Latent Growth Curves
I will progress through three models: linear, quadratic growth, and latent basis. In every example I use a sample of 400, 6 time points, and ‘affect’ as the variable of interest.
Don’t forget that multiplying by time
- \(0.6t\)
is different from describing over time
- \(0.6_t\).
1) Linear
The data generating process:
\[\begin{equation} y_{it} = 4 - 0.6t + e_{t} \end{equation}\]
library(tidyverse)
library(ggplot2)
library(MASS)
N <- 400
time <- 6
intercept <- 4
linear_growth <- -0.6
df_matrix <- matrix(, nrow = N*time, ncol = 3)
count <- 0
for(i in 1:400){
unob_het_affect <- rnorm(1,0,3)
for(j in 1:6){
count <- count + 1
if(j == 1){
df_matrix[count, 1] <- i
df_matrix[count, 2] <- j
df_matrix[count, 3] <- intercept + unob_het_affect + rnorm(1,0,1)
}else{
df_matrix[count, 1] <- i
df_matrix[count, 2] <- j
df_matrix[count, 3] <- intercept + linear_growth*j + unob_het_affect + rnorm(1,0,1)
}
}
}
df <- data.frame(df_matrix)
names(df) <- c('id', 'time', 'affect')
random_ids <- sample(df$id, 5)
random_df <- df %>%
filter(id %in% random_ids)
ggplot(df, aes(x = time, y = affect, group = id)) +
geom_point(color = 'gray85') +
geom_line(color = 'gray85') +
geom_point(data = random_df, aes(x = time, y = affect, group = id), color = 'blue') +
geom_line(data = random_df, aes(x = time, y = affect, group = id), color = 'blue')
Estimating the model:
Formatting the data:
df_wide <- reshape(df, idvar = 'id', timevar = 'time', direction = 'wide')
First, an intercept only (no change) model:
library(lavaan)
no_change_string <- '
# Latent intercept factor
intercept_affect =~ 1*affect.1 + 1*affect.2 + 1*affect.3 + 1*affect.4 + 1*affect.5 + 1*affect.6
# Mean and variance of latent intercept factor
intercept_affect ~~ intercept_affect
# Fix observed variable means to 0
affect.1 ~ 0
affect.2 ~ 0
affect.3 ~ 0
affect.4 ~ 0
affect.5 ~ 0
affect.6 ~ 0
# Constrain residual (error) variance of observed variables to equality across time
affect.1 ~~ res_var*affect.1
affect.2 ~~ res_var*affect.2
affect.3 ~~ res_var*affect.3
affect.4 ~~ res_var*affect.4
affect.5 ~~ res_var*affect.5
affect.6 ~~ res_var*affect.6
'
no_change_model <- growth(no_change_string, data = df_wide)
summary(no_change_model, fit.measures = T)
lavaan 0.6-9 ended normally after 20 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 8
Number of equality constraints 5
Number of observations 400
Model Test User Model:
Test statistic 2073.804
Degrees of freedom 24
P-value (Chi-square) 0.000
Model Test Baseline Model:
Test statistic 4042.863
Degrees of freedom 15
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.491
Tucker-Lewis Index (TLI) 0.682
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -5203.688
Loglikelihood unrestricted model (H1) -4166.786
Akaike (AIC) 10413.376
Bayesian (BIC) 10425.350
Sample-size adjusted Bayesian (BIC) 10415.831
Root Mean Square Error of Approximation:
RMSEA 0.462
90 Percent confidence interval - lower 0.445
90 Percent confidence interval - upper 0.479
P-value RMSEA <= 0.05 0.000
Standardized Root Mean Square Residual:
SRMR 0.194
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|)
intercept_affect =~
affect.1 1.000
affect.2 1.000
affect.3 1.000
affect.4 1.000
affect.5 1.000
affect.6 1.000
Intercepts:
Estimate Std.Err z-value P(>|z|)
.affect.1 0.000
.affect.2 0.000
.affect.3 0.000
.affect.4 0.000
.affect.5 0.000
.affect.6 0.000
intercept_ffct 2.067 0.153 13.509 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
intrcp_ 8.914 0.662 13.460 0.000
.affct.1 (rs_v) 2.698 0.085 31.623 0.000
.affct.2 (rs_v) 2.698 0.085 31.623 0.000
.affct.3 (rs_v) 2.698 0.085 31.623 0.000
.affct.4 (rs_v) 2.698 0.085 31.623 0.000
.affct.5 (rs_v) 2.698 0.085 31.623 0.000
.affct.6 (rs_v) 2.698 0.085 31.623 0.000Now, a linear growth model centered at time point 1. The intercept factor estimate, therefore, is the estimated average affect at time 1.
library(lavaan)
linear_change_string <- '
# Latent intercept and slope factors
intercept_affect =~ 1*affect.1 + 1*affect.2 + 1*affect.3 + 1*affect.4 + 1*affect.5 + 1*affect.6
slope_affect =~ 0*affect.1 + 1*affect.2 + 2*affect.3 + 3*affect.4 + 4*affect.5 + 5*affect.6
# Mean and variance of latent factors
intercept_affect ~~ intercept_affect
slope_affect ~~ slope_affect
# Covariance between latent factors
intercept_affect ~~ slope_affect
# Fix observed variable means to 0
affect.1 ~ 0
affect.2 ~ 0
affect.3 ~ 0
affect.4 ~ 0
affect.5 ~ 0
affect.6 ~ 0
# Constrain residual (error) variance of observed variables to equality across time
affect.1 ~~ res_var*affect.1
affect.2 ~~ res_var*affect.2
affect.3 ~~ res_var*affect.3
affect.4 ~~ res_var*affect.4
affect.5 ~~ res_var*affect.5
affect.6 ~~ res_var*affect.6
'
linear_change_model <- growth(linear_change_string, data = df_wide)
summary(linear_change_model, fit.measures = T)
lavaan 0.6-9 ended normally after 48 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 11
Number of equality constraints 5
Number of observations 400
Model Test User Model:
Test statistic 95.218
Degrees of freedom 21
P-value (Chi-square) 0.000
Model Test Baseline Model:
Test statistic 4042.863
Degrees of freedom 15
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.982
Tucker-Lewis Index (TLI) 0.987
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -4214.395
Loglikelihood unrestricted model (H1) -4166.786
Akaike (AIC) 8440.790
Bayesian (BIC) 8464.738
Sample-size adjusted Bayesian (BIC) 8445.700
Root Mean Square Error of Approximation:
RMSEA 0.094
90 Percent confidence interval - lower 0.075
90 Percent confidence interval - upper 0.114
P-value RMSEA <= 0.05 0.000
Standardized Root Mean Square Residual:
SRMR 0.034
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|)
intercept_affect =~
affect.1 1.000
affect.2 1.000
affect.3 1.000
affect.4 1.000
affect.5 1.000
affect.6 1.000
slope_affect =~
affect.1 0.000
affect.2 1.000
affect.3 2.000
affect.4 3.000
affect.5 4.000
affect.6 5.000
Covariances:
Estimate Std.Err z-value P(>|z|)
intercept_affect ~~
slope_affect 0.049 0.036 1.368 0.171
Intercepts:
Estimate Std.Err z-value P(>|z|)
.affect.1 0.000
.affect.2 0.000
.affect.3 0.000
.affect.4 0.000
.affect.5 0.000
.affect.6 0.000
intercept_ffct 3.806 0.154 24.653 0.000
slope_affect -0.696 0.011 -61.197 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
intrcp_ 8.992 0.674 13.336 0.000
slp_ffc -0.007 0.004 -1.712 0.087
.affct.1 (rs_v) 1.030 0.036 28.284 0.000
.affct.2 (rs_v) 1.030 0.036 28.284 0.000
.affct.3 (rs_v) 1.030 0.036 28.284 0.000
.affct.4 (rs_v) 1.030 0.036 28.284 0.000
.affct.5 (rs_v) 1.030 0.036 28.284 0.000
.affct.6 (rs_v) 1.030 0.036 28.284 0.000inspect(linear_change_model, 'cov.lv')
intrc_ slp_ff
intercept_affect 8.992
slope_affect 0.049 -0.007This model does an adequate job recovering the intercept and slope parameters.
If I wanted to center the model at time point 3 the latent intercept term would be interpreted as the estimated average affect at time 3 and the syntax would change to:
'
slope_affect =~ -2*affect.1 + -1*affect.2 + 0*affect.3 + 1*affect.4 + 2*affect.5 + 3*affect.6
'
2) Quadratic
The data generating process:
\[\begin{equation} y_{it} = 4 + 0.2t + 0.7t^2 + e_{t} \end{equation}\]
library(tidyverse)
library(ggplot2)
library(MASS)
N <- 400
time <- 6
intercept_mu <- 4
linear_growth2 <- 0.2
quad_growth <- 0.7
df_matrix2 <- matrix(, nrow = N*time, ncol = 3)
count <- 0
for(i in 1:400){
unob_het_affect <- rnorm(1,0,3)
for(j in 1:6){
count <- count + 1
if(j == 1){
df_matrix2[count, 1] <- i
df_matrix2[count, 2] <- j
df_matrix2[count, 3] <- intercept + rnorm(1,0,1) + rnorm(1,0,1)
}else{
df_matrix2[count, 1] <- i
df_matrix2[count, 2] <- j
df_matrix2[count, 3] <- intercept + linear_growth2*j + quad_growth*(j^2) + unob_het_affect + rnorm(1,0,1)
}
}
}
df2 <- data.frame(df_matrix2)
names(df2) <- c('id', 'time', 'affect')
random_ids2 <- sample(df2$id, 5)
random_df2 <- df2 %>%
filter(id %in% random_ids2)
ggplot(df2, aes(x = time, y = affect, group = id)) +
geom_point(color = 'gray85') +
geom_line(color = 'gray85') +
geom_point(data = random_df2, aes(x = time, y = affect, group = id), color = 'blue') +
geom_line(data = random_df2, aes(x = time, y = affect, group = id), color = 'blue') +
theme_wsj()
Estimating the model:
Quadratic growth model:
df_wide2 <- reshape(df2, idvar = 'id', timevar = 'time', direction = 'wide')
library(lavaan)
quad_change_string <- [1012 chars quoted with ''']
quad_change_model <- growth(quad_change_string, data = df_wide2)
summary(quad_change_model, fit.measures = T)
lavaan 0.6-9 ended normally after 82 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 15
Number of equality constraints 5
Number of observations 400
Model Test User Model:
Test statistic 609.350
Degrees of freedom 17
P-value (Chi-square) 0.000
Model Test Baseline Model:
Test statistic 3203.346
Degrees of freedom 15
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.814
Tucker-Lewis Index (TLI) 0.836
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -4603.247
Loglikelihood unrestricted model (H1) -4298.572
Akaike (AIC) 9226.494
Bayesian (BIC) 9266.409
Sample-size adjusted Bayesian (BIC) 9234.678
Root Mean Square Error of Approximation:
RMSEA 0.295
90 Percent confidence interval - lower 0.275
90 Percent confidence interval - upper 0.315
P-value RMSEA <= 0.05 0.000
Standardized Root Mean Square Residual:
SRMR 0.223
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|)
intercept_affect =~
affect.1 1.000
affect.2 1.000
affect.3 1.000
affect.4 1.000
affect.5 1.000
affect.6 1.000
slope_affect =~
affect.1 0.000
affect.2 1.000
affect.3 2.000
affect.4 3.000
affect.5 4.000
affect.6 5.000
quad_slope_affect =~
affect.1 0.000
affect.2 1.000
affect.3 4.000
affect.4 9.000
affect.5 16.000
affect.6 25.000
Covariances:
Estimate Std.Err z-value P(>|z|)
intercept_affect ~~
slope_affect 0.897 0.148 6.066 0.000
quad_slop_ffct -0.136 0.024 -5.748 0.000
slope_affect ~~
quad_slop_ffct -0.461 0.049 -9.468 0.000
Intercepts:
Estimate Std.Err z-value P(>|z|)
.affect.1 0.000
.affect.2 0.000
.affect.3 0.000
.affect.4 0.000
.affect.5 0.000
.affect.6 0.000
intercept_ffct 4.226 0.069 60.965 0.000
slope_affect 2.163 0.103 20.974 0.000
quad_slop_ffct 0.610 0.017 36.907 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
intrcp_ 0.626 0.146 4.288 0.000
slp_ffc 3.107 0.304 10.205 0.000
qd_slp_ 0.067 0.008 8.463 0.000
.affct.1 (rs_v) 1.579 0.064 24.495 0.000
.affct.2 (rs_v) 1.579 0.064 24.495 0.000
.affct.3 (rs_v) 1.579 0.064 24.495 0.000
.affct.4 (rs_v) 1.579 0.064 24.495 0.000
.affct.5 (rs_v) 1.579 0.064 24.495 0.000
.affct.6 (rs_v) 1.579 0.064 24.495 0.000This model recovers the intercept and quadratic parameters but not the linear growth parameter.
3) Latent Basis
This model allows us to see where a majority of the change occurs in the process. For example, does more change occur between time points 2 and 3 or 5 and 6? In this model we are not trying to recover the parameters, but describe the change process in detail.
Data generating process:
Time 1 - Time 3: \[\begin{equation} y_{it} = 4 + 0.2t + e_{t} \end{equation}\]
Time 4 - Time 6: \[\begin{equation} y_{it} = 4 + 0.8t + e_{t} \end{equation}\]
library(tidyverse)
library(ggplot2)
library(MASS)
N <- 400
time <- 6
intercept_mu <- 4
growth_1 <- 0.2
growth_2 <- 0.8
df_matrix3 <- matrix(, nrow = N*time, ncol = 3)
count <- 0
for(i in 1:400){
unob_het_affect <- rnorm(1,0,3)
for(j in 1:6){
count <- count + 1
if(j < 4){
df_matrix3[count, 1] <- i
df_matrix3[count, 2] <- j
df_matrix3[count, 3] <- intercept + growth_1*j + unob_het_affect + rnorm(1,0,1)
}else{
df_matrix3[count, 1] <- i
df_matrix3[count, 2] <- j
df_matrix3[count, 3] <- intercept + growth_2*j + unob_het_affect + rnorm(1,0,1)
}
}
}
df3 <- data.frame(df_matrix3)
names(df3) <- c('id', 'time', 'affect')
random_ids3 <- sample(df3$id, 5)
random_df3 <- df3 %>%
filter(id %in% random_ids3)
ggplot(df3, aes(x = time, y = affect, group = id)) +
geom_point(color = 'gray85') +
geom_line(color = 'gray85') +
geom_point(data = random_df3, aes(x = time, y = affect, group = id), color = 'blue') +
geom_line(data = random_df3, aes(x = time, y = affect, group = id), color = 'blue')
Estimating the model:
Latent basis:
Similar to a linear growth model but we freely estimate the intermediate basis coefficients. Remember to constrain the first basis coefficient to zero and the last to 1.
df_wide3 <- reshape(df3, idvar = 'id', timevar = 'time', direction = 'wide')
library(lavaan)
lb_string <- '
# Latent intercept and slope terms with intermediate time points freely estimated
intercept_affect =~ 1*affect.1 + 1*affect.2 + 1*affect.3 + 1*affect.4 + 1*affect.5 + 1*affect.6
slope_affect =~ 0*affect.1 + bc1*affect.2 + bc2*affect.3 + bc3*affect.4 + bc4*affect.5 + 1*affect.6
# Mean and variance of latent factors
intercept_affect ~~ intercept_affect
slope_affect ~~ slope_affect
# Covariance between latent factors
intercept_affect ~~ slope_affect
# Fix observed variable means to 0
affect.1 ~ 0
affect.2 ~ 0
affect.3 ~ 0
affect.4 ~ 0
affect.5 ~ 0
affect.6 ~ 0
# Constrain residual (error) variance of observed variables to equality across time
affect.1 ~~ res_var*affect.1
affect.2 ~~ res_var*affect.2
affect.3 ~~ res_var*affect.3
affect.4 ~~ res_var*affect.4
affect.5 ~~ res_var*affect.5
affect.6 ~~ res_var*affect.6
'
lb_model <- growth(lb_string, data = df_wide3)
summary(lb_model, fit.measures = T)
lavaan 0.6-9 ended normally after 64 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 15
Number of equality constraints 5
Number of observations 400
Model Test User Model:
Test statistic 17.043
Degrees of freedom 17
P-value (Chi-square) 0.451
Model Test Baseline Model:
Test statistic 4190.601
Degrees of freedom 15
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 1.000
Tucker-Lewis Index (TLI) 1.000
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -4202.053
Loglikelihood unrestricted model (H1) -4193.532
Akaike (AIC) 8424.107
Bayesian (BIC) 8464.021
Sample-size adjusted Bayesian (BIC) 8432.291
Root Mean Square Error of Approximation:
RMSEA 0.003
90 Percent confidence interval - lower 0.000
90 Percent confidence interval - upper 0.045
P-value RMSEA <= 0.05 0.973
Standardized Root Mean Square Residual:
SRMR 0.018
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|)
intercept_affect =~
affect.1 1.000
affect.2 1.000
affect.3 1.000
affect.4 1.000
affect.5 1.000
affect.6 1.000
slope_affect =~
affect.1 0.000
affect.2 (bc1) 0.056 0.015 3.799 0.000
affect.3 (bc2) 0.095 0.015 6.550 0.000
affect.4 (bc3) 0.666 0.013 49.749 0.000
affect.5 (bc4) 0.827 0.014 58.819 0.000
affect.6 1.000
Covariances:
Estimate Std.Err z-value P(>|z|)
intercept_affect ~~
slope_affect -0.086 0.158 -0.546 0.585
Intercepts:
Estimate Std.Err z-value P(>|z|)
.affect.1 0.000
.affect.2 0.000
.affect.3 0.000
.affect.4 0.000
.affect.5 0.000
.affect.6 0.000
intercept_ffct 3.921 0.167 23.461 0.000
slope_affect 4.647 0.069 67.762 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
intrcp_ 10.178 0.746 13.651 0.000
slp_ffc -0.111 0.074 -1.499 0.134
.affct.1 (rs_v) 0.996 0.035 28.284 0.000
.affct.2 (rs_v) 0.996 0.035 28.284 0.000
.affct.3 (rs_v) 0.996 0.035 28.284 0.000
.affct.4 (rs_v) 0.996 0.035 28.284 0.000
.affct.5 (rs_v) 0.996 0.035 28.284 0.000
.affct.6 (rs_v) 0.996 0.035 28.284 0.000bc1 represents the percentage of change for the average individual between time 1 and 2. bc2 represents the percentage change betwen time 1 and 3, bc4 is the percentage change between time 1 and 5, etc.
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