562 Journal of Agricultural Research vol. xx, No. 7 



coefficient for the path from A to X will be defined as the ratio of the 

 standard deviation of X due to A to the total standard deviation of X. 



Px-a = - 



"x 



0", 



Just as the regression of X on A is expressed by r^— the deviation 



0"a 

 of X directly caused by a unit deviation of A is given by the formula 



. °X °"x-A 



Px-A „ = rr~' 

 °A &A 



Another coefficient which it will be convenient to use, the coefficient 

 of determination of X by A, d x . A , measures the fraction of complete 

 determination for which factor A is directly responsible in the given 

 system of factors. This definition implies that the sum of such coefficients 

 must equal unity if all causes are accounted for. 



SYSTEMS OF INDEPENDENT CAUSES 



The degree of determination of one variable by another is most easily 

 found where the variables are connected by a mathematical relationship. 

 The simplest mathematical relationship is that between a sum and its 

 components. For the standard deviation of a sum the following relation 

 is well known : 



o"a+b = — -=* A + <r 2 B +2<r A 0- B r AB . 



If A and B are independent of each other, r AB = 0, and we have 



0- 2 A+B = 0- 2 A + ^ 2 B- 



The degree to which variation of the sum is determined by that of each 

 component is obvious. 



^x-A = _ 2^ and d x . B = -^> where X= A +B, 

 <* x ' o" X 



giving d x . A + d x . B = 1, as required by definition. 



For the standard deviation of X due to A we have in this case, c x . A = cr A . 



Thus, £x-a = — = — bv definition. 



°x °x ' 



. . 2(A' + B')A' 2A' 2 cr A 



Again, r XA =- L - '- — = =— • 



n<x x (T K na x a A a x 



Summing up, p x . A = ^d x . A = r XA . 



It can easily be shown that the same formulae hold in case we are 

 dealing with the sum of multiples of a number of independent factors 

 instead of with their own sum. 



We can pass at once from this case to cases in which variation of X is 

 caused in the physical or physiological sense by variation in several causes 



