176 
On Theories of Association 
" comedy of errors " has now ended, that comedy consisting in our overlooking the 
"fact" that it is " a very obvious matter" that 0 is applicable in its entirety to the 
2 x 2-fold table. That applicability is the very point to which Pearson took ex- 
ception in Dr Boas' use of cf>. Were it a fact, Mr Yule might throw both his Q 
and his co overboard, for what more is required than a method applicable in its 
entirety to a 2 x 2-fold table ? And the </> method at all points contradicts the 
Yulean results. But it is not true, for at best it would be a method of ranks, and 
the correlation of ranks is never a correlation of variates unless the ranked 
quantities proceed by absolute units of variation — as for example in the theoretical 
Mendelian case to which Pearson perfectly legitimately applied it — or in counting 
the teeth on the carapace of a prawn or the veins on a leaf. 
Mr Yule seems to consider that the inoculation with an antitoxin is equivalent 
to the addition of a unit of something to the individual ; we consider that this is 
wholly erroneous. To begin with, the dosage is not uniform, its repetition does not 
always occur at the same interval and the number of doses is not always the same ; 
further the interval between the onset of the disease and first inoculation is by no 
means the same ; lastly, apart from the resistance of the individual patients to the 
disease, the curative effect of the treatment depends on the relation of the antitoxin 
administered to the physiological individuality of the patient. It is idle therefore 
to consider this varying complex as a quantity undifferentiated from individual to 
individual. The group treated with antitoxin is not made up of identical indi- 
viduals but of a number of persons with increased power of resistance to the disease, 
which may vary from the case of a person who has gained nothing by it to that of 
a person who has immensely increased his power of recovery. In precisely the 
same way those who have not been treated can by no means be grouped into 
a single quantitative class ; it may be doubted, indeed, whether recuperative 
power when disease is incurred is really divided sharply by a line like treatment 
or non-treatment with antitoxin. It may be only the sharpest division we can 
take under the circumstances, and in our ignorance of the nature of the distribution 
a tetrachoric r t may be as effective a measure of association as Mr Yule's Q. At 
any rate no man of " ordinary intelligence " would believe that perfect association 
existed between treatment and recovery because out of 23 persons treated none 
died, while out of 977 not treated six died, yet this would be the result provided by 
Mr Yule's coefficient of association ! Clearly if only 0'6°/ o died without treatment, 
we should not expect any to die in a sample of 23 whether treated or not treated. 
The vanishing of association for a zero quadrant is a patent fallacy *. 
If the problem were presented to us as Mr Yule states it, i.e. the evaluation of 
a new treatment, we should certainly not use his Q for solving the problem. We 
should probably to-day not use a tetrachoric r t} except as a control. We should 
most likely use </>, although certainly not in the sense of a correlation of four points. 
We should question how far death and recovery are independent of treatment or non- 
treatment ; that is to say, we should ask what is the probability that recovery 
* This topic is of such importance that we have discussed it more at length in Appendix I. 
