---
title: "Growth Models on SSM Parameters"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Growth Models on SSM Parameters}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
has_glmmTMB <- requireNamespace("glmmTMB", quietly = TRUE)
set.seed(20260716)
library(ggplot2)
```

```{r setup}
library(circumplex)
```

## 1. The question growth modeling answers

A single Structural Summary Method (SSM) analysis describes one profile: an
elevation $e$, an amplitude $a$, and a displacement $d$. With repeated
measurements — the same persons assessed at several waves — a new question
opens up: *how does the profile change over time?* Does the group's
interpersonal style drift toward warmth? Does its distinctiveness
(amplitude) grow or fade?

Displacement is an angle, and angles resist ordinary growth modeling: a
trajectory drifting from 350° to 10° has moved 20°, not −340°, and a linear
model fit directly to raw displacements will get this wrong whenever a
trajectory crosses the 0°/360° boundary. This vignette presents the
package's recommended recipe, which avoids the boundary entirely by
modeling growth in the Cartesian coordinates $(x, y)$ — the same
coordinates the SSM estimator itself uses — and converting fitted
trajectories back to $(a(t), d(t))$ with circular-correct summaries at the
end.

The division of labor is deliberate and mirrors the package's Bayesian
vignette: **circumplex does not fit mixed models**. It prepares the
coordinate data on the way in (`ssm_parameters_id()`) and converts fitted
model draws on the way out (`ssm_draws()`); the growth model itself belongs
to a dedicated mixed-modeling package. The reference recipe below uses
**glmmTMB**; the same stacked-outcome formulation can be fit with `nlme`
(shipped with base R) using its `varIdent`/`corSymm` machinery.

```{r, echo = FALSE, results = "asis"}
if (!has_glmmTMB) {
  cat(
    "> **Note:** the glmmTMB package is not installed, so the model-fitting",
    "chunks below are not evaluated in this build of the vignette. Install",
    "glmmTMB to reproduce them.\n"
  )
}
```

## 2. From repeated measures to a coordinate table

The input to the growth model is a person-by-wave table of SSM coordinates.
`ssm_parameters_id()` computes each person's $(e, x, y)$ (and $a$, $d$,
fit) from their circumplex scale scores; applied per wave, it yields
exactly the tidy input a mixed model wants.

We simulate five waves of octant scores for 150 persons whose group-level
displacement drifts from 350° to 10° — deliberately crossing the 0°/360°
boundary — with amplitude near 0.6 throughout.

```{r simulate}
n <- 150
waves <- 0:4
theta <- as.numeric(octants()) * pi / 180

# Group trajectory: (x, y) linear between 0.6*(cos, sin)(350 deg) and
# 0.6*(cos, sin)(10 deg); elevation constant at 0.5
xy_start <- 0.6 * c(cos(350 * pi / 180), sin(350 * pi / 180))
xy_end <- 0.6 * c(cos(10 * pi / 180), sin(10 * pi / 180))
x_t <- seq(xy_start[1], xy_end[1], length.out = length(waves))
y_t <- seq(xy_start[2], xy_end[2], length.out = length(waves))

# Person effects: elevation shifts and profile tilts, stable across waves
u_e <- rnorm(n, 0, 0.30)
v_x <- rnorm(n, 0, 0.15)
v_y <- rnorm(n, 0, 0.15)

coord <- do.call(rbind, lapply(seq_along(waves), function(k) {
  mu <- 0.5 + (x_t[k] + v_x) %o% cos(theta) + (y_t[k] + v_y) %o% sin(theta)
  scores <- mu + u_e + matrix(rnorm(n * length(theta), 0, 0.40), n)
  colnames(scores) <- PANO()
  pp <- ssm_parameters_id(as.data.frame(scores), scales = PANO())
  data.frame(person = pp$id, wave = waves[k],
             e = pp$Elev, x = pp$Xval, y = pp$Yval)
}))
head(coord, 3)
```

## 3. One joint model, not three separate ones

The growth model treats the three coordinates as a *multivariate* outcome:
stack them into long format with an outcome indicator `dv`, give each
outcome its own intercept and slope, and let the person-level random
effects be **correlated across outcomes**.

```{r stack}
long <- reshape(coord, direction = "long",
                varying = list(c("e", "x", "y")), v.names = "value",
                timevar = "dv", times = c("e", "x", "y"),
                idvar = c("person", "wave"))
long$dv <- factor(long$dv, levels = c("e", "x", "y"))
long$person <- factor(long$person)
```

```{r fit, eval = has_glmmTMB}
fit <- glmmTMB::glmmTMB(
  value ~ 0 + dv + dv:wave + us(0 + dv | person),
  dispformula = ~ 0 + dv,
  data = long,
  REML = TRUE
)
glmmTMB::fixef(fit)$cond
```

Why joint? The displacement $d(t)$ is derived from $\hat{x}(t)$ and
$\hat{y}(t)$ *together*, so its uncertainty depends on their **joint**
sampling distribution — including the covariance
$\mathrm{Cov}(\hat{x}(t), \hat{y}(t))$. Fitting two (or three) univariate
mixed models instead is a tempting shortcut that produces valid-looking
output and **wrong $d(t)$ intervals**: separate fits have independent
covariance matrices, which silently sets
$\mathrm{Cov}(\hat{x}(t), \hat{y}(t)) = 0$. In our validation simulations,
a design with strongly correlated $x$–$y$ person effects (a realistic
situation — profile tilts are rarely axis-aligned) drops the shortcut's
pointwise $d(t)$ coverage from the nominal 95% to roughly 86%, while the
joint recipe stays at nominal. Do not fit the coordinates separately.

## 4. From fixed effects to $(a(t), d(t))$ with intervals

The fixed effects define the mean trajectory
$(\hat{e}(t), \hat{x}(t), \hat{y}(t))$. To carry uncertainty through the
nonlinear map to $(a(t), d(t))$, draw coefficient vectors from the
multivariate normal implied by the fixed-effect covariance matrix — the
same asymptotic move the package's Monte Carlo engine makes — evaluate each
draw at each wave, and hand the per-wave draws to `ssm_draws()`, which
applies the package's circular-statistics machinery (medians and circular
means, equal-tailed intervals with correct wrapping at the boundary).

```{r draws, eval = has_glmmTMB}
fe <- glmmTMB::fixef(fit)$cond
V <- as.matrix(vcov(fit)$cond)

# MVN draws of the coefficient vector (eigen root, robust to tiny
# negative eigenvalues from floating point)
mvn_draw <- function(n_draws, mu, sigma) {
  eig <- eigen(sigma, symmetric = TRUE)
  root <- eig$vectors %*% (sqrt(pmax(eig$values, 0)) * t(eig$vectors))
  sweep(matrix(rnorm(n_draws * length(mu)), nrow = n_draws) %*% root,
        2, mu, "+")
}
B <- mvn_draw(4000, fe, V)
colnames(B) <- names(fe)

per_wave <- lapply(waves, function(t) {
  draws_t <- cbind(
    e = B[, "dve"] + t * B[, "dve:wave"],
    x = B[, "dvx"] + t * B[, "dvx:wave"],
    y = B[, "dvy"] + t * B[, "dvy:wave"]
  )
  ssm_draws(draws_t, type = "parameters")
})

trajectory <- data.frame(
  wave = waves,
  a_est = sapply(per_wave, function(s) s$results$a_est),
  a_lci = sapply(per_wave, function(s) s$results$a_lci),
  a_uci = sapply(per_wave, function(s) s$results$a_uci),
  d_est = sapply(per_wave, function(s) as.numeric(s$results$d_est)),
  d_lci = sapply(per_wave, function(s) as.numeric(s$results$d_lci)),
  d_uci = sapply(per_wave, function(s) as.numeric(s$results$d_uci)),
  certified = sapply(per_wave, function(s) s$details$certified)
)
round_df <- trajectory
round_df[-1] <- lapply(round_df[-1], function(v) {
  if (is.numeric(v)) round(v, 2) else v
})
round_df
```

The displacement estimates hug the 0°/360° boundary by design — values
near 350° at early waves and near 10° at late waves — and the intervals
wrap correctly rather than spanning "the long way around." A table in this
shape — one row per time point, with `a_est`/`a_lci`/`a_uci`,
`d_est`/`d_lci`/`d_uci`, and optionally `certified` — is what
`ssm_plot_trajectory()` plots directly:

```{r plot, eval = has_glmmTMB, fig.width = 6.5, fig.height = 3.5}
ssm_plot_trajectory(trajectory, time = "wave")
```

The displacement panel is drawn on an *unwrapped* branch, so the trajectory
crosses the boundary as one continuous path instead of jumping a full turn,
and values there may legitimately fall outside $[0°, 360°)$. Each interval
is placed on its estimate's branch and keeps the width it was reported with
— including an interval that straddles the seam, which is stored with
`d_lci > d_uci`. Doing this by hand is easy to get subtly wrong: the natural
"shift each bound by its signed distance from the estimate" recipe silently
cannot represent an interval wider than a half-turn, which is exactly the
near-origin case Section 5 is about. The function handles both.

## 5. Certification: when $d(t)$ intervals are not interpretable

A direction is only meaningful when the trajectory is far enough from the
origin. As $a(t) \to 0$ the draws of $d(t)$ become diffuse or bimodal, and
a quantile interval of them is not a trustworthy statement about
direction. At each wave, `ssm_draws()` applies the package's scale-free
certification rule to the amplitude interval (lower bound at least 0.35
interval-widths above zero) and records the verdict in
`$details$certified` — the `certified` column above. **At any uncertified
wave, the $d(t)$ interval is not interpretable** and should be reported as
such, not narrated as a direction.

Every wave in our worked example is certified. Here is a trajectory where
that fails: the group crosses near the origin mid-study (its $x$
coordinate changes sign while $y$ stays near zero).

```{r lowamp, eval = has_glmmTMB}
x_t2 <- seq(0.5, -0.5, length.out = length(waves))
coord2 <- do.call(rbind, lapply(seq_along(waves), function(k) {
  mu <- 0.5 + (x_t2[k] + v_x) %o% cos(theta) + (0.02 + v_y) %o% sin(theta)
  scores <- mu + u_e + matrix(rnorm(n * length(theta), 0, 0.40), n)
  colnames(scores) <- PANO()
  pp <- ssm_parameters_id(as.data.frame(scores), scales = PANO())
  data.frame(person = pp$id, wave = waves[k],
             e = pp$Elev, x = pp$Xval, y = pp$Yval)
}))
long2 <- reshape(coord2, direction = "long",
                 varying = list(c("e", "x", "y")), v.names = "value",
                 timevar = "dv", times = c("e", "x", "y"),
                 idvar = c("person", "wave"))
long2$dv <- factor(long2$dv, levels = c("e", "x", "y"))
long2$person <- factor(long2$person)
fit2 <- glmmTMB::glmmTMB(
  value ~ 0 + dv + dv:wave + us(0 + dv | person),
  dispformula = ~ 0 + dv, data = long2, REML = TRUE
)
fe2 <- glmmTMB::fixef(fit2)$cond
B2 <- mvn_draw(4000, fe2, as.matrix(vcov(fit2)$cond))
colnames(B2) <- names(fe2)
mid <- ssm_draws(cbind(
  e = B2[, "dve"] + 2 * B2[, "dve:wave"],
  x = B2[, "dvx"] + 2 * B2[, "dvx:wave"],
  y = B2[, "dvy"] + 2 * B2[, "dvy:wave"]
), type = "parameters")
mid
```

The printed note is the certification rule firing: at this wave the
amplitude interval sits too close to zero, and the displacement interval —
here spanning more than half the circle — is not a statement about
direction. In our validation simulations of this regime, the caution fires
at the degraded wave in essentially every replicate while the origin-distal
waves remain certified and nominally covered.

Assembling this fit's full trajectory the same way, and carrying the
`certified` verdict into the table, lets the plot mark the verdict itself:

```{r lowamp-plot, eval = has_glmmTMB, fig.width = 6.5, fig.height = 3.5}
per_wave2 <- lapply(waves, function(t) {
  ssm_draws(cbind(
    e = B2[, "dve"] + t * B2[, "dve:wave"],
    x = B2[, "dvx"] + t * B2[, "dvx:wave"],
    y = B2[, "dvy"] + t * B2[, "dvy:wave"]
  ), type = "parameters")
})
trajectory2 <- data.frame(
  wave = waves,
  a_est = sapply(per_wave2, function(s) s$results$a_est),
  a_lci = sapply(per_wave2, function(s) s$results$a_lci),
  a_uci = sapply(per_wave2, function(s) s$results$a_uci),
  d_est = sapply(per_wave2, function(s) as.numeric(s$results$d_est)),
  d_lci = sapply(per_wave2, function(s) as.numeric(s$results$d_lci)),
  d_uci = sapply(per_wave2, function(s) as.numeric(s$results$d_uci)),
  certified = sapply(per_wave2, function(s) s$details$certified)
)
ssm_plot_trajectory(trajectory2, time = "wave")
```

Uncertified waves are drawn as **hollow** points on the displacement panel.
Read them as gaps in the argument, not as estimates with wide intervals: the
amplitude panel shows why, with the interval collapsing toward zero as the
group passes the origin. The displacement interval at such a wave can cover
most of the circle, and the panel draws it at that full width rather than
flattening it into something that looks precise.

The `certified` column is optional. A table without it plots the same way,
minus the hollow marking and its legend — the figure then makes no claim
about interpretability either way, which is the honest default when the
verdict was never computed.

## 6. A caution about REML intervals at small samples

The fixed-effect covariance matrix used for the draws conditions on the
estimated variance components: it ignores their uncertainty. At small
sample sizes this makes the resulting intervals anticonservative (too
narrow). This is a property of the mixed-model machinery, not of the SSM
transform, and the remedy is user-side: with modest N, prefer
degrees-of-freedom-adjusted inference — the Kenward–Roger adjustment for
`lme4` fits via **pbkrtest**, the approximate denominator degrees of
freedom `nlme` supplies, or $t$-quantile-based intervals — or a parametric
bootstrap of the fixed effects, over raw normal-approximation draws.

## 7. The unwrap alternative: `angle_unwrap()`

There is a second documented recipe: compute each person's displacement at
each wave, **unwrap** each person's sequence onto a continuous branch, and
fit an ordinary univariate growth model to the unwrapped angles.

```{r unwrap-demo}
# One person's displacement over five waves, drifting across the boundary
d_person <- c(350, 355, 2, 8, 12)
angle_unwrap(d_person)
```

`angle_unwrap()` wraps its input into [0°, 360°), then accumulates the
shortest signed rotation between successive waves (an exact 180° step
ascends, by the package's half-turn convention; an `NA` makes every later
wave branch-ambiguous, so `NA` propagates onward). The unwrapped values
live on an ordinary line, so any univariate mixed model applies — this
framing models the *mean of the person-level directions*, which is a
legitimate (and different) estimand from the direction of the mean
trajectory in Section 4.

Its failure modes are sharp, though, and they are the reason the $(x, y)$
recipe is the reference:

* **Fast movement between waves.** Unwrapping assumes successive waves
  move less than a half-turn. A trajectory sampled too sparsely (or a
  person who genuinely swings more than 180° between waves) is unwrapped
  onto the wrong branch with no warning — near-180° jumps are resolved by
  convention, not by information the data contain.
* **No common branch across persons.** When persons occupy genuinely
  heterogeneous locations around the circle (e.g., half the sample near
  90°, half near 270°), their unwrapped branches are not comparable, and
  the fixed-effect "mean trajectory" averages numbers that do not share a
  scale. The $(x, y)$ framing has no such requirement.
* **Low amplitude.** A person's observed displacement at a wave where
  their amplitude is near zero is mostly noise, and one noisy wave can
  throw the rest of that person's sequence onto a wrong branch (the same
  reason the Section 5 certification exists).

In the concentrated, common-branch regime — everyone well away from the
origin, trajectories moving slowly, directions clustered — the two recipes
agree closely (our validation simulations find mean trajectory differences
well under a degree), so the choice matters exactly when the unwrap
recipe's assumptions are in doubt.

## 8. Caveats and upgrades

Two statistical facts about the $(x, y)$ recipe deserve explicit
statement:

* **The derived $d(t)$ is the direction of the mean trajectory, not the
  mean of the person-level directions.** These differ whenever persons
  disperse directionally; neither is wrong, but they answer different
  questions, and Section 7's unwrap recipe targets the latter.
* **The derived $a(t)$ shrinks toward zero under directional
  dispersion.** The amplitude of an average profile is smaller than the
  average of individual amplitudes whenever persons point in different
  directions — the standard SSM aggregation fact, inherited intact by the
  growth setting.

Finally, the MVN-draw propagation used here is a large-sample
approximation that is defensible in the concentrated regime and guarded by
the Section 5 certification elsewhere. The fully model-based upgrade is
**projected-normal regression** — e.g., the **bpnreg** package — which
models circular outcomes directly with person-level structure and gives
exact posterior inference for $d(t)$; circumplex does not wrap it, but
`ssm_draws()` will happily summarize posterior draws produced by any such
model.

## References

* Girard, J. M., Zimmermann, J., & Wright, A. G. C. (2018). New tools for
  circumplex data analysis and visualization in R. *Assessment, 25*(1),
  3–20.
* 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.
* Cremers, J., & Klugkist, I. (2018). One direction? A tutorial for
  circular data analysis using R with examples in cognitive psychology.
  *Frontiers in Psychology, 9*, 2040.
