This package offers some helper functions to specify and analyse univariate and bivariate latent change score models (LCSM) using lavaan (Rosseel, 2012). For details about this method see for example McArdle (2009), Ghisletta (2012), Grimm et al. (2012), and Grimm, Ram & Estabrook (2017).

The lcsm package combines the strengths of other R packages like lavaan, broom, and semPlot by generating lavaan syntax that helps these packages work together. This is work in progress and feedback is very welcome!

Installation

You can install the released version of lcsm from CRAN with:

The development version can be installed from GitHub using:

# install.packages("devtools")
devtools::install_github("milanwiedemann/lcsm")

Overview of the functions

The lcsm package contains the functions listed below. A more detailed description of these functions is available in this README or vignettes. The interactive online application shinychange also illustrates some functions of this package.

How to use lcsm

Here are a few examples how to use the lcsm package.

# Load the package
library(lcsm)
#> 
#> ── This is lcsm 0.1.4 ──────────────────────────────────────────────────────────
#> ℹ Please report any issues or ideas at:
#> ℹ https://github.com/milanwiedemann/lcsm/issues

Visualise data

Longitudinal data can be visualised using the plot_trajectories() function. Here only 30% of the data is visualised using the argument random_sample_frac = 0.3. Only consecutive measures are connected by lines as specified in connect_missing = FALSE.

# Create plot for construct x
plot_x <- plot_trajectories(data = data_bi_lcsm,
                            id_var = "id", 
                            var_list = c("x1", "x2", "x3", "x4", "x5", 
                                         "x6", "x7", "x8", "x9", "x10"),
                            xlab = "Time", ylab = "X Score",
                            connect_missing = FALSE, 
                            random_sample_frac = 0.3)

# Create plot for construct y
plot_y <- plot_trajectories(data = data_bi_lcsm,
                            id_var = "id", 
                            var_list = c("y1", "y2", "y3", "y4", "y5", 
                                        "y6", "y7", "y8", "y9", "y10"),
                            xlab = "Time", ylab = "Y Score",
                            connect_missing = FALSE, 
                            random_sample_frac = 0.3)

# Arrange plots next to each other using patchwork
library(patchwork)
plot_x + plot_y + plot_annotation(tag_levels = 'A')
#> Warning: Removed 18 row(s) containing missing values (geom_path).
#> Warning: Removed 85 rows containing missing values (geom_point).
#> Warning: Removed 37 row(s) containing missing values (geom_path).
#> Warning: Removed 172 rows containing missing values (geom_point).

Fit LCSMs

The functions fit_uni_lcsm() and fit_bi_lcsm() can be used to fit univariate and bivariate LCSM with different model specifications. In a first step, these two function generate the user specified lavaan syntax by calling the specify_uni_lcsm() or specify_bi_lcsm() functions.

The following table describes some of the different model specifications that the model arguments can take. More detail can be found in the help files help(fit_uni_lcsm).

Fit univariate LCSMs

Model specification Description
alpha_constant Constant change factor
beta Proportional change factor
phi Autoregression of change scores

The example below shows how to specify a generic univariate latent change score model using the function specify_uni_lcsm(). A table of the description of all parameters that can be estimated is shown here.

specify_uni_lcsm(timepoints = 5,
                 var = "x",  
                 change_letter = "g",
                 model = list(alpha_constant = TRUE, 
                              beta = TRUE, 
                              phi = TRUE))
Click here to see the lavaan syntax specified above.

# Specify latent true scores 
lx1 =~ 1 * x1 
lx2 =~ 1 * x2 
lx3 =~ 1 * x3 
lx4 =~ 1 * x4 
lx5 =~ 1 * x5 
# Specify mean of latent true scores 
lx1 ~ gamma_lx1 * 1 
lx2 ~ 0 * 1 
lx3 ~ 0 * 1 
lx4 ~ 0 * 1 
lx5 ~ 0 * 1 
# Specify variance of latent true scores 
lx1 ~~ sigma2_lx1 * lx1 
lx2 ~~ 0 * lx2 
lx3 ~~ 0 * lx3 
lx4 ~~ 0 * lx4 
lx5 ~~ 0 * lx5 
# Specify intercept of obseved scores 
x1 ~ 0 * 1 
x2 ~ 0 * 1 
x3 ~ 0 * 1 
x4 ~ 0 * 1 
x5 ~ 0 * 1 
# Specify variance of observed scores 
x1 ~~ sigma2_ux * x1 
x2 ~~ sigma2_ux * x2 
x3 ~~ sigma2_ux * x3 
x4 ~~ sigma2_ux * x4 
x5 ~~ sigma2_ux * x5 
# Specify autoregressions of latent variables 
lx2 ~ 1 * lx1 
lx3 ~ 1 * lx2 
lx4 ~ 1 * lx3 
lx5 ~ 1 * lx4 
# Specify latent change scores 
dx2 =~ 1 * lx2 
dx3 =~ 1 * lx3 
dx4 =~ 1 * lx4 
dx5 =~ 1 * lx5 
# Specify latent change scores means 
dx2 ~ 0 * 1 
dx3 ~ 0 * 1 
dx4 ~ 0 * 1 
dx5 ~ 0 * 1 
# Specify latent change scores variances 
dx2 ~~ 0 * dx2 
dx3 ~~ 0 * dx3 
dx4 ~~ 0 * dx4 
dx5 ~~ 0 * dx5 
# Specify constant change factor 
g2 =~ 1 * dx2 + 1 * dx3 + 1 * dx4 + 1 * dx5 
# Specify constant change factor mean 
g2 ~ alpha_g2 * 1 
# Specify constant change factor variance 
g2 ~~ sigma2_g2 * g2 
# Specify constant change factor covariance with the initial true score 
g2 ~~ sigma_g2lx1 * lx1
# Specify proportional change component 
dx2 ~ beta_x * lx1 
dx3 ~ beta_x * lx2 
dx4 ~ beta_x * lx3 
dx5 ~ beta_x * lx4 
# Specify autoregression of change score 
dx3 ~ phi_x * dx2 
dx4 ~ phi_x * dx3 
dx5 ~ phi_x * dx4 

The function fit_uni_lcsm() can be used to fit a univariate LCSM using the sample data set data_uni_lcsm. This functions first writes the lavaan syntax specified in the model argument and passes it on to lavaaan::lavaan().

# Fit univariate latent change score model
fit_uni_lcsm(data = data_uni_lcsm, 
             var =  c("x1", "x2", "x3", "x4", "x5",
                      "x6", "x7", "x8", "x9", "x10"),
             model = list(alpha_constant = TRUE, 
                          beta = FALSE, 
                          phi = TRUE))
#> lavaan 0.6-8 ended normally after 66 iterations
#> 
#>   Estimator                                         ML
#>   Optimization method                           NLMINB
#>   Number of model parameters                        23
#>   Number of equality constraints                    16
#>                                                       
#>   Number of observations                           500
#>   Number of missing patterns                       273
#>                                                       
#> Model Test User Model:
#>                                                Standard      Robust
#>   Test Statistic                                 75.389      74.400
#>   Degrees of freedom                                 58          58
#>   P-value (Chi-square)                            0.062       0.072
#>   Scaling correction factor                                   1.013
#>        Yuan-Bentler correction (Mplus variant)

It is also possible to show the lavaan syntax that was created to fit the model by the function specify_uni_lcsm(). The lavaan syntax includes comments describing some parts of the syntax in more detail. To save the syntax in an object the argument return_lavaan_syntax has to be set to TRUE. This object looks a bit funny, it’s one very long line of text, but can be formatted to look more beautiful and readable using cat(syntax).

# Fit univariate latent change score model
syntax <- fit_uni_lcsm(data = data_uni_lcsm, 
                       var =  c("x1", "x2", "x3", "x4", "x5",
                                "x6", "x7", "x8", "x9", "x10"),
                       model = list(alpha_constant = TRUE, 
                                    beta = FALSE, 
                                    phi = TRUE),
                      return_lavaan_syntax = TRUE)

# Return lavaan syntax in easy to read format
cat(syntax)
Click here to see the lavaan syntax specified in syntax.

# Specify latent true scores 
lx1 =~ 1 * x1 
lx2 =~ 1 * x2 
lx3 =~ 1 * x3 
lx4 =~ 1 * x4 
lx5 =~ 1 * x5 
lx6 =~ 1 * x6 
lx7 =~ 1 * x7 
lx8 =~ 1 * x8 
lx9 =~ 1 * x9 
lx10 =~ 1 * x10 
# Specify mean of latent true scores 
lx1 ~ gamma_lx1 * 1 
lx2 ~ 0 * 1 
lx3 ~ 0 * 1 
lx4 ~ 0 * 1 
lx5 ~ 0 * 1 
lx6 ~ 0 * 1 
lx7 ~ 0 * 1 
lx8 ~ 0 * 1 
lx9 ~ 0 * 1 
lx10 ~ 0 * 1 
# Specify variance of latent true scores 
lx1 ~~ sigma2_lx1 * lx1 
lx2 ~~ 0 * lx2 
lx3 ~~ 0 * lx3 
lx4 ~~ 0 * lx4 
lx5 ~~ 0 * lx5 
lx6 ~~ 0 * lx6 
lx7 ~~ 0 * lx7 
lx8 ~~ 0 * lx8 
lx9 ~~ 0 * lx9 
lx10 ~~ 0 * lx10 
# Specify intercept of obseved scores 
x1 ~ 0 * 1 
x2 ~ 0 * 1 
x3 ~ 0 * 1 
x4 ~ 0 * 1 
x5 ~ 0 * 1 
x6 ~ 0 * 1 
x7 ~ 0 * 1 
x8 ~ 0 * 1 
x9 ~ 0 * 1 
x10 ~ 0 * 1 
# Specify variance of observed scores 
x1 ~~ sigma2_ux * x1 
x2 ~~ sigma2_ux * x2 
x3 ~~ sigma2_ux * x3 
x4 ~~ sigma2_ux * x4 
x5 ~~ sigma2_ux * x5 
x6 ~~ sigma2_ux * x6 
x7 ~~ sigma2_ux * x7 
x8 ~~ sigma2_ux * x8 
x9 ~~ sigma2_ux * x9 
x10 ~~ sigma2_ux * x10 
# Specify autoregressions of latent variables 
lx2 ~ 1 * lx1 
lx3 ~ 1 * lx2 
lx4 ~ 1 * lx3 
lx5 ~ 1 * lx4 
lx6 ~ 1 * lx5 
lx7 ~ 1 * lx6 
lx8 ~ 1 * lx7 
lx9 ~ 1 * lx8 
lx10 ~ 1 * lx9 
# Specify latent change scores 
dx2 =~ 1 * lx2 
dx3 =~ 1 * lx3 
dx4 =~ 1 * lx4 
dx5 =~ 1 * lx5 
dx6 =~ 1 * lx6 
dx7 =~ 1 * lx7 
dx8 =~ 1 * lx8 
dx9 =~ 1 * lx9 
dx10 =~ 1 * lx10 
# Specify latent change scores means 
dx2 ~ 0 * 1 
dx3 ~ 0 * 1 
dx4 ~ 0 * 1 
dx5 ~ 0 * 1 
dx6 ~ 0 * 1 
dx7 ~ 0 * 1 
dx8 ~ 0 * 1 
dx9 ~ 0 * 1 
dx10 ~ 0 * 1 
# Specify latent change scores variances 
dx2 ~~ 0 * dx2 
dx3 ~~ 0 * dx3 
dx4 ~~ 0 * dx4 
dx5 ~~ 0 * dx5 
dx6 ~~ 0 * dx6 
dx7 ~~ 0 * dx7 
dx8 ~~ 0 * dx8 
dx9 ~~ 0 * dx9 
dx10 ~~ 0 * dx10 
# Specify constant change factor 
g2 =~ 1 * dx2 + 1 * dx3 + 1 * dx4 + 1 * dx5 + 1 * dx6 + 1 * dx7 + 1 * dx8 + 1 * dx9 + 1 * dx10 
# Specify constant change factor mean 
g2 ~ alpha_g2 * 1 
# Specify constant change factor variance 
g2 ~~ sigma2_g2 * g2 
# Specify constant change factor covariance with the initial true score 
g2 ~~ sigma_g2lx1 * lx1
# Specify autoregression of change score 
dx3 ~ phi_x * dx2 
dx4 ~ phi_x * dx3 
dx5 ~ phi_x * dx4 
dx6 ~ phi_x * dx5 
dx7 ~ phi_x * dx6 
dx8 ~ phi_x * dx7 
dx9 ~ phi_x * dx8 
dx10 ~ phi_x * dx9 

Fit bivariate LCSMs

The function fit_bi_lcsm() allows to specify two univariate LCSMs using the arguments model_x and model_x. These two constructs can then be connected using the coupling argument. More details can be found in the help files help(fit_bi_lcsm).

Coupling specification Description
coupling_piecewise Piecewise coupling parameters
coupling_piecewise_num Changepoint of piecewise coupling parameters
delta_con_xy Change score x (t) determined by true score y (t)
delta_con_yx Change score y (t) determined by true score x (t)
delta_lag_xy Change score x (t) determined by true score y (t-1)
delta_lag_yx Change score y (t) determined by true score x (t-1)
xi_con_xy Change score x (t) determined by change score y (t)
xi_con_yx Change score y (t) determined by change score x (t)
xi_lag_xy Change score x (t) determined by change score y (t-1)
xi_lag_yx Change score y (t) determined by change score x (t-1)
fit_bi_lcsm(data = data_bi_lcsm, 
            var_x = c("x1", "x2", "x3", "x4", "x5",
                      "x6", "x7", "x8", "x9", "x10"),
            var_y = c("y1", "y2", "y3", "y4", "y5", 
                      "y6", "y7", "y8", "y9", "y10"),
            model_x = list(alpha_constant = TRUE, 
                           beta = TRUE, 
                           phi = FALSE),
            model_y = list(alpha_constant = TRUE, 
                           beta = TRUE, 
                           phi = TRUE),
            coupling = list(delta_lag_xy = TRUE, 
                            xi_lag_yx = TRUE))
#> lavaan 0.6-8 ended normally after 116 iterations
#> 
#>   Estimator                                         ML
#>   Optimization method                           NLMINB
#>   Number of model parameters                        87
#>   Number of equality constraints                    65
#>                                                       
#>   Number of observations                           500
#>   Number of missing patterns                       210
#>                                                       
#> Model Test User Model:
#>                                                Standard      Robust
#>   Test Statistic                                191.851     193.021
#>   Degrees of freedom                                208         208
#>   P-value (Chi-square)                            0.782       0.764
#>   Scaling correction factor                                   0.994
#>        Yuan-Bentler correction (Mplus variant)

Extract fit statistics and parmeters

The main underlying functions to extract parameters and fit statistics come from the broom package: broom::tidy() and broom::glance(). The functions extract_param() and extract_fit() offer some tools that I find helpful when running LCSMs in R, for example:

A table of the description of all parameters that can be estimated is shown here.

# First create a lavaan object
bi_lcsm_01 <- fit_bi_lcsm(data = data_bi_lcsm, 
                          var_x = c("x1", "x2", "x3", "x4", "x5",
                                    "x6", "x7", "x8", "x9", "x10"),
                          var_y = c("y1", "y2", "y3", "y4", "y5", 
                                    "y6", "y7", "y8", "y9", "y10"),
                          model_x = list(alpha_constant = TRUE, 
                                         beta = TRUE, 
                                         phi = FALSE),
                          model_y = list(alpha_constant = TRUE, 
                                         beta = TRUE, 
                                         phi = TRUE),
                          coupling = list(delta_lag_xy = TRUE, 
                                          xi_lag_yx = TRUE))

# Now extract parameter estimates
# Only extract first 7 columns for this example by adding [ , 1:7]
param_bi_lcsm_01 <- extract_param(bi_lcsm_01, printp = TRUE)[ , 1:7]

# Print table of parameter estimates
kable(param_bi_lcsm_01, digits = 3)
label estimate std.error statistic p.value std.lv std.all
gamma_lx1 21.066 0.036 588.187 < .001 30.014 30.014
sigma2_lx1 0.493 0.037 13.485 < .001 1.000 1.000
sigma2_ux 0.201 0.004 45.301 < .001 0.201 0.290
alpha_g2 -0.309 0.053 -5.834 < .001 -0.492 -0.492
sigma2_g2 0.395 0.028 14.330 < .001 1.000 1.000
sigma_g2lx1 0.155 0.022 7.017 < .001 0.351 0.351
beta_x -0.106 0.003 -30.818 < .001 -0.120 -0.120
gamma_ly1 5.025 0.029 172.786 < .001 11.009 11.009
sigma2_ly1 0.208 0.019 10.860 < .001 1.000 1.000
sigma2_uy 0.193 0.005 39.698 < .001 0.193 0.481
alpha_j2 -0.203 0.039 -5.217 < .001 -0.666 -0.666
sigma2_j2 0.093 0.008 11.766 < .001 1.000 1.000
sigma_j2ly1 0.017 0.008 2.156 .031 0.122 0.122
beta_y -0.197 0.005 -39.562 < .001 -0.293 -0.293
phi_y 0.144 0.029 4.963 < .001 0.126 0.126
sigma_su 0.009 0.003 2.581 .01 0.009 0.044
sigma_ly1lx1 0.185 0.021 8.905 < .001 0.577 0.577
sigma_g2ly1 0.072 0.016 4.437 < .001 0.251 0.251
sigma_j2lx1 0.093 0.012 7.916 < .001 0.437 0.437
sigma_j2g2 0.005 0.012 0.463 .643 0.029 0.029
delta_lag_xy 0.140 0.006 23.837 < .001 0.103 0.103
xi_lag_yx 0.360 0.037 9.634 < .001 0.640 0.640

Plot simplified path diagrams of LCSMs

This function is work in progress and can only plot univariate and bivariate LCSMs that were specified with fit_uni_lcsm() or fit_bi_lcsm(). Modified LCSMs will probably return errors as the layout matrix that gets created by this plot function only supports the specifications that can be modelled with this package. The input arguments for plotting a simplified path diagram are:

  • the estimated lavaan object lavaan_object,
  • the lavaan_syntax and ,
  • lcsm indicating whether the LCSMs is “univariate” or “bivariate”

Optional arguments can be used to change the look of the plot, for example:

  • lcsm_colours can be used to highlight the different parts of the latent change score model
    • white: observed scores
    • green: latent true scores
    • blue: latent change scores
    • yellow: latent change scores

Further arguments can be passed on to semPlot::semPaths(), for example:

  • what, “path” to show unweighted gray edges, “par” to show parameter estimates as weighted (green/red) edges
  • whatLabels, “label” to show edege names as label or “est” for parameter estimates, “hide” to hide edge labels

Univariate LCSM

# Fit bivariate lcsm and save the results 
uni_lavaan_results <- fit_uni_lcsm(data = data_uni_lcsm, 
                                   var = c("x1", "x2", "x3", "x4", "x5"),
                                   model = list(alpha_constant = TRUE, 
                                                beta = TRUE, 
                                                phi = TRUE)
                                  )
#> Warning in lav_data_full(data = data, group = group, cluster = cluster, : lavaan WARNING: some cases are empty and will be ignored:
#>   239

# Save the lavaan syntax that is used to create the layout matrix for semPlot
uni_lavaan_syntax <- fit_uni_lcsm(data = data_uni_lcsm, 
                                  var = c("x1", "x2", "x3", "x4", "x5"),
                                  model = list(alpha_constant = TRUE, 
                                               beta = TRUE, 
                                               phi = TRUE),
                                  return_lavaan_syntax = TRUE)

# Plot the results
plot_lcsm(lavaan_object = uni_lavaan_results,
          lavaan_syntax = uni_lavaan_syntax,
          edge.label.cex = .9,  
          lcsm_colours = TRUE,
          lcsm = "univariate")

Bivariate LCSM

# Fit bivariate lcsm and save the results 
bi_lavaan_results <- fit_bi_lcsm(data = data_bi_lcsm, 
                                 var_x = c("x1", "x2", "x3", "x4", "x5"),
                                 var_y = c("y1", "y2", "y3", "y4", "y5"),
                                 model_x = list(alpha_constant = TRUE, 
                                                beta = TRUE, 
                                                phi = FALSE),
                                 model_y = list(alpha_constant = TRUE, 
                                                beta = TRUE, 
                                                phi = TRUE),
                                 coupling = list(delta_lag_xy = TRUE, 
                                                 xi_lag_yx = TRUE))

# Save the lavaan syntax that is used to create the layout matrix for semPlot
bi_lavaan_syntax <- fit_bi_lcsm(data = data_bi_lcsm, 
                                var_x = c("x1", "x2", "x3", "x4", "x5"),
                                var_y = c("y1", "y2", "y3", "y4", "y5"),
                                model_x = list(alpha_constant = TRUE, 
                                               beta = TRUE, 
                                               phi = FALSE),
                                model_y = list(alpha_constant = TRUE, 
                                               beta = TRUE, 
                                               phi = TRUE),
                                coupling = list(delta_lag_xy = TRUE, 
                                                xi_lag_yx = TRUE),
                                return_lavaan_syntax = TRUE)

# Plot the results
plot_lcsm(lavaan_object = bi_lavaan_results, 
          lavaan_syntax = bi_lavaan_syntax,
          lcsm_colours = TRUE,
          whatLabels = "hide",
          lcsm = "bivariate")

Simulate data

The functions sim_uni_lcsm() and sim_bi_lcsm() simulate data based on some some parameters that can be specified. See the tables here for a full list of parameters that can be specified for the data simulation.

# Simulate some data 
sim_uni_lcsm(timepoints = 5, 
             model = list(alpha_constant = TRUE, beta = FALSE, phi = TRUE), 
             model_param = list(gamma_lx1 = 21, 
                                sigma2_lx1 = 1.5,
                                sigma2_ux = 0.2,
                                alpha_j2 = -0.93,
                                sigma2_j2 = 0.1,
                                sigma_j2lx1 = 0.2,
                                phi_x = 0.3),
             sample.nobs = 1000,
             na_pct = 0.3)
#> Parameter estimates for the data simulation are taken from the argument 'model_param'.
#> Warning: The following parameters are specified in LCS model but no parameter estimates have been entered in 'model_param':
#> -  alpha_g2
#> -  sigma2_g2
#> -  sigma_g2lx1
#> # A tibble: 1,000 x 6
#>       id    x1    x2    x3    x4    x5
#>    <int> <dbl> <dbl> <dbl> <dbl> <dbl>
#>  1     1  21.8  20.4  18.0  15.3  NA  
#>  2     2  22.5  22.7  NA    19.0  NA  
#>  3     3  22.3  21.4  20.5  18.8  18.5
#>  4     4  NA    NA    21.9  25.0  25.9
#>  5     5  18.8  18.4  18.9  18.8  NA  
#>  6     6  20.0  19.1  17.1  15.5  13.8
#>  7     7  20.1  18.6  16.9  NA    NA  
#>  8     8  22.3  23.2  22.6  23.6  25.3
#>  9     9  18.7  NA    19.3  19.5  19.8
#> 10    10  NA    22.3  NA    22.2  22.4
#> # … with 990 more rows

It is also possible to return the lavaan syntax instead of simulating data for further manual specifications. The modified object could then be used to simulate data using lavaan::simulateData().

# Return lavaan syntax based on the following argument specifications
simsyntax <- sim_bi_lcsm(timepoints = 5, 
                         model_x = list(alpha_constant = TRUE, beta = TRUE, phi = FALSE),
                         model_x_param = list(gamma_lx1 = 21,
                                              sigma2_lx1 = .5,
                                              sigma2_ux = .2,
                                              alpha_g2 = -.4,
                                              sigma2_g2 = .4,
                                              sigma_g2lx1 = .2,
                                              beta_x = -.1),
                         model_y = list(alpha_constant = TRUE, beta = TRUE, phi = TRUE),
                         model_y_param = list(gamma_ly1 = 5,
                                              sigma2_ly1 = .2,
                                              sigma2_uy = .2,
                                              alpha_j2 = -.2,
                                              sigma2_j2 = .1,
                                              sigma_j2ly1 = .02,
                                              beta_y = -.2,
                                              phi_y = .1),
                         coupling = list(delta_lag_xy = TRUE, 
                                         xi_lag_yx = TRUE),
                         coupling_param = list(sigma_su = .01,
                                               sigma_ly1lx1 = .2,
                                               sigma_g2ly1 = .1,
                                               sigma_j2lx1 = .1,
                                               sigma_j2g2 = .01,
                                               delta_lag_xy = .13,
                                               xi_lag_yx = .4),
                         return_lavaan_syntax = TRUE)
Click here to see the lavaan syntax specified in simsyntax.

# Specify parameters for construct x ----
# Specify latent true scores 
lx1 =~ 1 * x1 
lx2 =~ 1 * x2 
lx3 =~ 1 * x3 
lx4 =~ 1 * x4 
lx5 =~ 1 * x5 
# Specify mean of latent true scores 
lx1 ~ 21 * 1 
lx2 ~ 0 * 1 
lx3 ~ 0 * 1 
lx4 ~ 0 * 1 
lx5 ~ 0 * 1 
# Specify variance of latent true scores 
lx1 ~~ 0.5 * lx1 
lx2 ~~ 0 * lx2 
lx3 ~~ 0 * lx3 
lx4 ~~ 0 * lx4 
lx5 ~~ 0 * lx5 
# Specify intercept of obseved scores 
x1 ~ 0 * 1 
x2 ~ 0 * 1 
x3 ~ 0 * 1 
x4 ~ 0 * 1 
x5 ~ 0 * 1 
# Specify variance of observed scores 
x1 ~~ 0.2 * x1 
x2 ~~ 0.2 * x2 
x3 ~~ 0.2 * x3 
x4 ~~ 0.2 * x4 
x5 ~~ 0.2 * x5 
# Specify autoregressions of latent variables 
lx2 ~ 1 * lx1 
lx3 ~ 1 * lx2 
lx4 ~ 1 * lx3 
lx5 ~ 1 * lx4 
# Specify latent change scores 
dx2 =~ 1 * lx2 
dx3 =~ 1 * lx3 
dx4 =~ 1 * lx4 
dx5 =~ 1 * lx5 
# Specify latent change scores means 
dx2 ~ 0 * 1 
dx3 ~ 0 * 1 
dx4 ~ 0 * 1 
dx5 ~ 0 * 1 
# Specify latent change scores variances 
dx2 ~~ 0 * dx2 
dx3 ~~ 0 * dx3 
dx4 ~~ 0 * dx4 
dx5 ~~ 0 * dx5 
# Specify constant change factor 
g2 =~ 1 * dx2 + 1 * dx3 + 1 * dx4 + 1 * dx5 
# Specify constant change factor mean 
g2 ~ -0.4 * 1 
# Specify constant change factor variance 
g2 ~~ 0.4 * g2 
# Specify constant change factor covariance with the initial true score 
g2 ~~ 0.2 * lx1
# Specify proportional change component 
dx2 ~ -0.1 * lx1 
dx3 ~ -0.1 * lx2 
dx4 ~ -0.1 * lx3 
dx5 ~ -0.1 * lx4 
# Specify parameters for construct y ----
# Specify latent true scores 
ly1 =~ 1 * y1 
ly2 =~ 1 * y2 
ly3 =~ 1 * y3 
ly4 =~ 1 * y4 
ly5 =~ 1 * y5 
# Specify mean of latent true scores 
ly1 ~ 5 * 1 
ly2 ~ 0 * 1 
ly3 ~ 0 * 1 
ly4 ~ 0 * 1 
ly5 ~ 0 * 1 
# Specify variance of latent true scores 
ly1 ~~ 0.2 * ly1 
ly2 ~~ 0 * ly2 
ly3 ~~ 0 * ly3 
ly4 ~~ 0 * ly4 
ly5 ~~ 0 * ly5 
# Specify intercept of obseved scores 
y1 ~ 0 * 1 
y2 ~ 0 * 1 
y3 ~ 0 * 1 
y4 ~ 0 * 1 
y5 ~ 0 * 1 
# Specify variance of observed scores 
y1 ~~ 0.2 * y1 
y2 ~~ 0.2 * y2 
y3 ~~ 0.2 * y3 
y4 ~~ 0.2 * y4 
y5 ~~ 0.2 * y5 
# Specify autoregressions of latent variables 
ly2 ~ 1 * ly1 
ly3 ~ 1 * ly2 
ly4 ~ 1 * ly3 
ly5 ~ 1 * ly4 
# Specify latent change scores 
dy2 =~ 1 * ly2 
dy3 =~ 1 * ly3 
dy4 =~ 1 * ly4 
dy5 =~ 1 * ly5 
# Specify latent change scores means 
dy2 ~ 0 * 1 
dy3 ~ 0 * 1 
dy4 ~ 0 * 1 
dy5 ~ 0 * 1 
# Specify latent change scores variances 
dy2 ~~ 0 * dy2 
dy3 ~~ 0 * dy3 
dy4 ~~ 0 * dy4 
dy5 ~~ 0 * dy5 
# Specify constant change factor 
j2 =~ 1 * dy2 + 1 * dy3 + 1 * dy4 + 1 * dy5 
# Specify constant change factor mean 
j2 ~ -0.2 * 1 
# Specify constant change factor variance 
j2 ~~ 0.1 * j2 
# Specify constant change factor covariance with the initial true score 
j2 ~~ 0.02 * ly1
# Specify proportional change component 
dy2 ~ -0.2 * ly1 
dy3 ~ -0.2 * ly2 
dy4 ~ -0.2 * ly3 
dy5 ~ -0.2 * ly4 
# Specify autoregression of change score 
dy3 ~ 0.1 * dy2 
dy4 ~ 0.1 * dy3 
dy5 ~ 0.1 * dy4 
# Specify residual covariances 
x1 ~~ 0.01 * y1 
x2 ~~ 0.01 * y2 
x3 ~~ 0.01 * y3 
x4 ~~ 0.01 * y4 
x5 ~~ 0.01 * y5 
# Specify covariances betweeen specified change components (alpha) and intercepts (initial latent true scores lx1 and ly1) ----
# Specify covariance of intercepts 
lx1 ~~ 0.2 * ly1 
# Specify covariance of constant change and intercept within the same construct 
ly1 ~~ 0.1 * g2 
# Specify covariance of constant change and intercept within the same construct 
lx1 ~~ 0.1 * j2 
# Specify covariance of constant change factors between constructs 
g2 ~~ 0.01 * j2 
# Specify between-construct coupling parameters ----
# Change score x (t) is determined by true score y (t-1)  
dx2 ~ 0.13 * ly1 
dx3 ~ 0.13 * ly2 
dx4 ~ 0.13 * ly3 
dx5 ~ 0.13 * ly4 
# Change score y (t) is determined by change score x (t-1)  
dy3 ~ 0.4 * dx2 
dy4 ~ 0.4 * dx3 
dy5 ~ 0.4 * dx4