jan. 3,1921 Correlation and Causation 575 



The last formula brings out the close relation between the path coeffi- 

 cients and multiple regression. As already noted, the most probable 

 deviation of X for known deviations of A, B, C, etc., is given by the 

 formula 



X' A XA /1' A XB B' A/ £' 



°x A xx o- A A xx o- B 0- A <7 B 



As already stated, Pearson's coefficients of multiple correlation and 

 regression were not devised especially for the analysis of causal relations. 

 The formula for multiple regression, for example, gives the most proba- 

 ble value of one of the variates for given values of the others regardless 

 of causal relations. In cases in which all the correlations are known 

 in a system including an effect and a number of causes the method can 

 be used to find the path coefficients and the degrees of determination 

 of the effect by each cause in the sense used in this paper. Such cases 

 in which the direct methods can be used are, however, relatively 

 uncommon. Where the system of paths of influence is at all com- 

 plex, involving perhaps hypothetical factors, the causal relations can 

 be analyzed only by the indirect method of expressing the known cor- 

 relations in terms of the unknown path coefficients, making the sums of 

 the degrees of determination unity and solving the simultaneous equations. 



PART II. APPLICATION TO THE TRANSPIRATION OF PLANTS 



A large body of experimental data on the factors which affect the rate 

 of transpiration in plants has been published by Briggs and Shantz (2). 

 These data are well adapted for use in illustrating the methods of analyz- 

 ing causal relations presented in part I of this paper. 



The experiments which are used in this paper were conducted at 

 Akron, Colo. , in 1 91 4. A variety of crop plants were grown in sealed pots. 

 The total transpiration was measured each day. Among the environ- 

 mental factors studied were the total solar radiation during the day, the 

 wind velocity, the air temperature (in the shade), the rate of evaporation 

 from a shallow tank, and the wet-bulb depression (sheltered from sun but 

 not wind) . The correlations between the daily transpiration of each kind 

 of plant and the integrated values of the environmental factors were pub- 

 lished by Briggs and Shantz. In order to avoid the effect of seasonal 

 change in the plants, the logarithms of the ratios of the transpiration on 

 succeeding days were correlated with similar figures for the various factors. 

 The correlations between the various environmental factors for the 100 

 days from June 18 to September 25, 191 4, have been calculated by the 

 writer from the data presented by Briggs and Shantz. This period covers 

 all the crop periods but is longer than most of them. None of the corre- 

 lations appeared to depart much from linearity. 



