5 66 



Journal of Agricultural Research 



Vol. XX, No. 7 



under which r XY may equal zero is evidently that bb' = —cc'. An 

 example may be found in the absence of correlation between the sum 

 and difference of pairs of numbers picked at random from a table. 



In many cases a small actual correlation between variables will be 

 found on analysis to be the resultant of a balancing of very much more 

 important but opposed paths of influence leading from common causes. 



SYSTEMS OF CORRELATED CAUSES 



The discussion up to this point has dealt wholly with causes which 

 act independently of each other. It is necessary to consider the effects 

 of correlation among the causes. 



Let us consider the sum of £wo correlated variables (fig. 4). 



Let X=M + N 



o- 2 x = c 2 M + <r 2 N + 2a il a i{ r MN . 



We have defined o- x . M as the standard deviation of X when factors 

 other than M are constant, but M varies as much as before. The latter 

 qualification is important in the present case, since the making of N 

 constant tends to reduce the variation of M, reducing <r M to c^V 1 ""Aim- 



The definition of <r x . M implies that 

 not only is N made constant but 

 that there is such a readjustment 

 among the more remote causes, A, 

 B, and C, that o- M is unchanged. 

 Under the definition it is evident 

 that in this case <x x . M = <r M and (r x . N 



On 

 o"x °"x" 



In attempting to find the degrees 

 of determination of X by M and N 

 we meet a difficulty somewhat similar to that met in the case of non- 

 additive factors. The squared standard deviation is made up in part 

 of elements due wholly to M and N, respectively, but in part to a portion 

 which can not be divided between them. The term 2<r iS a N r WN is due 

 solely to the fact that the variations of X, which M and N determine, 

 tend to be in the same direction and so have greater effect than if varia- 

 tions M and N were combined at random. It seems best to define d x . M 



2 

 as the degree of determination of X due to M alone. Thus d x . M =-^> 



Fig. 4. — A system in which the value of variable 

 X is completely determined by causes M and N, 

 which are correlated with each other. 



Thus p x . M = ~* and />, 



dx-N = 



The remaining term may be considered as determination by 



M and N jointly and may be written d x -^= 2p x . M p x . N r m ,. 



These rules can be extended at once to the sums of more than two 

 variables, to sums of multiples of variables, and hence, as before, to 



