V.—MANIFOLD CLASSIFICATION. : 
For suppose that the sign of association in the elementary 
tetrads is positive, so that— 
(4 wh ”) (Amir Brtr) > (dn B nD (4,.B nt1) ! (1) 
and similarly, 
(An1Bo)(Ami2Busr) > (Amie Bu) (Amir Busr) / i 
Then multiplying up and cancelling we have 
(An Bp)(Ami2Bus1) > (Ams Br) (AmB) ” : (3) 
That is to say, the association is still positive though the two 
columns 4,, and 4,,,, are no longer adjacent. 
(2) An wsotropic distribution remains isotropic in whatever way 
ut may be condensed by grouping together adjacent rows or columns. 
Thus from (1) and (3) we have, adding— 
(4B) [(dns1Brsr) HE (Ams2Bns1)] > (AnBri)[(Adms1 Br) ot (4ms2Bn)); 
that is to say, the sign of the elementary association is unaffected 
by throwing the (m+ 1)th and (m+ 2)th columns into one. 
(3) As the extreme case of the preceding theorem, we may 
suppose both rows and columns grouped and regrouped until 
only a 2 x 2-fold table is left ; we then have the theorem— 
If an isotropic distribution be reduced to a fourfold distribution 
wn any way whatever, by addition of adjacent rows and columns, 
the sign of the association in such fourfold table is the same as in 
the elementary tetrads of the original table. 
The case of complete independence is a special case of isotropy. 
For if 
(AnBo) = (An) (BLN 
for all values of m and =, the association is evidently zero for 
every tetrad. Therefore the distribution remains independent 
in whatever way the table be grouped, or in whatever way the 
universe be limited by the omission of rows or columns. The 
expression ‘complete independence ” is therefore justified. 
From the work of the preceding section we may say that Table 
IL. is not isotropic as it stands, but may be regarded as a dis- 
arrangement of an isotropic distribution. It is best to rearrange 
such a table in isotropic order, as otherwise different reductions 
to fourfold form may lead to associations of different sign, though 
of course they need not necessarily do so. 
12. The following will serve as an illustration of a table that 
is not isotropic, and cannot be rendered isotropic by any rearrange- 
ment of the order of rows and columns. 
69 
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