Information Technology Reference
In-Depth Information
7.2.7 Analysis of variance and generalities
The basic weighted analysis of variance associated with a matrix
Y
is given by
W
1
YW
2
=
W
1
YW
2
+
W
1
(
Y
−
Y
)
W
2
2
2
2
(7.36)
or, in words, the 'total sum of squares' is the sum of the 'fitted sum of squares' and
the 'residual sum of squares', where
Y
is obtained by minimizing the residual sum of
squares. In our context, all the variants of CA discussed above are special cases of:
W
−
1
1
R
−
1
/
2
C
−
1
/
2
W
−
1
2
2
W
1
{
(
X
−
E
)
}
W
2
=
W
1
XW
2
X
}
W
2
2
+
W
1
{
W
−
1
R
−
1
/
2
(
X
−
E
)
C
−
1
/
2
W
−
1
2
2
,
−
(7.37)
1
W
−
1
1
C
−
1
/
2
W
−
1
2
R
−
1
/
2
which relates to (7.36) by setting
Y
={
(
X
−
E
)
}
, which is approx-
imated by
Y
=
X
. Substituting
R
−
1
/
2
(
X
−
E
)
C
−
1
/
2
=
U
V
into (7.37) gives
2
=
U
V
2
=
U
JV
2
+
U
(
I
−
J
)
V
2
,
χ
where
J
is zero apart from
r
units in its first diagonal places, giving a simple way of
expressing the
r
-dimensional approximation. Thus, when there are
q
nonzero singular
values, all methods give the same analysis of variance irrespective of the particular
weights
W
1
,
W
2
and the choice of
Y
:
2
2
1
2
2
2
q
2
1
2
2
2
r
2
r
+
1
2
r
+
2
2
q
χ
=
σ
+
σ
+···+
σ
=
(
σ
+
σ
+···+
σ
)
+
(
σ
+
σ
+···+
σ
).
In particular, we have
row predictivities
=
diag
(
U
J
U
)
[diag
(
U
U
)
]
−
1
=
diag
(
U
2
JU
)
[diag
(
U
2
U
)
]
−
1
.
(7.38)
Similarly,
2
JV
)
[diag
(
V
2
V
)
]
−
1
column predictivities
=
diag
(
V
.
(7.39)
The only things that change are the models being fitted,
U
V
W
−
1
W
−
1
R
−
1
/
2
(
X
−
E
)
C
−
1
/
2
W
−
1
=
W
−
1
,
1
2
1
2
1
U
JV
W
−
2
. That the weights
W
1
and
W
2
occur both
in the model being fitted and as the weights used in the least-squares criterion is confus-
ing, though not unique. Thus, for example, in multidimensional scaling one may wish
to approximate a set of distances by distances between points in a few dimensions,
weighting by some function of the observed, or even the fitted, distances so as to reduce
the importance of small or large distances (see Shepard and Carroll, 1966). What may
be questioned in CA is why the particular weights are those chosen, apart from the
convenience of always ensuring dependence on the SVD of
R
−
1
/
2
X
=
W
−
1
and the approximation
C
−
1
/
2
.
Table 7.1 summarizes the main results discussed above. We now turn to the implica-
tions of Table 7.1 for biplot displays.
(
X
−
E
)
