L. ISSERLIS 405 
The coefficient oi ^ySz' — x>iRz' on the left reduces to 
1 — 
xy 
.(68). 
^ I xij ^ ' xy ^ I xy 
yVz' ~ I'^ zy iyVx' ^"xy) / >'yz ^'xy 'xzV 
1 - r'xy 1 - yVx V 1 - / 
Similarly 
We shall still be correct to second order terms if when using (68) and (69) in 
(67) we replace 1 — ,,7]^" and 1 — ^rj,/ occurring in denominators by 1— so 
that 
( 77 2 _ I? 2\ g^-.'/^ ~ 1 
\xy-i-lz xy-l^z ) „ 
"^'xy^jfiy Qxy^ "/'yz '':cz 
xy 
^x-y'^'xz ^xy-f'yz 1 '"a;; 
^'xy^xy'^ ga;'-)/ ''0:^ '^'yz^'xy 
+ 
^xy-'^yz ([iS-yl'xz 1 ^' 
1 - ''"uv/ 
(70). 
§ 8. This result is of importance, as it shows that the heavy labour of the 
direct calculation of the generalised correlation ratio can be replaced by the 
calculation of four simple correlation ratios. 
The coefficients involved are the ordinary coefficients of linear regression 
denoted above by 73, 73' and expressions involving product moments of orders 
3 and 4. To these latter we may approximate by the methods of § 6. 
A good approximation for ^ is ^ . If greater accuracy be needed 
we may use 
_ (H/32+/32'-6) + 2)/% 
qxiy' - r\,j 1 + 1%, + i (/Ss + /S/ - 6) r\,j ' 
We saw in § 6 (equation 55) that 
qx'y = + "^xV/ - r%, V^,-/3i- 1 , 
approximately. /3i and /S/, which are zero in normal correlation, will in general 
be very small compared with ^j'?/ — r'xy and ,,7]^' — r-^y so that 
■rxyqxih, - qx y'^ ^ fxy '^{^■j-'^) ixVy' - r%,) - V (/3./ - 1 ) (,;7;/ - /%) 
9x^j/r^z - ga;y='> r,.^ V(/3, -!)(:, 7;/- ?%) - \/(;S/-l)(.,7/,.^- r^^j/) ' 
and 
^•a;>/grr.v^ - q^^y ^ ^Vv (^2' - 1) {yVx^ - - V (/3.. - 1) {xVy" " ^%/) 
^^y^^yz - qx^yTxz r,j, \/{l3:-l){yr)^'-7'%y) - '^W2-l){xVf-r%) ' 
Biometrika x 52 
