ON THE PARTIAL CORRELATION RATIO. 
PART I. THEORETICAL. 
By L. ISSERLIS, B.A. 
§ 1. The theory of non-linear regression in the case of two correlated variables 
is due to Prof Karl Pearson*. He shows that regression ceases to be linear 
when the correlation ratio 77 differs sensibly from the correlation coefficient r and 
establishes criteria for parabolic, cubic and higher forms of regression. 
The present paper deals with the regression surface of three correlated variables 
X, y, z, where, though the regression of z on x, y cannot be adequately represented 
by an equation of type 
= 7.-^— + 73^ (1)' 
the regression of z on x for a constant y and of z on y for a constant x is linear. 
A large proportion of the non-linear cases that occur in practice fall into this 
class. It will be remembered that z^y in (1) denoting the mean of the array of ^■'s 
for a given x and y the coefficients 73, 73' are partial regression coefficients and it 
will appear that just as it is necessary to introduce the correlation ratio 77 fur an 
adequate description of non-linear regression of two variables, there must be 
introduced multiple or partial rfs for the description of such regression in the 
case of more than two variables. 
We recall the definition and principal properties of xVy — the correlation ratio 
of y on X, being the square root mean weighted square standard deviations of 
the arrays of y : 
, S{n^a\) SS[n.^y{y-y^)-} 
(1 - 7/-) a,f = a-„^ = = (2), 
7]- = 
a-,/ Na-y 
•(3), 
and N (t - r') = S {y,,^ - Yf} (4). 
* Drapers' Company Research Memoirs. Mathematical Contributions to the Theory of Evolution. 
XIV. "On the General Theory of Skew Correlation and Non-linear Regression. " 1905. 
50—2 
