Jan. 3, 1921 



Correlation and Causation 



565 



Unfortunately we can not deal with chains of factors which involve 

 nonlinear relations by mere multiplication of the path coefficients of the 

 component links. In the present paper, unless otherwise stated, it will 

 be assumed that all correlations are m 



essentially linear. >^ rt 



EFFECTS OF COMMON CAUSES 



Suppose that two variables, X and Y, 

 are affected by a number of causes in 

 common, (B, C, D). Let A represent 

 causes affecting X alone and E causes 

 affecting Y alone (fig. 2). 



Let px-A = a 



Px-b = b 

 px-c = c 

 px-D = d 

 px-E = o 



B, C, and D are assumed 

 dependent of each other — thai 

 etc. 



Hence 



px-B = r X B, etc. 



rxY—bb' 



r 



B XY 



Fig. 2. — Diagram showing relations be- 

 tween two variables, X and Y, whose 

 values are determined in part by com- 

 mon causes, B, C, and D, which are in- 

 dependent of each other. 



V(i-& 2 ) (i-6 /2 ) 



B^XY B^XC B^YC 



fxY — bb' — cc' 



■yl(l-b 2 -C 2 )(l-b' 2 - c ' 2 ). 



X 



V(l— B^XC) (I — B^yc) 



When all common causes have been made constant, dcb^xy = o 



rxY = bb' + cc' + dd' = 2p x .BpY.B. 

 Thus, in those cases in which the causes are independent of each other, 

 the correlation between two variables equals the sum of the products of 



the pairs of path coefficients which con- 

 nect the two variables with each common 

 cause. An illustration of the use of this 

 principle was given in an earlier paper 

 (c?) in analyzing the nature of size factors 



X^ y \ ^~ in rabbits. 

 ^^ > C £y It may be deduced from the foregoing 



Fig. 3.— Diagram showing relations be- formula that two variables may even be 

 tween two variables, x and y, whose com pletelv determined by the same factors 



values are completely determined by 



common causes, b and c, which are in- and yet be uncorrelated with each other. 



dependent of each other. j^t variation of X be completely deter- 



mined by factors B and C, the path coefficients being b and c, respectively. 

 Let Y be completely determined by the same factors, the path coeffi- 

 cients being b' and c' (fig. 3). Then rxY = bb' + cc'. The condition 



