202 
On Theories of Association 
If we turn from surfaces of zero correlation to those of finite correlation, we 
find in the same way that Q takes innumerable values which have no mutual 
relationship. Heron has already demonstrated this as far as the Gaussian surface 
of frequency is concerned*. For example, in a Gaussian distribution Mr Yule's 
association can take every value from '37 to TO if the correlation be truly "3 
and the divisions be taken along the diagonal. These give for a Gaussian 
distribution the complete range of Q, but it by no means follows that this is 
true for other types of frequency. Mr Yule, however, makes no hypothesis as to 
the nature of his frequency surface. To test the kind of meaning Q conveys, 
suppose the frequency surface to be a rectangular block, length 2a, breadth 26, 
height h — that is to say, within a given rectangular area the frequency of all 
combinations of the two variates is equally probable. If the block slopes with its 
side 2a at an angle of 6 to the *"-variate, we have the correlation 
(a 2 — b 2 ) cos 6 sin 0 
Va 2 cos 2 0 + b 2 sin 2 0 Va 2 sin 2 0 + b 2 cos 2 0 ' 
The regression lines are built up of straight lines ; for a considerable distance 
they coincide with the axes of symmetry, but are afterwards bent round horizontally 
or vertically as the case may be. At the four corners Q = 0 and along one pair 
of parallel sides Q = + l, along the other Q = — 1. Thus two contour lines of 
zero association pass through the corners pair and pair. In the accompanying 
diagram (p. 203) the values of Q are given for the special case where 0=45°, a=l'5b, 
and accordingly ? , = , 3846. Going along the longer axis Q changes from — 10 to 
- - 295 and so to zero, then it rises to + '6 at the centre and falls to zero again, 
becoming negative and ultimately concluding with — TO at the boundary. Along 
the shorter axis Q varies from + TO to + '6 at the centre and rises to + l'O again 
below it. What can be learnt as to the real association of two variates by a 
coefficient which behaves in this way ? The case would be quite different if 
Mr Yule had indicated a type of surface for which Q was constant for all 
divisions and demonstrated that it represented, even with moderate approxi- 
mation, such distributions as occur in statistical practice. One such surface of 
" stable association " at any rate is known for the tetrachoric r t treated as a 
coefficient of association merely, and that surface is not widely divergent from a 
considerable number that we actually meet with. 
(b) The Fallacy of Mr Yule's Selection in the case of Continuous Variates. 
If Q be not even approximately comparable with itself when taken on the 
same surface with different dichotomies, how can it be comparable as a measure 
of any real relationship from one surface to a second ? Mr Yule will no doubt 
reply that a function of Q does measure certain percentages when the table 
is dressed in an equalised symmetrical form. Our reply is that that form has 
been obtained by a method of selection which makes very large changes in every 
other coefficient, including the Boas-Yulean, which has been used to measure 
* Biometrika, Vol. viii. p. 109. 
