Examples of Latent Growth Curve Models

The saved examples below were generated with nlpsem 1.0.0.9006; extdata/REGENERATED.txt records the versions and how the fits compare with the previous ones. Rendering this guide does not refit its disabled model-fitting chunks. Refit with the current version for a new analysis, use getDiagnostics() to check admissibility and uncertainty, and compare multiple starts for difficult nonlinear or mixture models. These examples are not evidence of numerical qualification.

Load nlpsem package, dependent packages and set CSOLNP as the optimizer

library(nlpsem)
mxOption(model = NULL, key = "Default optimizer", "CSOLNP", reset = FALSE)

Load pre-computed models

load(system.file("extdata", "getLGCM_examples.RData", package = "nlpsem"))

Load example data and preprocess data

# Load ECLS-K (2011) data
data("RMS_dat")
RMS_dat0 <- RMS_dat
# Re-baseline the data so that the estimated initial status is for the
# starting point of the study
baseT <- RMS_dat0$T1
RMS_dat0$T1 <- RMS_dat0$T1 - baseT
RMS_dat0$T2 <- RMS_dat0$T2 - baseT
RMS_dat0$T3 <- RMS_dat0$T3 - baseT
RMS_dat0$T4 <- RMS_dat0$T4 - baseT
RMS_dat0$T5 <- RMS_dat0$T5 - baseT
RMS_dat0$T6 <- RMS_dat0$T6 - baseT
RMS_dat0$T7 <- RMS_dat0$T7 - baseT
RMS_dat0$T8 <- RMS_dat0$T8 - baseT
RMS_dat0$T9 <- RMS_dat0$T9 - baseT
# Standardize time-invariant covariates (TICs)
## ex1 and ex2 are standardized growth TICs in models
RMS_dat0$ex1 <- scale(RMS_dat0$Approach_to_Learning)
RMS_dat0$ex2 <- scale(RMS_dat0$Attention_focus)
xstarts <- mean(baseT)

Example 1: Fit bilinear spline LGCMs with both a random and a fixed knot to evaluate the development of mathematics ability from Kindergarten to Grade 5. These models are then compared using a likelihood ratio test (LRT) utilizing the getLRT() function.

Math_LGCM_BLS_f <- getLGCM(
  dat = RMS_dat0, t_var = "T", y_var = "M", curveFun = "bilinear spline",
  intrinsic = TRUE, records = 1:9, growth_TIC = NULL, tries = 10
)
Math_LGCM_BLS_r <- getLGCM(
  dat = RMS_dat0, t_var = "T", y_var = "M", curveFun = "bilinear spline",
  intrinsic = FALSE, records = 1:9, growth_TIC = NULL, tries = 10
)

getLRT(
  full = Math_LGCM_BLS_f@mxOutput, reduced = Math_LGCM_BLS_r@mxOutput, boot = FALSE, rep = NA
  )
#>               # of Free Param -2loglik Degree of Freedom Diff in loglik
#> Full Model                 15 31261.60              4485             NA
#> Reduced Model              11 31347.39              4489       85.78892
#>               Diff in DoF    p.values      AIC      BIC
#> Full Model             NA          NA 31291.60 31354.82
#> Reduced Model           4 1.03177e-17 31369.39 31415.75

The stored comparison has four additional parameters for the random-knot model. Its likelihood and AIC/BIC summarize these particular fitted solutions. The ordinary chi-square p-value is not a validated significance test for the random-knot effect: the loading formulas use different smoothing, latent covariance admissibility must be checked, and the null model’s regularity conditions require justification. See ?getLRT for nesting, same-data and bootstrap requirements. Bootstrapping does not automatically correct non-nesting or an inadmissible fitted model. Use getDiagnostics() and repeated starts before interpreting either solution.

Example 2: Fit the full bilinear spline LGCM to assess the development of mathematics skill from Kindergarten to Grade 5. This model includes two growth time-invariant covariates (TICs), baseline values of teacher-reported approach to learning and attentional focus. Point estimates and corresponding standard errors (SEs) of all parameters are presented within the original parameter space. The plot of the growth status of mathematics ability is also provided.

Math_LGCM_TIC_BLS_f <- getLGCM(
  dat = RMS_dat0, t_var = "T", y_var = "M", curveFun = "bilinear spline",
  intrinsic = TRUE, records = 1:9, growth_TIC = c("ex1", "ex2"),
  tries = 10, paramOut = TRUE
  )

Math_LGCM_TIC_BLS_f@Estimates
#>         Name      Estimate          SE
#> 1     alpha0  3.946156e+01 0.485884890
#> 2     alpha1  1.761586e+00 0.016585418
#> 3     alpha2  6.786552e-01 0.015648003
#> 4     alphag  3.516124e+01 0.428935881
#> 5      psi00  1.017760e+02 7.479973627
#> 6      psi01  5.426046e-01 0.181099738
#> 7      psi02 -4.301106e-01 0.170415680
#> 8      psi0g -1.654138e+01 4.661110632
#> 9      psi11  8.735413e-02 0.008786026
#> 10     psi12 -4.589649e-03 0.005797510
#> 11     psi1g -5.309019e-01 0.177277756
#> 12     psi22  1.384221e-02 0.008202612
#> 13     psi2g -1.141548e-01 0.176318438
#> 14     psigg  2.954858e+01 6.074984601
#> 15 residuals  3.149364e+01 0.888291839
#> 16    beta10  2.311643e+00 0.752625285
#> 17    beta11 -4.392197e-02 0.026052725
#> 18    beta12 -1.901185e-02 0.023778156
#> 19    beta1g  2.186632e-01 0.645450511
#> 20    beta20  2.091795e+00 0.758076722
#> 21    beta21  1.149117e-01 0.026150147
#> 22    beta22 -1.487944e-02 0.023844276
#> 23    beta2g -9.864440e-01 0.642813795
#> 24      mux1 -3.662941e-07 0.044678221
#> 25      mux2 -3.481922e-07 0.044678045
#> 26     phi11  9.979989e-01 0.063082846
#> 27     phi12  7.759370e-01 0.056492149
#> 28     phi22  9.979986e-01 0.063082848
#> 29    mueta0  3.946155e+01 0.520103625
#> 30    mueta1  1.761586e+00 0.017017993
#> 31    mueta2  6.786552e-01 0.015711189
#> 32   mu_knot  3.516124e+01 0.430559175
Figure1 <- getFigure(
  model = Math_LGCM_TIC_BLS_f@mxOutput, nClass = NULL, cluster_TIC = NULL, sub_Model = "LGCM",
  y_var = "M", curveFun = "BLS", y_model = "LGCM", t_var = "T", records = 1:9,
  m_var = NULL, x_var = NULL, x_type = NULL, xstarts = xstarts, xlab = "Month",
  outcome = "Mathematics"
)
show(Figure1)
#> figOutput Object
#> --------------------
#> Trajectories: 1 
#> Figure 1:
#> `geom_smooth()` using method = 'gam' and formula = 'y ~ s(x, bs = "cs")'

Mathematics mean trajectory over months from a bilinear spline growth model with a random knot and time-invariant covariates, comparing the fitted mean and smoothed observed mean with pointwise 95 percent confidence limits.

In the provided output, the terms alpha0, alpha1, alpha2, and alphag denote the conditional mean values of the intercept, first slope, second slope, and knot on growth TICs. The terms mueta0, mueta1, mueta2, and mu_knot represent the mean values of the intercept, first slope, second slope, and knot. These corresponding values match as we’ve centered the growth TICs.

The mean values of growth factors suggest that the rate of progress in mathematical ability slowed down after approximately 35.16 months since the beginning of the study. Moreover, the psi terms indicate the unexplained variance or covariance of the growth factors, while residuals corresponds to residual variance.

Moreover, beta10, beta11, beta12, and beta1g are path coefficients from the first growth TIC (i.e., baseline teacher-reported approach to learning) to the intercept, first slope, second slope, and knot, respectively. Similarly, beta2’s correspond to the path coefficients of the baseline teacher-reported attentional focus.

The interpretations of these coefficients are rather intuitive. For instance, a beta1g value of \(0.2187\) implies that a one standardized unit increase in baseline teacher-reported approach to learning delays the transition time to slower development by \(0.2187\) months. Similarly, a beta2g value of \(-0.9865\) indicates that a one standardized unit increase in baseline teacher-reported attentional focus advances the transition time to slower development by \(0.9865\) months.

However, caution is required when considering potential colinearity between growth TICs. The graphic representation of mathematical development also supports the observation that developmental pace slows down in later stages.