Graphics (and numerics) for
univariate distributions
Nicholas J. Cox
Department of Geography
Durham University, UK
Klein and mine
Felix Klein (1848–1925) wrote a classic:
1908, 1925, 1928. Elementarmathematik
vom höheren Standpunkte aus.
Leipzig: Teubner; Berlin: Springer.
In this talk I look at elementary statistical graphics
from an intermediate standpoint.
2
Why is Stata graphics so complicated?
It offers
canned convenience commands for common
tasks (e.g. histograms, survival functions)
a framework for creating new kinds of graphs,
vital for programmers
cosmetic control of small details such as colours,
text and symbols
3
How to learn about Stata graphics?
The radical solution:
Read the documentation.
The friendly solution:
Read Michael Mitchell’s books.
Another solution:
Follow Statalist and the Stata Journal.
4
This talk
I will give a rag-bag of tips and tricks, including
some examples for official Stata commands
some examples of my own commands,
from the Stata Journal or SSC
(use net or ssc to install)
Code and datasets will be downloadable shortly.
5
Distributions
Most examples will show (fairly) raw data, but
there is plenty of scope to show distributions of
residuals, estimates, figures of merit, P-values,
q-values, and so forth.
Categorical variables will get short shrift, but my
best single tip is to check out catplot and
tabplot from SSC.
6
Small distributions with names
Bar charts need no introduction here.
graph hbar is a basic graph for showing
distributions with informative names attached.
hbar allows names to be written left to right.
20 or so values can be so shown fairly well, more if
the medium allows
(e.g. whole-page figure, poster).
7
25 largest cities in California
Los Angeles-Long Beach-Santa Ana
San Francisco-Oakland-Fremont
Riverside-San Bernardino-Ontario
San Diego-Carlsbad-San Marcos
Sacramento--Arden-Arcade--Roseville
San Jose-Sunnyvale-Santa Clara
Fresno
Bakersfield-Delano
Oxnard-Thousand Oaks-Ventura
Stockton
Modesto
Santa Rosa-Petaluma
Visalia-Porterville
Santa Barbara-Santa Maria-Goleta
Salinas
Vallejo-Fairfield
San Luis Obispo-Paso Robles
Santa Cruz-Watsonville
Merced
Chico
Redding
El Centro
Yuba City
Hanford-Corcoran
Madera-Chowchilla
0
2500 5000 7500 10000 12500
Population estimate for 1 July 2011 (000)
8
Small distributions with names
Less well known, graph dot is also a basic graph
for showing distributions with informative
names attached.
graph dot also allows names to be written left to
right.
Unlike bar graphs, graph dot also extends
naturally to cases in which logarithmic scales are
desired.
9
25 largest cities in California
Los Angeles-Long Beach-Santa Ana
San Francisco-Oakland-Fremont
Riverside-San Bernardino-Ontario
San Diego-Carlsbad-San Marcos
Sacramento--Arden-Arcade--Roseville
San Jose-Sunnyvale-Santa Clara
Fresno
Bakersfield-Delano
Oxnard-Thousand Oaks-Ventura
Stockton
Modesto
Santa Rosa-Petaluma
Visalia-Porterville
Santa Barbara-Santa Maria-Goleta
Salinas
Vallejo-Fairfield
San Luis Obispo-Paso Robles
Santa Cruz-Watsonville
Merced
Chico
Redding
El Centro
Yuba City
Hanford-Corcoran
Madera-Chowchilla
200
500 1000 2000 5000 10000
Population estimate for 1 July 2011 (000)
10
graph dot
This kind of graph is often called a dot chart or dot
plot.
There is scope for confusion, as the same name has
been applied to a different plot, on which more
later.
It is often named for William S. Cleveland, who
promoted it in various articles and books, as a
Cleveland dot chart.
11
Two or more distributions
graph dot is also good for comparing two
or more distributions with names
attached.
Here are some results on decadal population
change in urban areas of England and
Wales from the 2011 UK census.
Punch line: cities are growing!
12
urban areas in England and Wales outside London
with population > 275000 in 2011
Manchester
Leicester
Nottingham
Cardiff
Bradford
Bristol, City of
Kirklees
Birmingham
Sheffield
Sandwell
Liverpool
Bolton
Wigan
Newcastle upon Tyne
Leeds
Doncaster
Coventry
Wakefield
Dudley
Wirral
Stockport
Sunderland
-5
0
5
% change 1991 to 2001
10
15
20
% change 2001 to 2011
13
London boroughs
Tower Hamlets
Newham
Hackney
Hounslow
Greenwich
Waltham Forest
Redbridge
Islington
Haringey
Brent
Harrow
Wandsworth
Enfield
Southwark
Hillingdon
Barnet
Barking and Dagenham
Lambeth
Ealing
Camden
Westminster
Lewisham
Hammersmith and Fulham
Croydon
Richmond upon Thames
Kingston upon Thames
Havering
Bexley
Sutton
Merton
Bromley
Kensington and Chelsea
-5
0
5
10
% change 1991 to 2001
15
20
25
% change 2001 to 2011
14
graph dot small tips
Guide lines are best kept thin and a light colour.
MS Word users beware: dotted lines don’t transfer
well.
There is an undocumented vertical option, not
often needed but there if you really want it.
15
Larger distributions: histograms
At some point with larger distributions we have to
abandon naming every observation on a plot,
even if the names are known and informative.
Histograms remain very popular, despite the
possibility of graphic artefacts arising from
choices of bin width and bin origin.
Note that histogram and twoway histogram
are related but distinct commands.
16
Core based statistical areas, USA
200
Frequency
150
100
50
0
10000
100000
1m
Population estimate for 1 July 2011
10 m
17
Transformations and histograms
A twist in this example is use of a logarithmic scale.
Transform the variable first, here with log10().
Draw the histogram on a transformed scale.
Fix labels, e.g. 4 "10000", in xlabel().
Note that xsc(log) won’t do this for you.
18
Dividing histograms
Frequencies can be added.
So, for two subsets:
Lay down the frequency histogram for all.
Put the frequency histogram for a subset on top.
The difference is the other subset.
Use different colours, but the same bins.
Use the same (light) colour for blcolor().
This can be extended to three or more subsets.
19
Core based statistical areas, USA
200
Micropolitan
Metropolitan
Frequency
150
100
50
0
10000
100000
1m
Population estimate for 1 July 2011
10 m
20
Densities
If you really want to plot densities, kdensity is
the natural place to start.
Note that kdensity and twoway kdensity are
related but distinct commands.
21
Density estimation on transformed
scales
A longstanding but under-used idea is to estimate
densities on a transformed scale.
This will ensure that estimates are positive only
within the natural support and should help
stabilise estimates where data are thin on the
ground.
See Stata Journal 4: 66–88 (2004) for some
references.
22
Density estimation on transformed
scales
For density functions f of a variable x and a
monotone transform t(x),
estimate for f(x) =
estimate for f(t(x)) · | dt/dx |.
For example, estimate f(x) by f(ln x) · (1/x).
23
Example and code
Example data are lengths and widths of 158 glacial
cirques in the English Lake District. See more at
Earth Surface Processes and Landforms 32:
1902–1912 (2007).
tkdensity (SSC) is a convenience command that
does the estimation and graphing in one.
A paper with some photos of Romanian cirques
24
tkdensity with ln
.002
.0015
.001
0
0
.0005
Density
.0015
.001
.0005
Density
.002
kdensity
0
500
1000
length (m)
1500
2000
0
.0015
.001
0
0
.0005
Density
.001
.0005
Density
3000
biweight on ln scale .4
.0015
kernel = epanechnikov, bandwidth = 83.6216
1000
2000
length (m)
0
500
1000
width (m)
1500
kernel = epanechnikov, bandwidth = 96.1975
2000
0
500
1000
1500
width (m)
biweight on ln scale .4
2000
2500
25
Dot plots or strip plots
The main idea is to show each data point by one
marker symbol on a magnitude scale.
Usually, although not necessarily, there is binning
too and tied values are jittered or stacked to
show relative frequency.
In official Stata the command is dotplot.
stripplot from SSC is much more versatile.
First we look at some examples using the default
horizontal alignment.
26
50 m bins
width
length
0
500
1000
size (m)
boxes show 5 25 50 75 95 percentiles
1500
2000
27
Marginal box plots
stripplot (for that matter dotplot too) can
add box plots.
That way box plots do what they arguably do best:
summarize.
The fine structure of the data remains visible.
stripplot allows box plots with whiskers drawn
to specified percentiles, as well as those
following the Tukey rule that whiskers span data
points within 1.5 IQR of the nearer quartile.
28
An aside on box plots
If you like box plots, you will know that graph
box and graph hbox get you there…
… except in so far as they don’t.
Suppose you want to do something a bit different,
such as add points for means, or join medians.
See Stata Journal 9: 478-496 (2009) for details on
how to do box plots from first principles.
29
WA
VA
TN
OR
OK
OH
UT
AK
24
26
WY
SD
NH
MT
NV
MN WI
KS
RI
NM
ND
MI
VT
IA
WV
PA
MS
HI
KY
NC
IL
MD
NY
LA
TX
GA
IN
NE
DE ME MO
NJ
ID
SC CO
AZ
AL
CA AR MA
CT
28
30
Median age 1980
32
FL
34
0.5 year bins
30
How was that done?
This plot of median age in 1980 for US states also
used stripplot.
The main trick is very simple: make marker
symbols big enough and marker labels small
enough so they jointly act as small text boxes.
OH yes, I agree: 50 US states with two-letter
abbreviations AR an easy case, but WY not?
31
Panel or longitudinal data
Let’s change tack for a different kind of example,
with panel data.
The dataset is small: OECD data on percent regular
cigarette smoking at age 15 for 24 countries,
4 time periods and 2 genders.
Panel data can be seen as a series of distributions.
The distribution can serve as context for any
interesting case, just as a test score is reported as
a percentile rank.
32
boys
girls
40
30
20
10
0
1993-94
1997-98
2001-02
2005-06 1993-94
1997-98
2001-02
2005-06
33
Spaghetti plot, or a graphical pastiche
The usual kind of multiple time series plot
(here using line) is the usual kind of mess.
I suppressed the legend naming the countries.
There are ways of improving it as a time series plot,
such as using a by() option or some other
device for splitting out subsets.
But we will stick with the distribution theme.
First, look at a stripplot in which the USA is
highlighted.
34
Regular cigarette smoking in 15 year olds
girls
boys
40
percent
30
20
10
0
1993-94
1997-98
2001-02
2005-06
1993-94
1997-98
2001-02
2005-06
USA highlighted
35
stripplot for panel data
In principle, we lose some information on
individual trajectories.
In practice, a multiple time series plot is likely to
be too unattractive to invite detailed scrutiny.
As before, we could add box plots, or bars with
means and confidence intervals.
36
girls
boys
40
30
20
10
0
1993-94 1997-98
USA highlighted
2001-02
2005-06
1993-94 1997-98
2001-02
2005-06
37
devnplot
The previous graph was from devnplot (SSC).
devn here stands for deviation.
The values for each group are plotted as quantiles
or order statistics.
A subset of cases may be highlighted (here just one
panel).
A backdrop shows values as deviations from group
means.
38
devnplot
The choices shown are the defaults.
Other plotting orders are possible.
The backdrop can be removed, or tuned to show
deviations from any specified set of levels.
devnplot was first written with the aim of
showing data and summaries in anova style,
but I mostly use it to show sets of quantiles.
39
devnplot
A small but sometimes useful detail is that
devnplot adjusts the width for each group
according to the number of its values.
This can help if groups are of very different sizes.
40
Quantile plots
Quantile functions are also known as inverse
(cumulative) distribution functions. Quantiles
mean here the order statistics, as a function of
cumulative probability or fraction of the data.
For ranks i = 1, …, n, use a plotting position such as
(i − 0.5)/n as abscissa.
In official Stata the main command is quantile.
qplot from Stata Journal is much more versatile.
Code from SJ 12: 167 (2012).
41
girls
boys
40
30
20
10
0
1
.5
0
.5
0
1
fraction of the data
1993-94
1997-98
2001-02
2005-06
42
qplot options
What about smoothing?
qplot supports over() and by() options, to plot
quantiles by distinct groups within and between
graph panels.
In this example, some of the irregularity stems
from reporting values as integers. None of the
irregularity is easy to interpret.
So why not smooth the quantile functions?
43
Quantile smoothing
Quantile smoothing is less well known than kernel
density estimation.
The method of Harrell, F.E. and C.E. Davis. 1982.
A new distribution-free quantile estimator.
Biometrika 69: 635–640 turns out to be an exact
bootstrap estimator of the corresponding
population quantile.
hdquantile (SSC) offers an implementation.
44
girls
boys
40
30
20
10
1
.5
0
.5
0
1
fraction of the data
1993-94
1997-98
2001-02
2005-06
45
How much difference does quantile
smoothing make?
Quantile smoothing is conservative.
Here the difference between smoothed and
observed quantiles is ~1%.
So, smoothing mostly takes out noise, which is its
job.
46
3
2
1
0
-1
-2
0
10
20
30
Regular cigarette smoking in 15 year olds, %
40
47
Lord Rayleigh discovering argon
John William Strutt, Lord Rayleigh
(1842–1919) compared the mass of
nitrogen obtained by different methods
from a given container.
The marked difference led to the discovery of
argon with Sir William Ramsay and the award to
Rayleigh of the Nobel Prize for Physics in 1904.
The Rayleigh distribution is named for the same
Rayleigh.
48
method
chemical
air
2.31000
mass (g)
2.30500
2.30000
2.29500
ferrous
hydrate
hot
copper
hot
iron
ammonium nitric
oxide
nitrite
nitrous
oxide
49
Which plot?
devnplot works well here to show fine structure
in the data.
stripplot doesn’t work so well and a boxplot
just suppresses detail unnecessarily.
(Rayleigh was reporting extremely careful
experimental results to a resolution of 10 μg.)
50
Multiple quantile plots
For exploring a bundle of numeric variables, likely to have
very different ranges and units, multqplot is offered.
See also Stata Journal 12(3) (2012).
The recipe is just to produce a qplot for each variable and
then use graph combine.
A graph for each variable puts a premium on space. The
variable’s details go on top.
Values of selected quantiles are shown (by default
0(25)100, giving a box plot flavour).
51
Price
Mileage (mpg)
15906
Repair Record 1978
41
Headroom (in.)
5
5
4
3.5
3
6303
5079
4195
3291
24
3
20
18
2.5
12
0
.25
.5
.75
1
1
0
Trunk space (cu. ft.)
.25
.5
.75
1
1.5
0
Weight (lbs.)
.25
.5
.75
1
0
Length (in.)
.25
.5
.75
1
.75
1
Turn Circle (ft.)
23
4840
233
51
17
15
3670
204
193
43
3200
40
11
170
36
2240
5
1760
0
.25
.5
.75
1
142
0
Displacement (cu. in.)
.25
.5
.75
1
Gear Ratio
425
31
0
.25
.5
.75
1
.75
1
0
.25
.5
Car type
3.89
1
3.3
250
2.93
2.73
196
119
79
2.19
0
.25
.5
.75
1
0
0
.25
.5
.75
1
0
.25
.5
52
Features of quantile plots
Show well any outliers, gaps or granularity.
Scale well over a large range of possible sample sizes.
Entail a minimum of arbitrary choices.
Signal behaviour that might be awkward in modelling.
Behave reasonably with ordinal or binary variables.
For more propaganda: Stata Journal 5: 442–460 (2005).
53
Distribution and survival plots
Those who prefer distribution plots with axes
interchanged will find a command in distplot.
See discussion in Stata Technical Bulletin 51:
12–16 (1999). Get code from Stata Journal 10:
164 (2010).
The convention is to plot cumulative probability
against magnitude.
Those who plot survival functions are likely to be
working already with sts graph.
54
When to write a new graphics
command?
Sometimes you want a graph that is new to you.
After checking that no command exists, most often
you will plan to construct a graph using twoway
commands.
Less often, it will be an application for graph
dot, bar, hbar, box or hbox.
But play with do-files first.
Most advice is to plan program writing, but for
small projects it makes as much sense to see
what grows easily and naturally out of play.
55
Principles of laziness
Let official Stata do as much as possible.
Let other programs do as much as possible.
Don’t generalise programs or add features too
readily.
Don’t trust what you didn’t create.
Don’t plan too much; play and see what works.
56
Assessing normal probability plots
Suppose you are assessing fit to a normal or
Gaussian distribution.
qnorm is a dedicated official command for normal
probability plots (which are in fact quantilequantile plots).
How much departure from a straight line is
acceptable?
(If you really want a formal test, there are plenty
on offer.)
57
A plot is a sample statistic
Even in exploration, the attitude that a plot from a
sample is a sample statistic, just like a sample
mean or a slope estimate, is always salutary.
So we should be worrying about how the plot that
we do have — from our one sample — lies within
a sampling distribution of possible plots for
different samples.
The auto dataset gives a sample of 74 car weights.
58
5,000
4,000
3,000
2,000
1,000
1,000
2,000
3,000
Inverse Normal
4,000
5,000
59
Envelope curves
One recipe suggested is to get envelopes by
simulating several samples of the same size from
a Gaussian with the same mean and SD
sorting each sample from smallest to largest
reporting results for each order statistic as an
interval (e.g. spanning 95% of results)
60
One solution (mine)
Write a helper program, qenvnormal (SSC), that
calculates the envelopes. Mata is the work-horse.
qplot is already general enough to plot the
envelopes too, so there is no need for an extra
graphics program.
Stifle the urge to extend the first program to
include all my favourite distributions
(gamma, lognormal, and so forth).
61
95% reference intervals
6000
Weight, lb
4000
2000
0
0
.2
.4
.6
fraction of the data
.8
1
62
95% reference intervals
5000
Weight, lb
4000
3000
2000
1000
1000
2000
3000
Normal quantiles
4000
5000
63
qplot solutions
qplot is able to plot observed quantiles and the
envelopes, against both fraction of the data and
normal quantiles.
By the way, qenvnormal warns if there is a
reversal of order in the generated envelopes, best
taken to mean that the number of replications is
too small.
64
Going gray
A detail in several graphs worth flagging is the
usefulness of gray colours for less important
elements such as grid lines.
For more, see Stata Journal 9: 499–503 and
9: 648–651 (2009).
65
The aim
… delight lies somewhere between boredom
and confusion.
Sir Ernst Gombrich (1909–2001)
1984. The sense of order: A study in the
psychology of decorative art.
Oxford: Phaidon, p.9.
66
67
Bits and pieces follow
The ratings to follow are subjective ratings based
on how often I see such graphs compared with
how often they seem about the best possible.
A really good graph used when appropriate would
come in the middle on such a rating.
68
Graphs for measured variables
underrated
quantile plots
strip plots
distribution plots
survival plots
density plots
histograms
box plots
overrated
69
Graphs for categorical variables
underrated
dot charts
multi-way bar charts
side-by-side bar charts
stacked bar charts
spineplots
mosaic plots generally
pie charts
overrated
70
Presentation notes for font freaks
and similar strange people
The main font is Georgia.
Stata syntax is in Lucida Console.
The graphs use Arial. I nearly used Gill Sans MT.
The Stata graph scheme is s1color.
71