420 
R. A. FISHER ON THE CORRELATION BETWEEN 
Article 5. Then, since we are using the term environment formally for arbitrary 
external causes independent of heredity, the mean x of a group so chosen that y = t 
for each member will be simply t, but the mean y of a group so chosen that x = t for 
each member will be cp, where c x is a constant equal to the ratio of the variance 
with environment absolutely uniform to that when difference of environment also 
makes its contribution. Similarly for the group 2 = t, the mean value of y is t, but 
for the group y = t the mean z is c%t, where 
C 2 
0-2 _ A e 2 
(XXVII) 
Now, we may find the parental and grandparental correlations from the fact that 
the mean z of any sibship is the mean z of its parents ; but we shall obtain very 
different results in these as in other cases, according to the interpretation which we 
put upon the observed correlation between parents. For, in the .first place, this 
correlation may be simply the result of conscious selection. If the correlation for 
height stood alone this would be the most natural interpretation. But it is found 
that there is an independent association of the length of the forearm # : if it is due 
to selection it must be quite unconscious, and, as Professor Pearson points out, the 
facts may be explained if to some extent fertility is dependent upon genetic 
similarity. Thus there are two possible interpretations of marital correlations. 
One regards the association of the apparent characteristics as primary : there 
must, then, be a less intense association of the genotype y, and still less of z. 
The other regards the association as primarily in y or z, and as appearing somewhat 
masked by environmental effects in the observed correlation. In the first place, let 
us suppose the observed correlation in x to be primary. 
Then if n is the correlation for x, c x n will be that for y , and this must be written 
for n in the applications of the preceding paragraphs. Hence 
A = CjCj/x, 
and /*, cp * and A are the marital correlations for x, y , and z. 
Since the mean 2 of a sibship is equal to the mean z of its parents, we may 
calculate the parental and grandparental correlations thus : — For group chosen so 
that x = t : mean y, y = c\t ; mean z, z = CiC^t ; x of mate is A*tj z of mate is ciC 2 mL 
Therefore z of children is 
Hence, since there is no association except of z between parents and child, the 
parental correlation coefficient is 
1 + w 
c i c 2 2 ' 
Now. since we know the mean 2 of the children to be 
Pearson and Lee, “ On the Laws of Inheritance in Man,” Biometrika, id, 374. 
