568 



Journal of Agricultural Research 



Vol. XX, No. 7 



Thus, the correlation between two variables is equal to the sum of the 

 products of the chains of path coefficients along all of the paths by 

 which they are connected. 



If we know only the effects, X and Y, and correlated causes, such as 

 M and N, it will be well to substitute r MN for bb' in the foregoing formula. 



>xy = Px-mPy-u + />x-m*mn/>y-n+ Px-sPy-s + />x-n*WVm- 



We have reached a general formula expressing correlation in terms of 

 path coefficients. This is not the order in which knowledge of the coeffi- 

 cients must be obtained, but, nevertheless, by means of simultaneous 

 equations the values of the path coefficients in a system can often be 

 calculated from the known correlations. Additional equations are fur- 

 nished by the principle that the sum of the degrees of determination must 



Fig. 6. Diagram showing relations between two 

 variables. A' and V, whose values are de- 

 termined in part by common causes, M and 

 N, which are correlated with each other. 



Fig. 7. — Simplified diagram of factors which 

 determine birth weight in guinea pigs. 



equal unity. The fundamental equations can be written in general form 

 as follows: 



^X-A = PVa 

 ^X'ab = 2 Px- Arx-B^AB 



APPLICATION TO BIRTH WEIGHT OF GUINEA PIGS 



As a simple example, we may consider the factors which determine 

 birth weight in guinea pigs (fig. 7). 



Let X be birth weight; Q, prenatal growth curve; P, gestation period; 

 L, size of litter; A, hereditary and environmental factors which deter- 

 mine Q, apart from size of litter; C, factors determining gestation period 

 apart from size of litter. 



For the sake of simplicity, it will be assumed that the interval between 

 litters (if less than 75 days) accurately measures the gestation period 



