256 On Theories of Association 
recoveries, vaccinations and non-vaccinations (see Curves B of Diagrams XII and 
XIII). The fundamental result is obvious in all these diagrams, the selection, 
which does not change Mr Yule's coefficients, changes widely the true relationship 
as measured by the regression lines between the variates immunity conferred by 
vaccination and the severity of small-pox. Take a given grade of severity of the 
disease, and the grade of immunity which is associated with it is entirely altered by 
the selection, or take a definite grade of vaccination and the average severity of the 
disease is represented by quite different figures before and after selection. Mr Yule's 
most important property of his coefficients — the property that they remain 
unaltered by this selection — is the very property which forces us to the conviction 
that they are wholly unsuited to use in such a case as that which he asserts is 
eminently fitted for their use. 
If we consider the matter algebraically, we find : 
1 
r 
* 
Yulean 0 
Contingency 
Uncorrected 
Corrected 
Uncorrected 
Corrected 
Before selection 
•32 to -34 
•34 to -36 
•26 
•34 
•31 
•36 
After .selection 
•58 
■61 
•52 
•60 
•50 
•58 
We contend that a method which so substantially changes the real relationship 
between immunity and severity is wholly incapable of leading those who use it to 
any just inference as to the association of immunity and severity. It is only a 
logical quibble quite unworthy of the reputation of the man who uses it, and 
directed at an audience which had made no thorough study of the mathematics 
of statistics, to state as Mr Yule does that a man dies or does not die after 
incurring small-pox, and that he is vaccinated or not vaccinated. Death and 
vaccination are crude class-indices of severity and immunity, and Mr Yule's 
coefficients tell us nothing of the real relation between the two variates which 
is what we at least are seeking. 
(13) On the Stability of Coefiicients of Association. 
If we have, as we believe we certainly have, in this paper succeeded in 
demonstrating the idleness of Mr Yule's coefficients of association and colligation 
for any purpose of applied statistics ; if, as we hold, they are merely of interest 
from the standpoint of symbolic logic, i.e. in the discussion of verbal classifications, 
not in the treatment of the real things lying behind class-indices ; the reader may 
naturally ask how we propose to treat similar problems. The answer to our mind 
is fairly clear. When the variates truly advance by unit grades, then there is no 
difficulty whatever about the problem. The right method to use is the product- 
* Calculated by concentrating the contents of each cell at the Gaussian centroids of the subranges 
of the marginal totals and correcting for class-index correlations ; see Biometrika, Vol. ix. p. 119. 
