Chapter 19 Lavaan Lab 16: Latent Growth Models

In this lab, we will:

  • run and interpret a series of growth models (no growth, linear, quadratic, latent basis, spline growth);
  • compare nested models and identify the best possible shape for characterizing the growth patterns;
  • add predictors for the growth factors;
  • run growth models on latent variables.

Load up the lavaan library:

library(lavaan)

We will also need ggplot2, semPlot, and semTools. Install them if you haven’t:

#install.packages("ggplot2")
#install.packages("semPlot")
#install.packages("semTools")
library(ggplot2)
library(semPlot)
library(semTools)
  • For this lab, we will work with a simulated dataset # based on an example from McCoach & Kaniskan (2010).
  • The main DV is Oral Reading Fluency (ORF) and is measured over 4 time points (Fall and Spring, 2 consecutive years)
  • N = 277 Elementary students.
  • Let’s read in the dataset:
orf <- read.csv("readingSimData.csv", header = T)

Take a look at the dataset:

head(orf)
##   id treatmentDummy      orf1      orf2      orf3      orf4
## 1  1              0 133.93092 157.81992 178.74260 151.28058
## 2  2              0 191.84038 185.91922 192.02037 175.53109
## 3  3              1 144.80922 189.83059 201.97789 218.63097
## 4  4              0  22.58577  60.99331  63.06560  56.76576
## 5  5              0  46.27449  60.32287  35.20089  65.97300
## 6  6              0 166.39739 194.00734 166.40298 169.77082

sample size:

n <- nrow(orf)
n #277, just like the McCoach paper.
## [1] 277

sample means and cov matrix

orfNames <- paste0("orf", 1:4)
(samMeans <- round(apply(orf[,orfNames], 2, mean), 3))
##    orf1    orf2    orf3    orf4 
## 108.651 134.685 136.560 163.744
(samCov <- round(cov(orf[,orfNames])*((n-1)/n), 3))
##          orf1     orf2     orf3     orf4
## orf1 1989.243 1680.066 1718.200 1758.556
## orf2 1680.066 1817.077 1769.452 1823.106
## orf3 1718.200 1769.452 2157.290 2118.046
## orf4 1758.556 1823.106 2118.046 2525.285

19.1 PART I: Spaghetti Plot

For more details, check out https://www.r-bloggers.com/my-commonly-done-ggplot2-graphs/

First, let’s use reshape() to convert wide format to long format for plotting:

growthDataLong <- reshape(orf, varying = paste0("orf", 1:4), sep = "", direction = "long")
head(growthDataLong)
##     id treatmentDummy time       orf
## 1.1  1              0    1 133.93092
## 2.1  2              0    1 191.84038
## 3.1  3              1    1 144.80922
## 4.1  4              0    1  22.58577
## 5.1  5              0    1  46.27449
## 6.1  6              0    1 166.39739

Plot trajectory of individual with id=1

tspag_id1 = ggplot(growthDataLong[growthDataLong$id==1, ], aes(x=time, y=orf)) + 
  geom_line() + 
  xlab("Observation Time Point") +
  ylab("Y") + 
  ylim(-40, 300) + 
  ggtitle("Spaghetti plot") + 
  aes(colour = factor(id))
tspag_id1

plot trajectory of everyone

tspag = ggplot(growthDataLong, aes(x=time, y=orf)) + 
  geom_line(show.legend = FALSE) + 
  xlab("Observation Time Point") +
  ylab("Y") + 
  ylim(-40, 300) + 
  ggtitle("Spaghetti plot") + 
  aes(colour = factor(id))
tspag

19.2 PART II: Growth Models

19.2.1 1. No growth model

  • Let’s start by examining the hypothesis of no growth (intercept only)
  • Intercept loads on all variables with fixed loadings of 1.0
  • Use a*VAR1 to fix the coefficient of VAR1 at a:
noGrowthSyn <- "
    #Specify Latent Intercept
    I =~ 1*orf1 + 1*orf2 + 1*orf3 + 1*orf4
"
noGrowthFit <- growth(noGrowthSyn, data = orf, fixed.x = FALSE)
summary(noGrowthFit, fit.measures = T)
## lavaan 0.6-12 ended normally after 80 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                         6
## 
##   Number of observations                           277
## 
## Model Test User Model:
##                                                       
##   Test statistic                               622.276
##   Degrees of freedom                                 8
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                              1367.967
##   Degrees of freedom                                 6
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.549
##   Tucker-Lewis Index (TLI)                       0.662
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -5438.991
##   Loglikelihood unrestricted model (H1)      -5127.853
##                                                       
##   Akaike (AIC)                               10889.982
##   Bayesian (BIC)                             10911.726
##   Sample-size adjusted Bayesian (BIC)        10892.701
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.526
##   90 Percent confidence interval - lower         0.492
##   90 Percent confidence interval - upper         0.562
##   P-value RMSEA <= 0.05                          0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.263
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   I =~                                                
##     orf1              1.000                           
##     orf2              1.000                           
##     orf3              1.000                           
##     orf4              1.000                           
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .orf1              0.000                           
##    .orf2              0.000                           
##    .orf3              0.000                           
##    .orf4              0.000                           
##     I               134.926    2.559   52.728    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .orf1           1132.232  102.816   11.012    0.000
##    .orf2             98.891   28.254    3.500    0.000
##    .orf3            312.013   38.482    8.108    0.000
##    .orf4           1412.090  126.348   11.176    0.000
##     I              1746.714  154.599   11.298    0.000
semPaths(noGrowthFit, what='est', fade= F)

19.2.2 2. Linear growth model

  • Intercept loads on all variables with fixed loadings of 1.0
  • Slope loads on all variables with fixed loadings of t = 0, 1, 2, …, t-1
  • t must start from 0
linearGrowthSyn <- "
    #Specify Latent Intercept and Slope
    I =~ 1*orf1 + 1*orf2 + 1*orf3 + 1*orf4
    S =~ 0*orf1 + 1*orf2 + 2*orf3 + 3*orf4
"
linearGrowthFit <- growth(linearGrowthSyn, data = orf, fixed.x = FALSE)
summary(linearGrowthFit, fit.measures = T)
## lavaan 0.6-12 ended normally after 160 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                         9
## 
##   Number of observations                           277
## 
## Model Test User Model:
##                                                       
##   Test statistic                               159.802
##   Degrees of freedom                                 5
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                              1367.967
##   Degrees of freedom                                 6
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.886
##   Tucker-Lewis Index (TLI)                       0.864
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -5207.754
##   Loglikelihood unrestricted model (H1)      -5127.853
##                                                       
##   Akaike (AIC)                               10433.508
##   Bayesian (BIC)                             10466.124
##   Sample-size adjusted Bayesian (BIC)        10437.586
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.334
##   90 Percent confidence interval - lower         0.291
##   90 Percent confidence interval - upper         0.380
##   P-value RMSEA <= 0.05                          0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.076
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   I =~                                                
##     orf1              1.000                           
##     orf2              1.000                           
##     orf3              1.000                           
##     orf4              1.000                           
##   S =~                                                
##     orf1              0.000                           
##     orf2              1.000                           
##     orf3              2.000                           
##     orf4              3.000                           
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   I ~~                                                
##     S                -4.269   29.309   -0.146    0.884
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .orf1              0.000                           
##    .orf2              0.000                           
##    .orf3              0.000                           
##    .orf4              0.000                           
##     I               110.504    2.585   42.746    0.000
##     S                17.218    0.628   27.410    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .orf1            209.999   45.503    4.615    0.000
##    .orf2            257.156   30.447    8.446    0.000
##    .orf3            302.851   34.737    8.718    0.000
##    .orf4            168.261   48.428    3.474    0.001
##     I              1698.098  159.067   10.675    0.000
##     S                70.482   11.729    6.009    0.000
semPaths(linearGrowthFit, what='est', fade= F)

19.2.3 3. Quadratic growth model

  • Intercept loads on all variables with fixed loadings of 1.0
  • Slope loads on all variables with fixed loadings of t = 0, 1, 2, …, t-1
  • Quadratic loads on all variables with fixed loadings of t^2 = 0, 1, 4, …, (t-1)^2
  • Quadratic has no variance and covariances
quadGrowthSyn <- "
    #int and slope factors
    I =~ 1*orf1 + 1*orf2 + 1*orf3 + 1*orf4
    S =~ 0*orf1 + 1*orf2 + 2*orf3 + 3*orf4
    #quadratic factor = slope^2
    quadS =~ 0*orf1 + 1*orf2 + 4*orf3 + 9*orf4
"
quadGrowthFit <- growth(quadGrowthSyn, data = orf, fixed.x = FALSE)
## Warning in lav_object_post_check(object): lavaan WARNING: some estimated ov variances are negative
## Warning in lav_object_post_check(object): lavaan WARNING: some estimated lv variances are negative

If you get the following warning messages:

1: In lav_object_post_check(object) : lavaan WARNING: some estimated ov variances are negative

  • Use var1~~0*var2 to fix the (co)variances at 0
quadGrowthSyn_noQuad <- "
    #int and slope factors
    I =~ 1*orf1 + 1*orf2 + 1*orf3 + 1*orf4
    S =~ 0*orf1 + 1*orf2 + 2*orf3 + 3*orf4
    #quadratic factor = slope^2
    quadS =~ 0*orf1 + 1*orf2 + 4*orf3 + 9*orf4
    
    quadS ~~ 0*quadS #restrict quadratic variance to 0
    quadS ~~ 0*I #restrict quadratic covariance with I to 0
    quadS ~~ 0*S #restrict quadratic covariance with S to 0
"
quadGrowthNoquadFit <- growth(quadGrowthSyn_noQuad, 
                              data = orf, fixed.x = FALSE)
summary(quadGrowthNoquadFit, fit.measures = T)
## lavaan 0.6-12 ended normally after 170 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        10
## 
##   Number of observations                           277
## 
## Model Test User Model:
##                                                       
##   Test statistic                               159.759
##   Degrees of freedom                                 4
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                              1367.967
##   Degrees of freedom                                 6
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.886
##   Tucker-Lewis Index (TLI)                       0.828
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -5207.733
##   Loglikelihood unrestricted model (H1)      -5127.853
##                                                       
##   Akaike (AIC)                               10435.465
##   Bayesian (BIC)                             10471.705
##   Sample-size adjusted Bayesian (BIC)        10439.997
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.375
##   90 Percent confidence interval - lower         0.326
##   90 Percent confidence interval - upper         0.426
##   P-value RMSEA <= 0.05                          0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.077
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   I =~                                                
##     orf1              1.000                           
##     orf2              1.000                           
##     orf3              1.000                           
##     orf4              1.000                           
##   S =~                                                
##     orf1              0.000                           
##     orf2              1.000                           
##     orf3              2.000                           
##     orf4              3.000                           
##   quadS =~                                            
##     orf1              0.000                           
##     orf2              1.000                           
##     orf3              4.000                           
##     orf4              9.000                           
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   I ~~                                                
##     quadS             0.000                           
##   S ~~                                                
##     quadS             0.000                           
##   I ~~                                                
##     S                -4.081   29.338   -0.139    0.889
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .orf1              0.000                           
##    .orf2              0.000                           
##    .orf3              0.000                           
##    .orf4              0.000                           
##     I               110.574    2.619   42.216    0.000
##     S                16.827    1.544   10.897    0.000
##     quadS             0.129    0.460    0.281    0.779
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##     quadS             0.000                           
##    .orf1            210.600   45.691    4.609    0.000
##    .orf2            261.092   30.735    8.495    0.000
##    .orf3            298.679   34.435    8.674    0.000
##    .orf4            167.167   48.238    3.465    0.001
##     I              1697.894  159.144   10.669    0.000
##     S                70.527   11.730    6.013    0.000
semPaths(quadGrowthNoquadFit, what='est', fade= F)

19.2.4 4. Latent basis growth model (extension of linear growth model)

  • Intercept loads on all variables with fixed loadings of 1.0
  • Slope loads on all variables with free loadings between 0 and t-1
latentBasisSyn <- "
    #Int and slope specification
    I =~ 1*orf1 + 1*orf2 + 1*orf3 + 1*orf4
    
    #orf2 and orf3 are free in the latent basis specification
    S =~ 0*orf1 + alpha1*orf2 + alpha2*orf3 + 3*orf4
"
latentBasisFit <- growth(latentBasisSyn, 
                         data = orf, 
                         fixed.x = FALSE)
summary(latentBasisFit, fit.measures = T)
## lavaan 0.6-12 ended normally after 180 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        11
## 
##   Number of observations                           277
## 
## Model Test User Model:
##                                                       
##   Test statistic                                41.591
##   Degrees of freedom                                 3
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                              1367.967
##   Degrees of freedom                                 6
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.972
##   Tucker-Lewis Index (TLI)                       0.943
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -5148.648
##   Loglikelihood unrestricted model (H1)      -5127.853
##                                                       
##   Akaike (AIC)                               10319.297
##   Bayesian (BIC)                             10359.161
##   Sample-size adjusted Bayesian (BIC)        10324.281
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.215
##   90 Percent confidence interval - lower         0.160
##   90 Percent confidence interval - upper         0.276
##   P-value RMSEA <= 0.05                          0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.042
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   I =~                                                
##     orf1              1.000                           
##     orf2              1.000                           
##     orf3              1.000                           
##     orf4              1.000                           
##   S =~                                                
##     orf1              0.000                           
##     orf2    (alp1)    1.343    0.056   23.780    0.000
##     orf3    (alp2)    1.575    0.058   27.342    0.000
##     orf4              3.000                           
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   I ~~                                                
##     S                -1.364   30.564   -0.045    0.964
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .orf1              0.000                           
##    .orf2              0.000                           
##    .orf3              0.000                           
##    .orf4              0.000                           
##     I               108.867    2.633   41.354    0.000
##     S                18.294    0.644   28.418    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .orf1            232.136   45.754    5.074    0.000
##    .orf2            200.709   24.071    8.338    0.000
##    .orf3            215.666   25.245    8.543    0.000
##    .orf4            193.173   49.612    3.894    0.000
##     I              1688.625  159.099   10.614    0.000
##     S                67.573   12.843    5.261    0.000
semPaths(latentBasisFit, what='est', fade= F)

RMSEA failed us… one approach is to use model modification indices:

modindices(latentBasisFit,sort. = T)
##     lhs op  rhs     mi      epc  sepc.lv sepc.all sepc.nox
## 27 orf3 ~~ orf4 37.677  285.256  285.256    1.398    1.398
## 17 orf2 ~1      37.648   21.985   21.985    0.491    0.491
## 18 orf3 ~1      29.423  -20.326  -20.326   -0.447   -0.447
## 26 orf2 ~~ orf4 29.372 -191.644 -191.644   -0.973   -0.973
## 24 orf1 ~~ orf4 27.665  701.388  701.388    3.312    3.312
## 22 orf1 ~~ orf2 17.100  164.959  164.959    0.764    0.764
## 23 orf1 ~~ orf3 13.434 -113.046 -113.046   -0.505   -0.505
## 16 orf1 ~1      12.388  -47.984  -47.984   -1.095   -1.095
## 3     I =~ orf3 10.891   -0.077   -3.175   -0.070   -0.070
## 2     I =~ orf2  5.875    0.060    2.461    0.055    0.055
## 4     I =~ orf4  4.587    0.152    6.261    0.126    0.126
## 1     I =~ orf1  0.801    0.063    2.577    0.059    0.059
## 25 orf2 ~~ orf3  0.014   -4.660   -4.660   -0.022   -0.022
## 19 orf4 ~1       0.010   -1.525   -1.525   -0.031   -0.031

Another approach is to use Spline Growth Model.

19.2.5 5. Spline Growth Model

splineGrowthSyn <- "
    #Specify Latent Intercept and Slope
    I =~ 1*orf1 + 1*orf2 + 1*orf3 + 1*orf4
    S =~ 0*orf1 + 1*orf2 + 2*orf3 + 3*orf4

    #summer is the spline variable
    summer =~ 0*orf1 + 0*orf2 + 1*orf3 + 1*orf4

    #Summer gets a mean but no variance
    summer ~ 1
    summer ~~ 0*summer

    #Summer is uncorrelated with I and S
    summer ~~ 0*I 
    summer ~~ 0*S 
"
splineGrowthFit <- growth(splineGrowthSyn, data = orf, fixed.x = FALSE)
summary(splineGrowthFit, fit.measures = T)
## lavaan 0.6-12 ended normally after 167 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        10
## 
##   Number of observations                           277
## 
## Model Test User Model:
##                                                       
##   Test statistic                                 8.029
##   Degrees of freedom                                 4
##   P-value (Chi-square)                           0.091
## 
## Model Test Baseline Model:
## 
##   Test statistic                              1367.967
##   Degrees of freedom                                 6
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.997
##   Tucker-Lewis Index (TLI)                       0.996
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -5131.868
##   Loglikelihood unrestricted model (H1)      -5127.853
##                                                       
##   Akaike (AIC)                               10283.735
##   Bayesian (BIC)                             10319.975
##   Sample-size adjusted Bayesian (BIC)        10288.267
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.060
##   90 Percent confidence interval - lower         0.000
##   90 Percent confidence interval - upper         0.121
##   P-value RMSEA <= 0.05                          0.320
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.023
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   I =~                                                
##     orf1              1.000                           
##     orf2              1.000                           
##     orf3              1.000                           
##     orf4              1.000                           
##   S =~                                                
##     orf1              0.000                           
##     orf2              1.000                           
##     orf3              2.000                           
##     orf4              3.000                           
##   summer =~                                           
##     orf1              0.000                           
##     orf2              0.000                           
##     orf3              1.000                           
##     orf4              1.000                           
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   I ~~                                                
##     summer            0.000                           
##   S ~~                                                
##     summer            0.000                           
##   I ~~                                                
##     S                 3.511   28.391    0.124    0.902
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##     summer          -24.715    1.804  -13.697    0.000
##    .orf1              0.000                           
##    .orf2              0.000                           
##    .orf3              0.000                           
##    .orf4              0.000                           
##     I               108.303    2.579   41.998    0.000
##     S                26.616    0.986   26.987    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##     summer            0.000                           
##    .orf1            246.662   40.796    6.046    0.000
##    .orf2            165.717   21.986    7.537    0.000
##    .orf3            183.428   23.958    7.656    0.000
##    .orf4            219.241   42.313    5.181    0.000
##     I              1670.038  155.803   10.719    0.000
##     S                65.687   10.741    6.116    0.000
semPaths(splineGrowthFit, what='est', fade= F)

19.2.6 Model Comparison

lavTestLRT(noGrowthFit, linearGrowthFit)
## Chi-Squared Difference Test
## 
##                 Df   AIC   BIC  Chisq Chisq diff Df diff Pr(>Chisq)    
## linearGrowthFit  5 10434 10466 159.80                                  
## noGrowthFit      8 10890 10912 622.28     462.47       3  < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
  • Linear Growth Model fits significantly better than No Growth Model
lavTestLRT(linearGrowthFit, quadGrowthNoquadFit)
## Chi-Squared Difference Test
## 
##                     Df   AIC   BIC  Chisq Chisq diff Df diff Pr(>Chisq)
## quadGrowthNoquadFit  4 10436 10472 159.76                              
## linearGrowthFit      5 10434 10466 159.80   0.042546       1     0.8366
  • Linear Growth Model fits almost the same as the Quadratic Growth Model (keep linear model due to parsimony principle)
lavTestLRT(linearGrowthFit, latentBasisFit)
## Chi-Squared Difference Test
## 
##                 Df   AIC   BIC   Chisq Chisq diff Df diff Pr(>Chisq)    
## latentBasisFit   3 10319 10359  41.591                                  
## linearGrowthFit  5 10434 10466 159.802     118.21       2  < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
  • Latent Basis Model fits significantly better than Linear Growth Model
lavTestLRT(linearGrowthFit, splineGrowthFit)
## Chi-Squared Difference Test
## 
##                 Df   AIC   BIC    Chisq Chisq diff Df diff Pr(>Chisq)    
## splineGrowthFit  4 10284 10320   8.0292                                  
## linearGrowthFit  5 10434 10466 159.8018     151.77       1  < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
  • Spline Growth Model also fits significantly better than Linear Growth Model

19.2.7 6. Final Model: Spline Growth Model with a binary treatment predictor

splineGrowthTreatmentPredictorSyn <- "
    #Specify Latent Intercept and Slope
    I =~ 1*orf1 + 1*orf2 + 1*orf3 + 1*orf4
    S =~ 0*orf1 + 1*orf2 + 2*orf3 + 3*orf4

    #summer is the spline variable
    summer =~ 0*orf1 + 0*orf2 + 1*orf3 + 1*orf4

    #Summer gets a mean but no variance
    summer ~~ 0*summer

    #Summer is uncorrelated with I and S
    summer ~~ 0*I 
    summer ~~ 0*S 

    #Intercept, Slope, and Summer regressed on (predicted by) treatment
    I ~ treatmentDummy
    S ~ treatmentDummy
    summer ~ treatmentDummy
"
  • When including external predictors, we need to turn on fixed.x = T…
  • otherwise you’ll get a warning message and misleading model fit:
# do not do this:
splineGrowthTreatPredictorFit <- growth(splineGrowthTreatmentPredictorSyn, 
                                       data = orf, 
                                       fixed.x = F) # <- Here
## Warning in lav_data_full(data = data, group = group, cluster = cluster, : lavaan WARNING: some observed variances are (at least) a factor 1000 times larger than others; use varTable(fit) to investigate
## Warning in lav_partable_check(lavpartable, categorical = lavoptions$.categorical, : lavaan WARNING: automatically added intercepts are set to zero:
##     [treatmentDummy]

Instead, turn on fixed.x = T:

splineGrowthTreatPredictorFit <- growth(splineGrowthTreatmentPredictorSyn, 
                                       data = orf, 
                                       fixed.x = T) # <- Here
summary(splineGrowthTreatPredictorFit, fit.measures = T)
## lavaan 0.6-12 ended normally after 216 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        13
## 
##   Number of observations                           277
## 
## Model Test User Model:
##                                                       
##   Test statistic                                 8.288
##   Degrees of freedom                                 5
##   P-value (Chi-square)                           0.141
## 
## Model Test Baseline Model:
## 
##   Test statistic                              1604.488
##   Degrees of freedom                                10
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.998
##   Tucker-Lewis Index (TLI)                       0.996
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -5013.737
##   Loglikelihood unrestricted model (H1)      -5009.593
##                                                       
##   Akaike (AIC)                               10053.473
##   Bayesian (BIC)                             10100.585
##   Sample-size adjusted Bayesian (BIC)        10059.364
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.049
##   90 Percent confidence interval - lower         0.000
##   90 Percent confidence interval - upper         0.105
##   P-value RMSEA <= 0.05                          0.443
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.019
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   I =~                                                
##     orf1              1.000                           
##     orf2              1.000                           
##     orf3              1.000                           
##     orf4              1.000                           
##   S =~                                                
##     orf1              0.000                           
##     orf2              1.000                           
##     orf3              2.000                           
##     orf4              3.000                           
##   summer =~                                           
##     orf1              0.000                           
##     orf2              0.000                           
##     orf3              1.000                           
##     orf4              1.000                           
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   I ~                                                 
##     treatmentDummy    1.181    5.165    0.229    0.819
##   S ~                                                 
##     treatmentDummy   14.530    1.725    8.425    0.000
##   summer ~                                            
##     treatmentDummy    2.874    3.650    0.787    0.431
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##  .I ~~                                                
##    .summer            0.000                           
##  .S ~~                                                
##    .summer            0.000                           
##  .I ~~                                                
##    .S                -2.783   19.658   -0.142    0.887
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .orf1              0.000                           
##    .orf2              0.000                           
##    .orf3              0.000                           
##    .orf4              0.000                           
##    .I               107.754    3.565   30.223    0.000
##    .S                19.690    1.191   16.538    0.000
##    .summer          -26.082    2.520  -10.352    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .summer            0.000                           
##    .orf1            232.349   33.038    7.033    0.000
##    .orf2            171.092   20.837    8.211    0.000
##    .orf3            189.397   22.176    8.541    0.000
##    .orf4            206.315   31.878    6.472    0.000
##    .I              1678.279  155.946   10.762    0.000
##    .S                 5.773    6.112    0.944    0.345
semPaths(splineGrowthTreatPredictorFit, what='est', fade= F)

19.3 PART III: LGM on Latent Variables

19.3.1 Example

Please go over the checklist:

  1. Make sure the latent variables satisfy longitudinal measurement invariance at the level of scalar invariance or above
  2. Use the loadings over time (i.e., metric invariance)
  3. No need to correlate the latent factors
  4. Add intercepts for all indicators EXCEPT for marker indicators
  5. Add correlated residuals for repeated measures of the same indicators
  6. Use std.lv = TRUE as the scaling method in growth()

For this example, we will use the dataset exLong from package semTools:

data(exLong)
head(exLong)
##      sex       y1t1       y2t1        y3t1       y1t2        y2t2      y3t2      y1t3       y2t3       y3t3
## 1 female  2.7625423  2.2812510  2.49656014  2.9499400  1.12338865 2.1505466  1.912824  1.9625734  2.4403812
## 2 female  0.2707267 -0.7830365 -0.23554656  0.4631038  0.37536412 2.0283960  2.112440  0.4326280  2.6352259
## 3 female -0.2604141  0.3146881  1.21590069  0.3528803  0.00986991 1.0709696  1.472736  1.1951005  1.4287358
## 4 female -1.0227953 -1.3454733  0.04899156  1.9530137 -1.03357363 2.7817132  1.249376 -0.5369589  2.3371304
## 5 female  0.7385408 -0.6027341  0.42557808 -0.2027779 -0.34937717 1.1865013  1.425101  1.8539630  2.0861627
## 6 female  0.5864878 -0.4659974  0.10691644  0.7869085 -0.31982170 0.4825864 -1.201912 -2.0090700 -0.8366951
?exLong

The syntax for linear growth model with latent variables:

exLinearGrowthsyn <- "
  
  # Use the loadings over time (i.e., metric invariance)
  f_t1 =~ lamb1*y1t1 + lamb2*y2t1 + lamb3*y3t1
  f_t2 =~ lamb1*y1t2 + lamb2*y2t2 + lamb3*y3t2
  f_t3 =~ lamb1*y1t3 + lamb2*y2t3 + lamb3*y3t3

  #Int and slope specification
    I =~ 1*f_t1 + 1*f_t2 + 1*f_t3
    S =~ 0*f_t1 + 1*f_t2 + 2*f_t3
    
    # Add intercepts for all indicators EXCEPT for marker indicators
  y2t1 ~ 1
  y3t1 ~ 1
  y2t2 ~ 1
  y3t2 ~ 1
    y2t3 ~ 1
  y3t3 ~ 1

    # Add correlated residuals for repeated measures of the same indicators
    y1t1 ~~ y1t2
    y1t1 ~~ y1t3
    y1t2 ~~ y1t3
    y2t1 ~~ y2t2
    y2t1 ~~ y2t3
    y2t2 ~~ y2t3
    y3t1 ~~ y3t2
    y3t1 ~~ y3t3
    y3t2 ~~ y3t3
"

Use std.lv = TRUE as the scaling method in growth():

exLinearGrowthFit <- growth(exLinearGrowthsyn, 
                         data = exLong, 
                         fixed.x = FALSE,
                         std.lv = TRUE)
## Warning in lav_model_vcov(lavmodel = lavmodel, lavsamplestats = lavsamplestats, : lavaan WARNING:
##     The variance-covariance matrix of the estimated parameters (vcov)
##     does not appear to be positive definite! The smallest eigenvalue
##     (= 4.804423e-18) is close to zero. This may be a symptom that the
##     model is not identified.
summary(exLinearGrowthFit, fit.measures = T)
## lavaan 0.6-12 ended normally after 36 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        39
##   Number of equality constraints                     6
## 
##   Number of observations                           200
## 
## Model Test User Model:
##                                                       
##   Test statistic                                72.972
##   Degrees of freedom                                21
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                              1126.037
##   Degrees of freedom                                36
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.952
##   Tucker-Lewis Index (TLI)                       0.918
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -2209.468
##   Loglikelihood unrestricted model (H1)      -2172.981
##                                                       
##   Akaike (AIC)                                4484.935
##   Bayesian (BIC)                              4593.780
##   Sample-size adjusted Bayesian (BIC)         4489.232
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.111
##   90 Percent confidence interval - lower         0.084
##   90 Percent confidence interval - upper         0.140
##   P-value RMSEA <= 0.05                          0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.181
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   f_t1 =~                                             
##     y1t1    (lmb1)    0.595    0.027   22.333    0.000
##     y2t1    (lmb2)    0.371    0.021   18.042    0.000
##     y3t1    (lmb3)    0.513    0.025   20.776    0.000
##   f_t2 =~                                             
##     y1t2    (lmb1)    0.595    0.027   22.333    0.000
##     y2t2    (lmb2)    0.371    0.021   18.042    0.000
##     y3t2    (lmb3)    0.513    0.025   20.776    0.000
##   f_t3 =~                                             
##     y1t3    (lmb1)    0.595    0.027   22.333    0.000
##     y2t3    (lmb2)    0.371    0.021   18.042    0.000
##     y3t3    (lmb3)    0.513    0.025   20.776    0.000
##   I =~                                                
##     f_t1              1.000                           
##     f_t2              1.000                           
##     f_t3              1.000                           
##   S =~                                                
##     f_t1              0.000                           
##     f_t2              1.000                           
##     f_t3              2.000                           
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##  .y1t1 ~~                                             
##    .y1t2              0.112    0.049    2.315    0.021
##    .y1t3              0.083    0.045    1.851    0.064
##  .y1t2 ~~                                             
##    .y1t3              0.162    0.056    2.906    0.004
##  .y2t1 ~~                                             
##    .y2t2              0.117    0.034    3.452    0.001
##    .y2t3             -0.009    0.034   -0.283    0.777
##  .y2t2 ~~                                             
##    .y2t3              0.114    0.036    3.134    0.002
##  .y3t1 ~~                                             
##    .y3t2              0.158    0.046    3.392    0.001
##    .y3t3              0.029    0.042    0.690    0.490
##  .y3t2 ~~                                             
##    .y3t3              0.095    0.047    2.029    0.042
##   I ~~                                                
##     S                -0.079    0.119   -0.662    0.508
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .y2t1             -0.264    0.050   -5.288    0.000
##    .y3t1              0.540    0.058    9.269    0.000
##    .y2t2             -0.266    0.057   -4.689    0.000
##    .y3t2              0.291    0.068    4.287    0.000
##    .y2t3             -0.228    0.064   -3.584    0.000
##    .y3t3              0.534    0.071    7.542    0.000
##    .y1t1              0.000                           
##    .y1t2              0.000                           
##    .y1t3              0.000                           
##    .f_t1             -0.207    0.068   -3.036    0.002
##    .f_t2              0.119    0.080    1.484    0.138
##    .f_t3              0.262    0.050    5.201    0.000
##     I                 0.174    0.065    2.683    0.007
##     S                 0.643    0.071    9.078    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .y1t1              0.320    0.062    5.191    0.000
##    .y2t1              0.376    0.045    8.418    0.000
##    .y3t1              0.440    0.060    7.277    0.000
##    .y1t2              0.457    0.077    5.946    0.000
##    .y2t2              0.394    0.048    8.224    0.000
##    .y3t2              0.474    0.068    7.006    0.000
##    .y1t3              0.333    0.077    4.305    0.000
##    .y2t3              0.433    0.052    8.397    0.000
##    .y3t3              0.381    0.065    5.895    0.000
##    .f_t1              1.000                           
##    .f_t2              1.000                           
##    .f_t3              1.000                           
##     I                 1.000                           
##     S                 1.000
semPaths(exLinearGrowthFit, what='est', fade= F)

19.3.2 Exercise

Q: Could you fit the latent basis model to the same dataset and compare the fit of the two models?