408 
On the Partial Correlation Ratio 
The three identities connecting the six simple t/'s become in this notation 
{R - RJi%, = 4 {r,, ix' - VyX) {y/R' {r^,jfM' -X)- y,R (r^,\ - /)] • • -(79), 
(P - Pyr%, = 4 (r,, v' - {7/ P' (r,, v' - fi) - y,P (r,,/. - v')} . . .(80), 
( Q - QJr\,. = 4 (7-,., V - r^, v) {7/ Q' {r,^\' -v)-y,Q (r,^ v - \')] . . .(81). 
(i) We can deduce from (79) that if the correlations of both x and y on ^ be 
linear and equal, then the correlation ratio of a; on ?/ and y on x, i.e. ^Vy yVx 
are equal. Thus in biparental correlation, if the regression of the child on each 
parent be linear, then the correlation ratio of the father on the mother is equal to 
that of the mother on the father. 
For under the conditions stated 
xVz ~ yVz ~ 'f'zx ~ '^'zy 
Hence 73 = 7/ = — — — and (79) becomes 
7/ [A.- - yu,'=J= r\.„ = 4r„^ {^i - X) {- y./X- {')\-,,fjf - A,) + y.//jf- {r^yX - fx')], 
or 7j ( A- - jjff {X + fi')- r\y = ivy, {X - /z')'- {r„jXfj,' -X- - Xfu,' - /x'-}, 
(X, + f/)- — 4 (r^-yXfj, — XfM —X" — fjf-) 
= 0, 
or (X — /jfy- 
which reduces to 
But is numerically < 1. .". the factor in curved brackets is positive. Hence 
X = //, I.e. yUx = xUy. .• . y??^- = 0:^1 11 ■ 
(ii) An interesting deduction from the identities (79) — (81) is the following: 
" If any four of the six regression lines that occur in the mutual variation of three 
variables are linear, so are the other two." 
We have to prove that if any four of the six quantities X, /u,, v, X', fi, v vanish, 
then the remaining two vanish as well. 
First let A, = /X = 1/ = X' = 0. 
(79) gives r,./ (73 + v'-J = 4r,.,^' [yiv'-r^yij: - 7^73 Vj, 
(80) y:'v'ryi=-^v,yv''y,y;\ 
(81) /''r^/ = 0. 
From ( 80) 7/'^ v'' [47, Vy^ + 7/= r,,/] = 0. 
But 47,7-,., + y;H\„ ^ {'''■'^'''^^ " > 0. 
.-. ^.' = 0, 
and /Li' = 0, 
and these satisfy (79). 
There are three cases of this type. 
