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approaches demonstrates the subtleties engendered by weights appearing as part of the
model and weights being used in the least-squares criterion itself. Other least-squares
weights might be preferred, in which case the approximation would be based on a dif-
ferent SVD, but with no special difficulty.
7.2.3 Approximation to the contingency ratio
Surprisingly, plots based on interpretations of the Pearson residuals seem to be little used.
Rather, redefining
X
, we may rewrite (7.8) as the minimization of
R
1
/
2
{
R
−
1
(
X
−
E
)
C
−
1
X
}
C
1
/
2
2
−
.
(7.9)
Equation (7.9) represents a least-squares problem with weights
R
1
/
2
and
C
1
/
2
where now
the matrix
X
approximates
R
−
1
(
X
−
E
)
C
−
1
whose elements
x
ij
−
e
ij
e
ij
x
ij
e
ij
−
1
1
n
1
n
=
are proportional to the deviations of
x
ij
from independence relative to the approximation
e
ij
under independence. Note that the ratio
x
ij
/
e
ij
, sometimes called the contingency ratio
(see Greenacre, 2007) can be expanded as
x
ij
e
ij
=
x
ij
/
x
.
j
x
i
.
/
n
=
x
ij
/
x
i
.
x
.
j
/
n
=
n
x
ij
x
i
.
x
.
j
.
We have thus shown that
x
ij
−
nx
ij
x
i
.
x
.
j
−
1
e
ij
1
n
1
n
1
n
(
contigency ratio
−
1
).
=
=
(7.10)
e
ij
From(7.9)wehavethat
X
approximates
R
−
1
=
R
−
1
/
2
U
V
C
−
1
/
2
,so
(
X
−
E
)
C
−
1
2
as coordinates to give
visualizations of the departure from unity of the contingency ratios. Alternatively, we
get the same inner product by plotting the first
r
columns of
R
−
1
/
2
U
and
C
−
1
/
2
V
as
coordinates. Note that these biplots do not give the departure from unity directly but the
departure divided by
n
. However, as we have seen previously, this scaling factor does
not influence the shape of the biplot and can easily be provided for when constructing
the calibrations of the biplot axes.
The biplot resulting from using
R
−
1
/
2
U
and
C
−
1
/
2
V
as coordinates has an inter-
esting property, which we refer to as the
centroid property
and which follows from
noting that the rows of
R
−
1
X
give the proportions of each column category in each row.
Weighting the column coordinates by these proportions gives
1
/
2
1
/
we may plot the first
r
columns of
R
−
1
/
2
U
and
C
−
1
/
2
V
R
−
1
X
(
C
−
1
/
2
V
)
=
R
−
1
/
2
(
R
−
1
/
2
XC
−
1
/
2
)
V
R
−
1
/
2
V
+
R11
C
=
(
U
/
n
)
V
R
−
1
/
2
U
=
,
(7.11)
