THEORY OF STATISTICS. 
Grandparents and Grandchildren : Parents not-light-eyed. 
Proportion of light-eyed amongst the 
Ts of a lL | tie = oe =58'3 per cent. 
parents . . 
Proportion of light-eyed amongst the 
ais of nigh LL B08 503 mn 
grandparents ~~. : {ey LS 
In both cases the partial association is quite well-marked and 
positive ; the total association between grandparents and grand- 
children cannot, then, be due wholly to the total associations 
between grandparents and parents, parents and children, re- 
spectively. There is an ancestral heredity, as it is termed, as 
well as a parental heredity. 
We need not discuss the partial association between children and 
parents, as it is comparatively of little consequence. It may be 
noted, however, as regards the above results, that the most 
important feature may be brought out by stating three ratios 
only. 
If 4 and B are positively associated, (4.8)/(B)>(4)/N. 
If 4 and C are positively associated in the universe of Bs, 
(4BC)/(BC)>(4B)/(B). Hence (4)/N,(4B)/(B),and (4BC)/(BC) 
form an ascending series. Thus we have from the given data— 
ion of light- among 
Foi " 2 Ci > 2 = (yy = 116 peleent, 
Proportion of light- amongst the 
en of ht parents : ENABY(E) S321 ss 
Proportion of light-eyed amongst the 
children of light-eyed parents and | =(4B0)/(BC)=864 ,, 
grandparents : : 
If the great-grandparents, etc., etc., were also known, the series 
might be continued, giving (4BCD)/(BCD), (ABCDE)/(BCDE), 
and so forth. The series would probably ascend continuously 
though with smaller intervals, 4 and D being positively associated 
in the universe of BC’s, 4 and Z in the universe of BC D’s, etc. 
6. The above examples will serve to illustrate the practical 
application of partial associations to concrete cases. = The general 
nature of the fallacies involved in interpreting associations 
between two attributes as if they were necessarily due to the 
most obvious form of direct causation is more clearly exhibited 
by the following theorem :— 
If A and B are independent within the universe of C’s and also 
within the universe of v's, they will nevertheless be associated 
within the universe at large, unless C vs independent of either A 
or B or both. 
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