188 Probable Errors in Multiple Skeiv Regression 
But from the usual regression equation 
-^y-\ l-r\,')a,^y\ 1 - r\., ) a,' 
so that for Unear regression 
Px^vz - ( - 1 _ ^2 — ) ~ P^'v + ( -rzTr^ ) ^ P^'y- 
Further = 4 'Sf >S {n^^xhj) 
or Px'v = Px^xy— (25), 
while q^2y-2 = 1 + r^^v'^i^t — 1) (/S'2 — 1) approximately (26)*, 
so that (23) can be evaluated approximately by the use of simple moment 
coefficients and correlation coefficients only. 7/ in addition the distribution he 
normal, we know that 
Px-uj = ^■PxvPx-'- and ])y'y2= (I + 2r^^y)jp^,p,j2, 
so that for normal distributions 
Px-yz ("^xz '^xy'^yz\ ^ , 'j/z '^xz'^xy 1 ~l~ ^'^^xy 
PxvPxz VI ^ 'j'y ' ^ xz f "^^xv '^xy'^ xz 
or ^ = 2 + rJr,,r,, (27). 
'xv ' xz 
Similarly Ixv^- ^ i ^ 2r,,r^, ^^g), 
'^xy '^xy 
and ^-f^ = 1 + 2v,,A, (29), 
'xz 
Substituting these values in (23) we obtain, after some reduction and using 
(1 'f'^xy) (1 ~" ^'^a:z) -^r^yY^^ 
~ '^yz (1 *^a'i/) (1 "^^xz) i '^xy'^xz (1 *'^K3/ ' ^i/z ^^za; ~l~ 2?'j;j,f j^^f ^j;) 
(30), 
agreeing with (6) the value obtained by Pearson and Filon for normal distributions. 
As regards the more general case dealing with the correlation between deviations 
of r^y and those of r^i given by equation (22), we have ivhen the regression is linear 
Pxvzt = SSSS {n^y.t (xyzt)} 
X y z t 
= SSS{n„j^ {xyzi^y,)}, 
X y z 
* Biometrika, Vol. ix. p. 4. 
