Simulate Observed Data from a Two-Profile CULTA Model

Generates data from a two-profile longitudinal CULTA model where profile membership, trait components, and state dynamics are influenced by an observed covariate. The function produces simulated responses incorporating covariate effects, individual differences, and time-varying fluctuations.

Usage

GenCULTA2Profiles(
  n,
  m,
  mu_x,
  sigma_x,
  nu_0,
  kappa_0,
  alpha_0,
  beta_00,
  gamma_00,
  gamma_10,
  mu_t,
  psi_t,
  mu_p,
  psi_p,
  common_trait_loading,
  common_state_loading,
  phi_0,
  phi_1,
  psi_s0,
  psi_s,
  theta,
  mu_profile
)

Arguments

n

Positive integer. Number of individuals.

m

Positive integer. Number of measurement occasions.

mu_x

Numeric. Mean of the covariate \(\mu_X\).

sigma_x

Numeric. Variance of the covariate \(\sigma_X\).

nu_0

Numeric. Intercept \(\nu_{0}\) for the logistic model of initial profile membership.

kappa_0

Numeric. Covariate effect \(\kappa_{0}\) on initial profile membership.

alpha_0

Numeric. Intercept \(\alpha_{0}\) for the logistic model of profile transitions across time.

beta_00

Numeric. Effect \(\beta_{00}\) for self-persistence in profile 0 transitions.

gamma_00

Numeric. Covariate effect \(\gamma_{00}\) on remaining in profile 0.

gamma_10

Numeric. Covariate effect \(\gamma_{10}\) on transitioning from profile 1 to profile 0.

mu_t

Numeric or vector of length \(q\). Mean \(\mu_t\) of the common trait factor. If mu_t = NULL, defaults to zero.

psi_t

Numeric matrix of size \(q \times q\). Positive definite covariance matrix \(\Psi_t\) for the common trait factor.

mu_p

Numeric vector of length \(p\). Mean vector \(\boldsymbol{\mu}_p\) for unique trait components. If mu_p = NULL, defaults to zero.

psi_p

Numeric matrix of size \(p \times p\). Positive definite covariance matrix \(\Psi_p\) for unique trait components.

common_trait_loading

Numeric matrix of size \(p \times q\). Factor loading matrix specifying the influence of the common trait on each observed item.

common_state_loading

Numeric matrix of size \(p \times 1\). Factor loading matrix specifying the influence of the common state on each observed item.

phi_0

Numeric. Autoregressive coefficient \(\phi_0\) for the common state process in profile 0.

phi_1

Numeric. Autoregressive coefficient \(\phi_1\) for the common state process in profile 0.

psi_s0

Numeric. Variance \(\psi_{s0}\) of the initial common state.

psi_s

Numeric. Innovation variance \(\psi_{s}\) for the common state process.

theta

Numeric matrix of size \(p \times p\). Positive definite covariance matrix \(\boldsymbol{\Theta}\) for unique state components.

mu_profile

Numeric matrix of size \(p \times 2\). Profile-specific means for each observed item across two latent profiles.

Value

Returns an object of class simculta. which is a list with the following elements:

• call: Function call.

• fun: Function used ("GenCULTA2Profiles").

• args: Function arguments.

• id: Vector of ID numbers.

• covariate: Vector of covariate values.

• categorical: Latent profiles.

• common_trait: Common trait.

• unique_trait: Unique trait.

• common_state: Common state.

• trait: Common trait + unique trait.

• state: Common state + unique state.

• data: Generated data which is a matrix of observed variables generated from the CULTA model with two-profiles.

Details

The GenCULTA2Profiles() function generates data for a two-profile CULTA model with a covariate. The CULTA model incorporates a covariate, latent categorical variables, trait components, state components, and profile-specific means to simulate longitudinal data with latent profile transitions.

Let \(i \in \left\{ 1, \ldots, n \right\}\) denote the index for individuals, let \(t \in \left\{ 0, \ldots, m - 1 \right\}\) denote the index measurement occasions, let \(k \in \left\{ 1, \ldots, p \right\}\) denote the index items, and let \(c \in \left\{ 0, 1 \right\}\) be the index of the two latent profiles (profile 0 and profile 1). Let \(q\) be the trait dimension, \(q = 1\) in this context.

Covariate

The covariate is generated from a normal distribution with mean \(\mu_X\) and variance \(\sigma_X\).

Latent Categorical Variables

Latent categorical variables represent profile membership for each individual at each measurement occasion. In a two-profile model, profile membership is influenced by a covariate and previous profile status, following a logistic formulation. We distinguish between:

• Initial profile membership (baseline time point)

• Profile transitions across subsequent time points

We describe both components below.

Initial Profile Membership

For the first measurement occasion (\(t = 0\)), profile membership is determined by the following log-odds for belonging to profile 0 (with profile 1 as the reference category): $$ \left( \begin{array}{cc} \nu_{0} + \kappa_{0} \times \mathrm{Covariate} & 0 \\ \end{array} \right) . $$ The corresponding probability of belonging to each profile is given by: $$ \left( \begin{array}{cc} \frac{ \exp \left( \nu_{0} + \kappa_{0} \times \mathrm{Covariate} \right) }{ \exp \left( \nu_{0} + \kappa_{0} \times \mathrm{Covariate} \right) + 1 } & \frac{1}{ \exp \left( \nu_{0} + \kappa_{0} \times \mathrm{Covariate} \right) + 1 } \\ \end{array} \right) . $$ Profile membership at the first occasion is sampled based on these probabilities.

Profile Transitions

For subsequent occasions (\(t = 1, \ldots, m - 1\)), profile transitions depend on the profile at the previous occasion and the covariate. The log-odds for transitioning to profile 0 at time \(t\) are given by: $$ \left( \begin{array}{cc} \alpha_{0} + \beta_{00} + \gamma_{00} \times \mathrm{Covariate} & 0 \\ \alpha_{0} + \gamma_{10} \times \mathrm{Covariate} & 0 \\ \end{array} \right) . $$ The probability of transitioning to each profile is computed as: $$ \left( \begin{array}{cc} \frac{ \exp \left( \alpha_{0} + \beta_{00} + \gamma_{00} \times \mathrm{Covariate} \right) }{ \exp \left( \alpha_{0} + \beta_{00} + \gamma_{00} \times \mathrm{Covariate} \right) + 1 } & \frac{1}{ \exp \left( \alpha_{0} + \beta_{00} + \gamma_{00} \times \mathrm{Covariate} \right) + 1 } \\ \frac{ \exp \left( \alpha_{0} + \gamma_{10} \times \mathrm{Covariate} \right) }{ \exp \left( \alpha_{0} + \gamma_{10} \times \mathrm{Covariate} \right) + 1 } & \frac{1}{ \exp \left( \alpha_{0} + \gamma_{10} \times \mathrm{Covariate} \right) + 1 } \\ \end{array} \right) . $$ Profile membership for each subsequent time point is sampled using these transition probabilities, based on the individual's covariate value and previous profile.

Trait Components

The trait variate captures between-person differences and is composed of a shared (common) component and item-specific (unique) components. The full decomposition is given by: $$ \mathrm{Trait}_{i} = \mathrm{Common\ Trait\ Loading} \times \mathrm{Common\ Trait}_{i} + \mathrm{Unique\ Trait}_{i} . $$ We describe each component below.

Common Trait

The common trait \(\mathrm{Common\ Trait}_{i}\) represents shared individual differences that influence all items uniformly. It is drawn from a normal distribution with mean \(\mu_t\) and variance \(\psi_t\): $$ \mathrm{Common\ Trait}_{i} \sim \mathcal{N} \left( \mu_t, \psi_t \right) $$

The influence of the common trait on each item is determined by the \(p \times q\) common trait loading,

Unique Traits

The unique trait component \(\mathrm{Unique\ Trait}_{k, i}\) captures item-specific stable differences and is drawn from a multivariate normal distribution: $$ \mathrm{Unique\ Trait}_{i} \sim \mathcal{N} \left( \boldsymbol{\mu}_p, \boldsymbol{\Psi}_{p \times p} \right) $$

Combined Trait Variate

The trait variate for item \(k\) and individual \(i\) is obtained by combining the common and unique trait components: $$ \mathrm{Trait}_{k, i} = \mathrm{Common\ Trait\ Loading}_{k} \times \mathrm{Common\ Trait}_{i} + \mathrm{Unique\ Trait}_{k, i} . $$ The common trait component introduces shared variance across items, while the unique trait component allows for item-specific differences not explained by the common trait.

State Components

The state variate is composed of two parts: a common state shared across items, and unique states specific to each item. The full decomposition is given by: $$ \mathrm{State}_{k, i, t} = \mathrm{Common\ State\ Loading}_{k} \times \mathrm{Common\ State}_{i, t} + \mathrm{Unique\ State}_{k, i, t} . $$ We describe each component below.

Common State

The common state \(\mathrm{Common\ State}_{i, t}\) evolves over time following a first-order autoregressive process: $$ \mathrm{Common\ State}_{i, t} = \phi_{c} \times \mathrm{Common\ State}_{i, t - 1} + \zeta_{i, t} . $$ The initial common state is drawn from a normal distribution: $$ \mathrm{Common\ State}_{i, 0} \sim \mathcal{N} \left( 0, \psi_{s_{0}} \right) . $$ The innovation term \(\zeta_{i, t}\) is normally distributed: $$ \zeta_{i, t} \sim \mathcal{N} \left( 0, \psi_{s} \right) . $$ The autoregressive parameter \(\phi_{c}\) depends on latent profile membership \(c\): $$ \phi_{c} = \phi_{0} + \left( \phi_{1} - \phi_{0} \right) c . $$ Here, \(\phi_{0}\) and \(\phi_{1}\) represent the autoregressive coefficients for profiles coded as 0 and 1, respectively.

Unique State

The unique state \(\mathrm{Unique\ State}_{k, i, t}\) captures item-specific deviations and is drawn from a multivariate normal distribution: $$ \mathrm{Unique\ State}_{i, t} \sim \mathcal{N} \left( 0, \boldsymbol{\theta} \right) $$ where \(\boldsymbol{\theta}\) is the item-level covariance matrix for the unique state component.

Combined State Variate

The state variate for item \(k\), individual \(i\), and time \(t\) combines the common and unique state components: $$ \mathrm{State}_{k, i, t} = \mathrm{Common\ State\ Loading}_{k} \times \mathrm{Common\ State}_{i, t} + \mathrm{Unique\ State}_{k, i, t} $$ The common state loading parameter \(\mathrm{Common\ State\ Loading}_{k}\) controls the influence of the shared state on each item.

Observed Variables

The observed variable is given by $$ Y_{k, i, t} = \mu_{k, c} + \mathrm{Trait}_{k, i} + \mathrm{State}_{k, i, t} $$ where \(\mu_{k, c}\) is the profile specific mean, while \(\mathrm{Trait}_{k, i}\) and \(\mathrm{State}_{k, i, t}\) correspond to the trait and state components of the model.

See also

Other Data Generation Functions: GenData()

Author

Ivan Jacob Agaloos Pesigan

Examples

# complete list of R function arguments -------------------------------------

# random seed for reproducibility
set.seed(42)

# dimensions
n <- 10 # number of individuals
m <- 6 # measurement occasions
p <- 4 # number of items
q <- 1 # common trait dimension

# covariate parameters
mu_x <- 11.4009
sigma_x <- 24.67566

# profile membership and transition parameters
nu_0 <- -3.563
kappa_0 <- 0.122
alpha_0 <- -3.586
beta_00 <- 2.250
gamma_00 <- 0.063
gamma_10 <- 0.094

# trait parameters
psi_t <- diag(1)
mu_t <- 0
psi_p <- diag(p)
mu_p <- rep(x = 0, times = p)
common_trait_loading <- matrix(
  data = 1,
  nrow = p,
  ncol = q
)

# state parameters
common_state_loading <- matrix(
  data = 1,
  nrow = p,
  ncol = 1
)
phi_0 <- 0.000
phi_1 <- 0.311
psi_s0 <- 0.151
psi_s <- 0.290
theta <- diag(p)

# profile-specific means
mu_profile <- cbind(
  c(2.253, 1.493, 1.574, 1.117),
  c(-0.278, -0.165, -0.199, -0.148)
)

# data generation -----------------------------------------------------------
data <- GenCULTA2Profiles(
  n = n,
  m = m,
  mu_x = mu_x,
  sigma_x = sigma_x,
  nu_0 = nu_0,
  kappa_0 = kappa_0,
  alpha_0 = alpha_0,
  beta_00 = beta_00,
  gamma_00 = gamma_00,
  gamma_10 = gamma_10,
  mu_t = mu_t,
  psi_t = psi_t,
  mu_p = mu_p,
  psi_p = psi_p,
  common_trait_loading = common_trait_loading,
  common_state_loading = common_state_loading,
  phi_0 = phi_0,
  phi_1 = phi_1,
  psi_s0 = psi_s0,
  psi_s = psi_s,
  theta = theta,
  mu_profile = mu_profile
)