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intermediate cases where the rank-2 structure is retained, the so-called row and column
regression models which may be written as
β
+
λ
γβ
β
+
λ
αδ
,
11
+
α
1
+
11
+
α
1
+
X
=
µ
1
and
X
=
µ
1
in each of which one of the biadditive terms is proportional to a main effect parameter.
The first of these may be rewritten
X
=
(µ
1
+
α
)
1
+
(
1
+
λ
γ
)
β
,
µ
1
+
α
,
1
+
λ
γ
β
giving a biplot of coordinates (
) with a straight line for the
columns and general scatter for the rows; the row regression model does the opposite.
When
X
)and(
1
,
β
+
λ
αβ
we get two nonorthogonal straight lines. In this way
all these varieties of models with one biadditive term may be separated and identified
from the SVD of
X
.
Gower (1990) studied the three-dimensional geometry of these models, showing that
the row points of the biplot lay in one plane and the column points in another plane.
The projection of these two sets of points onto the intersection of the planes gives the
biadditive parameters, and projections onto lines, one within each plane, give the main
effects. Two-dimensional projections of this set-up give the diagnostic biplots discussed
above. In principle, the row and column planes of the three-dimensional geometry can be
extended to four dimensions including rank-2 interactions and could be used as a basis
for additional types of biplot, but this possibility has not yet been explored.
11
+
α
1
+
=
µ
1
