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In-Depth Information
6.2 A biadditive model
Consider the data given in Table 6.1. The measurements are the yields in grams per
square metre (g/m
2
of 12 varieties of winter wheat grown at seven sites at each of two
levels (low and high) of nitrogen. Varieties
Cappelle
(
Cap
),
Ranger
(
Ran
),
Huntsman
(
Hun
),
Te m p l a r
(
Tem
)and
Kinsman
(
Kin
) are classified as conventional varieties, while
Fundin
(
Fun
),
Durin
(
Dur
),
Hobbit
(
Hob
),
Sportsman
(
Spo
),
TJB 259/95
(
T95
),
TJB
325/464
(
T64
)and
TJB 368/268
(
T68
) are classified as semi-dwarf varieties. These data
are described in detail by Blackman
et al.
(1978) and are available as the R matrix object
wheat.data.
An early example of a biadditive biplot using these data was given by
Kempton (1984).
Apart from the column of row means and the row consisting of the column means,
the data contained in Table 6.1 appear to have the same form as the
samples
×
variables
data matrix
X
:
n
×
p
encountered in previous chapters. However, the body of
the table is quantitative, all elements referring to the same variable, unlike a data matrix
whose columns refer to different variables. Therefore it would be wrong to normalize the
columns as is often justifiable for a data matrix. Indeed, the two-way table is concerned
with only three variables: the row and column factors and the values in the body of the
table. Although there is possible interest in main effects, there is at least as much, if
not more, interest in the interaction of varieties with sites. The two-way table will be
written as
X
:
p
)
q
,where
p
denotes the number of levels of the row factor (the sites
in our example) and
q
the number of levels of the column factor (the varieties in our
example). The measurement
x
ij
can be thought of as a sample observation generated by
the model
×
r
X
ij
=
µ
+
α
i
+
β
j
+
1
γ
ik
δ
jk
+
ε
ij
,
r
i
=
1,
...
,
p
and
j
=
1,
...
,
q
.
(6.1)
k
=
The
{
ε
ij
}
in (6.1) are assumed to be independently distributed with equal variances. It is
often convenient to write (6.1) in matrix form:
r
X
=
µ
11
+
α
1
+
1
β
+
1
γ
k
δ
k
+
ε
.
(6.2)
k
=
6.3 Statistical analysis of the biadditive model
Two-way tables such as Table 6.1 may be analysed by fitting the biadditive model (6.1)
to the data contained therein. So, unlike PCA, we now have a dependent variable
X
with
p
+
q
independent dummy variables labelling the rows and columns in addition to
p
+
q
dummy variables for each of
r
multiplicative interaction terms. The parameters
α
are to be estimated. In its simplest form
the multiplicative terms may be omitted and the additive terms, known as
main effects
,
estimated by
,
β
,
γ
k
,and
δ
k
as well as the constant term
µ
ˆ
µ
=
x
,
α
i
=
x
i
.
−
x
..
,
β
j
=
x
.
j
−
x
..
,
r
i
=
1,
...
,
p
and
j
=
1,
...
,
q
,
(6.3)
