398 
On the Partial Correlation Ratio 
or 
. - ^,iH^' = - 2ai\_x - "2bry, - 2c(2x,jz + + b'^ + + c'q^f. 
+ 2cdrx,j + -labrxy + 2acqx2y + 2bcqxy2 (47), 
= a (- + a + br^y + cq^.y) 
+ h{- Vy, + b + + cq^yi) 
+ c ( - qxiiz + dr^y + aq^-Hj + bq^y^i + cq^iy?} 
+ d{d + cr^y) 
- ar,x - bvy, - cq^y, (48). 
The first four terms vanish by equs. (36)... (39), 
. •. xyH/ = + br,y + cq^^.y, (49). 
If we now insert the values of «, b, c from (43). ..(45) in (49) 
= (73 + cd) + (7/ + C(/>) r,y + cq^y^ 
= 73 '''zee + 73' ''yz + c ( 0r,^ + (f)ry, + q^y^ ) 
= (^^zx + <j^ry, + qxy,)- 
^ qa;ty6 + qxy"'<p + qx'-y-^ — I'-xy ' 
- — 2 {^^zi,^ '^%/) ^xti iXxx'^'xti ^'yz) qx'-i/O yz'^'xi/ ^'xz)l" (50) 
Vyx-,i/' ' xy) 
1 — 
It follows from (50) and (15) that 
qx--,!'' - r 
^ + q'x'^y 2q x ^jqxy2rxy (51) 
1 — r 
If we eliminate q^yz (which is a triple moment troublesome to calculate) 
between equations (45) and (49) we have 
= ar^x + bvy, + (qx^y^ - r"xy) + acqx^y + bcqxy^ 
= rzx (7» + cO) + 7\y (7/ + c0) + {qx-iy2 - r\y) 
+ (73 C + C-(9) qx^y + (73' C + C^^) 
= c'' [qx^yi - r-xy + ^f^,^,,, + ^qxA + c + 4>r,y + 73g.^..„ + 73' g^;;/'] 
+ 73'2.i^ + 73' ''zy 
= + [qxhf - r\y + 6qx-'y + ^qx,/^] by (46) and (11), 
Q'x-IJ "I" q'xy- 2qxyiq3SlyTxy 
Xl/Hj^ XyRz' — C" 
(]ix-y- 1' X. 
1 - f\ 
xy 
Hence 
c- = ■ 
xyH^' xyRz' 
q'^x'-y + Q~xy'^ ^qxy'-qx-y'^'xy 
.(52). 
1 - 
qx-y" ~ 1"xy 
This value of c- is positive by (51). 
Equation {ht) shows that xyHz = xy^z is a necessary condition for linear 
regression, which we have already proved in equation (26). 
