Jan. 3, 1921 



Correlation and Causation 



567 



X'A+"X'B+^X*C+"x-D !• 

 X-M + ^X-N + 2 rX-MPx-N r MN "T" 



linear relations of cause and effect in which the influence of the causes is 

 combined additively. It is also easy to show that the formulae apply 

 approximately for multiplying factors. 



Summing up, p x . M = V^x-m™ - ^ 



2d x . M + 22£ s . M £ x . N r MN = 1. 



The next problem is to find the 

 degree of determination of X by a 

 factor such as B, which is connect- 

 ed with X by more than one path 



(fig- 5). 



Assume that A, B, C, and D are 

 independent and completely deter- 

 mine X. d 

 But also d 

 d x . D =i. 



"x-B = "x-M — "x-A + "x-N — ^X-C + pi G . s _a system in which the value of X is af- 



2 /'x-M/ , X-N/ ) M-B/'N-B) rememrj eringthat fected by a factor, B, along two different paths, 

 _ , BMX and BNX. 



*MN — PM-BrN-B- 



Since d 1A . x + d M . B =i, etc., we have d x . M =d x . M d u . K + d x . M d u . B = d x . A + 



dji-ud-UL-BJ an d rf X 'K =< ^X'C + ( *X-N ( *N > B- 



Therefore c?x-b= ^ x -m^m-b +^x-A-b + ^Px-uPx-nPh-bPn-b 



= P 2 x-mP\vb + P 2 xsP\-b + 2 Px-uPx-nPu-bPx B 

 = (Px-mPm-b + Px-kPx-b) 2 



Px-B~ Px-T&PtA-B + Px-NrN-B- 



These results are easily extended to cases in which B acts on X through 

 any number of causes. If a path coefficient is assigned to each com- 

 ponent path, the combined path coefficient for all paths connecting an 

 effect with a remote cause equals the sum of the products of the path 

 coefficients along all the paths. Since B is independent of A , C, and 

 D, r x . B = p x . B = Px-mPm-b+Px-nPn-b- 



GENERAL FORMULA 



We are now in a position to express the correlation between any two 

 variables in terms of path coefficients. Let X and Y be two variables 

 which are affected by correlated causes M and N. Represent the various 

 path coefficients by small letters as in the diagram. Let A , B, and C be 

 hypothetical remote causes which are independent of each other (fig. 6). 



*'xy = / , x-a/ , y-a + />x-b/ , y-b + />-c/Vc 



= mam' a + (mb+nb') {m'b + n'b') + ncn'c 

 — mm' + mbb'n' + nn' + nb'bm' . 



