1909-10.] The Significance of the Correlation Coefficient, etc. 497 
exists in addition ; and even when reduced by the inclusion of the hybrid 
with the dominant they must approach those given by observation. 
39. The effect of parental selection on fraternal correlation remains to 
be considered. Referring to the parentages before given with reference to 
the correlation of offspring and parent, the two chief cases are given. 
Case I. Let the parentage on both sides be m (a, a) 4- 2, (a, b) + (b, b), 
and let the pure zygotes (a, a) be in excess or defect; and, 
Case II. Let the parentage on both sides be (a, a) + 2m, (a, b) + (b, b), 
and let the hybrid (a, b) be in excess or defect. 
Then if n = 2, i.e. if the family be 8 on an average, we have the 
fraternal correlation as in the accompanying table : — 
Fraternal Correlation. ( Hybrid distinct.) 
Value of m. 
Case I. 
Case II. 
•5 
•333 
•514 
1*0 
•428 
•428 
2*0 
•523 
•347 
3-0 
•572 
•314 
40. Thus, such selection as that when the dominant is in excess or the 
hybrid is in defect tends to raise the correlation, while the opposite condi- 
tion tends to lower it. If both conditions exist, and if the parentage be 
such that the dominant is twice as numerous and the hybrid half as 
numerous as in the stable population, we have r = - 70 when hybrid is 
distinct and r = ' 40 (product method) when dominant includes the hybrid. 
Consideration of Actual Cases. 
We have seen that many different factors affect the value of the 
correlation coefficients. What effect these have practically can only be 
estimated in a few cases. Professor Pearson has considered three cases of 
colour inheritance, namely : — 
1. Coat colour in horses.* 
2. Coat colour in cattle.f 
3. Coat colour in greyhounds.^ 
Each of these cases will be briefly discussed and the divergences of 
value in the correlation coefficients explained as far as possible on the basis 
of what has gone before. 
* Roy. Soc. Trans., vol. cxcv. p. 92. Biometrika, vol. i. p. 361 ; vol. ii. p. 230 et seq. 
t Biometrika, vol. iii. p. 245 et seq. J Ibid., vol. iv. p. 427 et seq. 
