Simulate Data from the Linear Growth Curve Model

Usage

SimSSMLinGrowth(
  n,
  time,
  mu0,
  sigma0_l,
  theta_l,
  type = 0,
  x = NULL,
  gamma = NULL,
  kappa = NULL
)

Arguments

n: Positive integer. Number of individuals.
time: Positive integer. Number of time points.
mu0: Numeric vector. A vector of length two. The first element is the mean of the intercept, and the second element is the mean of the slope.
sigma0_l: Numeric matrix. Cholesky factorization (t(chol(sigma0))) of the covariance matrix of the intercept and the slope.
theta_l: Numeric. Square root of the common measurement error variance.
type: Integer. State space model type. See Details for more information.
x: List. Each element of the list is a matrix of covariates for each individual i in n. The number of columns in each matrix should be equal to time.
gamma: Numeric matrix. Matrix linking the covariates to the latent variables at current time point ($\boldsymbol{\Gamma}$).
kappa: Numeric matrix. Matrix linking the covariates to the observed variables at current time point ($\boldsymbol{\kappa}$).

Value

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

call: Function call.
args: Function arguments.
data: Generated data which is a list of length n. Each element of data is a list with the following elements:
- id: A vector of ID numbers with length l, where l is the value of the function argument time.
- time: A vector time points of length l.
- y: A l by k matrix of values for the manifest variables.
- eta: A l by p matrix of values for the latent variables.
- x: A l by j matrix of values for the covariates (when covariates are included).
fun: Function used.

Details

Type 0

The measurement model is given by $$ Y_{i, t} = \left( \begin{array}{cc} 1 & 0 \\ \end{array} \right) \left( \begin{array}{c} \eta_{0_{i, t}} \\ \eta_{1_{i, t}} \\ \end{array} \right) + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( 0, \theta \right) $$ where $Y_{i, t}$, $\eta_{0_{i, t}}$, $\eta_{1_{i, t}}$, and $\boldsymbol{\varepsilon}_{i, t}$ are random variables and $\theta$ is a model parameter. $Y_{i, t}$ is the observed random variable at time $t$ and individual $i$, $\eta_{0_{i, t}}$ (intercept) and $\eta_{1_{i, t}}$ (slope) form a vector of latent random variables at time $t$ and individual $i$, and $\boldsymbol{\varepsilon}_{i, t}$ a vector of random measurement errors at time $t$ and individual $i$. $\theta$ is the variance of $\boldsymbol{\varepsilon}$.

The dynamic structure is given by $$ \left( \begin{array}{c} \eta_{0_{i, t}} \\ \eta_{1_{i, t}} \\ \end{array} \right) = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} \right) \left( \begin{array}{c} \eta_{0_{i, t - 1}} \\ \eta_{1_{i, t - 1}} \\ \end{array} \right) . $$

The mean vector and covariance matrix of the intercept and slope are captured in the mean vector and covariance matrix of the initial condition given by $$ \boldsymbol{\mu}_{\boldsymbol{\eta} \mid 0} = \left( \begin{array}{c} \mu_{\eta_{0}} \\ \mu_{\eta_{1}} \\ \end{array} \right) \quad \mathrm{and,} $$

$$ \boldsymbol{\Sigma}_{\boldsymbol{\eta} \mid 0} = \left( \begin{array}{cc} \sigma^{2}_{\eta_{0}} & \sigma_{\eta_{0}, \eta_{1}} \\ \sigma_{\eta_{1}, \eta_{0}} & \sigma^{2}_{\eta_{1}} \\ \end{array} \right) . $$

Type 1

The measurement model is given by $$ Y_{i, t} = \left( \begin{array}{cc} 1 & 0 \\ \end{array} \right) \left( \begin{array}{c} \eta_{0_{i, t}} \\ \eta_{1_{i, t}} \\ \end{array} \right) + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( 0, \theta \right) . $$

The dynamic structure is given by $$ \left( \begin{array}{c} \eta_{0_{i, t}} \\ \eta_{1_{i, t}} \\ \end{array} \right) = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} \right) \left( \begin{array}{c} \eta_{0_{i, t - 1}} \\ \eta_{1_{i, t - 1}} \\ \end{array} \right) + \boldsymbol{\Gamma} \mathbf{x}_{i, t} $$ where $\mathbf{x}_{i, t}$ represents a vector of covariates at time $t$ and individual $i$, and $\boldsymbol{\Gamma}$ the coefficient matrix linking the covariates to the latent variables.

Type 2

The measurement model is given by $$ Y_{i, t} = \left( \begin{array}{cc} 1 & 0 \\ \end{array} \right) \left( \begin{array}{c} \eta_{0_{i, t}} \\ \eta_{1_{i, t}} \\ \end{array} \right) + \boldsymbol{\kappa} \mathbf{x}_{i, t} + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( 0, \theta \right) $$ where $\boldsymbol{\kappa}$ represents the coefficient matrix linking the covariates to the observed variables.

The dynamic structure is given by $$ \left( \begin{array}{c} \eta_{0_{i, t}} \\ \eta_{1_{i, t}} \\ \end{array} \right) = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} \right) \left( \begin{array}{c} \eta_{0_{i, t - 1}} \\ \eta_{1_{i, t - 1}} \\ \end{array} \right) + \boldsymbol{\Gamma} \mathbf{x}_{i, t} . $$

References

Chow, S.-M., Ho, M. R., Hamaker, E. L., & Dolan, C. V. (2010). Equivalence and differences between structural equation modeling and state-space modeling techniques. Structural Equation Modeling: A Multidisciplinary Journal, 17(2), 303–332. doi:10.1080/10705511003661553

Author

Ivan Jacob Agaloos Pesigan

Examples

# prepare parameters
set.seed(42)
## number of individuals
n <- 5
## time points
time <- 5
## dynamic structure
p <- 2
mu0 <- c(0.615, 1.006)
sigma0 <- matrix(
  data = c(
    1.932,
    0.618,
    0.618,
    0.587
  ),
  nrow = p
)
sigma0_l <- t(chol(sigma0))
## measurement model
k <- 1
theta <- 0.50
theta_l <- sqrt(theta)
## covariates
j <- 2
x <- lapply(
  X = seq_len(n),
  FUN = function(i) {
    return(
      matrix(
        data = rnorm(n = j * time),
        nrow = j
      )
    )
  }
)
gamma <- diag(x = 0.10, nrow = p, ncol = j)
kappa <- diag(x = 0.10, nrow = k, ncol = j)

# Type 0
ssm <- SimSSMLinGrowth(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  theta_l = theta_l,
  type = 0
)

plot(ssm)


# Type 1
ssm <- SimSSMLinGrowth(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  theta_l = theta_l,
  type = 1,
  x = x,
  gamma = gamma
)

plot(ssm)


# Type 2
ssm <- SimSSMLinGrowth(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  theta_l = theta_l,
  type = 2,
  x = x,
  gamma = gamma,
  kappa = kappa
)

plot(ssm)