1 1 Small multiples, or the science and art of combining graphs Nicholas J. Cox Department of Geography Durham University, UK
Jan 09, 2016
11
Small multiples, or the science and art of combining graphs
Nicholas J. Cox Department of Geography Durham University, UK
2
Small multiples
Good graphics often exploit one simple design that is repeated for different parts of the data.
Edward Tufte called this the use of small multiples.
Well-designed small multiples are inevitably comparative, deftly multivariate, shrunken, high-
density graphics…. Edward Rolf Tufte (1942–)
3
…in Stata
In Stata, small multiples are supported for different subsets of the data with by() or over() options of many graph commands.
Users can emulate this in their own programs by writing wrapper programs that call twoway or graph bar and its siblings.
Otherwise, specific machinery offers repetition of a design for different variables, such as the graph matrix command.
4
Users can always put together their own composite graphs by saving individual graphs and then combining them using graph combine.
This presentation offers further modest automation of the same design repeated for different data.
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Original programs discussed
sparkline crossplot combineplot designplot
and with cameo roles aaplotsepscatter
All may be installed from SSC.
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What’s in a name? roseplot by any other name…A minor theme here is that definite names are
needed for programs, even if kinds of graphs do not have distinct agreed names.
As in advertising, a good name attracts and keeps users.
As in politics, a bad name can be fatal.
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sparkline
The purpose of visualization is insight, not picturesBen Shneiderman (1947–)
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Sparklines The name “sparkline” was suggested by Edward
Tufte for intense text-like graphics.
Sparklines are typically simple in design, sparing of space and rich in data, but they include several quite different kinds of graph otherwise.
The most common kind shows several time series stacked vertically.
sparkline is a Stata implementation.
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Sparklines have long been standard in several fields, including physics and chemistry (spectroscopy), seismology, climatology, ecology, archaeology and physiology (notably encephalography and cardiography).
Tufte provided an memorable and evocative new name and an excellent provocative discussion.
The Grunfeld data (webuse grunfeld) are a classic dataset in panel-based economics. Ten companies were monitored for 1935–54. They give us a simple sandpit.
1010
What are we doing here? The problem of time series graphicsComparisons of time series are a rich and
challenging area of statistical graphics.
The widespread term spaghetti plot hints immediately at the difficulties.
As always, we want to combine a grasp of general patterns with access to individual details.
With this in mind, we look at some sparklines of the Grunfeld dataset.
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257.7
1486.7
2792.2
6241.7
2.8
2226.3
invest
mvalue
kstock
1935 1940 1945 1950 1955year
company 1
1212
invest
mvalue
kstock
invest
mvalue
kstock
invest
mvalue
kstock
1935 1940 1945 1950 1955 1935 1940 1945 1950 1955
1935 1940 1945 1950 1955 1935 1940 1945 1950 1955
1 2 3 4
5 6 7 8
9 10
Graphs by company
1313
257.7
1486.72792.2
6241.72.8
2226.3
209.9
645.51362.4
2676.350.5
669.7
33.1
189.61170.6
2803.397.8
888.9
40.29
174.93410.9
1001.510.2
414.9
39.67
91.9151.2
398.4183.2
804.9
20.36
135.72197
927.36.5
238.7
23.21
89.51
210.1
98.1
100.2
511.3
12.93
90.08191.5
1193.5.8
213.5
20.89
66.11213.3
496162
468
.93
6.53
87.94
58.12
3.23
14.33
invest
mvalue
kstock
invest
mvalue
kstock
invest
mvalue
kstock
1935 1940 1945 1950 1955 1935 1940 1945 1950 1955
1935 1940 1945 1950 1955 1935 1940 1945 1950 1955
1 2 3 4
5 6 7 8
9 10
1414
Vertical and horizontal
By default sparkline stacks small graphs vertically.
If several graphs are combined, it is typical to cut down on axis labels and rely on differences in shape to convey information.
Horizontal stacking is also supported, which can be useful for archaeological or environmental problems focused on variations with depth or height.
Here is an archaeological dataset as example.
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leve
lscores blanks tools
3.8 17.7 25.6 74.7 18.6 56.9
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Nightingale’s data
Florence Nightingale (1820–1910) is well remembered for her nursing in the Crimean war and (within statistical science) for use of quantitative arguments.
Her most celebrated dataset is often reproduced using her polar diagram, but is easier to think about as time series.
Zymotic (loosely, infectious) disease mortality dominates other kinds, so much so that a square root scale helps comparison. (A logarithmic scale over-transforms here.)
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Source of image:
http://understandinguncertainty.org/coxcombs
1919
0
200
400
600
800
1000
18551854 1856
zymotic disease
wounds and injuriesall other causes
annualised rates per 1000
Nightingale's data on mortality in the Crimea
2020
0
200
400
600
800
1000
18551854 1856
zymotic disease
wounds and injuriesall other causes
annualised rates per 1000
Nightingale's data on mortality in the Crimea
0
25
100
225
400
625
900
18551854 1856
zymotic disease
wounds and injuriesall other causes
annualised rates per 1000
Nightingale's data on mortality in the Crimea
2121
Would sparkline help?
A sparkline display is useful to show relative shape, such as times of peaks.
We see that seasonality is only part of what is being seen.
The harsh winter of 1854–5 coincided with some of the hardest battles of the war, but 1855–6 was quite different.
But, as often happens, no one graph dominates others here.
2222
1.4
1022.8
.4
115.8
2.5
140.1
zymotic disease
wounds and injuries
all other causes
18551854 1856annualised rates per 1000
Nightingale's data on mortality in the Crimea
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crossplot
The scatter plot is the workhorse of statistical graphics.
John McKinley Chambers (1941– )
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crossplot
crossplot is designed as a quick-and-easy way to combine scatter plots.
The basic syntax is crossplot (yvarlist) (xvarlist) and the idea is to plot every y in yvarlist against every x in xvarlist.
The use of two varlists gives greater flexibility than does graph matrix, which produces every
possible scatter plot for a single varlist.
25
Scatter plot matrices
Scatter plot matrices are great, but they can be excessive.
Their main feature is also a limitation. p variables mean p2 plots all at once, so 10
means 100, and so forth.
(The half option just controls which plots you see. )
26
crossplot design
crossplot was developed in teaching, especially of regression, with the aim of encouraging focused comparisons.
Originally (1999) crossplot was called cpyxplot, cp meaning Cartesian product, but the name was ugly, cryptic and easily forgotten.
The syntax had to be as simple as possible.
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crossplot examples
Versions of a response variable versus a key predictor.
A response variable versus versions of a key predictor.
Each output versus each input. Principal components versus original variables.
First, let us look at four versions of mpg versus weight in the auto dataset.
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10
20
30
40
Mile
ag
e (
mp
g)
2,000 3,000 4,000 5,000Weight (lbs.)
34
56
7rt
_m
pg
2,000 3,000 4,000 5,000Weight (lbs.)
2.5
33
.54
ln_
mp
g
2,000 3,000 4,000 5,000Weight (lbs.)
24
68
rec_
mp
g
2,000 3,000 4,000 5,000Weight (lbs.)
29
Next we look at an audiometric dataset used as a multivariate example in the Stata manuals.
There are 8 response variables, 4 for left ears and 4 for right ears. Here we just focus on the 16 plots pairing left and right.
Another graph could be the 4 plots comparing left and right ears at the same frequency, the diagonal here.
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-10
-50
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01
5le
ft e
ar
at
50
0H
-10 0 10 20 30right ear at 500H
-10
-50
51
01
5le
ft e
ar
at
50
0H
-10 0 10 20right ear at 1000H
-10
-50
51
01
5le
ft e
ar
at
50
0H
-10 0 10 20 30 40right ear at 2000H
-10
-50
51
01
5le
ft e
ar
at
50
0H
-20 0 20 40 60 80right ear at 4000H
-10
01
02
0le
ft e
ar
at
10
00
H
-10 0 10 20 30right ear at 500H
-10
01
02
0le
ft e
ar
at
10
00
H-10 0 10 20
right ear at 1000H
-10
01
02
0le
ft e
ar
at
10
00
H
-10 0 10 20 30 40right ear at 2000H
-10
01
02
0le
ft e
ar
at
10
00
H
-20 0 20 40 60 80right ear at 4000H
-20
02
04
0le
ft e
ar
at
20
00
H
-10 0 10 20 30right ear at 500H
-20
02
04
0le
ft e
ar
at
20
00
H
-10 0 10 20right ear at 1000H
-20
02
04
0le
ft e
ar
at
20
00
H
-10 0 10 20 30 40right ear at 2000H
-20
02
04
0le
ft e
ar
at
20
00
H
-20 0 20 40 60 80right ear at 4000H
-20
02
04
06
08
0le
ft e
ar
at
40
00
H
-10 0 10 20 30right ear at 500H
-20
02
04
06
08
0le
ft e
ar
at
40
00
H
-10 0 10 20right ear at 1000H
-20
02
04
06
08
0le
ft e
ar
at
40
00
H
-10 0 10 20 30 40right ear at 2000H
-20
02
04
06
08
0le
ft e
ar
at
40
00
H
-20 0 20 40 60 80right ear at 4000H
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crossplot syntax for examples crossplot (mpg rt_mpg ln_mpg rec_mpg) weight, combine(imargin(small))
crossplot (lft*) (rght*), jitter(1)
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crossplot syntax extras
By default, crossplot is just calling twoway scatter followed by graph combine.
It follows that recast() is available to recast to twoway line or twoway connected.
crossplot has an extra sequence() option to label graphs to ease preparation of graphics for papers e.g. sequence(a b c d)
33
combineplot
The greatest value of a picture is when it forces us to notice what we never expected to see.
John Wilder Tukey (1915–2000)
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combineplot combineplot is a generalisation of crossplot,
more flexible and inevitably more complicated in syntax.
The general problem of combining plots of similar kind reduces to a loop producing individual plots and a call to graph combine. That is bound to be a challenge to beginning users.
The idea is to avoid that by encapsulating the predictable syntax within one command.
35
combineplot examples
We will look at a series of univariate examples followed by a series of bivariate examples.
A great variety is possible, as we can loop over user-written graphics commands as well as official commands.
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10
20
30
40
Mile
ag
e (
mp
g)
1 2 3 4 5
05
,00
01
0,0
00
15
,00
0P
rice
1 2 3 4 5
2,0
00
3,0
00
4,0
00
5,0
00
We
igh
t (l
bs.
)
1 2 3 4 5
12
34
5H
ea
dro
om
(in
.)
1 2 3 4 5
37
10
20
30
40
Mile
ag
e (
mp
g)
1 2 3 4 5Repair Record 1978
05
,00
01
0,0
00
15
,00
0P
rice
1 2 3 4 5Repair Record 1978
2,0
00
3,0
00
4,0
00
5,0
00
We
igh
t (l
bs.
)
1 2 3 4 5Repair Record 1978
1.0
2.0
3.0
4.0
5.0
He
ad
roo
m (
in.)
1 2 3 4 5Repair Record 1978
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05
,00
01
0,0
00
15
,00
0P
rice
0 5,000 10,000 15,000Inverse Normal
10
20
30
40
Mile
ag
e (
mp
g)
10 15 20 25 30 35Inverse Normal
1,0
00
2,0
00
3,0
00
4,0
00
5,0
00
We
igh
t (l
bs.
)
1,000 2,000 3,000 4,000 5,000Inverse Normal
14
01
60
18
02
00
22
02
40
Le
ng
th (
in.)
140 160 180 200 220 240Inverse Normal
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10
20
30
40
Mile
ag
e (
mp
g)
1 2 3 4 5
a
10
20
30
40
Mile
ag
e (
mp
g)
Domestic Foreign
b0
5,0
00
10
,00
01
5,0
00
Pri
ce
1 2 3 4 5
c
05
,00
01
0,0
00
15
,00
0P
rice
Domestic Foreign
d
40
05
,00
01
0,0
001
5,0
00
Pri
ce (
US
D)
10 20 30 40Mileage (mpg)
Domestic Foreign
05
,00
01
0,0
001
5,0
00
Pri
ce (
US
D)
2,000 3,000 4,000 5,000Weight (lbs.)
Domestic Foreign
05
,00
01
0,0
001
5,0
00
Pri
ce (
US
D)
140 160 180 200 220 240Length (in.)
Domestic Foreign
05
,00
01
0,0
001
5,0
00
Pri
ce (
US
D)
100 200 300 400 500Displacement (cu. in.)
Domestic Foreign
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A digression on sepscatter
The last example used sepscatter, a program automating separation of data points on a scatter plot by a categorical variable.
The repetition of the legend needs some kind of fix. In this and similar examples, the legend could be deleted and explaining symbols left as a task for the text caption.
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sepscatter and scatter plot matricescombineplot with sepscatter meets a felt need,
scatter plot matrices with categorisation of data points.
Here is an example with “size” variables from the auto dataset. The diagonal scatter plots have meaning, yet are not conventional. But not every graph need be immediately publishable.
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2,00
03,
000
4,00
05,
000
Wei
ght (
lbs.
)
2,000 3,000 4,000 5,000Weight (lbs.)
2,00
03,
000
4,00
05,
000
Wei
ght (
lbs.
)
140 160 180 200 220 240Length (in.)
2,00
03,
000
4,00
05,
000
Wei
ght (
lbs.
)
100 200 300 400 500Displacement (cu. in.)
140
160
180
200
220
240
Leng
th (
in.)
2,000 3,000 4,000 5,000Weight (lbs.)
140
160
180
200
220
240
Leng
th (
in.)
140 160 180 200 220 240Length (in.)
140
160
180
200
220
240
Leng
th (
in.)
100 200 300 400 500Displacement (cu. in.)
100
200
300
400
500
Dis
plac
emen
t (cu
. in.
)
2,000 3,000 4,000 5,000Weight (lbs.)
100
200
300
400
500
Dis
plac
emen
t (cu
. in.
)
140 160 180 200 220 240Length (in.)
100
200
300
400
500
Dis
plac
emen
t (cu
. in.
)100 200 300 400 500
Displacement (cu. in.)
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050
0010
000
1500
0P
rice
(U
SD
)
10 20 30 40Mileage (mpg)
n = 74 RMSE = 2,623.7
price = 11253 - 238.89 mpg R2 = 22.0%
050
0010
000
1500
0P
rice
(U
SD
)
2000 3000 4000 5000Weight (lbs.)
n = 74 RMSE = 2,502.3
price = -6.7074 + 2.0441 weight R2 = 29.0%
050
0010
000
1500
0P
rice
(U
SD
)
140 160 180 200 220 240Length (in.)
n = 74 RMSE = 2,678.7
price = -4584.9 + 57.202 length R2 = 18.6%
050
0010
000
1500
0P
rice
(U
SD
)
100 200 300 400 500Displacement (cu. in.)
n = 74 RMSE = 2,580.6
price = 3029 + 15.896 displace~t R2 = 24.5%
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A digression on aaplot
The last example used aaplot.
aaplot customises automatic annotation of scatter plots with fitted regressions with text for key results.
Originally, it was written following a request by my Ph.D. student Alona Armstrong.
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Back to combineplot
Some examples of its syntax will make clearer how it works. First look at a univariate example:
combineplot mpg price weight headroom: graph box @y, over(rep78)
Here we have one varlist and the syntax @y is a placeholder for the variable name.
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Next look at a bivariate example:
combineplot price (mpg weight length displacement): sepscatter @y @x, ytitle("Price (USD)") sep(foreign)
Here we have two varlists and the syntax elements @y and @x are placeholders for the variable names.
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The two varlists may each contain a single variable and they may be identical.
When both are presented, the combination is the Cartesian product of the varlists.
Naturally, you can reach through to control the options of graph combine as well as those of the particular graph command used.
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Quirk or quick?
The quirky syntax of combineplot might cause some queasiness.
Some might recall the obsolete for command.
Confident users would (should) be happy to write their own loops, topped by graph combine, and that is fine too.
The justification for combineplot is just convenience: it can be quicker than writing your own script.
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designplot
Real life is both complicated and short, and we make no mockery of honest adhockery.
Irving John Good (1916–2009)
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designplot
Here more than anywhere arbitrariness of names can bite.
If you have used S or S-Plus or R much, you may have come across “design plots”.
But as implemented there they do not look much like the graphs you are going to see. Nor are they plots showing fitted results; nor do they imply experimental design.
To understand designplot, we need to creep up on it step by step.
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0 10 20 30mean
5
4
3
2
1
(all)
Mileage (mpg)
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0 10 20 30mean
5 Foreign5 Domestic
4 Foreign4 Domestic
3 Foreign3 Domestic2 Domestic1 Domestic
ForeignDomestic
54321
(all)
Mileage (mpg)
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designplot syntax
Minimal syntax specifies a response first, then one or more predictors.
The predictors should in practice be categorical, meaning taking on only a small or moderate number of distinct levels (“factors”, if you like).
The examples were designplot mpg rep78 designplot mpg rep78 foreign
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designplot defaultThe statistics shown are means.
Given one, two, … predictors, the means are shown for all the data, each one-way breakdown, each two-way breakdown, ….
designplot uses a syntax of way being 0, 1, 2, …
graph dot is the default vehicle. statsby underpins calculations.
In essence, we can get a multiscale breakdown. In practice, we might want to restrict what is
shown.
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0 10 20 30mean
5
4
3
2
1
Foreign
Domestic
(all)
Mileage (mpg)
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Restricting designplot
Here we restricted the scope by
designplot mpg foreign rep78, maxway(1)
Let us look at a different dataset. The response variable for these data on the Titanic is a binary variable survived, so its mean is the fraction survived.
We restrict using maxway(2).
580 .2 .4 .6 .8 1
mean
adult maleadult female
child malechild female
third malethird female
second malesecond female
first malefirst femalecrew male
crew female
third adultthird child
second adultsecond child
first adultfirst child
crew adult
malefemale
adultchild
thirdsecond
firstcrew
(all)
survived
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So we have here:
the overall mean
one-way breakdowns for three predictors class, adult, male
two-way breakdowns for combinations class×adult, class×male,
adult×male
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This kind of graph is for detailed scrutiny, rather than delivering shock.
Logically similar displays are often used for reporting opinion poll or electoral results.
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That reminds us of…
The structure echoes analysis of variance, used descriptively.
Similar ideas appear in ANOVA and other literature going back to J.W. Tukey in 1977.
It also echoes the little used official command grmeanby.
By default, grmeanby also shows means. (Medians are allowed.) It allows one-way breakdowns only.
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female
male
child
adult
second
third
first
crew
.2.3
.4.5
.6.7
class adult male
Means of survived
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Foreign
Domestic
1
3
4
2
5
18
20
22
24
26
28
rep78 foreign
Means of mpg, Mileage (mpg)
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grmeanby
In these examples, grmeanby shows different means distinctly, but that is not guaranteed.
Using graph dot as a default within designplot ensures more readability, although that too has its limits.
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designplot can show other statisticsYou can show any summarize result.
In practice, you would only want to plot results sharing the same units of measurement (including none at all, as with skewness and kurtosis).
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0 10 20 30 40
5
4
3
2
1
Foreign
Domestic
(all)
Mileage (mpg)
min p25 median mean p75 max
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More to say…
Although based on graph dot by default, designplot can be recast to graph bar or graph hbar.
Although based on summarizing single variables, what could be simpler than putting different designplots side-by-side?
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0 10 20 30mean
missing54321
ForeignDomestic
(all)
Mileage (mpg)
0 2,000 4,000 6,000mean
missing54321
ForeignDomestic
(all)
Price
0 1,000 2,000 3,000 4,000mean
missing54321
ForeignDomestic
(all)
Weight (lbs.)
511
1830
82
2252
74
0 20 40 60 80count
missing54321
ForeignDomestic
(all)
counts
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Is this just a reinvention of graph dot? No.
graph dot and its siblings are restricted in offering only one-way or two-way or three-way breakdowns given, respectively, one or two or three “factors”.
designplot gives scope for saving results for separate graphing or tabulation.
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The main players againsparkline crossplot combineplot designplotAll may be installed from SSC.
Our attraction to images as a source of understanding
is both primal and pervasive.Stephen Jay Gould (1941–2002)