Small multiples, or the science and art of combining graphs

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Small multiples, or the science and art of
combining graphs

• Nicholas J. Cox
• Department of Geography
• Durham University, UK

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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)

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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. 

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• 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
• aaplot
• sepscatter
• All may be installed from SSC.

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Whats 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
  pictures
• Ben 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 193554. They give us a
  simple sandpit.

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What are we doing here? The problem of time
series graphics

• Comparisons 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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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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Nightingales data

• Florence Nightingale (18201910) 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

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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 18545 coincided with some of
  the hardest battles of the war, but 18556 was
  quite different.
• But, as often happens, no one graph dominates
  others here.

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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.

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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. )

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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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• 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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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)

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• combineplot
• The greatest value of a picture is when it forces
  us
• to notice what we never expected to see.
• John Wilder Tukey (19152000)

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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.

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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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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 matrices

• combineplot 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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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
  _at_y, over(rep78)
• Here we have one varlist and the syntax
• _at_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 _at_y _at_x, ytitle("Price
  (USD)") sep(foreign)
• Here we have two varlists and the syntax elements
  
• _at_y and _at_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 (19162009)

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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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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 default

• The 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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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).

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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
• classadult, classmale, adultmale

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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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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 statistics

• You 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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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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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 again

• sparkline
• crossplot
• combineplot
• designplot
• All may be installed from SSC.
• Our attraction to images as a source of
  understanding
• is both primal and pervasive.
• Stephen Jay Gould (19412002)

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