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, “The latent circumplex from a CPM fit”, draws the angles a cpm_fit() estimated, measure vectors, and the fitted correlation function. It closes with a joint confidence ellipse from posterior draws. Section 8, “Trajectories across occasions”, draws profiles estimated at several occasions. Section 9, “The angle axis for linear plots”, labels a linear axis with scale_x_circumplex(). Section 10, “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. The latent circumplex from a CPM fit

The canvases above place every scale at its theoretical angle. Those angles are an assumption. cpm_fit() estimates them instead. It fits Browne’s (1992) circular process model, which places each scale at an angle on a latent circle. The model describes the correlation between two scales as a function of their angular separation. The “Evaluating Circumplex Structure” vignette covers the model and its fit indices. This section draws what the fit estimated, with the layers the earlier sections used. Nothing here needs a new function.

The call below asks for analytic confidence intervals, but the figures draw only the point estimates. On this sample the fit sits at a boundary: the communality index of NO reached its upper limit, so print(cpm) notes a Heywood-type solution. The analytic intervals all come back NA. The Hessian is also ill-conditioned, so the call warns, and its condition number and some later digits differ across BLAS libraries. The “CPM Fits at a Boundary” vignette reads both the note and the warning in full.

cpm <- cpm_fit(jz2017, scales = PANO(), angles = octants(), ci_method = "analytic")
#> Warning: CPM Hessian is ill-conditioned (condition number 1.83e+14): angles
#> may be clustered or parameters weakly determined.
cpm$results[, c("Scale", "Angle_theory", "Angle", "Zeta")]
#>   Scale Angle_theory     Angle      Zeta
#> 1    PA           90  90.00000 0.7672831
#> 2    BC          135 125.07437 0.9313971
#> 3    DE          180 170.35282 0.7798895
#> 4    FG          225 195.42521 0.8609945
#> 5    HI          270 250.72074 0.9562548
#> 6    JK          315 269.49093 0.9421295
#> 7    LM          360 294.23003 0.8059460
#> 8    NO           45  11.30488 1.0000000

Angle_theory is the angle each scale was given. Angle is the angle the model estimated for it. The first scale, PA, is not estimated: it is fixed at its theoretical angle to set the rotation. Every other angle is read relative to it. Where the two differ, the instrument’s scales do not sit where the theory placed them, relative to PA, in this sample. Here every scale but PA sits clockwise of its theoretical angle. BC and DE sit 10 degrees clockwise and the rest 19 to 66 degrees clockwise. NO’s estimated angle of 11 degrees is near the theoretical position of LM. The three circle figures that follow draw the estimated angles alone. The ellipse figure at the end of the section returns to the theoretical octants() canvas. Zeta is each scale’s communality index, which the discussion of the correlation-function figure below uses.

The circle figures in this section follow Nagy, Etzel and Lüdtke (2019). Their Figure 6, and the two extension panels of their Figure 3, place the scales at their estimated angles. The canvas is a plain circle with a labelled crosshair. The canvas is ggcircumplex() with two arguments. grid = "cartesian" draws that circle and crosshair and no rings or spokes. angle_labels = TRUE writes each scale’s angle into its rim label.

Estimated angles as ticks on the rim

The first figure is the canvas alone. Its angles are the fit’s Angle column, so each rim tick sits at a scale’s estimated angle. Each label carries the scale’s name and its estimated angle rounded to the nearest degree. The crosshair marks the Cartesian coordinates of the score metric. The rim’s amplitude is 0.3, the value the measure figures below need.

canvas <- ggcircumplex(
  angles = cpm$results$Angle, labels = cpm$results$Scale,
  amax = 0.3, grid = "cartesian", angle_labels = TRUE
)
canvas

The ticks and their labels sit at the angles cpm_fit() estimated, and the theoretical angles appear only in the table above the figure. PA sits at 90 degrees by construction, because its angle was fixed. The other scales sit where the model placed them relative to PA. The tick for NO sits 11 degrees counterclockwise of LM’s theoretical position. DE and FG are 25 degrees apart where the theory puts them 45 apart. The crosshair’s labels are Cartesian coordinates and carry no meaning in this figure. They matter once a measure is placed inside the circle.

Measure points on the same rim

Nagy, Etzel and Lüdtke (2019, Figure 6) draw each external measure as a labelled point inside a circle whose rim carries the estimated angles. Their Figure 3 joins the point to the origin by a straight line. The point’s direction from the origin is the measure’s displacement and its distance from the origin is the measure’s amplitude. ssm_analyze() supplies both. The second figure adds five personality-disorder measures from jz2017 in that form, on the canvas above. Each measure is one point at a_est and d_est, one two-row path from amplitude 0 to that point, and one text label. The hjust and vjust values place each label beside its point, away from the other labels. They are chosen by hand and keyed by measure name.

ssm <- ssm_analyze(
  jz2017,
  scales = PANO(),
  measures = c("NARPD", "ASPD", "HISPD", "AVPD", "SCZPD")
)
measures <- ssm$results
hjust <- c(NARPD = -0.15, ASPD = 1.1, HISPD = -0.15, AVPD = 0.5, SCZPD = 1.15)
vjust <- c(NARPD = 1.4, ASPD = -0.3, HISPD = 0.5, AVPD = 1.6, SCZPD = 0.5)
measures$hjust <- hjust[measures$Label]
measures$vjust <- vjust[measures$Label]
spokes <- data.frame(
  Label = rep(measures$Label, each = 2),
  amplitude = c(rbind(0, measures$a_est)),
  displacement = rep(measures$d_est, each = 2)
)

canvas +
  geom_ssm_path(
    data = spokes,
    mapping = aes(
      amplitude = amplitude, displacement = displacement, group = Label
    )
  ) +
  geom_ssm_point(
    data = measures,
    mapping = aes(amplitude = a_est, displacement = d_est)
  ) +
  geom_text(
    data = measures,
    mapping = aes(
      x = d_est, y = a_est, label = Label, hjust = hjust, vjust = vjust
    )
  )

The rim and the points do not share a model. The rim ticks are the angles the circular process model estimated. The points are SSM displacements and amplitudes, and the SSM was computed with the theoretical angles, not the estimated ones. A point that lies in the direction of a tick does not mean that the measure loads on that scale’s latent position. Likewise, a measure’s distance from the origin is its SSM amplitude computed on the theoretical angles and not a loading on the latent circle. The amplitude is the cosine curve’s amplitude in the measure’s correlations with the eight scales, in the correlation metric that ssm_analyze() reports. To place a measure on the estimated circle itself, you would fit it inside the model, which cpm_fit() does not do. geom_text() places a label at x = d_est and y = a_est because coord_circumplex() reads x as displacement and y as amplitude. That is how every circumplex layer hands its positions to the coordinate system.

One measure with its amplitude and displacement

Nagy, Etzel and Lüdtke (2019, Figure 3) annotate a single measure. They draw the point, its line from the origin, and an arc that sweeps counterclockwise from 0 degrees to the measure’s displacement. The two values are written beside them. The third figure draws SCZPD that way. The arc is a path at one fixed amplitude whose displacement runs from 0 to d_est. It shows how the displacement is measured, counterclockwise from the positive horizontal axis. The two annotate() calls write d_est rounded to the nearest degree and a_est rounded to two decimals.

one <- measures[measures$Label == "SCZPD", ]
arc <- data.frame(
  amplitude = 0.08,
  displacement = seq(0, one$d_est, length.out = 100)
)

canvas +
  geom_ssm_path(
    data = spokes[spokes$Label == "SCZPD", ],
    mapping = aes(amplitude = amplitude, displacement = displacement)
  ) +
  geom_ssm_path(
    data = arc,
    mapping = aes(amplitude = amplitude, displacement = displacement)
  ) +
  geom_ssm_point(
    data = one,
    mapping = aes(amplitude = a_est, displacement = d_est)
  ) +
  geom_text(
    data = one,
    mapping = aes(x = d_est, y = a_est, label = Label),
    hjust = 1.2
  ) +
  annotate(
    "text", x = 60, y = 0.13, hjust = 0,
    label = paste0("d = ", round(one$d_est), "°")
  ) +
  annotate(
    "text", x = one$d_est, y = one$a_est / 2,
    label = paste0("a = ", sprintf("%.2f", one$a_est)), vjust = 1.5
  )

The displacement label starts above the arc at 60 degrees and runs to the right, clear of the vertical axis and its labels. The amplitude label sits at half the amplitude, below the line. SCZPD’s displacement is past 180 degrees, so the arc passes through the upper half of the circle and ends just below the negative horizontal axis. The values are the d_est and a_est columns of ssm$results, not values read off the figure.

The fitted correlation function

The fourth figure is linear. The fit’s corfun element is the estimated correlation function. It takes the angular separation between two scales, in degrees, and returns the correlation between their common parts. The figure draws it over separations from 0 to 180 degrees and places every pair of scales on the same axes. A pair’s separation is the absolute angular distance between its two estimated angles. The modular expression below computes it so that a pair straddling 0/360 gets the short way round. The filled points are the observed correlations from matrices$R. The hollow points are the correlations the model reproduces, from matrices$Phat.

curve <- data.frame(separation = 0:180)
curve$correlation <- cpm$corfun(curve$separation)

pair_idx <- which(upper.tri(cpm$matrices$R), arr.ind = TRUE)
a <- cpm$results$Angle[pair_idx[, 1]]
b <- cpm$results$Angle[pair_idx[, 2]]
observed <- data.frame(
  separation = abs(((a - b + 180) %% 360) - 180),
  observed = cpm$matrices$R[pair_idx],
  implied = cpm$matrices$Phat[pair_idx]
)

ggplot(observed, aes(x = separation)) +
  geom_line(data = curve, aes(y = correlation)) +
  geom_point(aes(y = observed, shape = "Observed"), size = 2) +
  geom_point(aes(y = implied, shape = "Reproduced"), size = 2) +
  scale_shape_manual(values = c(Observed = 16, Reproduced = 1), name = NULL) +
  scale_x_continuous(breaks = seq(0, 180, by = 45)) +
  labs(x = "Angular separation (degrees)", y = "Correlation") +
  theme_bw()

In this figure the points are observed correlations and the line is the fitted correlation function. The hollow points are the correlations the model reproduces, not observations. The filled points sit below the line at most separations, and part of that gap is attenuation, not misfit. The line is the correlation between two scales’ common parts. Under the default unit scaling, the model reproduces a pair’s correlation as the line’s value at the pair’s separation multiplied by both scales’ communality indices. Those indices are the Zeta column. A scale with a communality index below one pulls every point it belongs to below the line by that factor. The hollow points show where each pair lands after that attenuation. The filled point minus the hollow point is the pair’s residual, the entry of matrices$residuals. Read the line for the shape of the latent circumplex and the residuals for the fit.

A joint confidence ellipse from posterior draws

The wedge that geom_ssm_arc() draws is built from two intervals, one on amplitude and one on displacement. A profile’s uncertainty can also be shown as a joint region on the Cartesian coordinates under a normal approximation to the draws. geom_ssm_ellipse() draws that region as an ellipse. It takes a centre (x0, y0) and the three elements of a 2 by 2 covariance matrix as aesthetics. ssm_ellipse_data() computes those five columns from an ssm_draws() object, so the layer needs a result that retains draws. The draws below are the posterior draws that the “Bayesian SSM Analysis” vignette analyses. They ship with the package in bayesian_ssm_draws.rds, generated once by the seeded script data-raw/bayesian_ssm_draws.R, and that vignette states the model they come from.

draws <- readRDS("bayesian_ssm_draws.rds")
post <- ssm_draws(draws, type = "parameters")
ellipse <- ssm_ellipse_data(post)
ellipse
#>          x0         y0        var_x        var_y        cov_xy
#> 1 0.3523431 -0.3192543 0.0005952007 0.0005500366 -6.093168e-06

Nagy, Etzel and Lüdtke (2019, Figure 4 and Appendices B and C) read four lines off such an ellipse. Two are the tangents from the origin to the ellipse. Their angles bound the arc of directions the ellipse spans, read counterclockwise from the first tangent to the second. Two join the origin to the nearest and farthest points of the ellipse’s boundary. Their lengths are the smallest and largest amplitudes of any point in the ellipse. The function below computes all four from the one-row data frame above and the ellipse’s confidence level. It works from the covariance form that geom_ssm_ellipse() draws and solves the two problems numerically, rather than transcribing the appendices’ quartic and slope equations. A direction from the origin is tangent to the ellipse when the line along it meets the ellipse at exactly one point. That makes the quadratic for where the line meets the ellipse have a double root. Setting its discriminant to zero gives a quadratic in the direction’s slope, whose two roots are the two tangent directions. The two distances are the extremes of the boundary radius over the ellipse’s parametric angle, found on a fine grid and refined with optimize(). When the origin lies inside the ellipse there are no tangents, so those two angles are NA. The nearest distance is then the smallest boundary radius.

ellipse_lines <- function(ellipse, level = 0.95) {
  centre <- c(ellipse$x0, ellipse$y0)
  S <- matrix(
    c(ellipse$var_x, ellipse$cov_xy, ellipse$cov_xy, ellipse$var_y), 2, 2
  )
  r2 <- qchisq(level, df = 2)
  A <- solve(S)
  # A direction w is tangent when (w'Ac)^2 = (w'Aw)(c'Ac - r2), so w'Mw = 0.
  # The origin is outside the ellipse when c'Ac exceeds r2.
  outside <- drop(t(centre) %*% A %*% centre) - r2
  tangents <- c(NA_real_, NA_real_)
  if (outside > 0) {
    Ac <- drop(A %*% centre)
    M <- tcrossprod(Ac) - outside * A
    # The zero directions of w'Mw solve a quadratic in the slope. Use the
    # slope or its reciprocal, whichever gives the larger leading coefficient.
    # Both roots are real whenever the origin is outside (det(M) < 0), so
    # Re() drops only a zero imaginary part.
    if (abs(M[2, 2]) >= abs(M[1, 1])) {
      w <- rbind(1, Re(polyroot(c(M[1, 1], 2 * M[1, 2], M[2, 2]))))
    } else {
      w <- rbind(Re(polyroot(c(M[2, 2], 2 * M[1, 2], M[1, 1]))), 1)
    }
    # Point each direction towards the ellipse, then order the pair so that
    # the counterclockwise arc from the first to the second holds the centre.
    w <- sweep(w, 2, sign(colSums(w * Ac)), `*`)
    tangents <- (atan2(w[2, ], w[1, ]) * 180 / pi) %% 360
    if ((tangents[2] - tangents[1]) %% 360 > 180) tangents <- rev(tangents)
  }
  # The boundary radius as a function of the parametric angle.
  L <- t(chol(S)) * sqrt(r2)
  boundary <- function(theta) centre + L %*% rbind(cos(theta), sin(theta))
  radius <- function(theta) sqrt(colSums(boundary(theta)^2))
  grid <- seq(0, 2 * pi, length.out = 3601)[-3601]
  step <- grid[2] - grid[1]
  refine <- function(i, maximum) {
    optimize(radius, grid[i] + c(-step, step), maximum = maximum, tol = 1e-12)
  }
  nearest <- refine(which.min(radius(grid)), maximum = FALSE)
  farthest <- refine(which.max(radius(grid)), maximum = TRUE)
  angle_of <- function(theta) {
    v <- boundary(theta)
    (atan2(v[2], v[1]) * 180 / pi) %% 360
  }
  data.frame(
    line = c("tangent_1", "tangent_2", "nearest", "farthest"),
    displacement = c(
      tangents, angle_of(nearest$minimum), angle_of(farthest$maximum)
    ),
    amplitude = c(NA, NA, nearest$objective, farthest$objective)
  )
}
level <- 0.95
four_lines <- ellipse_lines(ellipse, level = level)
four_lines
#>        line displacement amplitude
#> 1 tangent_1     310.7611        NA
#> 2 tangent_2     324.8137        NA
#> 3   nearest     317.5519 0.4164769
#> 4  farthest     318.0873 0.5344676

The displacement column holds the two tangent angles and the directions of the nearest and farthest boundary points. The amplitude column holds the two distances. The tangents are listed so that the counterclockwise arc from tangent_1 to tangent_2 holds the ellipse, which matters for an ellipse that straddles 0/360 degrees.

The figure draws the wedge, the ellipse, the four lines and the point estimate for the same profile on one canvas. The wedge and the point come from post$results, as in Section 4. The ellipse comes from the one-row data frame above, drawn at the same level the four lines were computed at. The two tangent lines are dashed and run from the origin to the rim. The two distance lines are dotted and run from the origin to the nearest and farthest boundary points.

amax <- 0.6
tangents <- data.frame(
  line = rep(four_lines$line[1:2], each = 2),
  amplitude = rep(c(0, amax), times = 2),
  displacement = rep(four_lines$displacement[1:2], each = 2)
)
distances <- data.frame(
  line = rep(four_lines$line[3:4], each = 2),
  amplitude = c(rbind(0, four_lines$amplitude[3:4])),
  displacement = rep(four_lines$displacement[3:4], each = 2)
)

ggcircumplex(
  octants(), labels = PANO(), amax = amax,
  grid = "cartesian", angle_labels = TRUE
) +
  geom_ssm_arc(
    data = post$results,
    mapping = aes(
      amplitude_min = a_lci, amplitude_max = a_uci,
      displacement_min = d_lci, displacement_max = d_uci
    ),
    alpha = 0.15
  ) +
  geom_ssm_path(
    data = tangents,
    mapping = aes(
      amplitude = amplitude, displacement = displacement, group = line
    ),
    linetype = "dashed"
  ) +
  geom_ssm_path(
    data = distances,
    mapping = aes(
      amplitude = amplitude, displacement = displacement, group = line
    ),
    linetype = "dotted", color = "gray40"
  ) +
  geom_ssm_ellipse(
    data = ellipse,
    mapping = aes(
      x0 = x0, y0 = y0, var_x = var_x, var_y = var_y, cov_xy = cov_xy
    ),
    level = level
  ) +
  geom_ssm_point(
    data = post$results,
    mapping = aes(amplitude = a_est, displacement = d_est),
    size = 1.5
  ) +
  geom_text(
    data = post$results,
    mapping = aes(x = d_est, y = a_est, label = "Mean profile"),
    hjust = -0.3
  )

The label names what the draws describe: the mean octant profile of the subsample the “Bayesian SSM Analysis” vignette fits. The ellipse is small on this canvas because the posterior is tight, so the point is drawn small to keep the outline visible. The wedge is drawn light so that the ellipse and the four lines read over it. The tangent lines bound the directions the ellipse spans under the normal approximation and are not the displacement interval the wedge draws. The wedge’s displacement interval is a circular-quantile interval of the draws. The tangent angles come from the ellipse alone, so the two pairs of angles differ. Here the tangents at 311 and 325 degrees sit outside the wedge’s interval from 312 to 323 degrees.

Here the ellipse is centred on the posterior medians of x and y, not on the plotted point, and its covariance comes from the draws. The medians are the values ssm_draws() reports as x_est and y_est. The plotted point is a_est at d_est: the median amplitude at the circular mean displacement. Those are marginal summaries, so the point is not the polar form of the medians of x and y. The chunk below computes the distance between the centre and the point in the score metric. Here it is too small to see.

point <- post$results$a_est *
  c(cos(post$results$d_est * pi / 180), sin(post$results$d_est * pi / 180))
sqrt(sum((c(ellipse$x0, ellipse$y0) - point)^2))
#> [1] 0.0004661232

The covariance is the sample covariance of the draws of x and y. The wedge and the ellipse are two different summaries of the same draws. The ellipse is a joint region on the Cartesian coordinates under a normal approximation to the draws. By contrast, the wedge is the pair of marginal intervals on amplitude and displacement, a percentile interval on amplitude and a circular-quantile interval on displacement. The wedge draws those two intervals together as one region. Neither interval depends on a normal approximation. The two regions answer different questions, so the two need not coincide. In this figure the ellipse reaches farther than the wedge along both the amplitude axis and the displacement axis. The wedge’s four corners still fall outside the ellipse.

Read the ellipse against the origin with care. An ellipse at confidence level 1 − α that excludes the origin rejects zero amplitude in a Wald test at significance level α. That holds only under the normal approximation. A Wald test compares an estimate with its standard error, here the pair (x, y) with its covariance. The ellipse above is drawn at confidence level 0.95. It is the set of (x, y) values the test does not reject at significance level 0.05. Its covariance is a posterior covariance, so that reading also treats the posterior under the normal approximation as the sampling distribution of the estimate. The wedge makes no such test. A displacement interval that excludes some angle is not a significance test of that angle. An amplitude interval above zero is a statement about amplitude alone.

8. 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 329.28366 335.57815
#> 2       T2 0.5968632 354.84909 351.80498 357.66599
#> 3       T3 0.5959438  21.05834  17.83082  24.00907

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, and the line segments on either side of it are drawn dashed. A dashed segment touches a time point whose displacement is not interpretable, so the direction of change along it is not to be read. 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.

9. 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.

10. 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

• Browne, M. W. (1992). Circumplex models for correlation matrices. Psychometrika, 57(4), 469–497.

• Nagy, G., Etzel, J. M., & Lüdtke, O. (2019). Integrating covariates into circumplex structures: An extension procedure for Browne’s circular stochastic process model. Multivariate Behavioral Research, 54(3), 404–428.

• 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.