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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