454 JOURNAL OF the; WASHINGTON ACADEjMY OF SCONCES VOIv. 11, NO. 19 



Thus we shall have 



^J^j /a " a^a 



Pyz 



(8) 



r 





n 



J L 



where p is used to denote spurious correlation. 



All of the special cases considered under equation (1) will exhibit one 

 or more forms of spurious correlation. 



Xi Xi 



For example, when y = — and z = — ; we may have three different 



Xi Xi 



cases of spurious correlation: 



(a) when the fractions have the same denominator, 2. ^., ^^2 = ^i 



(b) when the denominator of one is the same as the numerator of the 

 other, i. e.,X2 = Xi' 



(c) when the fractions have common numerators, i. e.,Xi = Xi'. 

 The formulae for the spurious correlation in these three cases are 



(a) 



(b) 



(c) 



V2' 



Pyz 



Pyz 



P yz 



^ (^^1^ + ^2^) K'^ + ^2^) 



-"02^ 



V ivi' + v^') iv^' + ^.'2'^) 



Vi' 



(9) 



(10) 



(11) 



V (Di^ + 1)2-) (^1^ + ^'2'^) 



Of these three cases the first one is the one which usually arises in 

 practice. 



When the functions considered are products they may have a 

 common factor so that y = ^^1:^2 and z = Xx'xi. 

 Then 



2 



'01' 



Pyz — 



^{ih^- + V2') (vr- + c^2-). 



(12) 



