Graphics (and numerics) for univariate distributions

Nicholas J. Cox Department of Geography Durham University, UK

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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.

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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

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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.

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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)

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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, and so forth.

Categorical variables will get short shrift, but my best single tip is to check out catplot and tabplot from SSC.

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Small distributions with namesBar 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).

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0 0.5 m 1 m 1.5 m 2 mPopulation 2012

GotlandJämtlandBlekinge

KronobergKalmar

VästernorrlandNorrbotten

VästmanlandVästerbotten

VärmlandSödermanland

DalarnaGävleborg

ÖrebroHalland

JönköpingUppsala

ÖstergötlandSkåne

Västra GötalandStockholm

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0 50 100 150 200 250 300Population density per sq. km 2012

JämtlandNorrbotten

VästerbottenDalarna

VästernorrlandGävleborgVärmland

GotlandKalmar

KronobergJönköping

ÖrebroÖstergötland

UppsalaSödermanland

VästmanlandBlekingeHalland

Västra GötalandSkåne

Stockholm

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Small distributions with namesLess 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.

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2 5 10 20 50 100 200Population density per sq. km 2012

JämtlandNorrbotten

VästerbottenDalarna

VästernorrlandGävleborgVärmland

GotlandKalmar

KronobergJönköping

ÖrebroÖstergötland

UppsalaSödermanland

VästmanlandBlekingeHalland

Västra GötalandSkåne

Stockholm

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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.

Whatever the name, dot charts also work well for comparison of distributions.

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0 10 20 30 40 50

SwedenFinlandIrelandIceland

PortugalUnited Kingdom

SlovakiaBelgiumNorway

MaltaGermany

Czech RepublicFrance

NetherlandsItaly

RomaniaSwitzerland

SpainDenmarkHungaryCyprusAustriaGreecePoland

BulgariaLithuaniaSloveniaEstonia

Latvia

% daily smokers ~2002

females males

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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.

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Larger distributions: histogramsAt 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.

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0

50

100

150

200F

req

ue

ncy

10000 100000 1 m 10 mPopulation estimate for 1 July 2011

Core based statistical areas, USA

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Transformations and histogramsA 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.

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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.

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0

50

100

150

200F

req

ue

ncy

10000 100000 1 m 10 mPopulation estimate for 1 July 2011

MicropolitanMetropolitan

Core based statistical areas, USA

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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.

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Density estimation on transformed scalesA 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.

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Density estimation on transformed scalesFor 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).

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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

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kdensity tkdensity with ln0

.00

05

.00

1.0

01

5.0

02

De

nsi

ty

0 500 1000 1500 2000length (m)

kernel = epanechnikov, bandwidth = 83.6216

0.0

00

5.0

01

.00

15

.00

2D

en

sity

0 1000 2000 3000length (m)

biweight on ln scale .4

0.0

00

5.0

01

.00

15

De

nsi

ty

0 500 1000 1500 2000width (m)

kernel = epanechnikov, bandwidth = 96.1975

0.0

00

5.0

01

.00

15

De

nsi

ty

0 500 1000 1500 2000 2500width (m)

biweight on ln scale .4

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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.

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length

width

0 500 1000 1500 2000size (m)

boxes show 5 25 50 75 95 percentiles

50 m bins

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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.

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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.

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UT AK WY ID

LA

MS

NM

SC

TX

CO

GA

HI

ND

AZ

IN

KY

MI

MN

MT

SD

AL

NE

NC

VT

WI

CA

DE

IL

IA

KS

NV

NH

OH

OK

OR

TN

VA

WA

AR

ME

MD

WV

MA

MO

CT

NJ

NY

PA

RI

FL

24 26 28 30 32 34Median age 1980

0.5 year bins

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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?

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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.

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0

10

20

30

40

1993-94 1997-98 2001-02 2005-06 1993-94 1997-98 2001-02 2005-06

boys girls

Re

gu

lar

cig

are

tte

sm

oki

ng

in 1

5 y

ea

r o

lds,

%

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Spaghetti plot, or a graphical pasticheThe 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.

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0

10

20

30

40

1993-94 1997-98 2001-02 2005-06 1993-94 1997-98 2001-02 2005-06

boys girlsp

erc

en

t

USA highlighted

Regular cigarette smoking in 15 year olds

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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.

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0

10

20

30

40R

eg

ula

r ci

ga

rett

e s

mo

kin

g in

15

ye

ar

old

s, %

boys girls

1993-94 1997-98 2001-02 2005-06 1993-94 1997-98 2001-02 2005-06

USA highlighted

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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.

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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.

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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.

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Quantile plots

Quantile functions are also known as inverse (cumulative) distribution functions.

They are the order statistics, given 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 SJ is much more versatile.

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0

10

20

30

40

0 .5 1 0 .5 1

boys girls

1993-94 1997-98 2001-02 2005-06

Re

gu

lar

cig

are

tte

sm

oki

ng

in 1

5 y

ea

r o

lds,

%

fraction of the data

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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?

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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.

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10

20

30

40

0 .5 1 0 .5 1

boys girls

1993-94 1997-98 2001-02 2005-06

Re

gu

lar

cig

are

tte

sm

oki

ng

in 1

5 y

ea

r o

lds,

%

fraction of the data

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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.

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-2

-1

0

1

2

3

smo

oth

ed

qu

an

tile

- o

bse

rve

d q

ua

ntil

e,

%

0 10 20 30 40Regular cigarette smoking in 15 year olds, %

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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.

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2.29500

2.30000

2.30500

2.31000m

ass

(g

)

air chemicalmethod

ferroushydrate

hotcopper

hotiron

ammoniumnitrite

nitricoxide

nitrousoxide

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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.)

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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: 549-561 (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).

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3291419550796303

15906

0 .25 .5 .75 1

Price

12

1820

24

41

0 .25 .5 .75 1

Mileage (mpg)

1

33

4

5

0 .25 .5 .75 1

Repair Record 1978

1.5

2.5

3

3.5

5

0 .25 .5 .75 1

Headroom (in.)

5

11

1517

23

0 .25 .5 .75 1

Trunk space (cu. ft.)

1760

2240

3200

3670

4840

0 .25 .5 .75 1

Weight (lbs.)

142

170

193

204

233

0 .25 .5 .75 1

Length (in.)

31

36

40

43

51

0 .25 .5 .75 1

Turn Circle (ft.)

79119

196

250

425

0 .25 .5 .75 1

Displacement (cu. in.)

2.19

2.732.93

3.3

3.89

0 .25 .5 .75 1

Gear Ratio

000

11

0 .25 .5 .75 1

Car type

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Features of quantile plotsShow 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).

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Distribution and survival plotsThose who prefer distribution plots with

axes interchanged will find a command in distplot (SJ).

The convention is to plot cumulative probability against magnitude.

Those who plot survival functions are likely to be working already with sts graph.

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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.

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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.

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Assessing normal probability plotsSuppose you are assessing fit to a normal

or Gaussian distribution. qnorm is a dedicated official command for

normal probability plots (which are in fact quantile-quantile plots).

How much departure from a straight line is acceptable?

(If you really want a formal test, there are plenty on offer.)

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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.

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1,0

00

2,0

00

3,0

00

4,0

00

5,0

00

We

igh

t (lb

s.)

1,000 2,000 3,000 4,000 5,000Inverse Normal

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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)

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One solution (mine)

Write a helper program, qenvnormal (qenv package on 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, etc.)

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0

2000

4000

6000W

eig

ht,

lb

0 .2 .4 .6 .8 1fraction of the data

95% reference intervals

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1000

2000

3000

4000

5000W

eig

ht,

lb

1000 2000 3000 4000 5000Normal quantiles

95% reference intervals

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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.

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qenv on SSC

Maarten Buis (how to say his name) has extended the package beyond its original example programs for normal and gamma distributions.

Type ssc desc qenv in Stata for an up-to-date list.

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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).

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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.

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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.

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Graphs for measured variablesunderrated quantile plotsstrip plots distribution plotssurvival plotsdensity plots histogramsbox plotsoverrated

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Graphs for categorical variablesunderrated dot charts multi-way bar charts side-by-side bar charts stacked bar charts spineplots mosaic plots generally pie charts overrated

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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.