simulate_longitudinal_gompertz: Simulate Data From a Gompertz Change (Growth) Model

Description

Generates longitudinal data from a random-coefficients Gompertz change model: each unit (a person, an animal, a tree) follows an S-shaped curve that, unlike the logistic, is not symmetric about its point of inflection, the parameters vary randomly across units, and each measurement adds level-one error. The deterministic part of the four parameter Gompertz curve is $$\mu(t) = \alpha \exp\left(-\exp(-\gamma (t - \beta))\right) + \zeta,$$ the parameterization of Kelley (2005, 2008), which generalizes the three parameter Gompertz of the literature (Winsor, 1932; Ratkowsky, 1983) by adding $\zeta$ so the lower asymptote is itself a modeled quantity rather than fixed at zero.

Usage

simulate_longitudinal_gompertz(
  n,
  target_times = NULL,
  fixed_parameters,
  time_range = NULL,
  occasions = NULL,
  time_distribution = "uniform",
  random_variances = 0,
  random_correlation = NULL,
  error_variance = NULL,
  reliability = NULL,
  error_structure = c("independent", "ar1", "compound_symmetry", "toeplitz"),
  error_correlation = NULL,
  timing_sd = 0
)

Value

A long-format data.frame with columns id,

population, occasion, target_time, time,

true_score, and y, directly usable with

plot_trajectories and nonlinear mixed-model fitters such as nlme::nlme(). Attributes carry the model,

fixed_parameters, random_covariance,

error_variance, error_covariance,

reliability_by_occasion, and schedule

("shared" or "unit_specific").

Arguments

n

A single positive integer, the number of units (persons, animals, trees, classrooms) whose trajectories are drawn, or a vector giving the number of units for each population, one entry per parameter vector in fixed_parameters.

target_times

Numeric vector of the nominal measurement times, one shared schedule for every unit. Give either this or time_range.

fixed_parameters

The population parameters c(alpha = , beta = , gamma = , zeta = ) (an unnamed length-4 vector is taken in that order), or a list of such vectors, one per population (distinct data generating parameter vectors):

alpha: The total change: the curve travels alpha units from the lower asymptote zeta to the upper asymptote $\alpha + \zeta$ (for $\gamma > 0$).

beta

The point of inflection on the time axis. The Gompertz inflection is early and asymmetric: it occurs where the curve passes through $\alpha / e + \zeta$, about 36.8% of the total change, so growth accelerates briefly and decelerates over a long approach to the ceiling.

gamma

The curvature: how sharply the curve rises through its inflection. Negative values flip the curve to decreasing.

zeta

The lower asymptote (for $\gamma > 0$). The intercept is $\phi = \alpha \exp(-\exp(\gamma \beta)) + \zeta$.

time_range

Alternative to target_times: c(lower, upper) bounds from which each unit draws its own measurement times, so no two units share a schedule (e.g., age in weeks at testing rather than a fixed grade). Requires occasions; with time_range, the level-one error must be a single error_variance with the default independent structure, and timing_sd does not apply.

occasions

With time_range: a single positive integer (every unit measured the same number of times) or c(min, max), from which each unit's number of measurement times is drawn uniformly.

time_distribution

Distribution of the unit-specific times over time_range; currently "uniform".

random_variances

Between-unit variances of (alpha, beta, gamma, zeta), a single number recycled to all four or a length-4 vector. Default 0. A named vector over any subset of the parameter names (e.g., c(beta = 1.2)) varies only those named and leaves the rest fixed.

random_correlation

Optional 4-by-4 correlation matrix among the random parameters; default uncorrelated.

error_variance, reliability, error_structure, error_correlation, timing_sd

The level-one error and assessment-time machinery, with the same meaning as in simulate_longitudinal_polynomial: specify exactly one of error_variance or reliability (solved here through the first-order delta method true-score variance; because the curve is nonlinear in its parameters, that approximation, and the reliability_by_occasion attribute with it, can drift from the realized variance ratio when the random variances are large relative to the mean curve); error_structure and error_correlation set the across-occasion error correlation; timing_sd jitters the actual assessment times.

Author

Ken Kelley [email protected]

Details

The choice between the Gompertz and the logistic is substantive, not cosmetic: both are S-shaped, but the logistic spends equal time approaching floor and ceiling while the Gompertz commits to an early inflection (36.8% of total change) followed by a long deceleration. Processes with rapid early gains and slow consolidation, common in learning and development, are natural Gompertz candidates. The Gompertz is the $\delta \rightarrow 0$ limit of the Richards curve (simulate_longitudinal_richards), which frees the inflection entirely (Kelley, 2005, 2008).

References

Kelley, K. (2005). Estimating nonlinear change models in heterogeneous populations when class membership is unknown: Defining and developing the latent classification differential change model (Doctoral dissertation). University of Notre Dame.

Kelley, K. (2008). Nonlinear change models in populations with unobserved heterogeneity. Methodology, 4(3), 97--112.

Ratkowsky, D. A. (1983). Nonlinear regression modeling: A unified practical approach. Marcel Dekker.

Winsor, C. P. (1932). The Gompertz curve as a growth curve. Proceedings of the National Academy of Sciences, 18(1), 1--8.

Examples

Run this code

# The six-curve illustration from Kelley (2005): alpha = 0.75 and
# zeta = 0.25 throughout, so every curve crosses its inflection at
# the same height, 0.75 / exp(1) + 0.25, about 0.526. The three
# rising curves share the inflection time beta = 2 and differ only
# in curvature; the three falling curves share beta = 3. Each curve
# is its own population of size one, so a single call draws the whole panel.
panel <- simulate_longitudinal_gompertz(
  n = 1, target_times = seq(0, 6, by = 0.1),
  fixed_parameters = list(
    c(alpha = 0.75, beta = 2, gamma =  1.75, zeta = 0.25),
    c(alpha = 0.75, beta = 2, gamma =  1.00, zeta = 0.25),
    c(alpha = 0.75, beta = 2, gamma =  0.45, zeta = 0.25),
    c(alpha = 0.75, beta = 3, gamma = -0.35, zeta = 0.25),
    c(alpha = 0.75, beta = 3, gamma = -0.60, zeta = 0.25),
    c(alpha = 0.75, beta = 3, gamma = -2.00, zeta = 0.25)),
  error_variance = 0
)
plot_trajectories(panel, id = "id", time = "time",
                  outcome = "true_score", group = "population")

# Individual differences in a single parameter: only the inflection
# time varies (a named entry leaves every other variance at zero),
# so every trajectory shares the floor and the ceiling but reaches
# its fastest growth at its own moment.
set.seed(113)
d_beta <- simulate_longitudinal_gompertz(
  n = 25, target_times = seq(0, 8, by = 0.5),
  fixed_parameters = c(alpha = 75, beta = 3, gamma = 0.55, zeta = 10),
  random_variances = c(beta = 0.8), error_variance = 0
)
plot_trajectories(d_beta, id = "id", time = "time",
                  outcome = "true_score")

# Individual differences in every parameter at once, plus level-one
# error: the realistic sampling model.
set.seed(113)
d <- simulate_longitudinal_gompertz(
  n = 30, target_times = 0:12,
  fixed_parameters = c(alpha = 75, beta = 3, gamma = 0.55, zeta = 10),
  random_variances = c(alpha = 36, beta = 0.8, gamma = 0.01, zeta = 9),
  error_variance = 16
)
plot_trajectories(d, id = "id", time = "time", outcome = "y")

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