Jan. 3,1921 Correlation and Causation 563 



provided that these causes are independent of each other, have linear 

 relations to the dependent variable X, and that the deviations which they 

 determine are additive. They are independent of each other if there is 

 no correlation between their variations. A cause has a linear relation to 

 the effect and is combined additively with the other factors if a given 

 amount of change in it always determines the same change in the effect, 

 regardless of its own absolute value or that of the other causes. The con- 

 clusion is that, under these conditions, the path coefficient equals the 

 coefficient of correlation between cause and effect, and the degree of 

 determination equals the square of either of the preceding coefficients. 



CHAINS OF CAUSES 



If we know the extent to which a variable X is determined by a cer- 

 tain cause M, which is independent of other causes, combines with them 

 additively, and acts on X in a linear manner, and if we know the extent 

 to which M is determined by a more remote cause A , the degree of deter- 

 mination of X by A must be the product of the component degrees of 

 determination. 



Let X = M+N, and M = A+B 



O^m , _^!a a j _0~ 2 A 



° X u M °X 



Thus d x . A = d x . M d u . x 

 and p x . A = Px-mPk-a- 



NONADDITIVE FACTORS 



In cases in which a factor does not act additively with the other factors 

 in determining the variations in the dependent variable, its influence on 

 the latter can not be completely expressed apart from the other factors, 

 at least in terms of the ordinary measures of variability. This can be 

 made clearer by an illustration. Multiplying factors are among the most 

 important of those which do not combine by addition. 



I^et X=AB and assume that r AB = o 



v\ = M \o\ + M\a\ + —f- 



where A' and B' are deviations of A and B from their mean values M A 

 and M B . Putting B constant, we have o" 2 x . A = M\<r 2 A ; and similarly 

 putting A constant, we have o" 2 x . B = M 2 A <r 2 B . There remains a portion of <r 2 x 

 which is due to A and B jointly and which can not be separated into parts 



M 2 <7 2 



due to each alone. If we write d x . A = — f — as the degree of determi- 



X 



M 2 K <T 2 „ 



nation of X by variation of A alone, and d x . B = —2 — ~ as the corre- 



x 

 sponding degree of determination of X by variation of B alone, we must 



2A f2 B' 2 

 recognize an additional term 4-ab = 2 — ' i n order that the sum of the 



