, and appears in smallpox hospital statistics, then the values given 
above (Tables II. to IV. and XIV. — XXIII.) are apparent values of r corresponding 
to curtailed distributions. 
If any time-diminution occurs, then the number of persons sufficiently protected 
to escape the disease altogether will be much larger at the ages 0-10 than at 
higher ages, say 20 — 30. Hence, using the above notation, in Table XIV. the 
value of g is much greater than in Table XIX. Hence r — r' will be greater in 
the former case than in the latter. Although, therefore, the values of r' in the 
two cases are nearly equal, the values of r probably differ considerably. 
The above investigation of curtailed distributions also gives a possible explana- 
tion of the small correlation between severity of disease and number, size and 
foveation [i.e. depth] of the scars. 
For we have seen that if in any normal distribution, if one of the factors is 
divided into three classes, then leaving out either of the extreme classes from 
consideration will reduce the sixfold to a fourfold table. The resulting apparent 
correlation, however, will be less than the true value. Now in arranging our 
Tables I. — III. we have recognised only two degrees of vaccination, " vaccinated" 
and " unvaccinated." But if we split the vaccinated group into two, " four scars 
or more" and "less than four scars," we may arrange the whole as a sixfold table. 
Hence Table VIII. is seen to be a curtailed form of Table I.; and similar arguments 
apply to Tables VII. and IX. Macdonell, who discovered the lower correlation 
in these instances, remarks*: "It is obvious that in dealing with the last four 
tables we have descended to a much lower plane of correlation, and the results 
may possibly somewhat modify medical opinion as to the degree of significance of 
foveation, number of scars and scar area." If, however, the normal distribution 
held good for Table I., then a " lower plane of correlation " would necessarily follow 
in Tables VII. — IX., not as a medical, but as a mathematical phenomenon. 
* Biometrika, Vol. i. p. 383. 
