184 
On Theories of Association 
Then a = n m , b = q — n pq , c = p- n pq , d = N - p - q + n pq . 
Therefore ( x - 1) n 2 pq - n pq \(x - 1) (p + q) + N) + pq X = 0 
is a quadratic to find n m . It' we take as we always can arrange to do Q positive, 
then % lies between 1 and go or is also positive. The equation for n pq will have 
real roots if 
{(% - 1) (p + q) + N} 2 > 4% (% - i)pq, 
or, if 
(%-l)' 3 (p-?) 2 + (%-l)(p-g) 2 + (2^- j p-^) (x-l)(p + q) + W>0, 
which is true since %>1 and 2N > p + q. Hence, since XP9/(X~^) 1S a l wa ys 
positive, the quadratic has always one and only one real positive root. 
H? _ XI - n PQ (x ~ !) 
Further : = 
dp N + (p + q- 1n m ) ( X -1 ) ' 
but x is > %— l s §> npq, and p > r> pq , therefore it follows that both numerator and 
denominator are positive, or &n pq increases with p. Similarly it increases with q, 
or in subtracting n pq from either n p+ $ p , q or n Ptq+ & q we shall never reach a negative 
difference. Thus it is always possible to construct a surface for which Q is 
constant for every fourfold division. It seems to us that had Mr Yule realised 
the possibility of this surface and studied it*, he would have known more about 
the real properties of his Q and its bearing on such distributions as occur in 
practice. The fact that in every distribution of continuous variates we come 
across there is no approach to constancy in Q, that it varies continuously and 
almost in a predictable manner shows how very far the surface of constant Q is 
from representing the facts of experience. Still had Mr Yule fitted the best 
surface of constant Q to a known distribution of detailed data, and so ascertained 
his value of Q, he would have given us a coefficient which would have lived in 
statistical practice and theory, and he would have thrown real light on the relation- 
ship of association to correlation. We have not spent time in discussing the 
complicated equation to the surface of constant Q, but we have provided one 
illustration of such a surface. Taking the total frequencies of each eye-colour 
group in Father and Son, only adding 5 and 6 together, we have Table I given 
below for the eye-colour distribution categories ; this table would within the 
limits of our decimal places have the same coefficient of association, Q = 0'6, 
wherever we divide it into fourfold tables. For example, taking both divisions 
between 2 and 3 we have the fourfold 
$=•59996. 
191 -55 
143-45 
335 
166-45 
498-55 
665 
358 
642 
1000 
* We suggest that Mr Yule has not studied the matter, for he writes : "There is one case and one only 
where Q is independent of the axes chosen, and that is where the variables are strictly independent," 
Phil. Trans. Vol. 194, p. 278. For the equation to the surface see Appendix III. 
