Graphical Inference for Infovis
Hadley Wickham, Dianne Cook, Heike Hofmann, and Andreas Buja

Fig. 1. One of these plots doesn’t belong. These six plots show choropleth maps of cancer deaths in Texas, where darker colors =more deaths. Can you spot which of the six plots is made from a real dataset and not simulated under the null hypothesis of spatial
independence? If so, you’ve provided formal statistical evidence that deaths from cancer have spatial dependence. See Section 8 forthe answer.
Abstract— How do we know if what we see is really there? When visualizing data, how do we avoid falling into the trap of apophenia
where we see patterns in random noise? Traditionally, infovis has been concerned with discovering new relationships, and statistics
with preventing spurious relationships from being reported. We pull these opposing poles closer with two new techniques for rigorous
statistical inference of visual discoveries. The “Rorschach” helps the analyst calibrate their understanding of uncertainty and the “lineup” provides a protocol for assessing the significance of visual discoveries, protecting against the discovery of spurious structure.
Index Terms—Statistics, visual testing, permutation tests, null hypotheses, data plots.

1 I NTRODUCTION
What is the role of statistics in infovis? In this paper we try and answer that question by framing the answer as a compromise between
curiosity and skepticism. Infovis provides tools to uncover new relationships, tools of curiosity, and much research in infovis focuses onmaking the chance of finding relationships as high as possible. On theother hand, most statistical methods provide tools to check whether arelationship really exists: they are tools of skepticism. Most statistics
research focuses on making sure to minimize the chance of finding a
relationship that does not exist. Neither extreme is good: unfetteredcuriosity results in findings that disappear when others attempt to verify them, while rampant skepticism prevents anything new from being
discovered.
Graphical inference bridges these two conflicting drives to provide
a tool for skepticism that can be applied in a curiosity-driven context.
It allows us to uncover new findings, while controlling for apophenia,
the innate human ability to see pattern in noise. Graphical inference
helps us answer the question “Is what we see really there?”
The supporting statistical concepts of graphical inference are developed in [1]. This paper motivates the use of these methods for infovis
and shows how they can be used with common graphics to provide
users with a toolkit to avoid false positives. Heuristic formulations ofthese methods have been in use for some time. An early precursor
is [2], who evaluated new models for galaxy distribution by generating samples from those models and comparing them to the photo• Hadley Wickham is an Assistant Professor of Statistics at Rice University,
Email: hadley@rice.edu.
• Dianne Cook is a Full Professor of Statistics at Iowa State University.
• Heike Hofmann is an Associate Professor of Statistics at Iowa State
University.
• Andreas Buja is the Liem Sioe Liong/First Pacific Company Professor of
Statistics in The Wharton School at the University of Pennsylvania.
Manuscript received 31 March 2010; accepted 1 August 2010; posted online
24 October 2010; mailed on 16 October 2010.
For information on obtaining reprints of this article, please send
email to: tvcg@computer.org.

graphic plates of actual galaxies. This was a particularly impressive
achievement for its time: models had to be simulated based on tablesof random values and plots drawn by hand. As personal computers became available, such examples became more common.[3] compared
computer generated Mondrian paintings with paintings by the true
artist, [4] provides 40 pages of null plots, [5] cautions against overinterpreting random visual stimuli, and [6] recommends overlaying
normal probability plots with lines generated from random samples ofthe data. The early visualization system Dataviewer [7] implemented
some of these ideas.
The structure of our paper is as follows. Section 2 revises the basics
of statistical inference and shows how they can be adapted to work
visually. Section 3 describes the two protocols of graphical inference,
the Rorschach and the line-up, that we have developed so far. Section 4
discusses selected visualizations in terms of their purpose and associated null distributions. The selection includes some traditional statistical graphics and popular information visualization methods. Section 5
briefly discusses the power of these graphical tests. Section 8 tells you
which panel is the real one for all the graphics, and gives you somehints to help you see why. Section 7 summarizes the paper, suggests
directions for further research, and briefly discusses some of the ethical implications.
2

W HAT IS INFERENCE AND WHY DO WE NEED IT ?

The goal of many statistical methods is to perform inference, to draw
conclusions about the population that the data sample came from. This
is why statistics is useful: we don’t want our conclusions to apply only
to a convenient sample of undergraduates, but to a large fraction ofhumanity. There are two components to statistical inference: testing(is there a difference?) and estimation (how big is the difference?). In
this paper we focus on testing. For graphics, we want to address thequestion “Is what we see really there?” More precisely, is what we seein a plot of the sample an accurate reflection of the entire population?
The rest of this section shows how to answer this question by providing
a short refresher of statistical hypothesis testing, and describes how
testing can be adapted to work visually instead of numerically.
Hypothesis testing is perhaps best understood with an analogy tothe criminal justice system. The accused (data set) will be judged
guilty or innocent based on the results of a trial (statistical test). Eachtrial has a defense (advocating for the null hypothesis) and a prosecution (advocating for the alternative hypothesis). On the basis of how
evidence (the test statistic) compares to a standard (the p-value), thejudge makes a decision to convict (reject the null) or acquit (fail toreject the null hypothesis).
Unlike the criminal justice system, in the statistical justice system(SJS) evidence is based on the similarity between the accused andknown innocents, using a specific metric defined by the test statistic.
The population of innocents, called the null distribution, is generated
by the combination of null hypothesis and test statistic. To determine
the guilt of the accused we compute the proportion of innocents wholook more guilty than the accused. This is the p-value, the probability
that the accused would look this guilty if they actually were innocent.
There are two types of mistakes we can make in our decision: we
can acquit a guilty dataset (a type II error, or false negative), or falsely
convict an innocent dataset (a type I error, or false positive). Just as
in the criminal justice system, the costs of these two mistakes are notequal and vary based on the severity of the consequences (the risk ofletting a guilty shoplifter go free is not equal to the risk of letting a
guilty axe-murderer go free). Typically, as the consequences of ourdecisions become bigger, we want to become more cautious, and require more evidence to convict: an early-stage exploratory analysis is
free to make a few wrong decisions, but it is very important not to approve a possibly dangerous drug after a late-stage clinical trial. It is up
to the analyst to calculate and calibrate these costs.
To demonstrate these principles we use a small simulated example,
based on an experiment designed to compare the accuracy of condition
one vs. condition two in a usability study. Here, the defense argues
that there is no difference between the two groups, and the prosecution
argues that they are different. Statistical theory tells us to use the difference of the group means divided by the pooled standard deviation
as the measure of guilt (the test statistic), and that under this measure
the population of innocents will have (approximately) a t-distribution.
Figure 2 shows this distribution for a sample of 10,000 innocents, aone-side two-sample t-test. The value of the observed test statistic isrepresented as a vertical line on the histogram. Since we have no apriori notion of whether the difference between groups will be positiveor negative, it is better to compare the accused to the absolute value ofthe innocents, as shown in the bottom plot, a one-sided two-sample ttest. As you can see, there are few innocents (about 3%) who appear as
guilty as (or more guilty than) the accused and so the decision would
be to convict.

Fig. 2. (Top) Distribution of group means under null-hypothesis (distribution of innocents). Vertical line indicates value of observed test statistic.
This is a one-sided two-sample t-test. (Bottom) Distribution of absolute value of group means. This is more appropriate comparison for thetwo-sided test, where magnitude of difference is more important thandirection of difference. This is a two-sided two-sample t-test.

These principles remain the same with visual testing, except for two
aspects: the test statistic, and the mechanism of computing similarity.
The test statistic is now a plot of the data, and instead of a mathematical measurement of difference, we use a human judge, or even jury.
Figure 3 illustrates a graphical alternative to the traditional t-test.
The accused, a plot of the real data, is hidden among eight innocents,
plots of data generated from the null distribution. We need some newterminology to make this description more concise: A null dataset
is a sample from the null distribution, i.e. an example of an innocent
dataset, and a null plot is a plot of a null dataset, showing what an
innocent might look like. So Figure 3 hides the real plot amongst eightnull plots. Can you spot the suspect? If so, then there is some evidence
that the accused is different from the innocents, and we might move toconvict. (See Section 8 for the solution.)
This example shows the analogy between a traditional numericaltest and a new visual test, but the purpose of this work is not to supplant traditional tests. Traditional statistical tests are well studied,well-formulated and work best when data is well-behaved, following
a known distribution in relatively simple scenarios. But as researchers
in infovis have no doubt experienced, traditional statistical tests do notcover all of the complexities that arise when exploring data. The benefit of visual inference is that it can be used in complex data analysis
settings that do not have corresponding numerical tests.

Fig. 3. A visual t-test. For each data set, the observations are shownas points and the group means as crosses. The accused is hidden
amongst eight innocents. Can you spot him?

3 P ROTOCOLS OF GRAPHICAL INFERENCE
This section introduces two new rigorous protocols for graphical inference: the “Rorschach” and the “line-up”. The Rorschach is a calibrator, helping the analyst become accustomed to the vagaries of random
data, while the line-up provides a simple inferential process to produce
a valid p-value for a data plot. We describe the protocols and show examples of how they can be used, and refer the reader to [1] for moredetail.

self-administer such a test, particularly with appropriate software support, as described in Section 6.
Even when not administrated in a rigorous manner, this protocol isstill useful as a self-teaching tool to help learn which random features
we might spuriously identify. It is particularly useful when teachingdata analysis, as an important characteristic of a good analyst is theirability to discriminate signal from noise.

3.1 Rorschach
The Rorschach protocol is named after the Rorschach test, in which
subjects interpret abstract ink blots. The purpose is similar: readers
are asked to report what they see in null plots. We use this protocol tocalibrate our vision to the natural variability in plots in which the datais generated from scenarios consistent with the null hypothesis. Our
intuition about variability is often bad, and this protocol allows us toreduce our sensitivity to structure due purely to random variability.
Figure 4 illustrates the Rorschach protocol. These nine histograms
summarize the accuracy at which 500 participants perform nine tasks.
What do you see? Does it look like the distribution of accuracies is
the same for all of the tasks? How many of the histograms show aninteresting pattern? Take a moment to study these plots before youcontinue reading.

The SJS convicts based on difference between the accused and a set of
known innocents. Traditionally the similarity is measured numerically,
and the set of known innocents are described by a probability distribution. The line-up protocol adapts this to work visually: an impartial
observer is used to measure similarity with a small set of innocents.
The line-up protocol works like a police line-up: the suspect (teststatistic plot) is hidden in a set of decoys. If the observer, who has notseen the suspect, can pick it out as being noticeably different, there isevidence that it is not innocent. Note that the converse does not apply
in the SJS: failing to pick the suspect out does not provide evidence
they are innocent. This is related to the convoluted phraseology ofstatistics: we “fail to reject the null” rather than “accepting the alternative”.
The basic protocol of the line up is simple:

3.2

Line-up• Generate n − 1 decoys (null data sets).
• Make plots of the decoys, and randomly position a plot of thetrue data.
• Show to an impartial observer. Can they spot the real data?

Fig. 4. Nine histograms summarizing the accuracy at which 500 participants perform nine tasks. What do you see?

It is easy to tell stories about this data: in task 7 accuracy peaks
around 70% and drops off; in task 5, few people are 20-30% accurate; in task 9, many people are 60-70% accurate. But these stories
are all misleading. It may come as a surprise, but these results are
all simulations from a uniform distribution, that is, the distribution of
accuracy for all tasks is uniform between 0 and 1. When we display
a histogram of uniform noise, our expectation is that it should be flat.
We do not expect it to be perfectly flat (because we know it should
be a little different every time), but our intuition substantially underestimates the true variability in heights from one bar to the next. It is
fairly simple to work out the expected variability algebraically (using
a normal approximation): with 10 observations per bin, the bins will
have a standard error of 30%, with 100 observations 19% and 1000,
observations 6%. However, working through the math does not givethe visceral effect of seeing plots of null data.
To perform the Rorschach protocol an administrator produces nullplots, shows them to the analyst, and asks them what they see. To keep
the analyst on their toes and avoid the complacency that may arise if
they know all plots are null plots [8] the administrator might slip ina plot of the real data. For similar reasons, airport x-ray scanners
randomly insert pictures of bags containing guns, knives or bombs.
Typically, the administrator and participant will be different people,
and neither should know what the real data looks like (a doubleblinded scenario). However, with careful handling, it is possible toIn practice, we would typically set n = 19, so that if the accused is
innocent, the probability of picking the accused by chance is 1/20 =0.05, the traditional boundary for statistical significance. Comparing
20 plots is also reasonably feasible for a human observer. (The use ofsmaller numbers of n in this paper is purely for brevity.) More plots
would yield a smaller p-value, but this needs to be weighed against
increased viewer fatigue. Another way of generating more precise pvalues is to use a jury instead of a judge. If we recruit K jurors andk of them spot the real data, then the combined p-value is P(X ≤ k),where X has a binomial distribution B(K, p = 1/20). It can be as small
as 0.05K if all jurors spot the real data (k = K).
Like the Rorschach, we want the experiment to be double-blind neither the person showing the plots or the person seeing them should
know which is the true plot. The protocol can be self-administered,provided that it is the first time you’ve seen the data. After a firstviewing of the data, a test might still be useful, but it will not be inferentially valid because you are likely to have learned some of thefeatures of the data set and are more likely to recognize it. To maintain inferential validity once you have seen the data, you need to recruit
an independent observer.
The following section shows some examples of the line-up in use,with some discussion of how to identify the appropriate null hypothesis for a specific type of plot and figure out a method of generating
samples from the appropriate null distribution.
4

E XAMPLES

To use the line-up protocol, we need to:• Identify the question the plot is trying to answer.
• Characterize the null-hypothesis (the position of the defense).
• Figure out how to generate null datasets.
This section shows how to approach each of these tasks, and thendemonstrates the process in more detail for two examples. Section 4.1
shows a line-up of a tag cloud used to explore the frequency distribution of words in Darwin’s “Origin of Species” and Section 4.2 shows aline-up of a scatterplot used to explore the spatial distribution of threepoint throws in basketball.

Table 1. A selection of common statistical and infovis plots and thequestion that they might have been created to answer. Each plot maybe
be used to answer multiple questions and each question has a matchingnull hypothesis.

Plot

Question

Chloropleth maps
Tag cloudIs there a spatial trend?
Is this document the same as that document?
Is this document unusual?
Does the distribution of sizes follow a
power law?
Is the distribution within higher-level categories the same?
Is the underlying distribution smooth?
Is the underlying distribution uniform (ornormal or ...)?
Do the points lie along the line? (i.e. does
the data match the distribution?)
Are residuals normally distributed?
Are the two variables associated?
Are points clustered by colour?

Tag cloudTreemap
Treemap
Histogram
Histogram
QQPlotResidual plot
Scatterplot
Scatterplot withpoints colored
Facetted plots
Time series
Time series

Is the distribution of the data the same in all
facets?
Does the mean change over time?
Does the variability change over time?

It’s usually easy to identify the question a plot is trying to answer,
because certain types of plots are used for specific tasks and for a particular data set the analyst typically chooses to make a plot with aquestion in mind. Table 1 lists the questions associated with somecommon plots. Note that some plots can be used to answer multiple
questions, and so there may be different null hypotheses depending on
the circumstances.
The null hypothesis is the least interesting answer to a question,that is, an answer which is “what we see is really nothing”. In the language of the criminal justice system, the null hypothesis is the argument of the defense, that the suspect is innocent and nothing untoward
occurred. If we are using a scatterplot to answer the question “is therea relationship between x and y?”, then the null hypothesis is that there
is no relationship, or that the two variables are independent. We onlychange our minds if we have evidence to the contrary.
The next step of the process is to determine how to generate nulldatasets from the null hypothesis. There are two techniques that apply
in many circumstances:• Resampling. This is how null samples are generated for permutation tests [9; 10] and similarly for confidence intervals usingbootstrap samples [6]. In designed experiments it entails recreating the same randomization performed in the experiment. This
technique is used in Section 4.1.
• Simulation. We might be interested in a more specific set of hypothesis: does time increase linearly with distance from target?
Does accuracy decrease exponentially as number of distractors
increases? In those cases we have a probabilistic model and we
can generate null data sets by sampling from the distribution implied by the model. This approach is used in Section 4.2.
4.1 Tag clouds
A tag cloud (or text cloud) can be used to visualize frequency ofwords in a document (typically with common “stop” words removed).
Words are arranged in various ways, often alphabetically, with size
proportional to their frequency. Tag clouds are primarily descriptive, but if we look closely we can discover some inferential uses.

Firstly, there is an implicit question when we look at a tag cloud are there any unusually frequent (or infrequent) words? This question carries with it some notion of “usual” frequency distribution.
Secondly, tag clouds are also used for comparison. For example,
http://chir.ag/projects/preztags/ has a tag cloud foreach major presidential speech, with the implication that we can use
the tag clouds to spot the differences between speeches. Manyeyes[11] has an explicit comparison tool.
When comparing two texts with a tag cloud, we can imagine thedata as two columns (word and source) with a row for every word. In
the rawest form words are repeated reflecting their frequency. The null
hypothesis for a comparison tag cloud is that the two documents areequivalent, the frequency of words is the same in each document. In
other words, the word and source column are independent. It is simple
to generate new datasets under this null: we just randomly permute oneof the columns. Figure 5 uses this technique to compare the 1st and6th editions of Darwin’s “Origin of Species” (selected words displayed
due to space constraints). Word frequencies from the two editions are
displayed side-by-side, distinguished by colour (red = 1st ed, blue =6th ed). Can you spot the accused? (See Section 8 for the solution.)
Permutation of a single column can be used to address questions
about independence in other types of plots. The elegant feature of thisapproach to generating null data is that the marginal distribution ofeach variable is preserved while breaking any dependence. Here aresome common examples:• If we are interested in the spatial trend in a data map, then the null
hypothesis might be that location and value are independent. To
generate null datasets we permute the value column. This is how
Figure 1 was generated.
• In a scatterplot, an initial hypothesis might be that there is no
relationship between x and y. We can generate null hypotheses
by permuting either the x or y variables.
• If we have clustered the data and are displaying the results with acoloured scatterplot, we might be interested to know if the clusters are well separated. Hence the null hypothesis is that cluster
membership and position are independent, and we can generate
null datasets by permuting the cluster id column.
4.2

Scatterplot

A scatterplot displays the relationship between two continuous variables, and answers the question: are x and y related in some way? The
scatterplot can reveal many different types of relationships, e.g., linear
trends, non-linear relationships and clustering. A strong null hypothesis is that there is no relationship x and y variables, that is, they are
independent.
In many cases the assumption of independence is too strong: it isobvious that the two variables are related, and we want to investigate
a specific functional form. For example, we might believe that threepoint attempts in basketball follow a quadratic distribution in space:as the angle between the player and basket increases the player moves
closer to ensure success.
Figure 6 tests this hypothesis using data on all three pointers attempted by the Los Angeles Lakers in the 2008/09 season (data fromhttp://www.basketballgeek.com/). Can you spot the real
data? (See Section 8 for the solution.) It is embedded among null plots
generated under the hypothesis that the relationship really is quadratic.
These datasets are constructed by fitting the model, producing predictions and residuals, then adding rotated residuals [12] back on to thepredictions.
For more statistically-minded audiences, we could instead display
the residuals from the model. From the model definition, we expect
the residuals to be distributed normally with mean 0 and standard deviation 1, which makes null datasets easy to generate: sample from thestandard normal distribution. Figure 7 shows the results of this procedure. Is it easier or harder to spot the suspect compared to the previous
figure?

believe believe

case
case

closelyclosely descendants

believe believe

case
case

believe believe

believe believe

case

case
case

case

descendants few few

descendants few few

descendants few few

long long modified

long long modified
modified variations

long long modified
modified variations

long long modified

variations

variations

variations

very

very variations very

view view

case
case

closelycloselycloselycloselyclosely descendants closely descendants closely descendants
closely descendants

descendants few few

modified variations

believe believe

very view

view

very

modified variations

very view view very

descendants few few
long long modified
modified variations

very variations very

view view

very

view viewFig. 5. Five tag clouds of selected words from the 1st (red) and 6th (blue) editions of Darwin’s “Origin of Species”. Four of the tag clouds weregenerated under the null hypothesis of no difference between editions, and one is the true data. Can you spot it?

Fig. 6. Scatterplot of distance vs. angle for three pointers by the LA Lakers. True data is concealed in line-up of nine plots generated under the null
hypothesis that there is a quadratic relationship between angle and distance.

Fig. 7. Scatterplot of model residuals vs. angle for three point attempts by the LA Lakers. True data is concealed in line-up of nine plots generated
under the null hypothesis of standard normally distributed residuals.

Figures 6 and 7 both test the same null hypothesis. Which should
we use? The next section discusses the issue of power (the probabilityof spotting a real difference) in more detail.
5 P OWER
The power of a statistical test is the probability of correctly convicting
a guilty data set. The capacity to detect specific structure in plots candepend on many things, including an appropriate choice of plot, which
is where the psychology of perception is very important. Existing research [13; 14] provides good suggestions on how to map variables toperceptual properties to maximize the reader’s chances of accuratelyinterpreting structure. For example, we should map our most important continuous variables to position along a common scale, and to use
pre-attentive attributes, such as color, to represent categorical information like groups.
For large datasets, aggregation can make a big difference. Figure 8 displays a line-up for examining the relationship between
arrival delays and wind direction at Phoenix airport (data fromhttp://stat-computing.org/dataexpo/2009). The top
plot is a “natural” and elegant plot of the raw data, mapping wind direction to angle and arrival delay to radius. The bottom plot displays
the same data, but in the much aggregated form of a boxplot. It ismuch easier to spot the plot that is different from the others (panel 4),
which, indeed, corresponds to the real data: unusually high delays at
Phoenix airport are significantly associated with SW winds. One of
the reasons making the real data difficult to detect in the first plotis
that focus is on the outliers rather than the “hole” in the SW direction.
6 U SETo semi-automate the protocols we have created a
new R [15] package called nullabor, available fromhttp://github.com/hadley/nullabor.
It operates
simply. The user specifies how many decoy plots to create, and amechanism to generate null datasets. For the line-up, nullaborgenerates the decoys, labelled with a new .sample variable, appends
them to the real data set, and randomly chooses a position for theaccused. The Rorschach protocol is similar, but the true data is
only included with small probability. The package is bundled withmethods to generate null datasets from common null hypotheses(independence, specified model, and specified distribution), while
also allowing the user to add their own.
The following code shows two examples of nullabor in use. The
first line specifies the type of plot, in this case a scatterplot for both.
The second line specifies the protocol, the line-up, and the mechanism
for generating null datasets: permutation in the first example, and simulation from a model in the second. By default, the line-up will generate a plot with 20 panels, but this can be specified by the n argument
to the lineup function. The third line specifies the grid layout of thedecoys. The position of the true data is encrypted and output to thescreen so that the user can later decrypt the message and learn which
panel shows the true data.

analyst has seen the real data plot. At times the analyst will need toengage the services of an uninvolved observer. Services like Amazon
Mechanical Turk [16] might be useful here. This problem requires akeen pair of human eyes to evaluate the line-ups, and the Turk offers a
supply of workers who might even enjoy this type of task.
7

C ONCLUSION

This paper has described two protocols to bring rigorous statistical inference to freeform data exploration. Both techniques center around
identifying a null hypothesis, which then generates null datasets andnull plots. The Rorschach provides a tool for calibrating our expectations of null data, while the line-up brings the techniques of formal
statistical hypothesis testing to visualization.
Graphical inference is important because it helps us to avoid (or atleast calibrate the rate of) false convictions, when we decide a relationship is significant, when it is actually an artifact of our samplingor experimental process. These tools seem particularly important forvisualizations used in the VAST community, because the consequencesof false conviction of data can be so severe for the people involved.
We have provided a reference implementation of these ideas in the
R package nullabor. We hope others can build upon this work to
make tools that can be used in a wide variety of analytic settings.
8

S OLUTIONS

These are the solutions to each of the line-ups shown in the paper.
• Figure 1: the real data is in panel 3. The features that might clue
the reader in to this plot being different include spatial clustering,clumps of dark and light, and the prevalence of light polygons inthe south-west edge of the state. In the other plots the dark andlight coloring is scattered throughout the state.
• Figure 3: the real data is in panel 3. Features that the reader
might pick up include bigger differences between the means, andfairly consistent but shifted spread from one group to another.
• Figure 5: the real data is second from the right. Hint: look atbelieve, variations, view and very.
• Figure 6: the real data is panel 5. More outliers from center court
give the data away.
• Figure 7: the real data is panel 6. A few large outliers make thisplot different from the others.
ACKNOWLEDGMENTS
The authors wish to thank Robert Kosara for helpful discussion. This
work was partly supported by the National Science Foundation grant
DMS0706949. Graphics produced with ggplot2 [17].
R EFERENCES

qplot(radius, angle, data = threept) %+%lineup(permute("response"), threept, n = 9) +facet_wrap(˜ .sample)
qplot(radius, angle, data = threept) %+%lineup(model_null(radius ˜ angle), threept) +facet_wrap(˜ .sample)
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protocols as a normal step in their data analyses. We started with acommand-line user interface because it is what many statisticians aremost comfortable with, but it is not suitable for most analysts. We hopethat others will integrate graphical inference in to their tools, using theopen-source nullabor code to aid implementation.
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