Graphics Reference
In-Depth Information
Wewilldenote the oddsratio correspondingtothe setof cells
m
ij
, m
i
+
,j
, m
i, j
+
,
m
i
+
,j
+
as θ
ij
, i.e.,
m
ij
m
i
+
,j
+
m
i
+
,
m
i, j
+
θ
ij
=
for all
i
I
−
,
j
J
−
.
he differences in the highlighted heights, d
, d
and d
, are approximately linear in
the log odds ratio of the corresponding cells:
log
m
i
m
(
i
+)
−
ċ
d
i
log θ
i
=
m
(
i
+)
m
i
for i
=
,
,
.
hefactthat these differences donot change muchindicates alinear relationship (on
alogscale) between the percentage of highparental encouragement andthe IQlevel.
Obviously the two variables X and Y are independent if log θ
ij
foralli and j.
Byvisualizing low-dimensional relationships betweenvariables, mosaicplots provide
us with a way to find models graphically and to check the fits of existing models.
Earlier approaches by Friendly (
) and heus and Lauer (
) established the
linkbetweenloglinearmodelsandmosaicplotsbydisplayingfittedvaluesofloglinear
models together with their residuals.
=
Variants
13.3
By default, the sizes, shapes and locations of tiles in a mosaicplot are determined by
the mosaic's hierarchical construction process. However, it is of course possible to
vary these features, and this is a very useful technique for exploring the data, as each
of them tends to emphasize different aspects of the data. We are going to discuss two
different groups of variations: doubledecker plots are, as mentioned earlier, a variant
of the default structure of a mosaicplot. All other variations discussed here refer to
the layout of the plot.
Doubledecker Plots
13.3.1
Doubledecker plots were introduced in Hofmann et al. (
) as a way of visualiz-
ing an association rule within the framework of the whole contingency table. Dou-
bledeckerplotsareaspecialcaseofstandard mosaicplots. Insteadofsplitting thebins
alternately in horizontal and vertical directions as in a default mosaicplot, all of the
bins are split horizontally in a doubledecker plot. All of the bins in a doubledecker
plotarethe same height and aredrawnsidebyside.Doubledecker plots of p variables
X
,...,X
p
therefore have the structure X
, X
,...,X
p
−
, X
′
p
,whereX
p
is usually the
highlighting variable. hus, highlighting the heights in a p-dimensional mosaicplot
illustrates the conditional probabilities P
. One thing that distin-
guishes doubledecker plots from standard mosaicplots is that they are easy to label.
hisisdone by drawing a striped bar below the graphic for each of the variables. he
stripes show different shades of gray, with each shade representing one category of
(
X
p
X
,...,X
p
−
)