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where
p
q
q
p
1
pq
1
q
1
p
x
=
x
ij
,
x
i
.
=
x
ij
and
x
.
j
=
x
ij
.
i
=
1
j
=
1
j
=
1
i
=
1
α
i
and
ˆ
Note that any constant may be added to each ˆ
is accordingly
adjusted, fitted values remain unchanged. The identification constraints adopted give the
simple unique parameterization (6.3) in which the main effects sum to zero. When the
multiplicative terms are included, the main effects continue to be estimated by (6.3)
and the multiplicative terms, representing the interaction between the row and column
factors, are estimated by least squares from the singular value decomposition of the
residual matrix (also called the interaction matrix)
Z
:
p
×
q
with elements
β
j
but, provided ˆ
µ
ˆ
z
ij
=
x
ij
−
µ
−
α
i
−
β
j
=
x
ij
−
x
i
.
−
x
.
j
+
x
,
r
i
=
1,
...
,
p
and
j
=
1,
...
,
q
.
(6.4)
Defining centring matrices
P
=
11
/
p
and
Q
=
11
/
q
, we may write (6.1) in matrix terms
and in terms of the above estimates as
X
=
PXQ
+
(
I
−
P
)
XQ
+
PX
(
I
−
Q
)
+
(
I
−
P
)
X
(
I
−
Q
)
(6.5)
from which the following orthogonal analysis of variance is immediate:
2
2
2
2
||
X
−
PXQ
||
=||
(
I
−
P
)
XQ
||
+||
PX
(
I
−
Q
)
||
+||
(
I
−
P
)
X
(
I
−
Q
)
||
(6.6)
In particular,
Z
=
(
I
−
P
)
X
(
I
−
Q
)
(6.7)
which may be compared with the similar formula (7.2) arising in correspondence analysis.
The matrix
Z
may be analysed in the usual way by appealing to the Eckart-Young
theorem to give estimates of the multiplicative terms, presented in order of decreasing
importance according to the size of their corresponding singular values. Because this
computational process is shared, fitting biadditive models is often regarded as a version
of PCA. Above, we have stressed the fundamental statistical differences between fitting
a biadditive model and PCA, not the least of which is that one includes a dependent
variable and the other does not. Yet both depend on the same algebraic underpinning,
and this carries over into common algorithmic procedures.
Table 6.2 gives the values of the residuals or interactions (6.4), together with the
estimated main effects.
The whole analysis may be summarized in the orthogonal analysis of variance (6.6)
with entries for the main effects and each multiplicative term, together with degrees of
freedom. This is shown in Table 6.3 for the data of Table 6.1.
Our main purpose here is to discuss biplots associated with two-way tables like
Table 6.1, but first we make a few remarks commenting on Table 6.3. From this table
it is clear that both main effects are substantial relative to the Total Sites
×
Varieties
interaction. However, the first two dimensions of the interaction account for about 62%
of the total interaction and merit closer examination. The mean sum of squares for the
remaining nine terms with 99 degrees of freedom is 730.45 and can be considered as
