39 



ent reactions. The author feels that considerable information in an ab- 

 stract, concise, and workable form may be obtained from a study of the 

 coefficients of correlation. 



The Coefficient of Correlation. Where we are concerned merely 

 with the presence or absence of characters the coefficient of correlation be- 

 tween any two characters may easily be determined. Suppose that it is 

 desired to know if the characters X and Y are correlated and that a study 

 of a number of organisms showed that 'a' cultures are positive for both 

 X and Y; 'b' organisms positive for X but negative for Y; V cultures are 

 negative for X and positive for Y; and 'd' strains are negative for both X 

 and Y. The distribution of the organisms is first tabulated thus: 



Y 



The degree of association, or the coefficient of correlation, is then 

 expressed, according to Yule, by the formula 



' ad-bc (1) 



ad+bc 

 If 'ad' is equal to 'be' the coefficient becomes 



or 0; which 



ad _|_ be 



indicates that there is no correlation whatever. If either 'b' or 'c' is zero 



ad 



the formula becomes~j~-=l; indicating a perfect positive correlation. If 



be. 



c a' or 'd' is zero then we have ^~~= 1; showing a perfect negative cor- 

 relation. It should be observed that an absolute positive correlation exists 

 in reality only if both 'b' and 'c' are zero and an absolute negative cor- 

 relation when both 'a' and 'd' are zero. In order to avoid coefficients of 

 1 or 1 where only one group 'a', 'b', 'c', or 'd' is zero. Yule gives 

 the formula 



a (a + b + c + d) - (a + c) (a + b) (2) 



\/ (a + c) (b + d) (a + b) (c + d) 



In practice, however, a few strains are almost always found in each 

 of the four groups and Yule suggests the use of the simpler formula (1). 

 Some caution should therefore be employed in interpreting coefficients 

 of 1 or 1. 



It was assumed that if the coefficient between two characters is numeric- 

 ally greater than 0.5 they may be regarded as correlated, but if less than 



39 



/^es/ft 



Corre/' 



