Graphics Reference
In-Depth Information
Figure
.
.
hree
mosaicplots; the interaction between the variables increases from let to right
as for the middle mosaicplot, since d on the right hand side is about twice as big as
d in the middle diagram. he advantage of being able to read off the odds ratio from
a mosaicplot is that it can then be used in modeling: for two binary variables X and
Y, the model of independence can be written as
λ
i
λ
j
,
log m
ij
=
μ
+
+
, μ isthegrandmean, λ
i
istheeffectof the ith
levelof variable X,andλ
j
istheeffect of the jth level ofvariable Y (with
where m
ij
is the cell count of cell
(
i, j
)
i, j,
).
his model holds if the interaction term λ
XY
ij
is not significantly different from zero.
Using control group constraints, i.e.,all “first effects” are set to zero for identifiability,
the interaction term between X and Y is equal to the logarithm of the odds ratio
between the variables:
λ
XY
=
log θ .
A three-way interaction between variables X, Y,andZ is present if the (conditional)
two-way interaction between X and Y is different for different levels of Z.hetrick
thereforeistocompareoddsratios.Forthreevariables X, Y and Z,thecorresponding
(conditional) odds ratio between X and Y for a fixed level k of Z is defined as:
m
ijk
m
i
+
j
+
k
m
i
+
jk
m
ij
+
k
θ
ij
(
k
)
.
=
Figure
.
.
Mosaic plot of binary variables X, Y,andZ with structure Z, X, Y
′
.Bycomparingthetwo
conditional odds ratios θ
=
a
d
(
b
c
)
and θ
=
a
d
(
b
c
)
, conclusions can be made about the
three-way interaction between the variables visually