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Ta b l e 6 . 3
Analysis of variance for a biadditive model fitted to
the data of Table 6.1.
Source
Sum of squares
DF
Mean SS
Sites Main Effect
2200767.00
13
169289.80
Varieties Main Effect
196211.90
11
17837.45
Total S
×
V Interaction
188245.10
143
1316.40
Multiplicative term 1
65718.31
23
2857.32
Multiplicative term 2
50211.31
21
2391.02
Multiplicative term 3
21394.60
19
1126.03
Multiplicative term 4
15905.03
17
935.59
Multiplicative term 5
11115.25
15
741.02
Multiplicative term 6
7904.39
13
608.03
Multiplicative term 7
5753.79
11
523.07
Multiplicative term 8
3891.06
9
432.34
Multiplicative term 9
3669.92
7
524.27
Multiplicative term 10
2427.90
5
485.58
Multiplicative term 11
253.53
3
84.51
Mean
40368170.00
1
Mean corrected Total
2585224.00
167
Total
42953390.00
168
residual variation. The sums of squares for the interactions are given by the squares of the
singular values of
Z
(see Section 6.4). Note that the degrees of freedom for successive
interaction terms decrease by two and sum to the total of 143. This is a commonly used
approximation that deals sensibly with the degrees of freedom, but which ultimately
derives from canonical analysis (see Rao, 1952) and appears to have no firm theoretical
basis for use in other contexts.
6.4 Biplots associated with biadditive models
In what follows we assume that rank
(
X
)
=
q
≤
p
. There is no loss of generality because
if
q
>
p
we can take
X
since rows and columns of two-way tables have the same status.
It often happens that main effects swamp the interactions. Nevertheless the interac-
tions may be commercially important, showing for example that certain varieties perform
better than others at certain sites. The structure of the interactions may be presented as
a biplot of the interaction matrix
Z
based on the decomposition
Z
=
U
V
.
(6.8)
The biplot may be presented in several equivalent forms:
2
J
for the columns.
(ii) Plot
U
J
for the rows and
VJ
for the columns.
(iii) Plot
UJ
for the rows and
V
1
/
2
J
for the rows and
V
1
/
(i) Plot
U
J
for the columns.
Of the above, (i) treats rows and columns equally, so respecting the symmetric nature of
X
and
Z
. Unfortunately, it gives two sets of points and therefore suffers from the usual
