Chong Ho Yu, Ph.D. & Shawn
Stockford
Arizona State University
Paper presented at the Joint Statistical Meeting,
Altanta, GA (2001 August)
t
is commonly believed that visualization tools can help
researchers unveil hidden patterns and relationships
among variables, and also can help teachers and speakers
present abstract statistical concepts and complicated
data structures in a concrete manner. However,
higherdimension visualization techniques (depicting more
than three dimensions) can be confusing and even
misleading, especially when humaninstrument interface
and cognitive issues are underapplied. Furthermore,
statisticians, like other humans, are vulnerable to
visual illusions when viewing statistical graphs
(Wilkinson, 1993). Jacoby (1991, 1998) asserts that
multipledimension is not a problem to mathematics, but
remains a challenge to the school of data visualization.
From the standpoint of human perception and
understanding, the potentially extreme
multidimensionality of multivariate data presents
serious difficulties due to many cognitive limitations,
and is what many call the "curse of dimensionality"
(Bellman, 1961; Fox, 1997).
First, spatially speaking, humans live in a
threedimensional world. Four or more dimensions are out
of the scope of our spatial perception. Second,
traditional print media can depict twodimensional graphs
only. A socalled 3D graph that is rendered on paper
through a twodimensional window must involve nonlinear
projection or spatial compression, either of which
involves a certain degree of distortion, compromising the
viewer's ability to accurately perceive the multivariate
relationship therein (Wilkinson, 1999). With the advance
of computer technology, the rendering of
threedimensional graphs, such as the spin plot, becomes
more accessible than in the past. However, simultaneously
viewing more than three variables is still a challenge.
Nonetheless, researchers have been devoting tremendous
efforts to go beyond three dimensions in an attempt to
provide a tool that can capture rich associations among
variables whose relationships are too complex to be
considered with bivariate methods. This paper will
present a taxonomy of highdimensional data visualization
techniques, and further, evaluate an example from each
category (see Table 1).
Table 1. Taxonomy of higherdimension
visualization tools and examples evaluated

Spatialoriented

Temporaloriented

Datadriven

Splus2000's Trellis Conditioning plot

SAS/Insight's animated surface plot

Modeldriven

SyStat's 3D Triangular plot

Maple's animated 3D plot

In this study, attention is more focused on users'
mental and behavioral processes during graph
interpretation, rather than the rate of a successful
outcome. That is, we are more interested in how users
interact with these tools than whether they can arrive at
the right answer. Findings and recommendations from this
study are applied to the preceding categories of
visualization tools, as well as specific software
applications. The implications are concerned with two
aspectsresearch and teaching/presentations. Findings
based upon datadriven graphs are related to research
whereas findings based upon modeldriven plots are
applied to teaching and presentations.
The datadriven vs. modeldriven distinction is a
simple concept and thus will be explained briefly. In
datadriven graphics, raw data points make up the image
in the graph's presentation space, whereas modeldriven
plots show a mathematical function only. The former
approach is more appropriate at the early stage of data
analysis. The latter approach is bettersuited for
teaching and presentation when patterns and relationships
in one's data have been uncovered (Yu, 1994; Yu &
Behrens, 1995). Some graphs depict both observations and
a model, such as when a model is superimposed over raw
data points. These graphs can be considered datadriven
when the data points themselves determine the function
shape, and/or when the fourth variable updates the points
shown in the plot. Likewise, they can be considered
modeldriven when the mathematical function determines
the shape of the surface, and when the fourth dimension
informs the surface itself. In the next section, the
features of Spatialoriented and Temporaloriented
graphical displays will be discussed, and the example
graphs will be introduced.
Spatialoriented
Multiplesymbol vs. multipleview
Before the introduction of highpowered computers,
spatialoriented approaches were the dominant paradigms
for visualizing multivariate relationships.
Spatialoriented graphs are basically still graphs, in
which all relevant information is displayed at the same
time in a given space. Within this camp there are two
subcategories: Multiplesymbol and multipleview. In the
former, usually one display panel simultaneously shows
values of multiple variables that are represented by
different shapes, sizes, colors, and locations of symbols
(Tukey & Tukey, 1988). For example, although a 2D
scatterplot can display two variables only, the data
points can appear in different size to depict the third
variable. A "tail" can also be added to each data point,
in which the value of the fourth dimension is indicated
by the angle of the tail (Figure 4). Since the data
points represented by complex symbols are called
"glyphs," this type of display is termed as "glyph plot."
Chernoff face (Chernoff, 1973) is another example of a
multiplesymbol format. In a Chernoff face, multiple
variables are represented by different facial
features.However, the display can be very busy, and tends
to overload the viewer. Moreover, the subjective
assigning of facial features to variables has a marked
effect on the eventual shape of the face, and thus the
interpretation (du Toit, Steyn, & Stumpf, 1986). The
shortcomings of Chernoff face are also applied to other
types of graphs under the multiplesymbol paradigm.
Figure 1. Multiplesymbol display (glyph plot)
that uses symbol size and "tail" angle variations
Multipleview (Trellis) display
In the multipleview paradigm, usually only one type
of symbol is used but conditional relationships are
portrayed in multiple panels, and thus it is socalled
the multipleview approach. One major challenge of
multivariate visualization is to view all variables
simultaneously but avoid cognitive overloading. And thus,
some isolation of variables is essential. This mission is
paradoxical but the multipleview approach successfully
adopts a strategy of "divide and conquer." In this paper
discussion of spatialoriented visualization is centered
on this more promising paradigm.
There are several types of multipleview plots, such
as, caseman displays, coplots, and Trellis displays. The
Trellis display, which is available in Splus 2000
(MathSoft, 2001), is chosen to illustrate
spatialoriented/datadriven visualization (Becker,
Cleveland, & Shyu, 1996; Clark, Cleveland, Denby,
& Liu, 1999).At first glance, the Trellis display
looks like a scatterplot matrix because both utilize
multiple panels. However, a scatterplot matrix shows the
relationships in a pairwise fashion while a Trellis
display shows all relationships simultaneously. In a
Trellis display (Figure 2), the vertical axis shows a
dependent variable while the horizontal axis of each
panel (view) shows a "panel variable." The variables
appearing inside the "bars" of each panel are called
"conditioning variables." For our example, the first
panel of Figure 2 (lower left) shows the relationship
(simple slope) between Y and B while the values of A and
C are low. The second panel (bottom center) indicates the
relationship between Y and B when the value of C is low
and the value of A is medium. Relationships between Y and
B at different levels of A and C can be examined by the
multiple displays.
Using a movie as a metaphor, these multiple panels can
be thought of as frames of a filmstrip. The slope of B
against Y can be "animated" if the researcher stacks all
panels together and flips them through quickly. In this
example, since the slope of B against Y remains constant
in all nine panels, the relationship between Y and B must
be consistent across all levels of A and C. Thus, it is
concluded that there is no interaction effect among A, B,
and C. In addition, there are potentials for the Trellis
display to expand its usefulness. Users can control the
number of panels, and change the number, intervals and
layout of the conditioning variables.
Figure 2. A Trellis display showing no
interaction
Figure 3 shows a different scenario. The relationship
between Y and A appears to be consistent across different
levels of C, but seems to vary across the changes in B.
Thus, the Trellis plot suggests that there exists a 2way
interaction between A and B.
Figure 3. A Trellis display showing a 2way
interaction.
Following this strategy, a researcher could detect
whether a 3way interaction is present or not in Figure
4. In Figure 4 it is obvious that the relationship
between Y and C is inconsistent across different levels
of A, as well as different levels of B. Hence, a 3way
interaction is concluded.
Figure 4. A Trellis display showing a 3way
interaction.
In Wilkinson's (in press) view, the multiple panel
approach is less prone to erroneous perception than the
multiple symbol approach. Wilkinson uses the comparison
between bar charts in multiple panels and bar charts
using multiple symbols in fewer panels as an example.
Wilkinson argues that in the latter although the
collapsing of dimensions into fewer panels could save
space, it introduces a symbol choice problem. It is
difficult to find symbols that are easily distinguishable
for more than a few categories. On the other hand, bar
charts in separate panels, which are more similar to
Trellis displays, convey a higher degree of clarity.
One drawback to the Trellis display is that the
relationships depicted in each panel are bivariate. It
does not give a wholistic sense of the multivariate
relationship. We are not directly viewing the
fourvariable relationship in any one panel. This type of
display requires viewing the combination of the bivariate
plots to give the researcher a multivariate perspective.
Also, some multivariate relationships can be "hidden" in
the Trellis display. For example, a twoway interaction
between B and C could be virtually invisible in a Trellis
plot if the graph is created with the A variable on the
abscissa and both conditioning variables specified as B
and C (see figures x and x for illustration). Thus, the
user of Trellis displays must have an exploratory
mentality to exhaust all possible combinations of axis
and conditioning panel allocation.
3D triangular plot
The threedimensional triangular plot, which is
available in SyStat (SPSS, Inc., 2001), is used as an
example of a spatialoriented/modeldriven visualization
tool. Unlike the Trellis plot, raw data points are hidden
and only the function imposed on the data is shown in the
3D triangular plot (Figure 5). It is important to note
that the axes in this type of plot are collapsed using
triangular coordinates. In the graph, there are four
dimensionsthree variables are depicted in the
triangular coordinates on the "floor" of the data space,
while the Y variable is represented as a vertical axis as
in the Cartesian (rectangular) coordinate system. Since
this type of data space combines features of both
triangular and Cartesian coordinate systems, it is also
named 3D triangular/rectangular coordinate system
(Wilkinson, 1999).
Triangular coordinates are also known as Barycentric
coordinates, trilinear coordinates, and homogeneous
coordinates. The technique was introduced by August
Ferdinand Mobius in 1827 as a way to represent a point in
the plane with respect to a given triangle. Although this
new coordinate system was not appreciated at first, there
are many interesting and useful applications
(DanaPicard, 2000; Diamond, 2001). Usually there are
some constraints on the values of the three variables.
Each variable can have a relative concentration between
0% and 100%. If A is at 100%, B and C must both be at 0%,
and the point (100%, 0%, 0%) falls at one apex of the
triangle. As you notice, the three axes of three
variables in the SyStat's density plot do not range from
zero to one. A conversion takes place in the program that
allows the variables to be represented simultaneously in
the same data space. This results in a data space that
includes a limited range of values across the predictor
variables. Depending on the complexity of the variable
relationship, this restricted area of the data space can
be a major drawback of using this coordinate system.
As in some other higherdimensional graphs, in the
density plot using barycentric coordinates, the presence
or absence of interaction effects can be judged by seeing
whether the mesh surface is flat or curved. In Figure 5,
it is apparent that there is no interaction. Meanwhile,
Figure 6 is the depiction of a 2way interaction, while
Figure 7 shows a 3way interaction.
Figure 5. A 3D triangular plot showing no
interaction.

Figure 6. A 3D triangular plot showing a
2way interaction



Figure 7. A 3D triangular plot showing a
3way interaction.




The 3D triangular plot possesses a unique feature that
is not present in other visualization tools presented
here. A 3D triangular plot can display all four
dimensions at the same time in one view. In a Trellis
display, the user must swap the variables across each
axis panel to get a thorough view of the data. In Maple
3D animation and SAS/Insight, which will be introduced in
the next section, the fourth dimension is hidden unless
the user requests it. Nonetheless, this high degree of
condensation of dimensions comes at the expense of
clarity. Although this type of graph can clearly
distinguish no interaction, and 3way interactions, it
may be problematic to illustrate 2way interactions. To
be specific, even if there exists only an A*B
interaction, the graph also gives an illusion of an A*C
interaction because the slope of B against Y and the
slope of C against Y seem to be affected by A.
Temporaloriented
Temporaloriented visualization is also called
Kinematic displays (Tukey & Tukey, 1988). As the name
implies, temporaloriented visualization techniques
utilize variations across time to depict higher
dimensions. In other words, not all variables are shown
within the given space and time. The user must play an
animation module to unveil more information (Wainer &
Velleman, 2001). The "time" dimension can be designated
as a variable where the values of the variable are used
to illustrate change.
Animated graph in SAS/Insight
SAS/Insight's animated graph (SAS Institute, 2001) is
one example of a temporaloriented/datadriven plot. In
SAS/Insight, the fourth dimension is introduced as a
"time variable" (Figure 8). That is, the data points
representing a threevariable relationship suspended in a
threedimensional space rendered on a computer screen are
each highlighted as the values of a fourth variable are
added sequentially from its lowest to highest value. To
assist in the visualization process, SAS/Insight provides
several different visual fitting methods allowing the
researcher to examine the consistency between the data
and a model, namely, a parametric surface of the
researcher's choice (Figure 9a), a kernel density
smoothing surface (Figure 9b), and a spline smoothing
surface (Figure 9c). In the last two, there are slide
bars for the user to change the bandwidth in order to
adjust the level of smoothing. After the 3D plot is
drawn, animation of the data points on the graph
according to the value change of another variable gives
the point cloud the appearance of points dancing about on
the graph, allowing the researcher to detect patterns and
structure in the multivariate relationship (Cheung,
2001).
Figure 8. The fourth variable as the
temporal dimension

Figure 9a. Parametric surface



Figure 9b kernel density smoothing

Figure 9c Spline density smoothing



You
can view a QuickTime
movie
showing how data points "dance" by stepping
through the values of the fourth dimension
(Please use QuickTime Player, Microsoft Windows
Media Player cannot play the
movie)

However, this approach has at least three limitations.
First, in order to make the pattern amongst data points
emerge, a large data set is desirable because patterns
are clearer when the observations dance in clusters
across a dense cloud of points. A small data set may show
a scattering dance among sparse points, and thus may fail
to reveal any pattern at all. At first glance, this
notion seems contradictory with some experimental
findings. For example, Kareev, Lieberman and Lev (1997)
found that the use of small samples led to more accurate
detection of correlation. However, this is true if only a
pairwise relationship is displayed. Yu (1994) also found
empirically that the efficacy of visualization tools is a
function of both the sample size and the number of
dimensions. A large amount of data necessitates
featurerich visualization tools, and multiple dimensions
require more observations. Second, the function overlay
has been generated according to the first three variables
in the plot. Therefore, the addition of the fourth
variable does not alter the existing function. Although
the points are highlighted creating an illusion of
movement, the surface remains static. A third, related
limitation of the animated point cloud is that the
addition of the animation variable to a 3dimensional
plot is not the same as viewing a fourvariable
relationship. The dancing effect of the animation has a
different perceptual impact than that of the visual
impression created from the preexisting
threedimensional relationship. Further, it is the static
visual associations that most people are accustomed to
viewing and interpreting. Hence, the variable chosen as
the animation variable may have unrevealed relationships
with other variables involved.
Animated 3D plot in Maple
Maple offers an animated 3D plot procedure (Waterloo
Maple, 2001), which is one example of a
temporaloriented/modeldriven visualization tool. Like
SAS/Insight, in Maple the fourth variable is cast into a
"time" variable. After a 3D mesh surface plot is
generated, the mesh surface can be animated according to
the varying values of the fourth variable. But unlike
SAS/Insight, the surface is refitted based upon the
fourth dimension, and there are no data points shown in
the graph. Actually, Maple is capable of superimposing
data points on a smoothed function, resulting in a plot
very similar to the SAS/Insight plot prior to its
animation of points ^{1}.
However, the data points are fixed to the original three
dimensions in the Maple plot. The observations are not
animated or highlighted according to the fourth variable,
and it is only the mathematical function that has been
input with defined variable ranges (not specified values)
that determines the motion of the surface, which
represents the fourvariable relationship. Therefore,
Maple's animated 3D plot is classified under the
temporaloriented/modeldriven category of
higherdimension plots.
In a typical 3D plot, the shape of the mesh surface
determines the absence or presence of an interaction
effect. A flat plane indicates the absence of an
interaction effect while a warped surface is a sign of an
interaction. In an animated 3D plot, even if the mesh
surface is flat, one of the variables may still interact
with the fourth variable when the slope changes according
to the increment or decrement of the data value of the
fourth variable (Figure 10).
Figure 10. Animated 3D plot showing a 2way
interaction.


You can
press the
stop
button
on the browser to stop the
animation. To resume the animation,
press the reload button. You can
also view a QuickTime
version of this
animation.


When the mesh surface is curved, it is evident that
there is a 2way interaction. However, if the animated
graph shows a moving mesh surface conditioning upon the
fourth dimension, no doubt there is a 3way interaction
(Figure 11).
Figure 11. Animated 3D plot showing a 3way
interaction.


You can
press the
stop
button
on the browser to stop the
animation. To resume the animation,
press the reload button. You can
also view a QuickTime
version of this
animation.


Other research using the geometric features of these
displays includes Cleveland and McGill (1984), who argue
that Trellis displays are better than surface plots in
terms of interpretation error rates After they conducted
a series of experiments on the efficacy of different
graphical features, it was found that dots positioned
along a common scale are the most salient features, while
volume and color are more difficult to use as judgment
factors. In this view, it may be predicted that Trellis
displays are superior to functiondriven plots because
they use dots and each panel shares a common scale. Also,
Wilkinson (1999) argues that although surface plots
elicit a wholistic impression of a function, they are
less useful for decoding individual values. On another
occasion discussing surface plots, Wilkinson (1994) also
points out that researchers can usually gain more
information by displaying raw multivariate data directly,
rather than by smoothing the trends in the swarm of
observations. While we agree with Wilkinson's assessment
to surface plots, Cleveland and McGill's assertion may be
disputable. Our study will focus on viewers'
interpretation processes, and the overall effectiveness
of the different graph types identified above.
Method
In order to study the efficacy of the preceding
highdimensional displays, subjects were exposed to these
graphs with different scenarios. To generate different
scenarios, a series of equivalent regression equations
were developed that encompassed all possible 2way
interaction combinations among the 3 predictors labeled
A, B and C (A*B, A*C and B*C), a 3way interaction among
all of the predictor variables, as well as a function
that included the three predictors with no interaction.
Coefficients that were not associated with the
crossproduct predictor of interest were intentionally
given low values in order to create graphical images and
patterns of data that clearly depicted the interaction
that was sought. Intercepts were excluded from the
functions. Table 2 shows the regression equations that
were used to simulate the data and create the graphs.
Table 2. Functions used to create graphs and
simulate data
Graph
Shape

Regression
equation

No interaction

Y = .05(A).1(B)+.025(C)

3way interaction

Y =
.05(A).1(B)+.025(C)+.01(A*B)+.011(A*C).011(B*C)+.96(A*B*C)

2way (A*B)

Y = .05(A).1(B)+.025(C)+.96(A*B)

2way (A*C)

Y = .05(A).1(B)+.025(C)+.96(A*C)

2way (B*C)

Y = .05(A).1(B)+.025(C)+.96(B*C)

Modeldriven graphs were created using the above
functions in both the animate3d procedure in Maple
version 6.0, and the Function plot application in SyStat
version 10.0. For the datadriven graphs, a dataset was
randomly generated using standard normal curve parameters
for variables A, B and C. Five outcome variables were
then generated using the same regression functions above.
The data values for the three predictors and five outcome
variables for each scenario were then rounded to allow
for slight deviation from the models they were derived
(R2 values ranged from .9896 to .9995). Datadriven plots
were created using the Trellis display procedure in
SPlus 2000, and the fit procedure in SAS/insight.
Comparisons were made separately among the
functiondriven graphics produced by SyStat and Maple,
and among the datadriven plots derived in SAS/Insight
and SPlus. In an attempt to establish equivalency among
the sets of graphs across the graph types, graphs were
created such that all possible variable combinations were
represented for each model and for each program. For
example, since the Maple graphs were 3dimensional with
the fourth variable added into the image as an animation
variable, only two predictors could be represented on the
predictor plane at one time. Therefore, three graphs were
created for each outcome, one in which variables A and B
were on the predictor plane, another with A and C, and a
third with B and C. In each graph, the variable not
represented on the predictor plane was the animation or
conditioning variable.
Participants
Both student and faculty participants were asked to
view the graphs voluntarily. Only students with at least
2 graduate level statistics courses, and faculty members
who regularly conducted research and instructed
statistics courses were invited to participate. Student
participants included 1 female and 3 males, and ranged in
age from 22 to 32 years old. Students had completed
between 2 and 6 graduate level statistics courses.
Faculty participants included 1 female and 1 male, and
ranged in teaching experience from 5 years to 20
years.
Procedure
Participants completed the tasks individually. Each
was briefed of the project purpose, and asked four
preliminary questions regarding their statistical
background. Through immediate review of this
questionnaire, it was ensured that participants had
sufficient statistical knowledge and experience, and that
they understood the concept of interactions in a ordinary
leastsquares (OLS) regression context. Next, they were
oriented to the layout of graphical displays that they
were asked to interpret, and provided detailed directions
for using the graphical manipulation tools available in
each program, which they were encouraged to use as much
as they felt necessary. Graphs were prepared in advance
and randomly chosen for presentation. To compensate for a
carryover effect, the order of the graph types was also
randomized across participants.
The participants were instructed that they should
allow as much time as needed, and they may use any
combination of the graphs within a set to inform their
decision. Each subject was asked to view sets of graphs
from two programs, either the SAS/Insight and SPlus
plots (the two datadriven display types), or the SyStat
and Maple images (the two modeldriven display types).
Participants completed one set of graphs before moving on
to the next set, and they completed the sets of graphs
within a program before being guided on to the next
program by the observers. To account for fatigue and
prevent participant apathy, observers would not allow any
single graphical interpretation session to persist more
than 90minutes (although this constraint was not
communicated to participants because an unforced response
was most desired).
Quantitative measurements
Participants were asked to respond to the same
multiplechoice question for each set of graphs
corresponding to a particular outcome. The task expected
them to decide whether the set of graphs represented a
twoway interaction between A and B, a twoway
interaction between A and C, a twoway interaction
between B and C, a threeway interaction between A, B and
C, or a regression among the three variables with no
interaction. Upon making a choice and answering the
subsequent openended interview question, the subject was
presented with the next set of graphs. Correct decisions
were counted and comparisons across graph types (within
subjects) were made. Additionally, the time that
participants took to interpret each set of graphs was
recorded.
Qualitative measurements
We hesitated to do a mere performance comparison
across graph types based upon group means. Instead of
adopting a purely performance testbased approach, which
may only tell us which group achieved a higher score, but
cannot tell us why one treatment is superior to another,
(Yu, 1999), we approach this problem in a qualitative
fashion.
Upon choosing one of the five options, participants
were asked in an interview format to describe the factors
that lead to their decision. In addition to
interpretation questions, participants were also asked to
follow a "think aloud" protocol (Someren, Barnard, &
Sandberg, 1994) while going through the interpretation
process. In think aloud, the subject is expected to
verbalize each thought he/she has while performing the
interpretation tasks. The purpose was to attempt to flesh
out barriers and misconceptions that graph users may
encounter while viewing the images, and to attempt to
identify potential differences in interpretation
approaches that participants tend to implement from one
graph type to another. Student participants were provided
with a brief think aloud tutorial prior to the
presentation of the graphs, while faculty participants
were only asked if they understood the exercise.
Observers developed journals for each participant, and
data patterns among the different programs were analyzed
after data collection.
Results
Since the study is currently in its pilot stage, only
preliminary results from a small selective sample are
available, but nonetheless, are still enlightening. We
classified the higher dimension plots by the degree of
data reduction (datadriven vs. functiondriven), and
then by the degree of dimension reduction
(spatialoriented vs. temporaloriented). Comparisons
between the functiondriven and datadriven plots did not
seem appropriate since they have incompatible purposes. A
functiondriven plot is practically useless to the
researcher (exploratory or not) who hopes to find
meaningful patterns in his or her data. Plotting the
function superimposed over the data points can clearly be
beneficial to many people (and misleading to others), but
a geometrical picture of the mathematical function alone
does the researcher very little in the early stages of
his or her regression analysis. Datadriven plots that
show the observed relationship among the researchers'
variables seem to be more appropriate when the objective
is to explore and probe the data. A function plot becomes
useful when the purpose is to display a complex
relationship in a simple manner. For example, when one is
instructing the concept of interactions in regression, a
common way to graphically illustrate the interaction is
through plots of simple slopes (somewhat similar to a
crude Trellis display) along with an ANOVA relationship
demonstration. However, this requires some cognitive
resourcefulness for most novice learners as the simple
slope plots depict relationships that appear bivariate
but are actually multivariate.
Datadriven plots: Tools for data analysis
The SAS/Insight 3Dimensional Animated plot and the
SPlus Trellis displays were compared for their overall
userinterpretability and effectiveness, and for each,
the processes used by participants were examined. Thus
far in the pilot, four student participants and two
faculty participants have viewed the datadriven plots.
Next, we highlight some of the patterns that have emerged
from the student and faculty pilot samples.
Trellis Display in SPlus
None of the participants had previously been exposed
to any graphical tools in SPlus prior to this study. The
training time varied from four to fifteen minutes, and
the time spent on one set of graphs varied from five to
twentyone minutes. Different features of the Trellis
display attracted different users. One experienced
researcher said that the most salient feature of the plot
was the set of slopes, but he was initially confused
about how the slopes should be interpreted in this
context. He was not sure whether the gradient of the
slope or the direction of the slope should be used as a
criterion for judging the presence of interactions. For
example, if all conditioning panels showed positive
regression slopes but the magnitude for the slopes were
changing slightly across the panels, did it mean that the
relationship between the predictor and the outcome
variable was inconsistent across all levels of the
conditioning variables, indicating an interaction? As an
experienced researcher, he was clearly seeking more
subtle patterns in the data than most novices. For most
participants, the variations among the simple slopes were
the central focus.
Another experienced researcher spent quite a long time
(18 minutes) with the first set of Trellis plots. She
expressed that initially she tried to relate the
information shown on the 2D panels to a 3D rotating plot.
This extra mental processing slowed down the task and
lead to more confusion.
Another user initially paid a large portion of her
attention to the conditioning panels, but neglected the
relationship between the regressor and the dependent
variables. For example, while attempting to identify a
2way interaction between A and B, and when viewing
variable A plotted against variable Y with variables B
and C as the conditioning variables, she continually
confused the implications of the conditioning variable
values as being conditioned on each other rather than
with the AY relationship. The result was an incorrect
response of a B*C interaction.
The Trellis displays showed both the data points and
simple OLS regression lines superimposed, but none of the
users examined the residuals between the data and the
linear least squares function. They assumed that the
model was correct even though in some panels outliers
were present. This leads one to believe that users may
tend to focus attention on the superimposed function more
than the data points themselves. It may also be explained
by the hypothesis that since we only provided linear
interaction options from which participants were expected
to choose, they assumed that all relationships were
linear.
Observation of the first user's behaviors while
viewing the Trellis displays and SAS/Insight plots
confirmed our belief that usage of visualization tools
requires an exploratory spirit. With our initial design,
only one view of the trellis display was shown for each
set of relationships. Although we encouraged the user to
swap the positions of the predictor and the conditioning
variables, the user continued to use the default view.
However, a single view of the dataset could be very
misleading, as discussed earlier, because the
relationship may be concealed due to the variable layout.
Figure 12, which was seen by that user, did not show any
relationships since all slopes were flat.
Figure 12. One view of Trellis display shows
flat slopes across all levels of the conditioning
variables
However, Figure 13 tells a different story when the A
predictor became the regressor variable and B and C
became the conditioning variables. It clearly showed that
there was an interaction effect of A and B. Without
exploring the data thoroughly, the user failed to unveil
the interaction effect. The behavior of this participant
lead to an alteration in the research design whereby all
possible variable configurations were created to make up
a set of graphs for any one outcome and any one
program.
Figure 13. Another view of the Trellis display
shows an A* B interaction effect
3D animation surface plot in SAS/Insight
The threedimensional animated scatterplot in
SAS/Insight was an example of a datadriven plot that
simultaneously displayed the association among the first
three variables, and the fourth variable was added to
animate the plot. For the SAS/Insight module, both
faculty participants reported having used the program
extensively prior to the study for data analyses, but not
as often for its graphical tools. Only one student
participant had previously used the program, and she
reported only limited exposure to its graphical tools.
Probably due to the need for extra explanation of
nonparametric model surfaces, the student participants
took between eight and twentytwo minutes to be trained
on the SAS/Insight module. Time taken to make a decision
ranged from twelve to thirtyeight minutes.
A major disadvantage that appeared to affect this type
of plot quickly surfaced during the pilot study. Since
highlighting the points across the values of the
animation variable represented the association of the
fourth variable with the preexisting threeway
relationship, the fourth variable required a different
perceptual operation than that which was used to
interpret the initial threeway relationship. Evidently,
this was cognitively demanding for participants since it
seemed to require the viewer to simultaneously apply two
distinct firstorder factors of visual perception, a
general visualization ability and spatial relations
ability (see Carroll, 1993, for summaries of factor
analytic studies in human perception). Additionally,
given a continuous animation variable that included
numerous values each highlighted individually, the viewer
likely exceeded his or her shortterm memory capacity
prior to the completion of the animation effect and
before any pattern recognition was possible. The existing
relationship and the dancing of points that occurred in
that relationship could appear vastly different depending
on the variables chosen for the initial threevariable
plot. As a result, participants' interpretation accuracy
on the animated 3D plots in SAS/Insight was the poorer
between the two types of datadriven graphs, especially
with the threeway interaction graph.
Responses to interview questions and thinkaloud
procedures have thus far confirmed our hypotheses
regarding viewer impediments to identifying the patterns
in the SAS/Insight 3D animated plots. Some users
explicitly expressed that they had difficulties with the
hidden dimension (the fourth variable). For the
SAS/Insight sets of graphs, both the parametric fit and
the nonparametric fit plots showed three dimensions
only, the display of the fourth dimension must be
specially requested by the user. Obviously, this extra
step hindered users from further exploration. In
SAS/Insight, the user must go through a sequence of pull
down menus, choose among several options in subsequent
dialog windows, and make adjustments to the speed in
order to view the animation in a useable manner.
Additionally, the values of the animation variable are
flashing in a separate window while the points are
highlighted, which requires the rarely found ability of
focusing one's visual attention on two distinct stimuli
simultaneously. Thus, as with the Trellis plots, the
accessibility and explorative ease issues were similarly
problematic. The low visibility of the tool, in
conjunction with the hidden dimension (the fourth
variable), exacerbated the problem of ineffectively
depicting the multivariate relationship. This tended to
discourage many participants, leading them to "settle on"
a lessinformed decision.
Some users employed intuitive strategies to attempt to
detect the interaction relationships. For instance, one
user expected the regression mesh surface to update
according to the fourth dimension, and tended to focus on
the data point dance as it related to the preexisting
surface. The same user became further confused by viewing
the point dance with various smoothing thresholds
specified for the nonparametric surface in order to look
for additional evidence of interactions. For most
participants, the strategy of looking for relationships
using the animation effect was abandoned, and they simply
tried to interpret the static threeway relationships.
This resulted in a high error rate particularly for the
threeway interaction situation.
One experienced user was doubtful of the usefulness of
the surface plot in SAS. He had difficulty manipulating
the plot to his satisfaction, and becoming oriented with
the 3D space. It might be due to the lack of color as a
cue of depth. Also, the animation feature was not helpful
for him to reach his conclusions. Indeed, he based his
decisions on educated guesses informed by the static
threeway relationships rather than the pattern of data
"dance" representing the fourvariable relationship.
To this point in the study, among the datadriven
plots that were evaluated, the SPlus Trellis display
seems to be preferred by participants over the
SAS/Insight 3D animation plot. The interpretation
accuracy rate was also strikingly better with the Trellis
plots (see table 3).
Table 3. Accuracy rates for pilot samples on
the datadriven plots (percent of correct
interpretations).
*Caution:
small samples and pilot results only.
Graph
Type

Student Sample
(n=4)

Faculty Sample
(n=2)

SPlus Trellis Display (total)

10/12 (83%)

6/6 (100%)

No interaction

4/4 (100%)

2/2 (100%)

2way interaction

3/4 (75%)

2/2 (100%)

3way interaction

3/4 (75%)

2/2 (100%)

SAS/Insight 3D Animated Plot (total)

7/12 (58%)

4/5 (80%)

No interaction

3/4 (75%)

1/1 (100%)

2way interaction

3/4 (75%)

2/2 (100%)

3way interaction

1/4 (25%)

1/2 (50%)

Although each individual researcher may tend to find
comfort in using predominantly a few particular types of
display, these results suggest that those who are viewing
raw multivariate data in an attempt to explore complex
relationships among variables are more likely to uncover
distinguishable patterns in a multipanel, conditioning
type display rather than a threedimensional plot with
animation that shows change among observations. Interview
data further implies that researchers can benefit from
using the breakdown method of detecting patterns.
However, it is hypothesized that the animated plot may be
more appropriate and useful when the researcher wishes to
view the change in a relationship over time (when the
fourth variable is literally time), or when the animation
variable is a grouping variable. Improvements that were
recommended from participants for the SAS/Insight module
included issues of manipulation, and linking between the
models and data points allowing synchronized animation
effects between them.
Modeldriven graphs: Presenting ideas
clearly
The SyStat 3dimensional triangular display and the
Maple 3dimensional animation graphs were compared for
their overall userinterpretability and effectiveness,
and for each, the processes used by participants were
examined. For students, we sought a learner's
perspective, and for faculty we wanted a teaching
perspective. Three student participants and both faculty
participants viewed the modeldriven plots. The early
trends emerging from the pilot samples will be discussed
next.
3D Triangular plot in SyStat
No participants had previously been exposed to the 3D
triangular/rectangular coordinate system used in the
SyStat plots. Training for the tool was brief for both
student and faculty participants (explanation time varied
between two and eight minutes), which may have partially
been a function of the limited user controls that are
available for this graph. Interpretation time for one set
of graphs ranged from six to twentytwo minutes. All
users seemed to have difficulties interpreting the
coordinate system in the 3D Triangular plot. Like other
visualization tools, this type of graph allows the user
to examine the plot from different perspectives with a
rotation tool, but no other tools are available. . In
this case, not only accessibility of manipulation tools
is an issue, but also it seems that initial
incomprehension discouraged users from further
exploration.
Humans interpret unfamiliar things in terms of what we
are familiar with. For example, while using SyStat 3D
plot, some users attempted to view the 3D plot in a 2D
fashion by rotating the graph to a perspective that
isolates one particular variable. However, no matter how
they rotated the graph, they failed to see something like
a bivariate scatterplot and thus frustration set in. A
similar strategy observed involved some users trying to
track the change in the outcome as corresponding values
of the predictor variables varied. For example, the user
seemed to ask himself, "When A is low, B is low, and C is
high, what is the value of Y?" However, applying the
knowledge of Cartesian space into the triangular space
resulted in confusion and frustration.
Although each of the few participants who have viewed
the SyStat 3D triangular plot correctly identified the no
interaction and 3way interaction effects fairly easily,
all of them were confused by the 2way interaction
display. We hypothesize that this may be due to a lack of
experience with the triangular coordinate system.
Additional participants are necessary to substantiate
this claim.
3D animation plot in Maple
None of the participants had previously used or had
exposure to the Maple program or graphics. Training time
varied between five and twelve minutes. Interpretation
time ranged from eight to twentyseven minutes for one
set of graphs. In spite of the widespread availability
and application of 3D graphs, it seems that users were
still more comfortable with using a 2D approach to
interpret the 3D graphs. For example, although the Maple
3D animation plot shows a 3D surface, two users rotated
the box to a perspective in which only two axes were
seen. When the slope of one predictor against the outcome
variable appeared linear and constant during the
animation, they interpreted that there was no
interaction. When they did not see a straight line but
instead a warped surface or a shifting slope, they
concluded that an interaction effect was present. While
there was nothing wrong or improper in this approach, it
is similar to a "divide and conquer" approach to
understanding the relationship, and the same conclusions
could have been reached using a 3D graph in a wholistic
manner. Nevertheless, most users found the 3D animation
plot helpful in seeing interaction effects.
The results of the Maple module further emphasized the
notion that visualization tools require an exploratory
attitude. It was found that the degree of the user
exploration is strongly tied to the accessibility of the
features. In the Maple graph, all manipulation tools are
available by a rightmouse click and all movie control
buttons are visible in the top bar. Users tended to fully
use the animation features during the process.
One user found that actually she could reach the
correct conclusion by looking at the series of static
graphs alone. In Maple, the entire set of graphs was
shown within a single screen. It was easier for the user
to perform sidebyside comparisons. However, in other
graphical systems where different plots were not
presented simultaneously, it required users to switch
from one window to the other. In this case, the user had
to rely on his or her shortterm memory for
comparison.
One experienced researcher initially was confused by
the color of the surface plot. He asked whether the
variation of hues denote certain meanings. Just a moment
later he found that the color is simply for enhancing the
perception of depth. As mentioned before, in the multiple
symbol paradigm, different features of the symbol such as
size, shape, direction, and color indicate data values of
different dimensions. It is understandable that the
experienced researcher paid attention to the subtle
aspects of visualization. However, this is not a piece of
evidence to concur Cleveland and McGill's finding that
color as a graphical element might leads to erroneous
interpretation. Nevertheless, the confusion of that
experienced user was temporary and this confusion did not
hinder him from making the correct interpretation of the
interaction effect.
Table 4. Accuracy rates for pilot samples on
the modeldriven graphs (percent of correct
interpretations).
*Caution:
small samples and pilot results only.
Graph
Type

Student Sample
(n=3)

Faculty Sample
(n=2)

Maple 3D Animation Plot

8/9 (83%)

6/6 (100%)

No interaction

3/3 (100%)

2/2 (100%)

2way interaction

2/3 (75%)

2/2 (100%)

3way interaction

3/3 (100%)

2/2 (100%)

SyStat 3D Triangular Plot

6/9 (66%)

4/5 (80%)

No interaction

3/3 (100%)

1/1 (100%)

2way interaction

0/3 (0%)

1/2 (50%)

3way interaction

3/3 (100%)

2/2 (100%)

Discussion
The accuracy with which learners identified the sets
of graphs, and the qualitative data from all participants
have lead us to hypothesize that for teaching and
presentation purposes, the temporalbased displays, such
as the 3D animation plot in Maple, seem to have
advantages over the currently available spatialbased
graphs, such as the 3D triangular coordinate plot in
SyStat. It was apparent from the interview responses that
most participants were more familiar with the Cartesian
space and time than the Barycentric space, and thus
comprehension of the latter requires much more mental
processing (and figure manipulation controls, which
seemed limited and cumbersome in the SyStat example). The
Maple 3D animation plot, conversely, seemed to take
linked displays to another level. The smooth motion of
the animation along with mousebased control of the
rotation and easy access to other graph manipulations
made the Maple graph appealing to most users. Further, it
illustrated complex relationships among the four
variables in a highly perceptible, wholistic manner.
Users who attempted to comprehend the graphs by
rotating the plots into multiple 2D perspectives were
easily misled by the triangular plot. While in Maple's 3D
animation plot the information conveyed by the multiple
2D perspectives could easily be converted, users failed
to do so in the triangular plot. Also, the high degree of
accessibility of manipulation tools in Maple allows more
active exploration. For these reasons, it appears that
the Maple 3D animation plot is more helpful in
illustrating concepts such as regression interactions to
learners, and for presenting complex relationships than
the SyStat 3D triangular plot.
For research purposes, the spatialbased graphs, such
as Trellis displays in SPlus, are preferable over the
temporalbased displays, such as the 3D animated plot in
SAS/Insight. The multipleview strategy employed by
Trellis displays allows users to "divide and conquer" the
problem, allowing for the identification of complex
relationships. Multiple dimensions are displayed yet the
static graph allows users to examine the conditioning
panels one by one, and without any single variable being
at a disadvantage. The user is also easily able to keep
track of the changing values of the conditioning
variables. On the other hand, in SAS/Insight, the "dance"
of data points representing the fourvariable
relationship can be difficult to follow, especially since
the values of the conditioning variables are located in a
separate panel. One must follow the pattern and the
change in values simultaneously. As a result, the eyes
necessarily miss a split second of the animation effect.
Additional processing needed to perceive and concurrently
interpret the parametric fit, and the spline surface
tended to overwhelm users. Further, the variable chosen
to represent the fourth dimension is not viewed in an
equivalent, simultaneous manner with the first three
variables in the data space, thus giving it a
disadvantage.
We can not stress strongly enough that the results
reported herein are interim results of a small sample
pilot study, and should be viewed with appropriate
caution. The pilot study has allowed us to identify
numerous improvements in our research design, and several
research questions to explore in the next phase. In
addition, innovative ideas are accumulating for
improvements that may enhance each of the
higherdimension displays we have evaluated. We are
continually seeking alternative approaches to the graph
types we have identified, and will include them in the
comparison if possible.
Note
1. The function/data cloud overlay
involves creating the 3D point plot and the
animated function plot separately, and then
displaying the plots simultaneously. Hence,
these plots are not "linked", but only
rendered simultaneously in the same display
area.

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Appendix
Code used to create Maple 3D animation plots (in Maple version
6.0)
3way interaction
> with(plots):
animate3d(.05*a  .1*b + .025*c + .011*a*b + .011*a*c 
.011*b*c + .96*a*b*c, a=3..3,b=3..3,c=3..3);
A*B Interaction:
> with(plots):
animate3d(.05*a  .1*b + .025*c + .96*a*b,
a=3..3,b=3..3,c=3..3);
A*C Interaction:
> with(plots):
animate3d(.05*a  .1*b + .025*c + .96*a*c,
a=3..3,b=3..3,c=3..3);
B*C Interaction:
> with(plots):
animate3d(.05*a  .1*b + .025*c + .96*b*c,
a=3..3,b=3..3,c=3..3);

Procedure for creating SyStat triangular plots (SyStat version
10):
 Under the graph menu, choose function plot
 Type in model equation
 Under the coordinates option, choose triangular
 Other options can be chosen
 After graph is created, doubleclick to enter editmode
 Rotation tools are to the right
Procedure for creating 3D plots in SAS/Insight:
 Open the simulated dataset
 From the solutions menu, point to the analysis option, then
choose Interactive Data Analysis
 Choose the active dataset from the work directory
 From the analyze menu, choose Fit (Y,X)
 In the dialog window, choose three variables to begin the
display, two predictors (and their crossproduct if you wish)
should go in the X area, and the outcome variable should go in the
Y area.
 Once the graph is created, choose Edit, Windows, then
Animate
 Choose the predictor that is not in the current display as the
animation variable
Procedure for creating SPlus Trellis displays:
 Open the simulated dataset
 By pointing the arrow at the variable labels to choose them,
hold down the ctrl key and choose the predictor that you want on
the abscissa first and then the outcome variable of interest.
 Under the graph menu, choose 2D plot
 In the dialog window, choose a fit line if preferred (Other
options can also be altered if you wish)
 After graph is created, align the graph window such that the
variable labels in the data window is also visible.
 Press ctrl and highlight the remaining predictors
 By clicking once and holding in the shaded region, drag and
drop the selection into the graph
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