The Linear Growth Curve Model

Model

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

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 $\begin{equation} \boldsymbol{\mu}_{\boldsymbol{\eta} \mid 0} = \left( \begin{array}{c} \mu_{\eta_{0}} \\ \mu_{\eta_{1}} \\ \end{array} \right) \quad \mathrm{and,} \end{equation}$

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

Data Generation

Notation

Let $t = 5$ be the number of time points and $n = 10<sup>4</sup>$ be the number of individuals.

Let the measurement error variance $\theta$ be given by

$\begin{equation} \theta = 0.6 . \end{equation}$

Let the initial condition $\boldsymbol{\eta}_{0}$ be given by

$\begin{equation} \boldsymbol{\eta}_{0} \sim \mathcal{N} \left( \boldsymbol{\mu}_{\boldsymbol{\eta} \mid 0}, \boldsymbol{\Sigma}_{\boldsymbol{\eta} \mid 0} \right) \end{equation}$

$\begin{equation} \boldsymbol{\mu}_{\boldsymbol{\eta} \mid 0} = \left( \begin{array}{c} 0.615 \\ 1.006 \\ \end{array} \right) \end{equation}$

$\begin{equation} \boldsymbol{\Sigma}_{\boldsymbol{\eta} \mid 0} = \left( \begin{array}{ccc} 1.932 & 0.618 \\ 0.618 & 0.587 \\ \end{array} \right) . \end{equation}$

R Function Arguments

n
#> [1] 10000
time
#> [1] 5
mu0
#> [1] 0.615 1.006
sigma0
#>       [,1]  [,2]
#> [1,] 1.932 0.618
#> [2,] 0.618 0.587
sigma0_l
#>           [,1]      [,2]
#> [1,] 1.3899640 0.0000000
#> [2,] 0.4446158 0.6239525
theta
#> [1] 0.6
theta_l
#> [1] 0.7745967

Visualizing the Dynamics Without Measurement Error (n = 5 with Different Initial Condition)

Using the `SimSSMLinGrowth` Function from the `simStateSpace` Package to Simulate Data

library(simStateSpace)
sim <- SimSSMLinGrowth(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  theta_l = theta_l,
  type = 0
)
data <- as.data.frame(sim)
head(data)
#>   id time       y1
#> 1  1    0 1.723564
#> 2  1    1 2.148702
#> 3  1    2 3.384974
#> 4  1    3 6.225426
#> 5  1    4 5.993965
#> 6  2    0 1.951861
summary(data)
#>        id             time         y1          
#>  Min.   :    1   Min.   :0   Min.   :-10.7863  
#>  1st Qu.: 2501   1st Qu.:1   1st Qu.:  0.3943  
#>  Median : 5000   Median :2   Median :  2.1839  
#>  Mean   : 5000   Mean   :2   Mean   :  2.5978  
#>  3rd Qu.: 7500   3rd Qu.:3   3rd Qu.:  4.4536  
#>  Max.   :10000   Max.   :4   Max.   : 21.0250
plot(sim)

Model Fitting

Prepare Data

dynr_data <- dynr::dynr.data(
  dataframe = data,
  id = "id",
  time = "time",
  observed = "y1"
)

Prepare Initial Condition

dynr_initial <- dynr::prep.initial(
  values.inistate = mu0,
  params.inistate = c("mu0_0", "mu0_1"),
  values.inicov = sigma0,
  params.inicov = matrix(
    data = c(
      "sigma0_00", "sigma0_01",
      "sigma0_01", "sigma0_11"
    ),
    nrow = 2
  )
)

Prepare Measurement Model

dynr_measurement <- dynr::prep.measurement(
  values.load = matrix(data = c(1, 0), nrow = 1),
  params.load = matrix(data = "fixed", nrow = 1, ncol = 2),
  state.names = c("eta_0", "eta_1"),
  obs.names = "y1"
)

Prepare Dynamic Process

dynr_dynamics <- dynr::prep.formulaDynamics(
  formula = list(
    eta_0 ~ eta_0 + eta_1,
    eta_1 ~ eta_1
  ),
  isContinuousTime = FALSE
)
#> Warning in dynr::prep.formulaDynamics(formula = list(eta_0 ~ eta_0 + eta_1, :
#> You provided no start values: length(startval)==0. If you have no free
#> parameters, keep calm and carry on.

Prepare Process Noise

dynr_noise <- dynr::prep.noise(
  values.latent = matrix(data = 0, nrow = 2, ncol = 2),
  params.latent = matrix(data = "fixed", nrow = 2, ncol = 2),
  values.observed = theta,
  params.observed = "theta"
)

Prepare the Model

model <- dynr::dynr.model(
  data = dynr_data,
  initial = dynr_initial,
  measurement = dynr_measurement,
  dynamics = dynr_dynamics,
  noise = dynr_noise,
  outfile = "lin-growth.c"
)

Fit the Model

results <- dynr::dynr.cook(
  model,
  debug_flag = TRUE,
  verbose = FALSE
)
#> [1] "Get ready!!!!"
#> using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’
#> Optimization function called.
#> Starting Hessian calculation ...
#> Finished Hessian calculation.
#> Original exit flag:  3 
#> Modified exit flag:  3 
#> Optimization terminated successfully: ftol_rel or ftol_abs was reached. 
#> Original fitted parameters:  -0.5089635 0.60732 0.9952181 0.6477021 0.3152438 
#> -0.9463643 
#> 
#> Transformed fitted parameters:  0.6011183 0.60732 0.9952181 1.911144 0.6024764 
#> 0.5780766 
#> 
#> Doing end processing
#> Successful trial
#> Total Time: 1.115847 
#> Backend Time: 1.108694

Summary

summary(results)
#> Coefficients:
#>           Estimate Std. Error t value ci.lower ci.upper Pr(>|t|)    
#> theta     0.601118   0.004914  122.34 0.591488 0.610749   <2e-16 ***
#> mu0_0     0.607320   0.015112   40.19 0.577701 0.636939   <2e-16 ***
#> mu0_1     0.995218   0.007992  124.53 0.979554 1.010882   <2e-16 ***
#> sigma0_00 1.911144   0.032346   59.08 1.847747 1.974541   <2e-16 ***
#> sigma0_01 0.602476   0.013016   46.29 0.576966 0.627987   <2e-16 ***
#> sigma0_11 0.578077   0.009046   63.90 0.560347 0.595807   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> -2 log-likelihood value at convergence = 167712.56
#> AIC = 167724.56
#> BIC = 167777.48

#> [1] 0.6073200 0.9952181

Estimated Parameters

mu0_hat
#> [1] 0.6073200 0.9952181
sigma0_hat
#>           [,1]      [,2]
#> [1,] 1.9111442 0.6024764
#> [2,] 0.6024764 0.5780766
theta_hat
#>     theta 
#> 0.6011183

Linear Growth Curve Model in Structural Equation Modeling

Wide Data Set

data <- as.data.frame(sim, long = FALSE)
head(data)
#>   id        y1_0       y1_1        y1_2        y1_3       y1_4
#> 1  1  1.72356430  2.1487019  3.38497364  6.22542609  5.9939649
#> 2  2  1.95186051  2.6140623  5.60107919  5.59791920  7.6881040
#> 3  3  0.07909854 -1.1410633 -0.02940765  0.07103948 -0.4998955
#> 4  4 -0.14827866 -0.6478676 -0.33440620 -0.29683127 -0.6950588
#> 5  5  1.14625267  2.3844799  0.39101270 -0.76159166  1.6750324
#> 6  6  1.02604168  2.1852479  2.55168739  2.13223414  4.1499727

Model

model <- "
  # factor loadings
  eta0 =~ 1 * y1_0 + 1 * y1_1 + 1 * y1_2 + 1 * y1_3 + 1 * y1_4
  eta1 =~ 0 * y1_0 + 1 * y1_1 + 2 * y1_2 + 3 * y1_3 + 4 * y1_4
  # means of latent variables
  eta0 ~ mu0_0 * 1
  eta1 ~ mu0_1 * 1
  # variances and covariances of latent variables
  eta0 ~~ sigma0_00 * eta0
  eta0 ~~ sigma0_01 * eta1
  eta1 ~~ sigma0_11 * eta1
  # constrain error variance theta to be equal
  y1_0 ~~ theta * y1_0
  y1_1 ~~ theta * y1_1
  y1_2 ~~ theta * y1_2
  y1_3 ~~ theta * y1_3
  y1_4 ~~ theta * y1_4
"

fit <- lavaan::growth(
  model = model,
  data = data
)
lavaan::summary(fit)
#> lavaan 0.6-19 ended normally after 34 iterations
#> 
#>   Estimator                                         ML
#>   Optimization method                           NLMINB
#>   Number of model parameters                        10
#>   Number of equality constraints                     4
#> 
#>   Number of observations                         10000
#> 
#> Model Test User Model:
#>                                                       
#>   Test statistic                                 5.386
#>   Degrees of freedom                                14
#>   P-value (Chi-square)                           0.980
#> 
#> Parameter Estimates:
#> 
#>   Standard errors                             Standard
#>   Information                                 Expected
#>   Information saturated (h1) model          Structured
#> 
#> Latent Variables:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>   eta0 =~                                             
#>     y1_0              1.000                           
#>     y1_1              1.000                           
#>     y1_2              1.000                           
#>     y1_3              1.000                           
#>     y1_4              1.000                           
#>   eta1 =~                                             
#>     y1_0              0.000                           
#>     y1_1              1.000                           
#>     y1_2              2.000                           
#>     y1_3              3.000                           
#>     y1_4              4.000                           
#> 
#> Covariances:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>   eta0 ~~                                             
#>     eta1    (s0_0)    0.602    0.013   46.317    0.000
#> 
#> Intercepts:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>     eta0    (m0_0)    0.607    0.015   40.295    0.000
#>     eta1    (m0_1)    0.995    0.008  124.582    0.000
#> 
#> Variances:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>     eta0    (s0_0)    1.911    0.032   59.235    0.000
#>     eta1    (s0_1)    0.578    0.009   63.955    0.000
#>    .y1_0    (thet)    0.601    0.005  122.474    0.000
#>    .y1_1    (thet)    0.601    0.005  122.474    0.000
#>    .y1_2    (thet)    0.601    0.005  122.474    0.000
#>    .y1_3    (thet)    0.601    0.005  122.474    0.000
#>    .y1_4    (thet)    0.601    0.005  122.474    0.000

#> [1] 0.6073200 0.9952181

Estimated Parameters

mu0_hat
#> [1] 0.6073200 0.9952181
sigma0_hat
#>           [,1]      [,2]
#> [1,] 1.9111442 0.6024764
#> [2,] 0.6024764 0.5780766
theta_hat
#> [1] 0.6011183

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. https://doi.org/10.1080/10705511003661553

Ou, L., Hunter, M. D., & Chow, S.-M. (2019). What’s for dynr: A package for linear and nonlinear dynamic modeling in R. The R Journal, 11(1), 91. https://doi.org/10.32614/rj-2019-012

R Core Team. (2024). R: A language and environment for statistical computing. R Foundation for Statistical Computing. https://www.R-project.org/

Rosseel, Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48(2). https://doi.org/10.18637/jss.v048.i02

Ivan Jacob Agaloos Pesigan

2024-12-19

Model

Data Generation

Notation

R Function Arguments

Visualizing the Dynamics Without Measurement Error (n = 5 with Different Initial Condition)

Using the `SimSSMLinGrowth` Function from the `simStateSpace` Package to Simulate Data

Model Fitting

Prepare Data

Prepare Initial Condition

Prepare Measurement Model

Prepare Dynamic Process

Prepare Process Noise

Prepare the Model

Fit the Model

Summary

Estimated Parameters

Linear Growth Curve Model in Structural Equation Modeling

Wide Data Set

Model

Estimated Parameters

References

The Linear Growth Curve Model

Ivan Jacob Agaloos Pesigan

2024-12-19

Model

Data Generation

Notation

R Function Arguments

Visualizing the Dynamics Without Measurement Error (n = 5 with Different Initial Condition)

Using the SimSSMLinGrowth Function from the simStateSpace Package to Simulate Data

Model Fitting

Prepare Data

Prepare Initial Condition

Prepare Measurement Model

Prepare Dynamic Process

Prepare Process Noise

Prepare the Model

Fit the Model

Summary

Estimated Parameters

Linear Growth Curve Model in Structural Equation Modeling

Wide Data Set

Model

Estimated Parameters

References

Using the `SimSSMLinGrowth` Function from the `simStateSpace` Package to Simulate Data