visualizing independence using extended association and mosaic … › ~zeileis › papers ›...
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![Page 1: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/1.jpg)
Visualizing Independence UsingExtended Association and Mosaic Plots
Achim Zeileis David Meyer Kurt Hornik
![Page 2: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/2.jpg)
Overview
❆ The independence problem in 2-way contingency tables
❖ Standard approach: χ2 test
❖ Alternative approach: max test
❆ Visualizing the independence problem
❖ Association plots
❖ Mosaic plots
❆ Extensions
❖ Visualization & significance testing
❖ HCL instead of HSV colors
❖ Implementation in grid
❖ Multi-way tables
❆ The vcd package
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The independence problem
Standard approach:
❆ Analyze the relationship between two categorical variables
based on the associated 2-way contingency table.
❆ Measure the discrepancy between observed frequencies{nij
}and expected frequencies under independence
{n̂ij
}by the
Pearson residuals:
rij =nij − n̂ij√
n̂ij
.
❆ Use the Pearson X2 statistic for testing:
X2 =∑ij
r2ij,
which has an asymptotic χ2 distribution.
![Page 4: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/4.jpg)
The independence problem
Alternative approach(es):
❆ There are many conceivable functionals λ(·) which lead toreasonable test statistics λ
({rij
}).
❆ In particular:
M = maxij
∣∣∣rij
∣∣∣ .Then, every residual exceeding the critical value cα violatesthe null hypothesis at level α.
❆ Instead of relying on unconditional limiting distributions, per-form a permutation test, either by simulating or computingthe conditional permutation distribution of λ
({rij
}).
![Page 5: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/5.jpg)
The independence problem
Relationship between hair color and eye color among 328 female
students:
Eye colorHair color Brown Blue Hazel Green Total
Black 36 9 5 2 52Brown 81 34 29 14 158
Red 16 7 7 7 37Blond 4 64 5 8 181Total 137 114 46 31 328
X2 = 112.30 p = 0
M = 6.76 p = 0
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The independence problem
Home and away goals in the Bundesliga in 1995:
Away goalsHome goals 0 1 2 3 4 5 6
0 26 16 13 5 0 1 01 19 58 20 5 4 0 12 27 23 20 5 1 1 13 14 11 10 4 2 0 04 3 5 3 0 0 0 05 4 1 0 1 0 0 06 1 0 0 1 0 0 0
X2 = 46.07 p = 0.121
M = 2.87 p = 0.355
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The independence problem
Treatment for rheumatoid arthritis:
TreatmentImprovement Placebo Treated Total
None 29 13 42Some 7 7 14
Marked 7 21 28Total 43 41 84
X2 = 13.06 p = 0.001
M = 1.98 p = 0.001
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Visualization
Association plot: display for the Pearson residuals{rij
}and the
raw residuals{nij − n̂ij
}in an rectangular array.
Mosaic plot: display in which the sizes of the mosaic tiles is
proportional to the observed frequencies{nij
}.
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Visualization
Blo
ndR
edB
row
nB
lack
Brown Blue Hazel Green
Hai
rEye
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Visualization
Brown Blue Hazel GreenB
lack
Bro
wn
Red
Blo
nd
EyeH
air
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HSV colors
Colors are commonly used to enhance these plots. In particular,
Friendly (1994) suggested shadings for mosaic displays.
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HSV colors
Colors are commonly used to enhance these plots. In particular,
Friendly (1994) suggested shadings for mosaic displays.
In R these are implemented based on HSV colors.
The HSV color space is one of the most common implementa-
tions of color in many computer packages. Hue, saturation and
value range in [0,1].
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HSV colors
The hue is typically used to code the sign of the residuals.
0.0 0.2 0.4 0.6 0.8 1.0hue
saturation = 1value = 1
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HSV colors
The hue is typically used to code the sign of the residuals.
0.0 0.2 0.4 0.6 0.8 1.0hue
saturation = 1value = 1
rij < 0 rij > 0
![Page 15: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/15.jpg)
HSV colors
Friendly’s extended mosaic displays use the saturation to code
the absolute size of the residuals.
0.0 0.2 0.4 0.6 0.8 1.0saturation
h = 0
h = 2/3
value = 1
![Page 16: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/16.jpg)
HSV colors
Friendly’s extended mosaic displays use the saturation to code
the absolute size of the residuals.
0.0 0.2 0.4 0.6 0.8 1.0saturation
h = 0
h = 2/3
value = 1
rij < 2 2 < rij < 4 rij > 4
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HSV colors
Value is currently not used for coding, always set to 1.
0.0 0.2 0.4 0.6 0.8 1.0value
h = 0
h = 2/3
saturation = 1
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HSV colors
Value is currently not used for coding, always set to 1.
0.0 0.2 0.4 0.6 0.8 1.0value
h = 0
h = 2/3
saturation = 1
![Page 19: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/19.jpg)
HSV colors
Blo
ndR
edB
row
nB
lack
Brown Blue Hazel Green
Hai
r
Eye
−4
−2
0
2
4
6
Pearsonresiduals:
![Page 20: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/20.jpg)
HSV colors
−4
−2
0
2
4
6
Pearsonresiduals:
Brown Blue Hazel Green
Bla
ckB
row
nR
edB
lond
Eye
Hai
r
![Page 21: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/21.jpg)
Visualization & testing
−2
−1
0
1
2
Pearsonresiduals:
0 1 2 3 4 5 6
01
23
45
6
HomeGoals
Aw
ayG
oals
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Visualization & testing
Intuition: colored cells convey the impression that there is sig-
nificant dependence.
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Visualization & testing
Intuition: colored cells convey the impression that there is sig-
nificant dependence.
Currently this is not true. But it can be achieved by using the
90% and 99% critical values for the max statistic M instead of
2 and 4.
Advantage:
❆ color ⇔ significance
❆ highlights the cells which “cause” the dependence (if any).
Disadvantage:
❆ does not work for the χ2 test (or any other functional λ(·)).
![Page 24: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/24.jpg)
Visualization & testing
−4
−2
0
2
4
6
Pearsonresiduals:
p−value:< 2.22e−16
Brown Blue Hazel Green
Bla
ckB
row
nR
edB
lond
Eye
Hai
r
![Page 25: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/25.jpg)
Visualization & testing
−2
−1
0
1
2
Pearsonresiduals:
0 1 2 3 4 5 6
01
23
45
6
HomeGoals
Aw
ayG
oals
![Page 26: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/26.jpg)
Visualization & testing
Use value to code the result of a significance test for indepen-
dence.
0.0 0.2 0.4 0.6 0.8 1.0value
h = 0
h = 2/3
saturation = 1
![Page 27: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/27.jpg)
Visualization & testing
Use value to code the result of a significance test for indepen-
dence.
0.0 0.2 0.4 0.6 0.8 1.0value
h = 0
h = 2/3
saturation = 1
non−significant significant
![Page 28: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/28.jpg)
Visualization & testing
−4
−2
0
2
4
6
Pearsonresiduals:
p−value:< 2.22e−16
Brown Blue Hazel Green
Bla
ckB
row
nR
edB
lond
Eye
Hai
r
![Page 29: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/29.jpg)
Visualization & testing
−2
−1
0
1
2
Pearsonresiduals:
p−value:0.12133
0 1 2 3 4 5 6
01
23
45
6
HomeGoals
Aw
ayG
oals
![Page 30: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/30.jpg)
HCL colors
Disadvantages of HSV colors:
❆ device dependent,
❆ not copierproof,
❆ flashy colors good for drawing attention to a plot, but hard
to look at.
![Page 31: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/31.jpg)
HCL colors
Disadvantages of HSV colors:
❆ device dependent,
❆ not copierproof,
❆ flashy colors good for drawing attention to a plot, but hard
to look at.
Alternative: use HCL colors instead (see Ihaka, 2003).
HCL colors are defined by hue (in [0,360]), chroma and lumi-
nance (in [0,100]). HCL space essentially looks like a double
cone.
![Page 32: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/32.jpg)
HCL colors
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HCL colors
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HCL colors
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HCL colors
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HCL colors
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HCL colors
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HCL colors
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HCL colors
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HCL colors
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HCL colors
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HCL colors
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HCL colors
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HCL colors
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HCL colors
chroma
025
5075
100
025
5075
100
100 75 50 25 0 25 50 75 100
hue = 0 hue = 260
lum
inan
ce
![Page 46: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/46.jpg)
HCL colors
5075
100
5075
100
100 75 50 25 0 25 50 75 100
hue = 0 hue = 260chroma
lum
inan
ce
![Page 47: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/47.jpg)
HCL colors
5075
100
5075
100
100 75 50 25 0 25 50 75 100
hue = 0 hue = 260chroma
lum
inan
ce
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HCL colors
5075
100
5075
100
100 75 50 25 0 25 50 75 100
hue = 0 hue = 260chroma
lum
inan
ce
![Page 49: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/49.jpg)
HCL colors
5075
100
5075
100
100 75 50 25 0 25 50 75 100
hue = 0 hue = 260chroma
lum
inan
ce
![Page 50: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/50.jpg)
HCL colors
5075
100
5075
100
100 75 50 25 0 25 50 75 100
hue = 0 hue = 260chroma
lum
inan
ce
significant
![Page 51: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/51.jpg)
HCL colors
5075
100
5075
100
100 75 50 25 0 25 50 75 100
hue = 0 hue = 260chroma
lum
inan
ce
significant
non−significant
![Page 52: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/52.jpg)
HCL colors
−4
−2
0
2
4
6
Pearsonresiduals:
p−value:< 2.22e−16
Brown Blue Hazel Green
Bla
ckB
row
nR
edB
lond
Eye
Hai
r
![Page 53: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/53.jpg)
HCL colors
−4
−2
0
2
4
6
Pearsonresiduals:
p−value:< 2.22e−16
Brown Blue Hazel Green
Bla
ckB
row
nR
edB
lond
Eye
Hai
r
![Page 54: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/54.jpg)
HCL colors
−2
−1
0
1
2
Pearsonresiduals:
p−value:0.12133
0 1 2 3 4 5 6
01
23
45
6
HomeGoals
Aw
ayG
oals
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HCL colors
−2
−1
0
1
2
Pearsonresiduals:
p−value:0.12133
0 1 2 3 4 5 6
01
23
45
6
HomeGoals
Aw
ayG
oals
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HCL colors
−1
0
1
Pearsonresiduals:
p−value:0.0014626
Placebo Treated
Non
eS
ome
Mar
ked
Treatment
Impr
oved
![Page 57: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/57.jpg)
HCL colors
−1
0
1
Pearsonresiduals:
p−value:0.001
Placebo Treated
Non
eS
ome
Mar
ked
Treatment
Impr
oved
![Page 58: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/58.jpg)
Implementation in grid
The graphics engine grid overcomes the old R concept of plots
with a plot region surrounded by a margin. grid is
❆ based on generic drawing regions (viewports),
❆ allows for plotting to relative coordinates,
❆ is also the basis for an implementation of Trellis graphics
called lattice.
(see Murrell, 2002)
Thus, the new implementation of mosaic and association plots
makes them easily reusable, e.g., in Trellis-like layouts.
![Page 59: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/59.jpg)
Implementation in grid
Furthermore, graphics parameters for the rectangles, e.g.,
❆ fill color,
❆ line type,
❆ line color,
can be specified for each cell individually by the user. Each
graphics parameter can be an object of the same dimenionality
as the original table.
→ new shadings can easily be implemented.
![Page 60: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/60.jpg)
Multi-way tablesG
ende
r
Admit
Dept = AF
emal
eM
ale
Admit Reject
Dept = B
Fem
ale
Mal
e
Admit Reject
Dept = C
Fem
ale
Mal
e
Admit Reject
Dept = D
Fem
ale
Mal
e
Admit Reject
Dept = E
Fem
ale
Mal
e
Admit Reject
Dept = F
Fem
ale
Mal
e
Admit Reject
![Page 61: Visualizing Independence Using Extended Association and Mosaic … › ~zeileis › papers › OeSG-2003.pdf · 2018-01-19 · Visualizing Independence Using Extended Association](https://reader033.vdocuments.pub/reader033/viewer/2022042315/5f0365897e708231d409022c/html5/thumbnails/61.jpg)
The vcd package
New methods will be available in the package vcd for visualizing
categorical data.
Currently only in development version. The released version is
available from the Comprehensive R Archive Network
http://CRAN.R-project.org/
and it already offers some functionality for
❆ fitting & graphing of discrete distributions,
❆ plots for independence and agreement,
❆ visualization of log-linear models.