NOV. 10, 1921 reed: CORRELATION BETWEEN FUNCTIONS 453 



It should be noted that the standard deviation of the variable x/ 

 does not enter this equation directly, the only factor involving Xi' 

 being the correlation coefficient r.T„.Ti'- Equation (6) which was 

 derived as a special case of formula (5) might have been obtained 

 directly from (7) . 



SPURIOUS CORRELATION 



Attention was first called to the subject of spurious correlation in a 

 paper by Pearson^ where he considered the case of the spurious cor- 



Xi Xo 



relation between two ratios of the form — and - . Pearson showed 



Xs Xs 



that although Xi, x-i, and x^ were uncorrelated variables there would 



Xl X'y ... 



be correlation between the two ratios — and — , this correlation arising 



Xs Xs 



from the fact that Xs is common to the two indices. Since this cor- 

 relation exists where there is no correlation between Xu Xi and Xz he 

 gave it the name of spurious corr elation. The arguments used by 

 Pearson in connection with the spurious correlation between ratios 

 will hold in the case of correlation between any two functions, and a 

 general definition of spurious correlation might be given as follows. 



Though no correlation exists between any two of a set of variables 

 there will still exist correlation between any two functions of these 

 variables whenever these two functions have any of the variables in 

 common. The correlation existing under these conditions will be 

 called spurious correlation. 



A general formula for spurious correlation may be derived directly 



from equation (1). Using, as before, y = f{xi, JC2, Xn) and 



2 = F{xi', Xt' Xk') we shall have spurious correlation only 



when some of the variables X\, x-i, Xn are identical with some of 



the variables x^' , x-i Xk'- 



Let Xi = X\ 



Xo = x-->' 



^ .... ^ 



^h = ^h where h <k and h < n. 

 Then, from the definition of spurious correlation, rxaX0= except 

 for the cases where a = P < hin which event Txax^ = 1 • 



^ Op. cit. 



