jan. 3,1921 Correlation and Causation 573 



The formulae for degree of determination by residual factors may be 

 written as follows : 



d x -o = <i>(XA) in system XA. 



4(XAB) . „, D 



x -° = 4>(ab) m s y stem XAB - 



<t>(XABC) . vat,^ 



x-0 = 4>{ABC) m s y stem XABC - 



<t>(XABCD) . „ , D _ n 



x '° = <j>(ABCD) m s >" stem X ^ 5CI? 



The degree of determination by the known causes is now easily cal- 

 culated. When all causes of variation in X are constant except A, 

 variation of X is measured by o-'-cb^x arj d variation of A is meas- 

 ured by o-'-cb^aj writing the constant factors as subscripts to the left. 

 Assuming that the relation between A and X is linear, the deviation of 

 X determined by a unit deviation of A should be constant, whatever the 

 amount of variation in A . Thus : 



. (T X = (T X-A = 0-'-CB <T X 



a A °A 0"*CB°"a 



In the case of the residual factor O, assumed to be independent of the 

 known factors A, B, C, etc., ... CBA (r = (r , 

 and we have cr x . = ...cba^x 



d v . n = 



<t>(XABC...)_*\. ... CBA <7 2 x 



" x '° 4>(ABC.) <j\ 

 Thus: 



2 <j>(XABC.) 2 

 " -cba^x ^ABC.) ° x * 



This should be the general formula for the squared standard deviation 

 with a number of constant factors. 

 Hence : 



r 2 



4>{XBC...O) 2 / <f>(ABC...O) 

 '' 4>(BC...O) <7x / <p(BC...O) ( 



<t>(XBC...O) 



4>{ABC...O) 1 



U{XBC...O) 

 Px ' A ^ 4>(ABC...O) 



4>(XBC...O) 4>{XBC...)-d x . 4>(BC...) 

 x ' K ~(j>{ABC...O) ~ 4>(ABC.) 



17777°— 21 5 



