[ 205 ] 



XXV. The Variation of the Multiple Correlation Coefficient 

 in Samples drawn from an Infinite Population with Normal 

 Distribution. By L. Isserlis, B.A., D.Sc, Head of the 

 Mathematical Department, West Ham Municipal Technical 

 Institute *. 



§ 1. TI^HE Multiple Correlation Coefficient Ri-23... 7i is 

 X usually defined as the correlation coefficient of 

 the variable X\ with a linear function of the variables x 2} 

 x z , . . . x n , the constants in the linear function being so 

 chosen as to make the correlation coefficient a maximum. 

 It is essentially positive, and when the regression of the 

 first variable on the remaining n — 1 variables is linear, the 

 multiple correlation coefficient measures the dependence of 

 the first variable on the others. When the regression is not 

 linear this dependence must be measured by the Multiple 

 Correlation Ratio H^.y.. I have given elsewhere formulae by 

 which H x . yz can be approximately calculated when R z . yz is 

 known f- The direct calculation of H^.^ is very laborious. 



In practice Ri-23...n is calculated as a derived statistical 

 constant. In the case of three variables we have 



-"l'23 = V12 + 7 13 ~" * r \2 T lZ r 23)IO- — T 23 ^ 



and Ri-23 is the positive root, or we may use the relation 



l-BJ„-(l-f£)(l-*k). 



where r i2 -3 is the partial correlation coefficient of the first 

 two variables for constant values of the third, and is itself 

 expressed in terms of ordinary correlation coefficients by the 

 relation 



12-3 



= (^-V 23 )l^/{(l-rl 3 )(l-rl s )}. 



Thus Ri-23 is not calculated directly from the distribution. 

 The determination of the Probable Error of a multiple 

 correlation coefficient, and more generally a discussion of 

 the nature of the distribution of its values in many samples, 

 is a problem of considerable difficulty. It has been assumed, 

 by analogy presumably with a well-known formula for the 

 probable error of an ordinary correlation coefficient calcu- 



* Communicated bv the Author. 



+ Biumetrika, vols. x. & xi. (1914 & 1915). 



