100 The Correlation Coefficient of a Polychoric Table 
Mean {dn^^dn^.) = - ^^^^ (28 e), 
Mean {dn,,.dn, .) = n,,- (^1 - (28/). 
Hence squaring both sides, summing for all possible samples and dividing 
by the number of samples we have, 
(Xn - Xl2 - X2I + X22)^ (^r^ = (^12 + ^21)- 'hi + ^2I^ «21 + (a22 + ^2l)^ %1 
+ "12^ '^.12 + »^22 + a!:2^%2 + (ai2 + jSoo)^ + ^22^1123 + (ttj. + 1832)^ n^s 
^ ((ai2 + ^21) + Ai»2i + (a22 + ^21) "31 ) ^ 
+ 012%, + + a22W32 I (29). 
I + ("12 + ^£2) ^13 + Al« 23 + («22 + ^22) '''22 J 
The expression within the large brackets 
= aia^i- + 022^3. + ^21 '^-•i + ^i2'>^-z + ^22- 
Calling this Nm we have 
(Xii - X12 - X21 + X22)^ (^r^ = {0.12 + ^21 - '")^ 'hi + (^21 - "0^ 'hi 
+ ("22 + ^21 - '^31 + ("12 - '«')^ '^2 + (1 - "0^ «22 + (^22 " 'Hz 
+ {0-12 + ^22 ~ "^Y ''13 + i^22 - »*)^ '^23 + (a22 + ^22 - "0^ »^33 (30). 
The following form for m is instructive although giving an apparently less 
symmetrical form than the above, 
m = ^ («i2%- + «22%- + 1821^.1 + ^22"-3 + "22) 
N 
\_ 
N 
iA,2 - ^11) I'll. + {B22 - B,2) {N - m.2) + (^21 - 5,1) m.i 
= 4 \Al2"h- + Bi2m.2 — ^12 - ^11^1. - ^ll'J^-l + Wll - ^22'»2- 
+ (522 - ^12) - ^-2) + "''22 - ™2l - ^^12 + »hl 
1 
N 
^22^. 2 + m22 + ^2i»i2. + -BjjHt.i - ni2i + N {A22 - A21 + ^22 - -812) j 
Pn ~~ P12 ~~ P21 + ^22 
022 + P32 
N 
= a22 + ^22-§ (31). 
We may then write 
(Xll - X12 - XU + X22)^ (^r^ 
■ = (^ai2 - ^22 + ^21 - ^22 + ^ j »hl + [0- a22 + ^21 " ^22 +^) ''2I 
/ m\2 / 9a\2 
+ I 1^21 - 1^22 + ^ j >^31 + ( "12 - a22 + 0 - ^22 + ^ ) ''l2 
+ - ^22 + ^ - ^22 + f )' '"22 + (0 - h2 + f )' "32 
+ (a^2 - "22 + f )' »i3 + (0 - a22 + H23 + (f )' %3 (32). 
