Information Technology Reference
In-Depth Information
Fair
Clerical
Dark
Myfanwy
Scotland
Alisdair
Wales
F
University
School
Jane
Harriet
Brown
Ivor
George
Manual
M
Professional
England
Jeremy
Grey
Postgrad
Figure 8.3
The row chi-squared MCA biplot
of
Figure 8.1 but
with row point
p
−
1
U
.
coordinates
reduced
to
Columns
are
plotted
as
in
Figure
8.1
by
using
Z
=
p
−
1
/
2
L
−
1
/
2
V
.
English people in Figure 8.1? The coordinates of all such points are given by the rows of
L
−
1
G
Z
0
,
(8.6)
which we term the
category centroids
. Here,
G
Z
0
sums the relevant coordinates, while
L
−
1
ensures that each sum is divided by the correct frequency. From (8.3) - (8.5) we
have that
(
L
−
1
/
2
G
GL
−
1
/
2
L
−
1
G
Z
0
=
L
−
1
G
(
p
−
1
/
2
GL
−
1
/
2
V
)
=
p
−
1
/
2
L
−
1
/
2
)
V
=
p
1
/
2
L
−
1
/
2
2
V
)
V
=
p
Z
2
(
V
.
(8.7)
2
the category centroids would be at
p
times the actual
category-level points; rescaling would make them coincide. Although the singular values
interfere with this nice property, (8.7) encourages us to rescale as is shown in Figure 8.3.
It also encourages us to consider replacing the projected CLPs by the category centroids
as is shown in Figures 8.5 and 8.6.
If chi-squared distance can be controversial for a contingency table (see Chapter 7), it
is even more so for a categorical data matrix
G
. This is because the only contribution to
If it were not for the factor
