212 
On Theories of Association 
We have at once from this table Mr Yule's coefficient of association 
N' (21 + m) 
Q 
N' (21 + m) + m (21' T m') ' 
But the correlation r, as found by the product-moment method for discrete unit 
characters, is 
Vwi' + 2n' (21 + m) 
V(m + 2n) (I + m) [*NN' - (m + 2n) (in' + 2n')J ' 
Now examine Mr Yule's coefficient: 
(i) There is perfect association, Q = l, when we take no heteromorphic 
fathers (m = 0). 
m, , c /I {in + 2h') 
Ihe value of r = A / ^, , T , 7 — r , 
V UN' + n (2V + m')' 
and takes, as it should do, all sorts of values according to the nature of the 
mothers, and the proportions of true dominant and true recessive fathers in 
the whole paternity. Thus, for all fathers recessive, it is zero, and for all fathers 
true dominant (n = 0) with all mothers purely recessive (V = m' = 0) it is unity. 
Thus the true correlation under the influence of selection can take every possible 
value while Mr Yule's coefficient gives perfect association throughout ! 
(ii) There is perfect association, Q = 1, when we take only recessive mothers, 
i.e. I' = m' = 0. 
The value of r = A / , — ^ ^,7^ 
(m + 2n) (I + m) ' 
and depends entirely on the distribution of fathers. When there are no hetero- 
morphic fathers (m = 0), it is unity. When there are no recessive fathers (n = 0), 
it is zero. That is to say, while Mr Yule's coefficient shows perfect relationship 
throughout, the true correlation or real association can run through the whole 
range from zero to unit}'. 
It is not too much to say that those who suggest that Mr Yule's coefficients 
of association and colligation will be of service in Mendelian problems cannot have 
had any acquaintance with the nature of those problems at all. Mendelian theory 
relates to discrete units and every coefficient which is uninfluenced by selection is 
on that very ground wholly unsuited for use with such units. Selection modifies 
correlation when we deal with discrete units just as much as when we deal with 
continuous characters, and any coefficient is valueless which directly starts with 
the property that it will not be modified by selection. 
Mendelian practice classifies under unit designations individuals which, as we 
have just indicated, often show no sharp line of division at all. In such cases to 
treat the difference of two classes as a unit is juggling with class names, not 
dealing with the things so classed. Mr Yule has failed in these matters because 
