Advanced Circumplex Visualization

library(circumplex)
library(ggplot2)

Level: Advanced. Read “Structure Tests and Ipsatization” first.

1. Overview

The ssm_plot_circle(), ssm_plot_curve(), ssm_plot_contrast(), and ssm_plot_trajectory() functions cover the most common circumplex figures. But they each produce a finished plot with a fixed set of layers. Sometimes you want more control. You may want to overlay individual respondents on a group profile, or to zoom in on a band of amplitudes. Or you may want to restyle the points, or to place several circumplex panels side by side. vignette("introduction-to-ssm-analysis") defines the SSM terms used here, such as amplitude and displacement.

To make that possible, circumplex exposes the building blocks that the built-in plots are themselves made of. These are ordinary ggplot2 components, so you compose them with + and combine them freely with any other ggplot2 layers, scales, and themes:

• coord_circumplex() is the coordinate system. It maps the displacement aesthetic (degrees) onto the angle and the amplitude aesthetic onto the radius. It also owns the amplitude-to-radius scaling for the whole plot.
• ggcircumplex() assembles the empty circular canvas: the coordinate system plus the amplitude rings, displacement spokes, and scale labels.
• geom_ssm_point() and geom_ssm_arc() are the layers that place profile points and their confidence regions in the circle, taking amplitude and displacement directly as aesthetics.
• theme_circumplex() is the theme the canvas is drawn with. The rings and spokes are ordinary panel gridlines, so further theming restyles them.
• scale_x_circumplex() is a scale for the angle axis of linear circumplex plots. An example is the score-by-angle curve, with scale angle on a straight x-axis and score on the y-axis.

This vignette works through each of these and then combines them. Section 2, “The circular canvas”, draws the empty canvas with ggcircumplex(). Section 3, “The coordinate system”, builds a figure from scratch with coord_circumplex(). Section 4, “Placing SSM results in the circle”, adds profiles with geom_ssm_point() and geom_ssm_arc(). Section 5, “Restyling the canvas”, themes it. Section 6, “Composing custom layers”, adds respondents behind a group profile with ssm_score(). Section 7, “Trajectories across occasions”, draws profiles estimated at several occasions. Section 8, “The angle axis for linear plots”, labels a linear axis with scale_x_circumplex(). Section 9, “Relationship to the built-in plots”, says how the built-in plots use these parts. The Wrap-up lists what the page covered and names the next page, and the References list the sources cited.

2. The circular canvas

ggcircumplex() returns a ggplot2 object containing just the circular backdrop, with no data drawn on it yet. By default it uses octant scales (eight scales placed 45° apart), labeled by their angular position in degrees:

ggcircumplex()

You can label the scales however you like. Passing a character vector labels the spokes in the order of the angles:

ggcircumplex(octants(), labels = PANO())

The labels need not be abbreviations. The octant scales also have full interpersonal names, which you can put on the spokes instead:

ggcircumplex(octants(), labels = csip$Scales$Label)

You may be working with one of the instruments bundled with the package. If so, you can pass it directly with ggcircumplex(instrument = csip). Its scale angles and abbreviations are then taken from the instrument rather than typed by hand.

Throughout, displacement runs counterclockwise from the right, and the 0/360 degree position is labeled 360.

3. The coordinate system

ggcircumplex() is a convenience wrapper. Underneath it, the piece that makes a circumplex plot circular is coord_circumplex(). You can add that to a bare ggplot() yourself when you want to build a figure from scratch. On top of the coordinate system you supply three things: an x-scale carrying the spoke breaks and labels, a data layer, and the theme.

results <- ssm_analyze(
  jz2017,
  scales = PANO(),
  measures = c("NARPD", "ASPD")
)

The table below shows five columns of results$results: the profile label, the amplitude and displacement estimates, and the amplitude interval. (The code that selects these columns is omitted.)

#>   Label    a_est    d_est     a_lci     a_uci
#> 1 NARPD 0.189244 108.9667 0.1537900 0.2271848
#> 2  ASPD 0.226159 115.9267 0.1905403 0.2640428

ggplot(results$results) +
  coord_circumplex(amax = 0.3) +
  scale_x_continuous(breaks = octants(), labels = PANO()) +
  geom_ssm_point(aes(amplitude = a_est, displacement = d_est, fill = Label)) +
  theme_circumplex()

The scale_x_continuous() line is the one that tells the coordinate system where the scale angles are. Without it, the spokes would fall on ggplot2’s default breaks rather than on the octants. Supplying those breaks and labels, along with the theme, is what ggcircumplex() does on top of the coordinate system. Build from the parts when you want to vary one of those pieces. Reach for ggcircumplex() when you do not.

The coordinate system owns the amplitude-to-radius mapping. So amax is set exactly once per plot, and the canvas and the data layers cannot disagree about what a given radius means. (Earlier versions of the package took an amax argument on each layer. Those arguments are now deprecated and ignored, with a one-time note.) Leaving amax = NULL trains it from the data, as ssm_plot_circle() does.

Moving the center

By default, the center of the circle is amplitude 0. So radial distance is proportional to amplitude, and the origin means “no differentiation among the scales.” The center argument moves that inner limit. This is useful when every profile sits in a narrow band of amplitudes and the interesting variation is squeezed against the rim:

ggplot(results$results) +
  coord_circumplex(amax = 0.28, center = 0.15) +
  scale_x_continuous(breaks = octants(), labels = PANO()) +
  geom_ssm_point(aes(amplitude = a_est, displacement = d_est, fill = Label)) +
  theme_circumplex()

This is a zoom, and it changes how the figure should be read. With a nonzero center, radial distance is no longer proportional to amplitude. The origin no longer represents zero amplitude. So differences in radius are exaggerated relative to the default view. The amplitude ring labels still report the true amplitudes, and they are what the reader should be directed to. Use a nonzero center to resolve closely spaced profiles, and say so in the caption.

Moving the amplitude axis

The amplitude (radial) axis and its tick labels are placed automatically in the widest gap between the displacement spokes. So they never collide with a spoke label. ssm_plot_circle() and plot() for a CPM fit go one step further and use the widest gap that holds no plotted point. You can override the placement with r_axis_angle, given as a displacement in degrees:

ggplot(results$results) +
  coord_circumplex(amax = 0.3, r_axis_angle = 67.5) +
  scale_x_continuous(breaks = octants(), labels = PANO()) +
  geom_ssm_point(aes(amplitude = a_est, displacement = d_est, fill = Label)) +
  theme_circumplex()

Note that these examples build the canvas from its parts: the coordinate system, an x-scale carrying the spoke breaks and labels, and the theme. They do not add a second coordinate system on top of ggcircumplex(). If they did, ggplot2 would replace the existing coordinate system and print a message.

4. Placing SSM results in the circle

Let’s draw the two-measure profile from above on a labeled canvas ourselves, rather than calling ssm_plot_circle().

geom_ssm_point() places a point for each profile at its amplitude (a_est) and displacement (d_est). geom_ssm_arc() draws a wedge for each profile. The wedge spans the profile’s amplitude confidence interval radially and its displacement confidence interval angularly. Both take the SSM parameters directly as aesthetics and handle the conversion into circular coordinates internally. That includes wrap-around when a displacement interval crosses the 0/360 degree boundary.

ggcircumplex(octants(), labels = PANO(), amax = 0.3) +
  geom_ssm_arc(
    data = results$results,
    mapping = aes(
      amplitude_min = a_lci, amplitude_max = a_uci,
      displacement_min = d_lci, displacement_max = d_uci,
      fill = Label
    ),
    alpha = 0.4, color = NA
  ) +
  geom_ssm_point(
    data = results$results,
    mapping = aes(amplitude = a_est, displacement = d_est, fill = Label)
  )

Each arc displays two separate confidence intervals for one profile at once. Its radial extent is the amplitude interval, and its angular extent is the displacement interval. It is a convenient way to show both intervals together. It is not a single joint confidence region with its own coverage level, and it is not a hypothesis test.

The angular extent in particular is a range of plausible directions. Zero degrees is an arbitrary reference direction rather than a null value. So, unlike a confidence interval for a linear parameter (such as elevation), the angular extent should not be read as a significance test. Displacement is only worth interpreting at all when the amplitude interval is clearly above zero and the model fits reasonably well. See the “Introduction to SSM Analysis” vignette and ?ssm_analyze.

5. Restyling the canvas

theme_circumplex() is the theme ggcircumplex() applies. Because the rings, spokes, and labels are themed panel furniture rather than drawn geometry, any further theming reaches them. Adjust the base font size through the theme, and restyle the gridlines with an ordinary theme() call:

ggcircumplex(octants(), labels = PANO(), amax = 0.3) +
  geom_ssm_point(
    data = results$results,
    mapping = aes(amplitude = a_est, displacement = d_est, fill = Label)
  ) +
  theme_circumplex(base_size = 14) +
  theme(
    panel.grid.major = element_line(color = "steelblue", linetype = "dotted"),
    legend.position = "bottom"
  )

6. Composing custom layers

Because the canvas and geoms are ordinary ggplot2 objects, you can add anything else to them. A common request is to show where individual respondents fall relative to a summary. We can compute each person’s own amplitude and displacement with ssm_score() and draw them as a faint cloud behind a group-level point.

# Per-person SSM parameters for a subset of the sample
people <- ssm_score(
  jz2017[1:100, ],
  scales = PANO(),
  append = FALSE
)

A respondent whose scores are flat has no displacement. ssm_score() returns NA for that person, with a warning. So we drop the rows of people whose Disp is NA and keep only the well-defined profiles. (That code is omitted.)

# Group-level profile for the same subset
group <- ssm_analyze(jz2017[1:100, ], scales = PANO())

# The group amplitude is shorter than a typical individual amplitude
c(group = group$results$a_est, median_individual = median(people$Ampl))
#>             group median_individual 
#>         0.3651863         0.5189425

ggcircumplex(octants(), labels = PANO(), amax = 1.75) +
  geom_ssm_point(
    data = people,
    mapping = aes(amplitude = Ampl, displacement = Disp),
    fill = "grey70", size = 1.5, alpha = 0.6
  ) +
  geom_ssm_point(
    data = group$results,
    mapping = aes(amplitude = a_est, displacement = d_est),
    fill = "#0072B2", size = 4
  )

The individual points spread widely around the circle, while the group summary sits close to the origin. That contrast is not an artifact. The group profile is the SSM of the mean scale scores. So its position is the average of the individual positions in (x, y), the Cartesian coordinates of each profile’s point in the circle. Averaging vectors that point in different directions yields an average vector shorter than the typical individual vector. The two amplitudes printed above show this. A group amplitude smaller than a typical person’s therefore indicates disagreement about direction among the respondents, not that each person’s profile is flat. None of the built-in functions produce this picture directly. Any other ggplot2 layer (text annotations, additional geoms, faceting) can be added the same way.

7. Trajectories across occasions

Sometimes the same people are measured on the same scales at two or more occasions. Then ssm_analyze_long() (for long data) or ssm_analyze(occasions = ) (for wide data) estimates one SSM profile per occasion. It resamples persons so that within-person dependence across occasions is respected. ssm_plot_trajectory() then draws each SSM parameter against time.

The package ships a small simulated three-wave data set, simulated_occasions. Its group profile rotates counterclockwise across the 0/360 degree boundary, which is the case worth seeing drawn. The data frame has one row per person per wave. Its columns are an id, a wave factor with levels T1, T2 and T3, and the eight PANO() scales. ?simulated_occasions states how it was simulated. We load it and estimate one profile per wave with ssm_analyze_long():

data("simulated_occasions")
results_long <- ssm_analyze_long(
  simulated_occasions,
  scales = PANO(),
  id = "id",
  occasion = "wave"
)

The table below shows five columns of results_long$results: the occasion, the amplitude and displacement estimates, and the displacement interval. (The code that selects these columns is omitted.)

#>   Occasion     a_est     d_est     d_lci     d_uci
#> 1       T1 0.5707423 332.41252 329.10979 335.70188
#> 2       T2 0.5968632 354.84909 351.81938 357.84738
#> 3       T3 0.5959438  21.05834  17.95188  24.08043

ssm_plot_trajectory(results_long, drop_xy = TRUE)

Two things about the displacement panel are worth reading carefully. First, it is drawn on an unwrapped branch, which lets angles go past 360 (or below 0) so that the line stays continuous. The profile crosses the 0/360 boundary between the second and third wave. Rather than jumping a full turn, the panel continues past 360, so values outside [0, 360) are expected there.

Second, the occasion order comes from the data rather than from the plot. For a character occasion column, it is first-appearance order. For a factor, it is the factor’s level order. Note that factor() sorts its levels alphabetically by default, which would place T10 before T2. So if your occasion column is a factor, set its levels in temporal order.

The unwrap carries an assumption that no data can check: that the profile rotates less than a half-turn between consecutive occasions. Waves that are far apart in time, or a series with a gap, could rotate further than that. Such a rotation would be drawn as the shorter rotation regardless. So read widely spaced occasions with that in mind.

A time point’s amplitude interval can be too close to zero for its displacement to be interpretable. Such a time point is drawn as a hollow point. The hollow point marks an interpretability precondition, not a significance test.

drop_xy = TRUE above omits the X-value and Y-value panels (the $x$ and $y$ coordinates of each profile), leaving elevation, amplitude, and displacement.

The bands are the per-occasion confidence intervals, one per time point. They are not a simultaneous confidence band for the trajectory as a whole. Overlap (or its absence) between two occasions’ bands is not a test of change between them. For that, estimate the contrast directly (see ?ssm_analyze and ssm_plot_contrast()).

ssm_plot_trajectory() also accepts a trajectory table. This is a data frame of a_est/a_lci/a_uci and d_est/d_lci/d_uci triples at numeric time points. With it, you plot a model-based trajectory evaluated from a fitted growth model, rather than one estimated separately at each wave. That workflow is the subject of the “Growth Models on SSM Parameters” vignette.

The same change as movement on the circle

The panels above show each parameter against time separately. That is the right figure for reading a confidence interval, but a poor one for seeing motion. The amplitude and displacement of a single occasion are split across two panels. geom_ssm_path() draws the same series as a path on the circular canvas, so a change in (amplitude, displacement) reads as movement through circumplex space.

ggplot() +
  # The amplitude axis goes in the 45-90 gap, clear of the three occasions
  coord_circumplex(amax = 0.8, r_axis_angle = 67.5) +
  scale_x_continuous(breaks = octants(), labels = PANO()) +
  theme_circumplex() +
  geom_ssm_point(
    data = results_long$results,
    mapping = aes(amplitude = a_est, displacement = d_est),
    size = 2
  ) +
  # Drawn after the points so the terminal arrowhead is not covered by the
  # final occasion's marker, and sized to clear it
  geom_ssm_path(
    data = results_long$results,
    mapping = aes(amplitude = a_est, displacement = d_est),
    arrow = arrow(length = unit(0.18, "inches"), type = "closed"),
    linewidth = 0.7
  )

The arrowhead marks the direction of time. Note what the layer does at the boundary. The estimated profile moves from about 332 to 355 to 21 degrees. The step from the second to the third wave is drawn as the short arc of about 26 degrees across the 0/360 pole. It is not drawn as a sweep of about 334 degrees the long way round. The path is curved because coord_circumplex() munches each segment along the polar geodesic: it splits the segment into short pieces that bend with the circle. The layer supplies the ordering, not the drawing.

Occasions are connected in the order the rows appear in the data, exactly as geom_path() does. Mapping group draws one path per series. When you assemble a data frame by hand, sort it into time order first. For the reason noted above, sorting occasion labels as text puts T10 before T2 and silently reverses time. ssm_plot_circle(), shown next, does that sorting for you.

The same figure is available ready-made from ssm_plot_circle(), which adds the path to its usual points and confidence wedges:

ssm_plot_circle(results_long, path = TRUE)

An occasion whose displacement is undefined (a flat or zero-amplitude profile) breaks the path rather than being interpolated through. The segment after the gap is still drawn on the correct branch. A path that skipped such an occasion would draw a movement that never happened.

8. The angle axis for linear plots

Not every circumplex figure is circular. The score-by-angle curve drawn by ssm_plot_curve() is a linear plot whose x-axis runs through the scale angles. scale_x_circumplex() labels that axis consistently with the circular canvas: by default with the angle in degrees, or with custom labels or an instrument’s abbreviations.

The example below draws a made-up profile at the octant angles:

angles <- octants()

The data frame curve has one row per angle in angles, in its angle column. Its score column follows a cosine curve with elevation 1, amplitude 0.8 and displacement 135 degrees. (The code that builds curve is omitted.)

ggplot(curve, aes(x = angle, y = score)) +
  geom_line() +
  geom_point(size = 2) +
  scale_x_circumplex(angles, labels = PANO()) +
  labs(x = "Scale", y = "Score") +
  theme_bw()

Pass the same labels (or the same instrument) to both ggcircumplex() and scale_x_circumplex(). This guarantees that a circular figure and a linear one label their scales identically.

9. Relationship to the built-in plots

The built-in plotting functions are implemented on exactly these components: ssm_plot_circle() is ggcircumplex() plus geom_ssm_arc() and geom_ssm_point(). It also moves the amplitude axis to a gap that holds no point. ssm_plot_curve() uses scale_x_circumplex() for its angle axis. So you can always start from a built-in plot and add to it, or rebuild it from the pieces when you need finer control. Whichever route you take, the coordinates are computed the same way, so the results line up. Only the amplitude axis can differ: to put it where ssm_plot_circle() puts it, pass r_axis_angle to coord_circumplex(), as in the path figure above.

Wrap-up

Every built-in circumplex figure is a composition of the same parts. ggcircumplex() or coord_circumplex() draws the canvas, and geom_ssm_point() and geom_ssm_arc() draw the profiles. scale_x_circumplex() labels a linear angle axis, and theme_circumplex() styles the canvas. Start from a built-in plot and add to it, or rebuild it from the pieces. No page follows this one. To estimate how a profile moves across waves before you draw it, read “Growth Models on SSM Parameters”.

References

• Gurtman, M. B. (1992). Construct validity of interpersonal personality measures: The interpersonal circumplex as a nomological net. Journal of Personality and Social Psychology, 63(1), 105–118.

• Wright, A. G. C., Pincus, A. L., Conroy, D. E., & Hilsenroth, M. J. (2009). Integrating methods to optimize circumplex description and comparison of groups. Journal of Personality Assessment, 91(4), 311–322.

• Zimmermann, J., & Wright, A. G. C. (2017). Beyond description in interpersonal construct validation: Methodological advances in the circumplex Structural Summary Approach. Assessment, 24(1), 3–23.