96 The Correlation Coej^cient of a Polychoric Table 
Let l-r„ = To', l-e,^-B,'. 
Mean in a, 
m a 1 ' ^ 
^ = To + - 1 + ^ 
= To + ^0 - 1 + -To'^o' + -TiOir + T^dJr^ + ... 
== To + ^?o - 1 - (1 - Tn) (1 - e,) + (H. (t', r, r) 
^r,d, + VI{T',9',r) (2). 
Mean in b, 
m c 
N^^''~ N 
= To - (to^o' + r,e,' (- T) - r,d,' (- r)» + ...^ 
= To - To (1 - ^?„) - ® (t, 0', ~ r) 
= Toeo-® (t, -r) (3). 
Mean in c, 
= ^0 - (ro'^o + r^9, (- r) + t,'^/ (- r)^ + ...) 
= ^o-(i-To) e,-VL{T', e,-r) 
=^r,d,-'^{r',e,~r) (4). 
Mean in d, ^ = r^B^ -\- (r, 6, r) (5). 
In place of taking a quadrant we may take a marginal or internal block. I shall 
only consider the latter as the case of a marginal block may be deduced from that 
of an interna] block by removing one of the bounding planes to an infinite distance. 
In discussing the central block we in effect reduce any table to a 3 x 3 (ennea- 
choric) table as we consider it to be constituted of a central block (or group of cells) 
and 4 marginal and 4 corner blocks. I may therefore use the nomenclature for a 
3x3 table without any loss of generality. 
.-. f = .,7-0 + ® {,T, ,e, r) - iT„ A - ^ ii-r, A r) 
- 2^c 1^0 - ® (2T, i9, r) + iTo + ^ (iT, ,9, r) 
= (2-^0 - i-^o) (2^0 - 1^0) + (2-^1 - i^-i) (2^1 - 1^1) ^ 
+ (2^2 - i^i) (2^2 - 1^2) '>'^ + <>tc. 
Ill fVi 
= + iir, - iTi) (A - A) r + {,T, - ,r,) (A - A) + etc. 
(6). 
As an example of the rearrangement of the formula for computation consider 
the case when the mean is in ^22: 
(mean in a) = .^Tq 2^0 + 2^1 2^1' ^ + 2'^2' 2^2' ''^ + 2^3 2^3' ''^ + 
