402 
On the Partial Correlation Ratio 
xtjilz ~ yVz = SSS 
xy z 
n^, /a (x - Xy) ^ cy {x - Xy)\ / ^ _^ ^ x + Xy ^^hj 
N \ ax (Jy <Ty J \ (Ty 
cy X + x\ 
+ u + 
Cjj\ fx + Xi 
II (V 0-;,//\ O-yJ X <T^ JSl ] ^y [ ai \ (Ty) N) 
Now S [nxyx"] = S {{x - Xy + XyY n^y] = iiy {a^^' + 0 + ,«,/), 
xy H f 
SUa+'~l)\\-yrj^) 
N 
lacy ^ c^f\) ^ 
cTy a-y'J)' 
:. xyU^ - y?;/ = (1 - y-rix') ia' + c') (59). 
Similarly ^y H/ - ^.t?/ = ( 1 - ^77/) (^'^ + c^) (60). 
Remembering that a = 73 + c^, 6 = 73' + c^ we get from (.59) and (60) 
1 y Vx 
73 
XljH/ xVz'\ '/ ± I Q\ 
1— —2 = 7;i73 (73</> - 73 G) 
-I xVy ' 
+ & {rs'<f> - 73^ - 0^ (73</> - 7/ (61 ). 
From the values of 73, 73', 6, (f) in (9), (10), (41), (42) we obtain easily 
J, ^, ' a — ' ^2?a;2.</ — 't'xzlxy"- 
739-73(7 = — 
xy 
73'</>-^"3< 
73> - 73^ - (730 - 73'^) 
^'xz<ix'y '>'yz9xy - 
1 - 1%, 
_ ^ xz^x^!/ fyz'ilxy '' i'^'xy({xy'' (Ix^y) 0'xy<]xh/ ' - qxy') (»' 'yzflxiy '>'xz 9_xy'') 
xy 
{^-r\yy 
and is given by (52). Hence (61) may be written 
i^xyH^" xyRz') 
iXyz Ifxz^'xy) iXxy^xy^ ~ (jx'^y) O'xz ^yz^'.ri/) 0'xy<ixy'^ 9x"ri) 
" (1 - r\yr (1 - yVx')~ (1 - r\yy (1 - ^Vy'Y 
{'^'xzQx'iy ~ "^^yzQxy ) ('''xy^xy- Qx^y) i^xy^ai-y Qxy-) (Tyz^x-y ~ "^^xzfjxy^) 
1 — r^. 
(l-r%,)^ 
{q^,y. - r%) - 
(9V-1/ + 9 a^.'/- ^'Jxy-fjx'^y'^'xy) 
xy 
