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where r = 1 x , the total of the new row, and w 1 is its weight. Similarly, to add a new
column x : p × 1 requires
y : p × 1 = W 1
R 1 / 2
( x c R1 / n ) c 1 / 2 w 1
(7.42)
1
2
where c denotes the total of the new column.
These settings may be inserted into (7.40) to give the new row and column coordinates
a and b to give:
a :1 × q
y B ( B B ) 1 ,
=
(7.43)
( A A ) 1 A y .
b : p ×
1
=
This is all rather general and simplifies when we consider the actual values of A and B
used in the various forms of correspondence analysis that we have considered. In every
case we have
A = W 1
1
U α J ,
which may be written
A = W 1
1
( U V ) V β J = W 1
1
R 1 / 2
( X E ) C 1 / 2 V β J .
Now, from the orthogonality relationships of the SVD we have that EC 1 / 2 V
=
0 ,so
A = W 1
1
( R 1 / 2 XC 1 / 2
) V β J .
Interpolating a new row x :1
×
q now gives
r 1 / 2 x C 1 / 2 V
a =
w 1
1
β J
.
(7.44)
Similarly, for interpolating a new column x : p × 1,
( C 1 / 2 X R 1 / 2
B = W 1
) U α J ,
2
so that
c 1 / 2 x R 1 / 2 U
b =
w 1
2
α J
(7.45)
provides the coordinates for interpolating the new column x .
Thus, (7.44) and (7.45) are our basic formulae which cover all the variants of Table 7.1
where the settings of W 1 and W 2 are listed; the choices of α and β must satisfy
α + β = 1, but otherwise are free. These formulae can be derived from (7.40) by plug-
ging in the settings of A and B , but the direct approach is simpler. If λ -scaling is used
then a and b will have to be scaled appropriately.
Measures of fit for an interpolated row (column) follow directly from (7.38) and
(7.39).
7.4 Other CA related methods
It was pointed out in Section 7.1 that although CA was developed for two-way contin-
gency tables, it remains computationally feasible whenever the margins are positive and,
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