5°4 



Journal of Agricultural Research 



Vol. XX, No. 7 



coefficients of determination may equal unity. Regression is linear and 



M 2 a 2 



— I — -• Thus d x . A = r\ A as in the case of independent 



additive factors. The term 



)A' 2 B' 



is small unless the amounts of 



variation in A and B are large in comparison with the mean values. In 

 many cases it is safe to deal with path coefficients and degrees of deter- 

 mination in the case of multiplying factors just as in the case of addi- 

 tive factors. 



As a concrete illustration of these points take two independent vari- 

 ables, for each of which the values 1, 2, and 3 occur in the frequencies 

 1,2, and 1, respectively. Below is the correlation table between one of 

 these factors and their product. 



Product (X). 



M A = 2 a A =^JTf2 r AX =-y[8/Tj 



</> 



8/17 



., ,— - 2A' 2 B' 2 . d x . B =8/i 7 

 v ^ n<r 2 x 1 i/i7 



"X-AB 



I 



In this case the amounts of variation in the factors are relatively large 

 compared with their mean values, making the distribution surface mark- 

 edly heteroscedastic, yet the degree of determination by either factor 

 comes out only slightly less than one-half. 



NONLINEAR RELATIONS 



<t( m ) 

 Pearson's definition of the correlation ratio, t? x . a = > has already 



fx 



been given. The variations of the mean value of X for different values 

 of A are the variations which can be attributed to the direct influence of 

 A, assuming that A is cause, X effect, and that other causes are com- 

 bined with A additively. Thus o- x .a = o-( a m x ) and we have at once 



Again, as the total variation of X is composed of the variation of its 

 mean values for different values of A, plus the variation about these 

 mean values, we have o- 2 x = <r 2 ( A M x ) + A(T 2 X , giving A o r2 x = o- 2 x (1— v 2 x.a), as 

 already noted. 



Thus r? 2 x . A measures the portion of a\ lost by making A constant, so 

 that as before d x .A = v 2 x-A = p 2 x-A. 



