simulateData_lavaan
2023-03-22
First part of this SEM exercise introduces data simulation for both singe and multi-group analysis. Second, an example of multi-group analysis is given (including measurement invariance testing across groups).
Simulate SEM data based on previous research
We will create a simulation data based on the previous SEM exercise results with simstandard package.
First, let's define the model we are going to use. Details are available in SEM_R.Rmd and SEM_R.html.
We will use the marker variable method (for other scaling options, see Beaujean, 2014, pp. 41) to scale the latent variables: One indicator (manifest) variable's loading for each latent variable is constrained to an arbitrary value (usually 1).
library(lavaan)
sem_model <- '
# Factors
ENC =~ NA*v5 + v7 + 1*v17 # Encouraging leadership
BUI =~ 1*v42r + v43 # Build-up of work requirements
PSY =~ 1*v44 + v45 # Psychic stress of the work
# Regressions
PSY ~ ENC + BUI # Predict PSY with ENC and BUI
# Covariances
ENC ~~ BUI # Let ENC and BUI correlate
'Second, we need to fit the model:
# read in dfttu4 data (n = 447)
dfttu4 <- read.csv("data1_csv.dat", header = T)
sem_model_fit <- sem(sem_model,
data=dfttu4,
mimic="mplus",
group = NULL, # no grouping variable (e.g., gender, control/intervention,...)
cluster = NULL, # no clustering variable (e.g., multilevel data may contain several values for each participant)
missing = "fiml.x" # full information maximum likelihood approach
)Is our model under/just/overidentified?
# how many parameters are to be estimated in the model?
inspect(sem_model_fit) # 24## $lambda
## ENC BUI PSY
## v5 1 0 0
## v7 2 0 0
## v17 0 0 0
## v42r 0 0 0
## v43 0 3 0
## v44 0 0 0
## v45 0 0 4
##
## $theta
## v5 v7 v17 v42r v43 v44 v45
## v5 8
## v7 0 9
## v17 0 0 10
## v42r 0 0 0 11
## v43 0 0 0 0 12
## v44 0 0 0 0 0 13
## v45 0 0 0 0 0 0 14
##
## $psi
## ENC BUI PSY
## ENC 15
## BUI 7 16
## PSY 0 0 17
##
## $beta
## ENC BUI PSY
## ENC 0 0 0
## BUI 0 0 0
## PSY 5 6 0
##
## $nu
## intrcp
## v5 18
## v7 19
## v17 20
## v42r 21
## v43 22
## v44 23
## v45 24
##
## $alpha
## intrcp
## ENC 0
## BUI 0
## PSY 0# the model is overidentified (good!) as there are 28 non-redundant elements [7(7+1)/2 = 56/2 = 28]
# and 24 parameters to be estimated (28 - 24 = 4):
#
# 4 manifest variable loadings on three factors (lambda matrix)
# 7 error variances of manifest variables (theta matrix)
# 3 latent variances and 1 covariance (psi matrix)
# 2 regressions (beta matrix)
# 7 intercepts(nu matrix)We may proceed as the model is overidentified.
Third, we'll check the results.
Note that “Estimate” column reports unstandardized values, while “Std.lv” column reports values when latent variables are standardized and “Std.all” column values reflect situation when both latent and observed (manifest) values are standardized.
summary(sem_model_fit,
standardized=TRUE,
rsq=TRUE,
fit.measures = T)## lavaan 0.6.14 ended normally after 41 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 24
##
## Number of observations 447
## Number of missing patterns 10
##
## Model Test User Model:
##
## Test statistic 62.390
## Degrees of freedom 11
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 1131.720
## Degrees of freedom 21
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.954
## Tucker-Lewis Index (TLI) 0.912
##
## Robust Comparative Fit Index (CFI) 0.954
## Robust Tucker-Lewis Index (TLI) 0.912
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -4334.157
## Loglikelihood unrestricted model (H1) -4302.962
##
## Akaike (AIC) 8716.315
## Bayesian (BIC) 8814.776
## Sample-size adjusted Bayesian (SABIC) 8738.610
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.102
## 90 Percent confidence interval - lower 0.078
## 90 Percent confidence interval - upper 0.128
## P-value H_0: RMSEA <= 0.050 0.000
## P-value H_0: RMSEA >= 0.080 0.938
##
## Robust RMSEA 0.103
## 90 Percent confidence interval - lower 0.079
## 90 Percent confidence interval - upper 0.128
## P-value H_0: Robust RMSEA <= 0.050 0.000
## P-value H_0: Robust RMSEA >= 0.080 0.940
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.062
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC =~
## v5 0.896 0.048 18.766 0.000 0.962 0.814
## v7 0.765 0.043 17.629 0.000 0.822 0.768
## v17 1.000 1.074 0.886
## BUI =~
## v42r 1.000 0.777 0.672
## v43 0.798 0.089 8.924 0.000 0.620 0.571
## PSY =~
## v44 1.000 1.118 0.901
## v45 0.592 0.068 8.702 0.000 0.662 0.551
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## PSY ~
## ENC -0.104 0.063 -1.653 0.098 -0.100 -0.100
## BUI 1.223 0.152 8.028 0.000 0.850 0.850
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC ~~
## BUI -0.253 0.062 -4.092 0.000 -0.303 -0.303
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 3.818 0.056 68.091 0.000 3.818 3.229
## .v7 3.436 0.051 67.756 0.000 3.436 3.211
## .v17 3.280 0.058 56.971 0.000 3.280 2.705
## .v42r 2.992 0.055 54.600 0.000 2.992 2.584
## .v43 3.803 0.052 73.253 0.000 3.803 3.501
## .v44 3.277 0.059 55.754 0.000 3.277 2.639
## .v45 2.387 0.057 42.027 0.000 2.387 1.990
## ENC 0.000 0.000 0.000
## BUI 0.000 0.000 0.000
## .PSY 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.473 0.047 9.978 0.000 0.473 0.338
## .v7 0.470 0.041 11.380 0.000 0.470 0.410
## .v17 0.317 0.048 6.588 0.000 0.317 0.216
## .v42r 0.736 0.080 9.226 0.000 0.736 0.549
## .v43 0.795 0.067 11.829 0.000 0.795 0.674
## .v44 0.291 0.119 2.458 0.014 0.291 0.189
## .v45 1.002 0.079 12.747 0.000 1.002 0.696
## ENC 1.153 0.105 10.934 0.000 1.000 1.000
## BUI 0.604 0.098 6.186 0.000 1.000 1.000
## .PSY 0.269 0.132 2.035 0.042 0.215 0.215
##
## R-Square:
## Estimate
## v5 0.662
## v7 0.590
## v17 0.784
## v42r 0.451
## v43 0.326
## v44 0.811
## v45 0.304
## PSY 0.785# Latent Variables:
# Estimate Std.Err z-value P(>|z|) Std.lv Std.all
# ENC =~
# v5 0.896 0.048 18.766 0.000 0.962 0.814
# v7 0.765 0.043 17.629 0.000 0.822 0.768
# v17 1.000 1.074 0.886
# BUI =~
# v42r 1.000 0.777 0.672
# v43 0.798 0.089 8.924 0.000 0.620 0.571
# PSY =~
# v44 1.000 1.118 0.901
# v45 0.592 0.068 8.702 0.000 0.662 0.551
#
# Regressions:
# Estimate Std.Err z-value P(>|z|) Std.lv Std.all
# PSY ~
# ENC -0.104 0.063 -1.653 0.098 -0.100 -0.100
# BUI 1.223 0.152 8.028 0.000 0.850 0.850
#
# Covariances:
# Estimate Std.Err z-value P(>|z|) Std.lv Std.all
# ENC ~~
# BUI -0.253 0.062 -4.092 0.000 -0.303 -0.303
# Variances:
# Estimate Std.Err z-value P(>|z|) Std.lv Std.all
# .v5 0.473 0.047 9.978 0.000 0.473 0.338
# .v7 0.470 0.041 11.380 0.000 0.470 0.410
# .v17 0.317 0.048 6.588 0.000 0.317 0.216
# .v42r 0.736 0.080 9.226 0.000 0.736 0.549
# .v43 0.795 0.067 11.829 0.000 0.795 0.674
# .v44 0.291 0.119 2.458 0.014 0.291 0.189
# .v45 1.002 0.079 12.747 0.000 1.002 0.696
# ENC 1.153 0.105 10.934 0.000 1.000 1.000
# BUI 0.604 0.098 6.186 0.000 1.000 1.000
# .PSY 0.269 0.132 2.035 0.042 0.215 0.215There is also a third type of standardization (“Std.nox”) where variables are standardized except for the exogenous observed (manifest) variables:
head(parameterEstimates(sem_model_fit, standardized = T, ci = F))## lhs op rhs est se z pvalue std.lv std.all std.nox
## 1 ENC =~ v5 0.896 0.048 18.766 0 0.962 0.814 0.814
## 2 ENC =~ v7 0.765 0.043 17.629 0 0.822 0.768 0.768
## 3 ENC =~ v17 1.000 0.000 NA NA 1.074 0.886 0.886
## 4 BUI =~ v42r 1.000 0.000 NA NA 0.777 0.672 0.672
## 5 BUI =~ v43 0.798 0.089 8.924 0 0.620 0.571 0.571
## 6 PSY =~ v44 1.000 0.000 NA NA 1.118 0.901 0.901# lhs op rhs est se z pvalue std.lv std.all std.nox
# 1 ENC =~ v5 0.896 0.048 18.766 0 0.962 0.814 0.814
# 2 ENC =~ v7 0.765 0.043 17.629 0 0.822 0.768 0.768
# 3 ENC =~ v17 1.000 0.000 NA NA 1.074 0.886 0.886
# 4 BUI =~ v42r 1.000 0.000 NA NA 0.777 0.672 0.672
# 5 BUI =~ v43 0.798 0.089 8.924 0 0.620 0.571 0.571
# 6 PSY =~ v44 1.000 0.000 NA NA 1.118 0.901 0.901This is the figure based on standardized results:
library(semPlot)
semPlot::semPaths(sem_model_fit,
layout="tree",
"std", # display standardized parameter estimates
edge.label.cex=0.7,
edge.width=0.2,
edge.color="black",
optimizeLatRes=TRUE,
label.scale=TRUE,
normalize=TRUE,
fade=FALSE,
# filetype = "jpg", # save file
# filename = "SEM_figure",
as.expression=c("nodes"))
library(lavaan)
# we'll use the same syntax than before, but upgraded with estimates taken from the results of the earlier SEM model (above)
sem_model_simulation <- '
# Factors
ENC =~ 0.81*v5 + 0.77*v7 + 0.87*v17 # standardized loadings from "Std.all" output above
BUI =~ 0.67*v42r + 0.57*v43
PSY =~ 0.90*v44 + 0.55*v45
# Regressions
PSY ~ -0.10*ENC + 0.85*BUI # standardized path coefficients from "Std.all" output above
# Covariances
ENC ~~ -.30*BUI # covariance from "Std.all" output above
'
# install.packages("simstandard")
library(simstandard)
# simulate data
set.seed(2023)
simulated_data <- simstandard::sim_standardized(sem_model_simulation, n = 100000)# Schneider's tutorial
# https://cran.r-project.org/web/packages/simstandard/vignettes/simstandard_tutorial.html
# has a really nice "ggcor" function to display matrices:
ggcor <- function(d) {
require(ggplot2)
as.data.frame(d) %>%
tibble::rownames_to_column("rowname") %>%
tidyr::pivot_longer(-rowname, names_to = "colname", values_to = "r") %>%
dplyr::mutate(rowname = forcats::fct_inorder(rowname) %>% forcats::fct_rev()) %>%
dplyr::mutate(colname = factor(colname,
levels = rev(levels(rowname)))) %>%
ggplot(aes(colname, rowname, fill = r)) +
geom_tile(color = "lightgray") +
geom_text(aes(label = formatC(r, digits = 2, format = "f") %>%
stringr::str_replace_all("0\\.",".") %>%
stringr::str_replace_all("1.00","1")),
color = "white",
fontface = "bold",
family = "serif") +
scale_fill_gradient2(NULL,
na.value = "darkgrey",
limits = c(-1.01, 1.01),
high = "darkred",
low = "darkblue") +
coord_equal() +
scale_x_discrete(NULL,position = "top") +
scale_y_discrete(NULL) +
theme_light(base_family = "serif", base_size = 12) +
theme(axis.text.x = element_text(angle = 90))
}
# check covariances
# cov(simulated_data) # quite good match with the original results!
# a way lot easier to read
cov(simulated_data) %>% ggcor() +
labs(title="covariance matrix of the observed and latent variables")
# inspect the matrices (A, S) that simstandard uses to create the data
sem_model_simulation_matrices <- sim_standardized_matrices(sem_model_simulation)
# Reticular Action Model (RAM) by McArdle (1978)
# See Ferrer et al. (2019, pp. 126-142) chapter
#
# A matrix contains all the asymmetric path coefficients (i.e., the loadings and the structural coefficients).
sem_model_simulation_matrices$RAM_matrices$A %>% ggcor() +
labs(title="A matrix: loadings and the structural coefficients")
# S matrix contains all the symmetric path coefficients (i.e., the variances and correlations of the observed and latent variables)
sem_model_simulation_matrices$RAM_matrices$S %>% ggcor() +
labs(title="S matrix: variances and correlations of the observed \n and latent variables")
We can see that the simulated data is close to the “sem_model_simulation” with fixed parameters:
# ENC =~ 0.81*v5 + 0.77*v7 + 0.87*v17
# BUI =~ 0.67*v42r + 0.57*v43
# PSY =~ 0.90*v44 + 0.55*v45
#
# PSY ~ -0.10*ENC + 0.85*BUI
#
# ENC ~~ -.30*BUI Next, we'll see how the lavaan output with simulated data and non-fixed model compares to the simulation results.
# # this was the original "sem_model_simulation" model:
# #
# sem_model_simulation <- '
#
# # Factors
# ENC =~ 0.81*v5 + 0.77*v7 + 0.87*v17 # standardized loadings from "Std.all" output above
# BUI =~ 0.67*v42r + 0.57*v43
# PSY =~ 0.90*v44 + 0.55*v45
#
# # Regressions
# PSY ~ -0.10*ENC + 0.85*BUI # standardized path coefficients from "Std.all" output above
#
# # Covariances
# ENC ~~ -.30*BUI # covariance from "Std.all" output above
# '
# we'll make a new version of model where all fixed parameters are set free
sem_model_simulation_free <- simstandard::fixed2free(sem_model_simulation)
# # check
# cat(sem_model_simulation_free)
# ENC ~~ BUI
# PSY ~ ENC + BUI
# BUI =~ v42r + v43
# ENC =~ v5 + v7 + v17
# PSY =~ v44 + v45
# actually, the resulting model is identical to our original model:
# sem_model <- '
# # Factors
# ENC =~ NA*v5 + v7 + 1*v17 # Encouraging leadership
# BUI =~ 1*v42r + v43 # Build-up of work requirements
# PSY =~ 1*v44 + v45 # Psychic stress of the work
# # Regressions
# PSY ~ ENC + BUI # Predict PSY with ENC and BUI
# # Covariances
# ENC ~~ BUI # Let ENC and BUI correlate
# '
# we'll use lavaan to see if the observed data in "sem_model_simulation" conform to the model in "sem_model_simulation_free"
# create simulated data
set.seed(2023)
simulated_data2 <- simstandard::sim_standardized(sem_model_simulation, # or use "sem_model"
latent = F, # Note: only following vars are included: "v5", "v7", "v17", "v42r", "v43", "v44", "v45"
errors = F,
n = 100000)
# evaluate the fit of free model on simulated data
lavaan_fit1 <- sem(model = sem_model_simulation_free,
data = simulated_data2)
# see the results
summary(lavaan_fit1,
standardized = T,
fit.measures = T)## lavaan 0.6.14 ended normally after 29 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 17
##
## Number of observations 100000
##
## Model Test User Model:
##
## Test statistic 10.908
## Degrees of freedom 11
## P-value (Chi-square) 0.451
##
## Model Test Baseline Model:
##
## Test statistic 235459.035
## Degrees of freedom 21
## 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) -875614.620
## Loglikelihood unrestricted model (H1) -875609.166
##
## Akaike (AIC) 1751263.240
## Bayesian (BIC) 1751424.959
## Sample-size adjusted Bayesian (SABIC) 1751370.933
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.000
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.003
## P-value H_0: RMSEA <= 0.050 1.000
## P-value H_0: RMSEA >= 0.080 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.001
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## BUI =~
## v42r 1.000 0.672 0.671
## v43 0.850 0.006 140.143 0.000 0.571 0.573
## ENC =~
## v5 1.000 0.811 0.811
## v7 0.952 0.004 251.582 0.000 0.772 0.770
## v17 1.074 0.004 269.927 0.000 0.871 0.870
## PSY =~
## v44 1.000 0.897 0.897
## v45 0.608 0.004 136.737 0.000 0.545 0.547
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## PSY ~
## ENC -0.116 0.004 -27.082 0.000 -0.105 -0.105
## BUI 1.139 0.009 125.759 0.000 0.853 0.853
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## BUI ~~
## ENC -0.162 0.003 -62.765 0.000 -0.297 -0.297
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v42r 0.551 0.004 142.768 0.000 0.551 0.549
## .v43 0.669 0.004 181.689 0.000 0.669 0.672
## .v5 0.341 0.002 147.360 0.000 0.341 0.342
## .v7 0.410 0.002 168.832 0.000 0.410 0.408
## .v17 0.244 0.002 107.338 0.000 0.244 0.243
## .v44 0.196 0.005 41.525 0.000 0.196 0.196
## .v45 0.696 0.004 195.755 0.000 0.696 0.701
## BUI 0.452 0.005 94.492 0.000 1.000 1.000
## ENC 0.658 0.005 144.705 0.000 1.000 1.000
## .PSY 0.167 0.006 28.828 0.000 0.208 0.208# extract RAM paths
sem_model_simulation_free_matrices <- simstandard::lav2ram(lavaan_fit1)
# display asymmetric paths (i.e., single-headed arrows for loadings and structure coefficients)
sem_model_simulation_free_matrices$A %>% ggcor() +
labs(title="A matrix: loadings and the structural coefficients")
# display symmetric paths (i.e., curved double-headed arrows exogenous variances, error variances, disturbance variances, and any covariances among these)
sem_model_simulation_free_matrices$S %>% ggcor() +
labs(title="S matrix: variances and correlations of the observed \n and latent variables")
lavaan results with the simulation data match the results of model with fixed parameters:
# ENC =~ 0.81*v5 + 0.77*v7 + 0.87*v17
# BUI =~ 0.67*v42r + 0.57*v43
# PSY =~ 0.90*v44 + 0.55*v45
#
# PSY ~ -0.10*ENC + 0.85*BUI
#
# ENC ~~ -.30*BUI So, we are happy!
Next, let's look how lavaan::simulateData can produce simulated data.
library(lavaan)
# compare to simstandard::sim_standardized
# define simulated measurement model (same as before)
sem_model_simulation <- '
# Factors
ENC =~ 0.81*v5 + 0.77*v7 + 0.87*v17 # standardized loadings from "Std.all" output above
BUI =~ 0.67*v42r + 0.57*v43
PSY =~ 0.90*v44 + 0.55*v45
# Regressions
PSY ~ -0.10*ENC + 0.85*BUI # standardized path coefficients from "Std.all" output above
# Covariances
ENC ~~ -.30*BUI # covariance from "Std.all" output above
'
# simulate 100 data frame rows
set.seed(2013)
simData_lavaan <- lavaan::simulateData(
sem_model_simulation,
standardized = T, # known issue to produce variances exceeding 2 if set to TRUE
# see https://github.com/yrosseel/lavaan/issues/46
sample.nobs=1000000
)
nrow(simData_lavaan)## [1] 1000000# define measurement model
sem_model <- '
# Factors
ENC =~ NA*v5 + v7 + 1*v17 # Encouraging leadership
BUI =~ 1*v42r + v43 # Build-up of work requirements
PSY =~ 1*v44 + v45 # Psychic stress of the work
# Regressions
PSY ~ ENC + BUI # Predict PSY with ENC and BUI
# Covariances
ENC ~~ BUI # Let ENC and BUI correlate
'
# fit the model on simulated data
lavaan_fit2 <- sem(sem_model, data=simData_lavaan)
# print the summary
summary(lavaan_fit2, standardized = T, fit.measures = T)## lavaan 0.6.14 ended normally after 30 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 17
##
## Number of observations 1000000
##
## Model Test User Model:
##
## Test statistic 13.945
## Degrees of freedom 11
## P-value (Chi-square) 0.236
##
## Model Test Baseline Model:
##
## Test statistic 2349574.554
## Degrees of freedom 21
## 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) -8757642.574
## Loglikelihood unrestricted model (H1) -8757635.601
##
## Akaike (AIC) 17515319.147
## Bayesian (BIC) 17515520.011
## Sample-size adjusted Bayesian (SABIC) 17515465.984
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.001
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.001
## P-value H_0: RMSEA <= 0.050 1.000
## P-value H_0: RMSEA >= 0.080 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.001
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC =~
## v5 0.930 0.001 852.061 0.000 0.810 0.810
## v7 0.885 0.001 817.830 0.000 0.771 0.771
## v17 1.000 0.871 0.871
## BUI =~
## v42r 1.000 0.668 0.669
## v43 0.853 0.002 439.447 0.000 0.570 0.570
## PSY =~
## v44 1.000 0.899 0.900
## v45 0.611 0.001 431.722 0.000 0.549 0.549
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## PSY ~
## ENC -0.103 0.001 -80.620 0.000 -0.099 -0.099
## BUI 1.145 0.003 392.682 0.000 0.851 0.851
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC ~~
## BUI -0.175 0.001 -201.168 0.000 -0.300 -0.300
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.345 0.001 469.008 0.000 0.345 0.344
## .v7 0.407 0.001 532.417 0.000 0.407 0.406
## .v17 0.243 0.001 337.703 0.000 0.243 0.242
## .v42r 0.552 0.001 451.761 0.000 0.552 0.553
## .v43 0.675 0.001 575.043 0.000 0.675 0.675
## .v44 0.190 0.002 126.262 0.000 0.190 0.190
## .v45 0.698 0.001 616.892 0.000 0.698 0.698
## ENC 0.759 0.002 501.915 0.000 1.000 1.000
## BUI 0.446 0.002 296.434 0.000 1.000 1.000
## .PSY 0.174 0.002 94.011 0.000 0.216 0.216# Latent Variables:
# Estimate Std.Err z-value P(>|z|) Std.lv Std.all
# ENC =~
# v5 0.930 0.001 852.061 0.000 0.810 0.810
# v7 0.885 0.001 817.830 0.000 0.771 0.771
# v17 1.000 0.871 0.871
# BUI =~
# v42r 1.000 0.668 0.669
# v43 0.853 0.002 439.447 0.000 0.570 0.570
# PSY =~
# v44 1.000 0.899 0.900
# v45 0.611 0.001 431.722 0.000 0.549 0.549
#
# Regressions:
# Estimate Std.Err z-value P(>|z|) Std.lv Std.all
# PSY ~
# ENC -0.103 0.001 -80.620 0.000 -0.099 -0.099
# BUI 1.145 0.003 392.682 0.000 0.851 0.851
#
# Covariances:
# Estimate Std.Err z-value P(>|z|) Std.lv Std.all
# ENC ~~
# BUI -0.175 0.001 -201.168 0.000 -0.300 -0.300# extract RAM paths
lavaan_sem_model_simulated_matrices <- simstandard::lav2ram(lavaan_fit2)
# display asymmetric paths (i.e., single-headed arrows for loadings and structure coefficients)
sem_model_simulation_free_matrices$A %>% ggcor() +
labs(title="A matrix (simstandard): loadings and the structural coefficients")
lavaan_sem_model_simulated_matrices$A %>% ggcor() +
labs(title="A matrix (lavaan): loadings and the structural coefficients")
# display symmetric paths (i.e., curved double-headed arrows exogenous variances, error variances, disturbance variances, and any covariances among these)
sem_model_simulation_free_matrices$S %>% ggcor() +
labs(title="S matrix (simstandard): variances and correlations of the observed \n and latent variables")
lavaan_sem_model_simulated_matrices$S %>% ggcor() +
labs(title="S matrix (lavaan): variances and correlations of the observed \n and latent variables")
lavaan simulation matches the simstandard results.
Next, we proceed with grouped SEM simulation using lavaan::simulateData.
library(lavaan)
# create three groups, each with 200 observations
# define population model for the simulated data (we'll use the same model as before)
sem_model_simulation <- '
# Factors
ENC =~ 0.81*v5 + 0.77*v7 + 0.87*v17
BUI =~ 0.67*v42r + 0.57*v43
PSY =~ 0.90*v44 + 0.55*v45
# Regressions
PSY ~ -0.10*ENC + 0.85*BUI
# Covariances
ENC ~~ -.30*BUI
'
# create simulated multi-group data based on the population model
set.seed(2023)
sim_multiGroupData_lavaan <- lavaan::simulateData(sem_model_simulation,
sample.nobs=c(200,200,200), # add "group" variable
standardized = T,
return.type = "data.frame")
sem_multiGroup_model <- '
# Factors
ENC =~ NA*v5 + v7 + 1*v17 # Encouraging leadership
BUI =~ 1*v42r + v43 # Build-up of work requirements
PSY =~ 1*v44 + v45 # Psychic stress of the work
# Regressions
PSY ~ ENC + BUI # Predict PSY with ENC and BUI
# Covariances
ENC ~~ BUI # Let ENC and BUI correlate
'# need to check that variances do not exceed 1
lavaan_mg_mi_fit1 <- sem(sem_multiGroup_model,
# group = "group",
data = sim_multiGroupData_lavaan)
# extract RAM paths
lavaan_mg_simulation_free_matrices <- simstandard::lav2ram(lavaan_mg_mi_fit1)
# display asymmetric paths (i.e., single-headed arrows for loadings and structure coefficients)
lavaan_mg_simulation_free_matrices$A %>% ggcor() +
labs(title="A matrix: loadings and the structural coefficients")
# display symmetric paths (i.e., curved double-headed arrows exogenous variances, error variances, disturbance variances, and any covariances among these)
lavaan_mg_simulation_free_matrices$S %>% ggcor() +
labs(title="S matrix: variances and correlations of the observed \n and latent variables")
# fit multi-group model and test measurement invariance (MI) using CFA
cfa_multiGroup_model <- '
# Factors
ENC =~ NA*v5 + v7 + 1*v17 # Encouraging leadership
BUI =~ 1*v42r + v43 # Build-up of work requirements
PSY =~ 1*v44 + v45 # Psychic stress of the work
# Correlations
ENC ~~ BUI
ENC ~~ PSY
BUI ~~ PSY
'
# -------------------------
# 1. configural invariance: the same factor structure is imposed on all groups
lavaan_mg_mi_fit1conf <- cfa(cfa_multiGroup_model,
group = "group",
data = sim_multiGroupData_lavaan)
summary(lavaan_mg_mi_fit1conf, fit.measures = TRUE, standardized = TRUE)## lavaan 0.6.14 ended normally after 48 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 72
##
## Number of observations per group:
## 1 200
## 2 200
## 3 200
##
## Model Test User Model:
##
## Test statistic 33.939
## Degrees of freedom 33
## P-value (Chi-square) 0.422
## Test statistic for each group:
## 1 13.813
## 2 11.823
## 3 8.303
##
## Model Test Baseline Model:
##
## Test statistic 1504.518
## Degrees of freedom 63
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.999
## Tucker-Lewis Index (TLI) 0.999
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -5187.488
## Loglikelihood unrestricted model (H1) -5170.518
##
## Akaike (AIC) 10518.975
## Bayesian (BIC) 10835.554
## Sample-size adjusted Bayesian (SABIC) 10606.974
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.012
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.054
## P-value H_0: RMSEA <= 0.050 0.925
## P-value H_0: RMSEA >= 0.080 0.001
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.031
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [1]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC =~
## v5 0.917 0.080 11.421 0.000 0.744 0.816
## v7 0.859 0.081 10.592 0.000 0.697 0.735
## v17 1.000 0.812 0.854
## BUI =~
## v42r 1.000 0.803 0.734
## v43 0.730 0.119 6.150 0.000 0.587 0.555
## PSY =~
## v44 1.000 0.847 0.867
## v45 0.592 0.099 5.998 0.000 0.501 0.527
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC ~~
## BUI -0.157 0.065 -2.409 0.016 -0.241 -0.241
## PSY -0.267 0.064 -4.175 0.000 -0.388 -0.388
## BUI ~~
## PSY 0.592 0.086 6.867 0.000 0.870 0.870
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 -0.130 0.064 -2.019 0.044 -0.130 -0.143
## .v7 -0.055 0.067 -0.818 0.413 -0.055 -0.058
## .v17 -0.001 0.067 -0.022 0.983 -0.001 -0.002
## .v42r 0.074 0.077 0.957 0.339 0.074 0.068
## .v43 0.163 0.075 2.178 0.029 0.163 0.154
## .v44 0.153 0.069 2.211 0.027 0.153 0.156
## .v45 0.072 0.067 1.069 0.285 0.072 0.076
## ENC 0.000 0.000 0.000
## BUI 0.000 0.000 0.000
## PSY 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.277 0.046 6.093 0.000 0.277 0.334
## .v7 0.412 0.052 7.871 0.000 0.412 0.459
## .v17 0.244 0.049 4.958 0.000 0.244 0.270
## .v42r 0.554 0.108 5.137 0.000 0.554 0.462
## .v43 0.772 0.092 8.425 0.000 0.772 0.692
## .v44 0.236 0.098 2.418 0.016 0.236 0.248
## .v45 0.652 0.073 8.912 0.000 0.652 0.722
## ENC 0.659 0.097 6.805 0.000 1.000 1.000
## BUI 0.646 0.141 4.577 0.000 1.000 1.000
## PSY 0.717 0.132 5.421 0.000 1.000 1.000
##
##
## Group 2 [2]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC =~
## v5 1.004 0.076 13.231 0.000 0.877 0.887
## v7 0.797 0.074 10.793 0.000 0.697 0.705
## v17 1.000 0.874 0.856
## BUI =~
## v42r 1.000 0.727 0.760
## v43 0.763 0.106 7.184 0.000 0.554 0.601
## PSY =~
## v44 1.000 0.838 0.753
## v45 0.731 0.092 7.926 0.000 0.612 0.609
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC ~~
## BUI -0.181 0.061 -2.955 0.003 -0.285 -0.285
## PSY -0.410 0.078 -5.265 0.000 -0.560 -0.560
## BUI ~~
## PSY 0.601 0.085 7.057 0.000 0.988 0.988
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.070 0.070 0.994 0.320 0.070 0.070
## .v7 0.049 0.070 0.707 0.480 0.049 0.050
## .v17 0.023 0.072 0.318 0.751 0.023 0.022
## .v42r -0.094 0.068 -1.394 0.163 -0.094 -0.099
## .v43 -0.003 0.065 -0.040 0.968 -0.003 -0.003
## .v44 -0.013 0.079 -0.171 0.864 -0.013 -0.012
## .v45 -0.005 0.071 -0.077 0.938 -0.005 -0.005
## ENC 0.000 0.000 0.000
## BUI 0.000 0.000 0.000
## PSY 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.209 0.047 4.450 0.000 0.209 0.214
## .v7 0.492 0.057 8.633 0.000 0.492 0.503
## .v17 0.279 0.050 5.523 0.000 0.279 0.267
## .v42r 0.387 0.072 5.356 0.000 0.387 0.423
## .v43 0.544 0.065 8.371 0.000 0.544 0.639
## .v44 0.537 0.089 6.056 0.000 0.537 0.433
## .v45 0.635 0.074 8.601 0.000 0.635 0.629
## ENC 0.764 0.109 7.015 0.000 1.000 1.000
## BUI 0.528 0.103 5.130 0.000 1.000 1.000
## PSY 0.702 0.132 5.315 0.000 1.000 1.000
##
##
## Group 3 [3]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC =~
## v5 0.935 0.077 12.111 0.000 0.761 0.800
## v7 0.981 0.081 12.034 0.000 0.798 0.794
## v17 1.000 0.814 0.872
## BUI =~
## v42r 1.000 0.779 0.741
## v43 0.794 0.117 6.768 0.000 0.618 0.598
## PSY =~
## v44 1.000 0.916 0.923
## v45 0.598 0.097 6.156 0.000 0.547 0.549
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC ~~
## BUI -0.162 0.062 -2.616 0.009 -0.256 -0.256
## PSY -0.228 0.064 -3.577 0.000 -0.306 -0.306
## BUI ~~
## PSY 0.595 0.085 7.014 0.000 0.835 0.835
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.067 0.067 0.996 0.319 0.067 0.070
## .v7 -0.023 0.071 -0.331 0.741 -0.023 -0.023
## .v17 0.084 0.066 1.268 0.205 0.084 0.090
## .v42r -0.064 0.074 -0.861 0.389 -0.064 -0.061
## .v43 -0.113 0.073 -1.544 0.123 -0.113 -0.109
## .v44 -0.124 0.070 -1.770 0.077 -0.124 -0.125
## .v45 -0.064 0.070 -0.906 0.365 -0.064 -0.064
## ENC 0.000 0.000 0.000
## BUI 0.000 0.000 0.000
## PSY 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.325 0.047 6.974 0.000 0.325 0.360
## .v7 0.373 0.052 7.119 0.000 0.373 0.369
## .v17 0.209 0.043 4.836 0.000 0.209 0.240
## .v42r 0.499 0.092 5.433 0.000 0.499 0.451
## .v43 0.688 0.084 8.172 0.000 0.688 0.643
## .v44 0.147 0.107 1.371 0.171 0.147 0.149
## .v45 0.692 0.079 8.774 0.000 0.692 0.698
## ENC 0.662 0.093 7.143 0.000 1.000 1.000
## BUI 0.607 0.125 4.848 0.000 1.000 1.000
## PSY 0.838 0.144 5.825 0.000 1.000 1.000# fitMeasures(lavaan_mg_mi_fit1conf)
# ----------------------------
# 2. metric (weak) invariance: the factor loadings are constrained to be equal across groups
lavaan_mg_mi_fit2metric <- cfa(cfa_multiGroup_model,
group = "group",
group.equal = c("loadings"),
data = sim_multiGroupData_lavaan)
summary(lavaan_mg_mi_fit2metric, fit.measures = TRUE, standardized = TRUE)## lavaan 0.6.14 ended normally after 41 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 72
## Number of equality constraints 8
##
## Number of observations per group:
## 1 200
## 2 200
## 3 200
##
## Model Test User Model:
##
## Test statistic 41.022
## Degrees of freedom 41
## P-value (Chi-square) 0.470
## Test statistic for each group:
## 1 14.436
## 2 15.373
## 3 11.212
##
## Model Test Baseline Model:
##
## Test statistic 1504.518
## Degrees of freedom 63
## 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) -5191.029
## Loglikelihood unrestricted model (H1) -5170.518
##
## Akaike (AIC) 10510.058
## Bayesian (BIC) 10791.462
## Sample-size adjusted Bayesian (SABIC) 10588.279
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.002
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.048
## P-value H_0: RMSEA <= 0.050 0.959
## P-value H_0: RMSEA >= 0.080 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.038
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [1]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC =~
## v5 (.p1.) 0.955 0.045 21.303 0.000 0.760 0.826
## v7 (.p2.) 0.876 0.045 19.326 0.000 0.698 0.736
## v17 1.000 0.796 0.845
## BUI =~
## v42r 1.000 0.789 0.723
## v43 (.p5.) 0.766 0.066 11.643 0.000 0.604 0.568
## PSY =~
## v44 1.000 0.822 0.846
## v45 (.p7.) 0.650 0.055 11.859 0.000 0.534 0.555
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC ~~
## BUI -0.153 0.063 -2.436 0.015 -0.243 -0.243
## PSY -0.254 0.061 -4.165 0.000 -0.388 -0.388
## BUI ~~
## PSY 0.578 0.081 7.141 0.000 0.891 0.891
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 -0.130 0.065 -2.001 0.045 -0.130 -0.141
## .v7 -0.055 0.067 -0.818 0.413 -0.055 -0.058
## .v17 -0.001 0.067 -0.022 0.982 -0.001 -0.002
## .v42r 0.074 0.077 0.961 0.337 0.074 0.068
## .v43 0.163 0.075 2.165 0.030 0.163 0.153
## .v44 0.153 0.069 2.222 0.026 0.153 0.157
## .v45 0.072 0.068 1.056 0.291 0.072 0.075
## ENC 0.000 0.000 0.000
## BUI 0.000 0.000 0.000
## PSY 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.268 0.042 6.376 0.000 0.268 0.317
## .v7 0.411 0.051 8.107 0.000 0.411 0.458
## .v17 0.255 0.043 5.862 0.000 0.255 0.287
## .v42r 0.568 0.099 5.749 0.000 0.568 0.477
## .v43 0.765 0.090 8.499 0.000 0.765 0.677
## .v44 0.268 0.079 3.407 0.001 0.268 0.284
## .v45 0.641 0.071 8.991 0.000 0.641 0.692
## ENC 0.634 0.081 7.825 0.000 1.000 1.000
## BUI 0.622 0.121 5.160 0.000 1.000 1.000
## PSY 0.676 0.111 6.098 0.000 1.000 1.000
##
##
## Group 2 [2]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC =~
## v5 (.p1.) 0.955 0.045 21.303 0.000 0.838 0.866
## v7 (.p2.) 0.876 0.045 19.326 0.000 0.769 0.743
## v17 1.000 0.878 0.861
## BUI =~
## v42r 1.000 0.725 0.758
## v43 (.p5.) 0.766 0.066 11.643 0.000 0.555 0.602
## PSY =~
## v44 1.000 0.868 0.769
## v45 (.p7.) 0.650 0.055 11.859 0.000 0.564 0.574
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC ~~
## BUI -0.183 0.061 -3.006 0.003 -0.288 -0.288
## PSY -0.425 0.078 -5.438 0.000 -0.558 -0.558
## BUI ~~
## PSY 0.620 0.083 7.455 0.000 0.985 0.985
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.070 0.068 1.017 0.309 0.070 0.072
## .v7 0.049 0.073 0.675 0.500 0.049 0.048
## .v17 0.023 0.072 0.318 0.751 0.023 0.022
## .v42r -0.094 0.068 -1.394 0.163 -0.094 -0.099
## .v43 -0.003 0.065 -0.040 0.968 -0.003 -0.003
## .v44 -0.013 0.080 -0.168 0.866 -0.013 -0.012
## .v45 -0.005 0.069 -0.079 0.937 -0.005 -0.006
## ENC 0.000 0.000 0.000
## BUI 0.000 0.000 0.000
## PSY 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.234 0.041 5.757 0.000 0.234 0.250
## .v7 0.479 0.057 8.375 0.000 0.479 0.447
## .v17 0.269 0.045 5.929 0.000 0.269 0.259
## .v42r 0.389 0.069 5.623 0.000 0.389 0.425
## .v43 0.542 0.064 8.500 0.000 0.542 0.637
## .v44 0.520 0.093 5.568 0.000 0.520 0.408
## .v45 0.646 0.072 8.919 0.000 0.646 0.670
## ENC 0.770 0.096 7.984 0.000 1.000 1.000
## BUI 0.526 0.094 5.620 0.000 1.000 1.000
## PSY 0.754 0.132 5.693 0.000 1.000 1.000
##
##
## Group 3 [3]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC =~
## v5 (.p1.) 0.955 0.045 21.303 0.000 0.791 0.814
## v7 (.p2.) 0.876 0.045 19.326 0.000 0.726 0.753
## v17 1.000 0.829 0.882
## BUI =~
## v42r 1.000 0.789 0.748
## v43 (.p5.) 0.766 0.066 11.643 0.000 0.604 0.588
## PSY =~
## v44 1.000 0.892 0.901
## v45 (.p7.) 0.650 0.055 11.859 0.000 0.579 0.575
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC ~~
## BUI -0.166 0.063 -2.631 0.009 -0.254 -0.254
## PSY -0.232 0.064 -3.631 0.000 -0.314 -0.314
## BUI ~~
## PSY 0.596 0.082 7.290 0.000 0.847 0.847
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.067 0.069 0.974 0.330 0.067 0.069
## .v7 -0.023 0.068 -0.344 0.731 -0.023 -0.024
## .v17 0.084 0.066 1.259 0.208 0.084 0.089
## .v42r -0.064 0.075 -0.857 0.391 -0.064 -0.061
## .v43 -0.113 0.073 -1.553 0.120 -0.113 -0.110
## .v44 -0.124 0.070 -1.776 0.076 -0.124 -0.126
## .v45 -0.064 0.071 -0.895 0.371 -0.064 -0.063
## ENC 0.000 0.000 0.000
## BUI 0.000 0.000 0.000
## PSY 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.318 0.046 6.954 0.000 0.318 0.337
## .v7 0.403 0.050 8.085 0.000 0.403 0.433
## .v17 0.197 0.040 4.901 0.000 0.197 0.223
## .v42r 0.491 0.089 5.530 0.000 0.491 0.441
## .v43 0.693 0.082 8.473 0.000 0.693 0.655
## .v44 0.183 0.081 2.251 0.024 0.183 0.187
## .v45 0.679 0.076 8.968 0.000 0.679 0.669
## ENC 0.687 0.085 8.077 0.000 1.000 1.000
## BUI 0.623 0.114 5.444 0.000 1.000 1.000
## PSY 0.795 0.120 6.626 0.000 1.000 1.000# fitMeasures(lavaan_mg_mi_fit2weak)
# ------------------------------
# 3. scalar (strong) invariance: the factor loadings and intercepts are constrained to be equal across groups
lavaan_mg_mi_fit3scalar <- cfa(cfa_multiGroup_model,
group = "group",
group.equal = c("intercepts", "loadings"),
data = sim_multiGroupData_lavaan)
summary(lavaan_mg_mi_fit3scalar, fit.measures = TRUE, standardized = TRUE)## lavaan 0.6.14 ended normally after 48 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 78
## Number of equality constraints 22
##
## Number of observations per group:
## 1 200
## 2 200
## 3 200
##
## Model Test User Model:
##
## Test statistic 53.124
## Degrees of freedom 49
## P-value (Chi-square) 0.318
## Test statistic for each group:
## 1 19.518
## 2 18.604
## 3 15.001
##
## Model Test Baseline Model:
##
## Test statistic 1504.518
## Degrees of freedom 63
## 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) -5197.080
## Loglikelihood unrestricted model (H1) -5170.518
##
## Akaike (AIC) 10506.160
## Bayesian (BIC) 10752.388
## Sample-size adjusted Bayesian (SABIC) 10574.604
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.021
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.051
## P-value H_0: RMSEA <= 0.050 0.939
## P-value H_0: RMSEA >= 0.080 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.041
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [1]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC =~
## v5 (.p1.) 0.963 0.045 21.257 0.000 0.763 0.825
## v7 (.p2.) 0.879 0.046 19.276 0.000 0.696 0.736
## v17 1.000 0.792 0.841
## BUI =~
## v42r 1.000 0.783 0.719
## v43 (.p5.) 0.778 0.066 11.806 0.000 0.609 0.571
## PSY =~
## v44 1.000 0.824 0.848
## v45 (.p7.) 0.645 0.054 11.967 0.000 0.531 0.553
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC ~~
## BUI -0.152 0.062 -2.444 0.015 -0.245 -0.245
## PSY -0.254 0.061 -4.173 0.000 -0.389 -0.389
## BUI ~~
## PSY 0.575 0.081 7.138 0.000 0.892 0.892
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 (.21.) -0.072 0.061 -1.184 0.237 -0.072 -0.078
## .v7 (.22.) -0.079 0.058 -1.361 0.174 -0.079 -0.084
## .v17 (.23.) -0.041 0.063 -0.654 0.513 -0.041 -0.044
## .v42r (.24.) 0.101 0.073 1.385 0.166 0.101 0.093
## .v43 (.25.) 0.117 0.062 1.887 0.059 0.117 0.110
## .v44 (.26.) 0.147 0.068 2.170 0.030 0.147 0.151
## .v45 (.27.) 0.092 0.054 1.720 0.085 0.092 0.096
## ENC 0.000 0.000 0.000
## BUI 0.000 0.000 0.000
## PSY 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.273 0.043 6.372 0.000 0.273 0.319
## .v7 0.411 0.051 8.089 0.000 0.411 0.459
## .v17 0.259 0.044 5.923 0.000 0.259 0.293
## .v42r 0.574 0.098 5.847 0.000 0.574 0.484
## .v43 0.766 0.090 8.464 0.000 0.766 0.674
## .v44 0.266 0.079 3.365 0.001 0.266 0.282
## .v45 0.642 0.071 9.008 0.000 0.642 0.695
## ENC 0.627 0.080 7.798 0.000 1.000 1.000
## BUI 0.613 0.119 5.148 0.000 1.000 1.000
## PSY 0.679 0.111 6.102 0.000 1.000 1.000
##
##
## Group 2 [2]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC =~
## v5 (.p1.) 0.963 0.045 21.257 0.000 0.841 0.867
## v7 (.p2.) 0.879 0.046 19.276 0.000 0.768 0.743
## v17 1.000 0.874 0.857
## BUI =~
## v42r 1.000 0.721 0.755
## v43 (.p5.) 0.778 0.066 11.806 0.000 0.561 0.606
## PSY =~
## v44 1.000 0.870 0.770
## v45 (.p7.) 0.645 0.054 11.967 0.000 0.561 0.572
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC ~~
## BUI -0.183 0.061 -3.018 0.003 -0.290 -0.290
## PSY -0.424 0.078 -5.435 0.000 -0.558 -0.558
## BUI ~~
## PSY 0.618 0.083 7.456 0.000 0.986 0.986
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 (.21.) -0.072 0.061 -1.184 0.237 -0.072 -0.074
## .v7 (.22.) -0.079 0.058 -1.361 0.174 -0.079 -0.077
## .v17 (.23.) -0.041 0.063 -0.654 0.513 -0.041 -0.040
## .v42r (.24.) 0.101 0.073 1.385 0.166 0.101 0.106
## .v43 (.25.) 0.117 0.062 1.887 0.059 0.117 0.126
## .v44 (.26.) 0.147 0.068 2.170 0.030 0.147 0.130
## .v45 (.27.) 0.092 0.054 1.720 0.085 0.092 0.094
## ENC 0.114 0.090 1.275 0.202 0.131 0.131
## BUI -0.182 0.095 -1.921 0.055 -0.253 -0.253
## PSY -0.158 0.101 -1.561 0.119 -0.182 -0.182
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.233 0.041 5.678 0.000 0.233 0.248
## .v7 0.479 0.057 8.367 0.000 0.479 0.449
## .v17 0.277 0.046 6.040 0.000 0.277 0.266
## .v42r 0.393 0.069 5.719 0.000 0.393 0.431
## .v43 0.541 0.064 8.461 0.000 0.541 0.633
## .v44 0.519 0.094 5.528 0.000 0.519 0.407
## .v45 0.647 0.072 8.934 0.000 0.647 0.673
## ENC 0.763 0.096 7.954 0.000 1.000 1.000
## BUI 0.520 0.093 5.617 0.000 1.000 1.000
## PSY 0.757 0.133 5.697 0.000 1.000 1.000
##
##
## Group 3 [3]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC =~
## v5 (.p1.) 0.963 0.045 21.257 0.000 0.795 0.816
## v7 (.p2.) 0.879 0.046 19.276 0.000 0.726 0.751
## v17 1.000 0.826 0.880
## BUI =~
## v42r 1.000 0.783 0.742
## v43 (.p5.) 0.778 0.066 11.806 0.000 0.608 0.589
## PSY =~
## v44 1.000 0.894 0.903
## v45 (.p7.) 0.645 0.054 11.967 0.000 0.576 0.573
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC ~~
## BUI -0.165 0.063 -2.630 0.009 -0.255 -0.255
## PSY -0.231 0.064 -3.624 0.000 -0.313 -0.313
## BUI ~~
## PSY 0.594 0.081 7.289 0.000 0.849 0.849
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 (.21.) -0.072 0.061 -1.184 0.237 -0.072 -0.074
## .v7 (.22.) -0.079 0.058 -1.361 0.174 -0.079 -0.082
## .v17 (.23.) -0.041 0.063 -0.654 0.513 -0.041 -0.044
## .v42r (.24.) 0.101 0.073 1.385 0.166 0.101 0.096
## .v43 (.25.) 0.117 0.062 1.887 0.059 0.117 0.113
## .v44 (.26.) 0.147 0.068 2.170 0.030 0.147 0.149
## .v45 (.27.) 0.092 0.054 1.720 0.085 0.092 0.092
## ENC 0.119 0.087 1.361 0.174 0.144 0.144
## BUI -0.204 0.099 -2.056 0.040 -0.261 -0.261
## PSY -0.268 0.097 -2.776 0.005 -0.300 -0.300
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.317 0.046 6.899 0.000 0.317 0.334
## .v7 0.406 0.050 8.100 0.000 0.406 0.435
## .v17 0.199 0.040 4.947 0.000 0.199 0.226
## .v42r 0.500 0.089 5.647 0.000 0.500 0.449
## .v43 0.698 0.083 8.449 0.000 0.698 0.653
## .v44 0.181 0.082 2.207 0.027 0.181 0.184
## .v45 0.680 0.076 8.987 0.000 0.680 0.672
## ENC 0.682 0.085 8.063 0.000 1.000 1.000
## BUI 0.612 0.113 5.415 0.000 1.000 1.000
## PSY 0.799 0.120 6.634 0.000 1.000 1.000# fitMeasures(lavaan_mg_mi_fit3scalar)
# --------------------------------
# 4. strict (residual) invariance: the factor loadings, intercepts and residuals are constrained to be equal across groups
lavaan_mg_mi_fit4strict <- cfa(cfa_multiGroup_model,
group = "group",
group.equal = c("intercepts", "loadings", "residuals"),
data = sim_multiGroupData_lavaan)
summary(lavaan_mg_mi_fit4strict, fit.measures = TRUE, standardized = TRUE)## lavaan 0.6.14 ended normally after 47 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 78
## Number of equality constraints 36
##
## Number of observations per group:
## 1 200
## 2 200
## 3 200
##
## Model Test User Model:
##
## Test statistic 73.614
## Degrees of freedom 63
## P-value (Chi-square) 0.170
## Test statistic for each group:
## 1 22.616
## 2 30.028
## 3 20.970
##
## Model Test Baseline Model:
##
## Test statistic 1504.518
## Degrees of freedom 63
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.993
## Tucker-Lewis Index (TLI) 0.993
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -5207.325
## Loglikelihood unrestricted model (H1) -5170.518
##
## Akaike (AIC) 10498.650
## Bayesian (BIC) 10683.321
## Sample-size adjusted Bayesian (SABIC) 10549.983
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.029
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.054
## P-value H_0: RMSEA <= 0.050 0.914
## P-value H_0: RMSEA >= 0.080 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.043
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [1]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC =~
## v5 (.p1.) 0.962 0.046 21.131 0.000 0.764 0.826
## v7 (.p2.) 0.874 0.046 19.158 0.000 0.694 0.726
## v17 1.000 0.794 0.848
## BUI =~
## v42r 1.000 0.815 0.757
## v43 (.p5.) 0.784 0.066 11.859 0.000 0.639 0.616
## PSY =~
## v44 1.000 0.797 0.814
## v45 (.p7.) 0.649 0.054 11.982 0.000 0.517 0.538
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC ~~
## BUI -0.151 0.062 -2.439 0.015 -0.234 -0.234
## PSY -0.249 0.061 -4.098 0.000 -0.394 -0.394
## BUI ~~
## PSY 0.574 0.080 7.133 0.000 0.883 0.883
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 (.21.) -0.072 0.061 -1.177 0.239 -0.072 -0.078
## .v7 (.22.) -0.077 0.058 -1.313 0.189 -0.077 -0.080
## .v17 (.23.) -0.042 0.063 -0.665 0.506 -0.042 -0.045
## .v42r (.24.) 0.101 0.073 1.387 0.165 0.101 0.094
## .v43 (.25.) 0.117 0.062 1.874 0.061 0.117 0.113
## .v44 (.26.) 0.146 0.068 2.150 0.032 0.146 0.149
## .v45 (.27.) 0.092 0.053 1.732 0.083 0.092 0.096
## ENC 0.000 0.000 0.000
## BUI 0.000 0.000 0.000
## PSY 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 (.11.) 0.272 0.027 10.016 0.000 0.272 0.317
## .v7 (.12.) 0.434 0.031 13.855 0.000 0.434 0.474
## .v17 (.13.) 0.247 0.028 8.906 0.000 0.247 0.282
## .v42r (.14.) 0.493 0.051 9.670 0.000 0.493 0.426
## .v43 (.15.) 0.665 0.046 14.348 0.000 0.665 0.620
## .v44 (.16.) 0.323 0.052 6.175 0.000 0.323 0.337
## .v45 (.17.) 0.656 0.043 15.213 0.000 0.656 0.710
## ENC 0.631 0.080 7.858 0.000 1.000 1.000
## BUI 0.664 0.111 5.963 0.000 1.000 1.000
## PSY 0.635 0.102 6.244 0.000 1.000 1.000
##
##
## Group 2 [2]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC =~
## v5 (.p1.) 0.962 0.046 21.131 0.000 0.840 0.850
## v7 (.p2.) 0.874 0.046 19.158 0.000 0.763 0.757
## v17 1.000 0.873 0.869
## BUI =~
## v42r 1.000 0.671 0.691
## v43 (.p5.) 0.784 0.066 11.859 0.000 0.526 0.542
## PSY =~
## v44 1.000 0.936 0.855
## v45 (.p7.) 0.649 0.054 11.982 0.000 0.608 0.600
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC ~~
## BUI -0.184 0.060 -3.042 0.002 -0.314 -0.314
## PSY -0.416 0.077 -5.383 0.000 -0.509 -0.509
## BUI ~~
## PSY 0.611 0.082 7.453 0.000 0.973 0.973
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 (.21.) -0.072 0.061 -1.177 0.239 -0.072 -0.073
## .v7 (.22.) -0.077 0.058 -1.313 0.189 -0.077 -0.076
## .v17 (.23.) -0.042 0.063 -0.665 0.506 -0.042 -0.042
## .v42r (.24.) 0.101 0.073 1.387 0.165 0.101 0.104
## .v43 (.25.) 0.117 0.062 1.874 0.061 0.117 0.120
## .v44 (.26.) 0.146 0.068 2.150 0.032 0.146 0.133
## .v45 (.27.) 0.092 0.053 1.732 0.083 0.092 0.091
## ENC 0.110 0.090 1.229 0.219 0.126 0.126
## BUI -0.182 0.095 -1.917 0.055 -0.271 -0.271
## PSY -0.158 0.101 -1.561 0.119 -0.169 -0.169
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 (.11.) 0.272 0.027 10.016 0.000 0.272 0.278
## .v7 (.12.) 0.434 0.031 13.855 0.000 0.434 0.427
## .v17 (.13.) 0.247 0.028 8.906 0.000 0.247 0.245
## .v42r (.14.) 0.493 0.051 9.670 0.000 0.493 0.523
## .v43 (.15.) 0.665 0.046 14.348 0.000 0.665 0.707
## .v44 (.16.) 0.323 0.052 6.175 0.000 0.323 0.269
## .v45 (.17.) 0.656 0.043 15.213 0.000 0.656 0.640
## ENC 0.762 0.095 8.041 0.000 1.000 1.000
## BUI 0.450 0.088 5.112 0.000 1.000 1.000
## PSY 0.877 0.126 6.970 0.000 1.000 1.000
##
##
## Group 3 [3]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC =~
## v5 (.p1.) 0.962 0.046 21.131 0.000 0.792 0.835
## v7 (.p2.) 0.874 0.046 19.158 0.000 0.719 0.737
## v17 1.000 0.823 0.856
## BUI =~
## v42r 1.000 0.786 0.746
## v43 (.p5.) 0.784 0.066 11.859 0.000 0.616 0.603
## PSY =~
## v44 1.000 0.835 0.826
## v45 (.p7.) 0.649 0.054 11.982 0.000 0.542 0.556
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC ~~
## BUI -0.162 0.063 -2.592 0.010 -0.251 -0.251
## PSY -0.232 0.064 -3.627 0.000 -0.338 -0.338
## BUI ~~
## PSY 0.587 0.082 7.197 0.000 0.894 0.894
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 (.21.) -0.072 0.061 -1.177 0.239 -0.072 -0.076
## .v7 (.22.) -0.077 0.058 -1.313 0.189 -0.077 -0.079
## .v17 (.23.) -0.042 0.063 -0.665 0.506 -0.042 -0.043
## .v42r (.24.) 0.101 0.073 1.387 0.165 0.101 0.096
## .v43 (.25.) 0.117 0.062 1.874 0.061 0.117 0.114
## .v44 (.26.) 0.146 0.068 2.150 0.032 0.146 0.145
## .v45 (.27.) 0.092 0.053 1.732 0.083 0.092 0.095
## ENC 0.120 0.087 1.374 0.170 0.146 0.146
## BUI -0.205 0.099 -2.067 0.039 -0.261 -0.261
## PSY -0.265 0.097 -2.742 0.006 -0.318 -0.318
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 (.11.) 0.272 0.027 10.016 0.000 0.272 0.302
## .v7 (.12.) 0.434 0.031 13.855 0.000 0.434 0.456
## .v17 (.13.) 0.247 0.028 8.906 0.000 0.247 0.268
## .v42r (.14.) 0.493 0.051 9.670 0.000 0.493 0.444
## .v43 (.15.) 0.665 0.046 14.348 0.000 0.665 0.637
## .v44 (.16.) 0.323 0.052 6.175 0.000 0.323 0.317
## .v45 (.17.) 0.656 0.043 15.213 0.000 0.656 0.691
## ENC 0.677 0.085 7.930 0.000 1.000 1.000
## BUI 0.618 0.106 5.813 0.000 1.000 1.000
## PSY 0.697 0.108 6.462 0.000 1.000 1.000# fitMeasures(lavaan_mg_mi_fit4resid)# test for significant differences between groups with chi-square test
# first, lavaan::lavTestLRT
lavaan::lavTestLRT(lavaan_mg_mi_fit1conf,lavaan_mg_mi_fit2metric,lavaan_mg_mi_fit3scalar,lavaan_mg_mi_fit4strict)##
## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff RMSEA Df diff
## lavaan_mg_mi_fit1conf 33 10519 10836 33.939
## lavaan_mg_mi_fit2metric 41 10510 10792 41.022 7.0831 0.000000 8
## lavaan_mg_mi_fit3scalar 49 10506 10752 53.124 12.1020 0.050633 8
## lavaan_mg_mi_fit4strict 63 10499 10683 73.614 20.4899 0.048144 14
## Pr(>Chisq)
## lavaan_mg_mi_fit1conf
## lavaan_mg_mi_fit2metric 0.5277
## lavaan_mg_mi_fit3scalar 0.1467
## lavaan_mg_mi_fit4strict 0.1154# Chi-Squared Difference Test
#
# Df AIC BIC Chisq Chisq diff RMSEA Df diff Pr(>Chisq)
# lavaan_mg_mi_fit1conf 33 10519 10836 33.939
# lavaan_mg_mi_fit2metric 41 10510 10792 41.022 7.0831 0.000000 8 0.5277
# lavaan_mg_mi_fit3scalar 49 10506 10752 53.124 12.1020 0.050633 8 0.1467
# lavaan_mg_mi_fit4strict 63 10499 10683 73.614 20.4899 0.048144 14 0.1154
# second, more specific results from semTools::compareFit
library(semTools)
LRT_MI_result <- semTools::compareFit(lavaan_mg_mi_fit1conf,lavaan_mg_mi_fit2metric,lavaan_mg_mi_fit3scalar,lavaan_mg_mi_fit4strict)
summary(LRT_MI_result)## ################### Nested Model Comparison #########################
##
## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff RMSEA Df diff
## lavaan_mg_mi_fit1conf 33 10519 10836 33.939
## lavaan_mg_mi_fit2metric 41 10510 10792 41.022 7.0831 0.000000 8
## lavaan_mg_mi_fit3scalar 49 10506 10752 53.124 12.1020 0.050633 8
## lavaan_mg_mi_fit4strict 63 10499 10683 73.614 20.4899 0.048144 14
## Pr(>Chisq)
## lavaan_mg_mi_fit1conf
## lavaan_mg_mi_fit2metric 0.5277
## lavaan_mg_mi_fit3scalar 0.1467
## lavaan_mg_mi_fit4strict 0.1154
##
## ####################### Model Fit Indices ###########################
## chisq df pvalue rmsea cfi tli srmr aic
## lavaan_mg_mi_fit1conf 33.939† 33 .422 .012 0.999 0.999 .031† 10518.975
## lavaan_mg_mi_fit2metric 41.022 41 .470 .002† 1.000† 1.000† .038 10510.058
## lavaan_mg_mi_fit3scalar 53.124 49 .318 .021 0.997 0.996 .041 10506.160
## lavaan_mg_mi_fit4strict 73.614 63 .170 .029 .993 .993 .043 10498.650†
## bic
## lavaan_mg_mi_fit1conf 10835.554
## lavaan_mg_mi_fit2metric 10791.462
## lavaan_mg_mi_fit3scalar 10752.388
## lavaan_mg_mi_fit4strict 10683.321†
##
## ################## Differences in Fit Indices #######################
## df rmsea cfi tli srmr
## lavaan_mg_mi_fit2metric - lavaan_mg_mi_fit1conf 8 -0.010 0.001 0.001 0.007
## lavaan_mg_mi_fit3scalar - lavaan_mg_mi_fit2metric 8 0.019 -0.003 -0.004 0.003
## lavaan_mg_mi_fit4strict - lavaan_mg_mi_fit3scalar 14 0.009 -0.005 -0.004 0.002
## aic bic
## lavaan_mg_mi_fit2metric - lavaan_mg_mi_fit1conf -8.917 -44.092
## lavaan_mg_mi_fit3scalar - lavaan_mg_mi_fit2metric -3.898 -39.073
## lavaan_mg_mi_fit4strict - lavaan_mg_mi_fit3scalar -7.510 -69.067# Chi-Squared Difference Test
#
# Df AIC BIC Chisq Chisq diff RMSEA Df diff Pr(>Chisq)
# lavaan_mg_mi_fit1conf 33 10519 10836 33.939
# lavaan_mg_mi_fit2metric 41 10510 10792 41.022 7.0831 0.000000 8 0.5277 # metric MI hold
# lavaan_mg_mi_fit3scalar 49 10506 10752 53.124 12.1020 0.050633 8 0.1467 # scalar MI holds
# lavaan_mg_mi_fit4strict 63 10499 10683 73.614 20.4899 0.048144 14 0.1154 # strict MI holds
#
# ####################### Model Fit Indices ###########################
# chisq df pvalue rmsea cfi tli srmr aic bic
# lavaan_mg_mi_fit1conf 33.939† 33 .422 .012 0.999 0.999 .031† 10518.975 10835.554 # configural MI holds
# lavaan_mg_mi_fit2metric 41.022 41 .470 .002† 1.000† 1.000† .038 10510.058 10791.462 # metric MI holds
# lavaan_mg_mi_fit3scalar 53.124 49 .318 .021 0.997 0.996 .041 10506.160 10752.388 # scalar MI holds
# lavaan_mg_mi_fit4strict 73.614 63 .170 .029 .993 .993 .043 10498.650† 10683.321† # strict MI holds
#
# ################## Differences in Fit Indices #######################
# df rmsea cfi tli srmr aic bic
# lavaan_mg_mi_fit2metric - lavaan_mg_mi_fit1conf 8 -0.010 0.001 0.001 0.007 -8.917 -44.092 # RMSEA and CFI delta ok
# lavaan_mg_mi_fit3scalar - lavaan_mg_mi_fit2metric 8 0.019 -0.003 -0.004 0.003 -3.898 -39.073 # CFI delta ok
# lavaan_mg_mi_fit4strict - lavaan_mg_mi_fit3scalar 14 0.009 -0.005 -0.004 0.002 -7.510 -69.067 # RMSEA and DFI delta ok
# Note: Less than -.01 change in CFI, combined with less than .015 change in RMSEA is used as evidence for non-invariance
# Chen, F. F. (2007). Sensitivity of goodness of fit indexes to lack of measurement invariance. Structural Equation Modeling, 14(3), 464–504. Conclusion from measurement invariance tests: All four types on invariance are satisfied with this data.
Test if the three groups differ by their regressions
As we have established measurement invariance across the three groups in the mg simulation data, we may proceed to test if the these groups differ by their PSY ~ ENC and PSY ~ BUI regressions.
First, we will estimate the free model:
# earlier we created a simulated mg data ("sim_multiGroupData_lavaan") with three groups (each n = 200)
# we'll do it again:
# define population model for the simulated data
sem_model_simulation <- '
# Factors
ENC =~ 0.81*v5 + 0.77*v7 + 0.87*v17
BUI =~ 0.67*v42r + 0.57*v43
PSY =~ 0.90*v44 + 0.55*v45
# Regressions
PSY ~ -0.10*ENC + 0.85*BUI
# Covariances
ENC ~~ -.30*BUI
'
# create simulated multi-group data based on the population model
set.seed(2023)
sim_multiGroupData_lavaan <- lavaan::simulateData(sem_model_simulation,
sample.nobs=c(200,200,200), # add "group" variable
standardized = T,
return.type = "data.frame")
# define SEM model
sem_multiGroup_model <- '
# Factors
ENC =~ NA*v5 + v7 + 1*v17 # Encouraging leadership
BUI =~ 1*v42r + v43 # Build-up of work requirements
PSY =~ 1*v44 + v45 # Psychic stress of the work
# Regressions
PSY ~ ENC + BUI # Predict PSY with ENC and BUI
# Covariances
ENC ~~ BUI # Let ENC and BUI correlate
'
# fit the "free" model where the coefficients are allowed to vary across three groups
lavaan_mg_free <- sem(sem_multiGroup_model,
group = "group",
data = sim_multiGroupData_lavaan)
# see the results
summary(lavaan_mg_free)## lavaan 0.6.14 ended normally after 52 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 72
##
## Number of observations per group:
## 1 200
## 2 200
## 3 200
##
## Model Test User Model:
##
## Test statistic 33.939
## Degrees of freedom 33
## P-value (Chi-square) 0.422
## Test statistic for each group:
## 1 13.813
## 2 11.823
## 3 8.303
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [1]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ENC =~
## v5 0.917 0.080 11.421 0.000
## v7 0.859 0.081 10.592 0.000
## v17 1.000
## BUI =~
## v42r 1.000
## v43 0.730 0.119 6.150 0.000
## PSY =~
## v44 1.000
## v45 0.592 0.099 5.998 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## PSY ~
## ENC -0.197 0.084 -2.345 0.019
## BUI 0.869 0.147 5.932 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ENC ~~
## BUI -0.157 0.065 -2.409 0.016
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .v5 -0.130 0.064 -2.019 0.044
## .v7 -0.055 0.067 -0.818 0.413
## .v17 -0.001 0.067 -0.022 0.983
## .v42r 0.074 0.077 0.957 0.339
## .v43 0.163 0.075 2.178 0.029
## .v44 0.153 0.069 2.210 0.027
## .v45 0.072 0.067 1.069 0.285
## ENC 0.000
## BUI 0.000
## .PSY 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.277 0.046 6.093 0.000
## .v7 0.412 0.052 7.871 0.000
## .v17 0.244 0.049 4.958 0.000
## .v42r 0.554 0.108 5.137 0.000
## .v43 0.772 0.092 8.425 0.000
## .v44 0.236 0.098 2.417 0.016
## .v45 0.652 0.073 8.912 0.000
## ENC 0.659 0.097 6.805 0.000
## BUI 0.646 0.141 4.577 0.000
## .PSY 0.150 0.113 1.332 0.183
##
##
## Group 2 [2]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ENC =~
## v5 1.004 0.076 13.231 0.000
## v7 0.797 0.074 10.793 0.000
## v17 1.000
## BUI =~
## v42r 1.000
## v43 0.763 0.106 7.184 0.000
## PSY =~
## v44 1.000
## v45 0.731 0.092 7.926 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## PSY ~
## ENC -0.291 0.078 -3.735 0.000
## BUI 1.039 0.154 6.753 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ENC ~~
## BUI -0.181 0.061 -2.955 0.003
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.070 0.070 0.994 0.320
## .v7 0.049 0.070 0.707 0.480
## .v17 0.023 0.072 0.318 0.751
## .v42r -0.094 0.068 -1.394 0.163
## .v43 -0.003 0.065 -0.040 0.968
## .v44 -0.013 0.079 -0.171 0.864
## .v45 -0.005 0.071 -0.077 0.938
## ENC 0.000
## BUI 0.000
## .PSY 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.209 0.047 4.450 0.000
## .v7 0.492 0.057 8.633 0.000
## .v17 0.279 0.050 5.523 0.000
## .v42r 0.387 0.072 5.355 0.000
## .v43 0.544 0.065 8.371 0.000
## .v44 0.537 0.089 6.056 0.000
## .v45 0.635 0.074 8.601 0.000
## ENC 0.764 0.109 7.015 0.000
## BUI 0.528 0.103 5.130 0.000
## .PSY -0.042 0.087 -0.483 0.629
##
##
## Group 3 [3]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ENC =~
## v5 0.935 0.077 12.111 0.000
## v7 0.981 0.081 12.034 0.000
## v17 1.000
## BUI =~
## v42r 1.000
## v43 0.794 0.117 6.768 0.000
## PSY =~
## v44 1.000
## v45 0.598 0.097 6.156 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## PSY ~
## ENC -0.111 0.085 -1.309 0.190
## BUI 0.951 0.145 6.541 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ENC ~~
## BUI -0.162 0.062 -2.616 0.009
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.067 0.067 0.996 0.319
## .v7 -0.023 0.071 -0.331 0.741
## .v17 0.084 0.066 1.268 0.205
## .v42r -0.064 0.074 -0.861 0.389
## .v43 -0.113 0.073 -1.544 0.123
## .v44 -0.124 0.070 -1.770 0.077
## .v45 -0.064 0.070 -0.906 0.365
## ENC 0.000
## BUI 0.000
## .PSY 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.325 0.047 6.974 0.000
## .v7 0.373 0.052 7.119 0.000
## .v17 0.209 0.043 4.836 0.000
## .v42r 0.499 0.092 5.433 0.000
## .v43 0.688 0.084 8.172 0.000
## .v44 0.147 0.107 1.371 0.171
## .v45 0.692 0.079 8.774 0.000
## ENC 0.662 0.093 7.143 0.000
## BUI 0.607 0.125 4.848 0.000
## .PSY 0.247 0.125 1.971 0.049# our interest is to see if there are differences across three groups in these regressions:
# Group 1
# Regressions:
# Estimate Std.Err z-value P(>|z|)
# PSY ~
# ENC -0.197 0.084 -2.345 0.019
# BUI 0.869 0.147 5.932 0.000
# Group 2
# Regressions:
# Estimate Std.Err z-value P(>|z|)
# PSY ~
# ENC -0.291 0.078 -3.735 0.000
# BUI 1.039 0.154 6.753 0.000
# Group 3
# Regressions:
# Estimate Std.Err z-value P(>|z|)
# PSY ~
# ENC -0.111 0.085 -1.309 0.190
# BUI 0.951 0.145 6.541 0.000Second, we'll estimate the constrained model:
# fit a "constrained" model where both intercepts and path coefficients are fixed to be the same in all groups
lavaan_mg_constrained <- sem(sem_multiGroup_model,
group = "group",
group.equal = c("intercepts", "regressions"),
data = sim_multiGroupData_lavaan)
# see the results
summary(lavaan_mg_constrained)## lavaan 0.6.14 ended normally after 54 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 78
## Number of equality constraints 18
##
## Number of observations per group:
## 1 200
## 2 200
## 3 200
##
## Model Test User Model:
##
## Test statistic 50.528
## Degrees of freedom 45
## P-value (Chi-square) 0.264
## Test statistic for each group:
## 1 19.363
## 2 17.440
## 3 13.725
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [1]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ENC =~
## v5 0.918 0.081 11.349 0.000
## v7 0.864 0.082 10.574 0.000
## v17 1.000
## BUI =~
## v42r 1.000
## v43 0.770 0.116 6.610 0.000
## PSY =~
## v44 1.000
## v45 0.581 0.096 6.082 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## PSY ~
## ENC (.p8.) -0.205 0.048 -4.291 0.000
## BUI (.p9.) 0.980 0.089 11.005 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ENC ~~
## BUI -0.145 0.057 -2.529 0.011
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .v5 (.21.) -0.066 0.060 -1.093 0.274
## .v7 (.22.) -0.076 0.058 -1.318 0.187
## .v17 (.23.) -0.033 0.061 -0.538 0.591
## .v42r (.24.) 0.105 0.071 1.482 0.138
## .v43 (.25.) 0.122 0.061 2.000 0.046
## .v44 (.26.) 0.147 0.068 2.176 0.030
## .v45 (.27.) 0.094 0.055 1.714 0.086
## ENC 0.000
## BUI 0.000
## .PSY 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.284 0.046 6.165 0.000
## .v7 0.411 0.053 7.819 0.000
## .v17 0.247 0.049 5.019 0.000
## .v42r 0.604 0.087 6.909 0.000
## .v43 0.782 0.092 8.488 0.000
## .v44 0.230 0.099 2.319 0.020
## .v45 0.655 0.073 8.958 0.000
## ENC 0.654 0.096 6.793 0.000
## BUI 0.566 0.099 5.740 0.000
## .PSY 0.115 0.109 1.058 0.290
##
##
## Group 2 [2]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ENC =~
## v5 1.015 0.077 13.246 0.000
## v7 0.807 0.074 10.908 0.000
## v17 1.000
## BUI =~
## v42r 1.000
## v43 0.738 0.094 7.868 0.000
## PSY =~
## v44 1.000
## v45 0.771 0.094 8.225 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## PSY ~
## ENC (.p8.) -0.205 0.048 -4.291 0.000
## BUI (.p9.) 0.980 0.089 11.005 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ENC ~~
## BUI -0.212 0.060 -3.509 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .v5 (.21.) -0.066 0.060 -1.093 0.274
## .v7 (.22.) -0.076 0.058 -1.318 0.187
## .v17 (.23.) -0.033 0.061 -0.538 0.591
## .v42r (.24.) 0.105 0.071 1.482 0.138
## .v43 (.25.) 0.122 0.061 2.000 0.046
## .v44 (.26.) 0.147 0.068 2.176 0.030
## .v45 (.27.) 0.094 0.055 1.714 0.086
## ENC 0.109 0.088 1.234 0.217
## BUI -0.191 0.096 -1.994 0.046
## .PSY 0.060 0.082 0.728 0.467
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.205 0.048 4.279 0.000
## .v7 0.492 0.057 8.594 0.000
## .v17 0.288 0.051 5.629 0.000
## .v42r 0.382 0.065 5.846 0.000
## .v43 0.539 0.064 8.412 0.000
## .v44 0.546 0.086 6.318 0.000
## .v45 0.631 0.075 8.456 0.000
## ENC 0.758 0.108 6.991 0.000
## BUI 0.565 0.092 6.169 0.000
## .PSY -0.025 0.075 -0.333 0.739
##
##
## Group 3 [3]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ENC =~
## v5 0.943 0.077 12.230 0.000
## v7 0.972 0.081 12.035 0.000
## v17 1.000
## BUI =~
## v42r 1.000
## v43 0.862 0.113 7.599 0.000
## PSY =~
## v44 1.000
## v45 0.601 0.090 6.688 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## PSY ~
## ENC (.p8.) -0.205 0.048 -4.291 0.000
## BUI (.p9.) 0.980 0.089 11.005 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ENC ~~
## BUI -0.129 0.056 -2.316 0.021
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .v5 (.21.) -0.066 0.060 -1.093 0.274
## .v7 (.22.) -0.076 0.058 -1.318 0.187
## .v17 (.23.) -0.033 0.061 -0.538 0.591
## .v42r (.24.) 0.105 0.071 1.482 0.138
## .v43 (.25.) 0.122 0.061 2.000 0.046
## .v44 (.26.) 0.147 0.068 2.176 0.030
## .v45 (.27.) 0.094 0.055 1.714 0.086
## ENC 0.108 0.085 1.266 0.205
## BUI -0.207 0.095 -2.177 0.030
## .PSY -0.046 0.085 -0.537 0.591
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.325 0.047 6.925 0.000
## .v7 0.380 0.052 7.250 0.000
## .v17 0.209 0.043 4.855 0.000
## .v42r 0.521 0.079 6.623 0.000
## .v43 0.679 0.086 7.918 0.000
## .v44 0.158 0.103 1.544 0.122
## .v45 0.688 0.078 8.826 0.000
## ENC 0.658 0.092 7.175 0.000
## BUI 0.558 0.096 5.816 0.000
## .PSY 0.234 0.115 2.033 0.042Third, we'll compare the free and constrained models:
anova(lavaan_mg_free,lavaan_mg_constrained)##
## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff RMSEA Df diff
## lavaan_mg_free 33 10519 10836 33.939
## lavaan_mg_constrained 45 10512 10775 50.528 16.59 0.04373 12
## Pr(>Chisq)
## lavaan_mg_free
## lavaan_mg_constrained 0.1657As the two models are not significantly different, it indicates that the regression paths do not vary by the three groups (this data could be analysed as pooled).
What if we modify the third group's population model a little bit (so that it differs from the group 1 and 2 models) and create a new version of the simulated data?
# earlier we created a simulated mg data ("sim_multiGroupData_lavaan") with three groups (each n = 200)
# we'll do it again:
# define a different population model for the third group data simulation
sem_model_simulation_group3 <- '
# Factors
ENC =~ 0.81*v5 + 0.77*v7 + 0.87*v17
BUI =~ 0.67*v42r + 0.57*v43
PSY =~ 0.90*v44 + 0.55*v45
# Regressions
PSY ~ .70*ENC + 0.85*BUI
# Covariances
ENC ~~ -.30*BUI
'
# create simulated multi-group data for the group 3
set.seed(2023)
sim_multiGroupData_lavaan_group3 <- lavaan::simulateData(sem_model_simulation_group3,
sample.nobs=200,
standardized = T,
return.type = "data.frame")
# add "group" var into df with fixed value of "3"
sim_multiGroupData_lavaan_group3$group <- 3
# create simulated multi-group data for the groups 2&3
set.seed(2023)
sim_multiGroupData_lavaan_groups2_3 <- lavaan::simulateData(sem_model_simulation,
sample.nobs=c(200,200), # add "group" variable
standardized = T,
return.type = "data.frame")
# put the two data frames together
sim_multiGroupData_lavaan_v2 <- rbind(sim_multiGroupData_lavaan_groups2_3,sim_multiGroupData_lavaan_group3)Now that we have a new simulated multi-group data, we can repeat the model fitting/comparison procedure.
First, fit the free model:
# define SEM model
sem_multiGroup_model <- '
# Factors
ENC =~ NA*v5 + v7 + 1*v17 # Encouraging leadership
BUI =~ 1*v42r + v43 # Build-up of work requirements
PSY =~ 1*v44 + v45 # Psychic stress of the work
# Regressions
PSY ~ ENC + BUI # Predict PSY with ENC and BUI
# Covariances
ENC ~~ BUI # Let ENC and BUI correlate
'
# fit the "free" model v2 where the coefficients are allowed to vary across three groups
lavaan_mg_free_v2 <- sem(sem_multiGroup_model,
group = "group",
data = sim_multiGroupData_lavaan_v2)
# see the results
summary(lavaan_mg_free_v2)## lavaan 0.6.14 ended normally after 57 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 72
##
## Number of observations per group:
## 1 200
## 2 200
## 3 200
##
## Model Test User Model:
##
## Test statistic 38.937
## Degrees of freedom 33
## P-value (Chi-square) 0.220
## Test statistic for each group:
## 1 13.813
## 2 11.823
## 3 13.301
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [1]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ENC =~
## v5 0.917 0.080 11.421 0.000
## v7 0.859 0.081 10.592 0.000
## v17 1.000
## BUI =~
## v42r 1.000
## v43 0.730 0.119 6.150 0.000
## PSY =~
## v44 1.000
## v45 0.592 0.099 5.998 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## PSY ~
## ENC -0.197 0.084 -2.345 0.019
## BUI 0.869 0.147 5.932 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ENC ~~
## BUI -0.157 0.065 -2.409 0.016
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .v5 -0.130 0.064 -2.019 0.044
## .v7 -0.055 0.067 -0.818 0.413
## .v17 -0.001 0.067 -0.022 0.983
## .v42r 0.074 0.077 0.957 0.339
## .v43 0.163 0.075 2.178 0.029
## .v44 0.153 0.069 2.210 0.027
## .v45 0.072 0.067 1.069 0.285
## ENC 0.000
## BUI 0.000
## .PSY 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.277 0.046 6.093 0.000
## .v7 0.412 0.052 7.871 0.000
## .v17 0.244 0.049 4.958 0.000
## .v42r 0.554 0.108 5.137 0.000
## .v43 0.772 0.092 8.425 0.000
## .v44 0.236 0.098 2.417 0.016
## .v45 0.652 0.073 8.912 0.000
## ENC 0.659 0.097 6.805 0.000
## BUI 0.646 0.141 4.577 0.000
## .PSY 0.150 0.113 1.332 0.183
##
##
## Group 2 [2]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ENC =~
## v5 1.004 0.076 13.231 0.000
## v7 0.797 0.074 10.793 0.000
## v17 1.000
## BUI =~
## v42r 1.000
## v43 0.763 0.106 7.184 0.000
## PSY =~
## v44 1.000
## v45 0.731 0.092 7.926 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## PSY ~
## ENC -0.291 0.078 -3.735 0.000
## BUI 1.039 0.154 6.753 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ENC ~~
## BUI -0.181 0.061 -2.955 0.003
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.070 0.070 0.994 0.320
## .v7 0.049 0.070 0.707 0.480
## .v17 0.023 0.072 0.318 0.751
## .v42r -0.094 0.068 -1.394 0.163
## .v43 -0.003 0.065 -0.040 0.968
## .v44 -0.013 0.079 -0.171 0.864
## .v45 -0.005 0.071 -0.077 0.938
## ENC 0.000
## BUI 0.000
## .PSY 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.209 0.047 4.450 0.000
## .v7 0.492 0.057 8.633 0.000
## .v17 0.279 0.050 5.523 0.000
## .v42r 0.387 0.072 5.355 0.000
## .v43 0.544 0.065 8.371 0.000
## .v44 0.537 0.089 6.056 0.000
## .v45 0.635 0.074 8.601 0.000
## ENC 0.764 0.109 7.015 0.000
## BUI 0.528 0.103 5.130 0.000
## .PSY -0.042 0.087 -0.483 0.629
##
##
## Group 3 [3]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ENC =~
## v5 0.900 0.078 11.501 0.000
## v7 0.859 0.080 10.786 0.000
## v17 1.000
## BUI =~
## v42r 1.000
## v43 0.698 0.112 6.240 0.000
## PSY =~
## v44 1.000
## v45 0.645 0.093 6.922 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## PSY ~
## ENC 0.874 0.107 8.189 0.000
## BUI 1.137 0.201 5.642 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ENC ~~
## BUI -0.154 0.061 -2.534 0.011
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .v5 -0.074 0.067 -1.105 0.269
## .v7 -0.030 0.067 -0.442 0.659
## .v17 -0.174 0.065 -2.654 0.008
## .v42r -0.015 0.074 -0.196 0.845
## .v43 -0.057 0.067 -0.848 0.396
## .v44 -0.111 0.072 -1.555 0.120
## .v45 -0.158 0.075 -2.095 0.036
## ENC 0.000
## BUI 0.000
## .PSY 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.363 0.048 7.515 0.000
## .v7 0.430 0.053 8.150 0.000
## .v17 0.210 0.042 4.969 0.000
## .v42r 0.586 0.096 6.113 0.000
## .v43 0.658 0.075 8.720 0.000
## .v44 0.212 0.087 2.433 0.015
## .v45 0.801 0.087 9.155 0.000
## ENC 0.646 0.091 7.120 0.000
## BUI 0.513 0.120 4.277 0.000
## .PSY -0.037 0.123 -0.303 0.762# our interest (again) is to see if there are differences across three groups in these regressions:
# Group 1
# Regressions:
# Estimate Std.Err z-value P(>|z|)
# PSY ~
# ENC -0.197 0.084 -2.345 0.019
# BUI 0.869 0.147 5.932 0.000
# Group 2
# Regressions:
# Estimate Std.Err z-value P(>|z|)
# PSY ~
# ENC -0.291 0.078 -3.735 0.000
# BUI 1.039 0.154 6.753 0.000
# Group 3
# Regressions:
# Estimate Std.Err z-value P(>|z|)
# PSY ~
# ENC 0.874 0.107 8.189 0.000 # NOTE: Now we have quite different estimate here!!!
# BUI 1.137 0.201 5.642 0.000Second, estimate the constrained model with the new simulated data:
# fit a "constrained" model where both intercepts and path coefficients are fixed to be the same in all groups
lavaan_mg_constrained_v2 <- sem(sem_multiGroup_model,
group = "group",
group.equal = c("intercepts", "regressions"),
data = sim_multiGroupData_lavaan_v2)
# see the results
summary(lavaan_mg_constrained_v2)## lavaan 0.6.14 ended normally after 85 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 78
## Number of equality constraints 18
##
## Number of observations per group:
## 1 200
## 2 200
## 3 200
##
## Model Test User Model:
##
## Test statistic 129.197
## Degrees of freedom 45
## P-value (Chi-square) 0.000
## Test statistic for each group:
## 1 43.169
## 2 41.240
## 3 44.788
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [1]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ENC =~
## v5 0.927 0.083 11.239 0.000
## v7 0.876 0.083 10.523 0.000
## v17 1.000
## BUI =~
## v42r 1.000
## v43 0.733 0.149 4.931 0.000
## PSY =~
## v44 1.000
## v45 0.597 0.102 5.850 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## PSY ~
## ENC (.p8.) 0.577 0.160 3.612 0.000
## BUI (.p9.) 2.833 0.467 6.060 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ENC ~~
## BUI -0.229 0.046 -5.009 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .v5 (.21.) -0.066 0.060 -1.103 0.270
## .v7 (.22.) -0.024 0.056 -0.423 0.672
## .v17 (.23.) -0.070 0.062 -1.117 0.264
## .v42r (.24.) 0.104 0.069 1.508 0.132
## .v43 (.25.) 0.118 0.057 2.062 0.039
## .v44 (.26.) 0.152 0.067 2.285 0.022
## .v45 (.27.) 0.069 0.055 1.252 0.211
## ENC 0.000
## BUI 0.000
## .PSY 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.286 0.046 6.168 0.000
## .v7 0.408 0.053 7.741 0.000
## .v17 0.260 0.049 5.260 0.000
## .v42r 0.955 0.099 9.600 0.000
## .v43 0.992 0.098 10.114 0.000
## .v44 0.248 0.098 2.532 0.011
## .v45 0.648 0.073 8.865 0.000
## ENC 0.639 0.095 6.701 0.000
## BUI 0.233 0.051 4.598 0.000
## .PSY -0.649 0.247 -2.628 0.009
##
##
## Group 2 [2]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ENC =~
## v5 1.037 0.078 13.237 0.000
## v7 0.817 0.076 10.769 0.000
## v17 1.000
## BUI =~
## v42r 1.000
## v43 0.734 0.120 6.109 0.000
## PSY =~
## v44 1.000
## v45 0.709 0.090 7.857 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## PSY ~
## ENC (.p8.) 0.577 0.160 3.612 0.000
## BUI (.p9.) 2.833 0.467 6.060 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ENC ~~
## BUI -0.301 0.056 -5.406 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .v5 (.21.) -0.066 0.060 -1.103 0.270
## .v7 (.22.) -0.024 0.056 -0.423 0.672
## .v17 (.23.) -0.070 0.062 -1.117 0.264
## .v42r (.24.) 0.104 0.069 1.508 0.132
## .v43 (.25.) 0.118 0.057 2.062 0.039
## .v44 (.26.) 0.152 0.067 2.285 0.022
## .v45 (.27.) 0.069 0.055 1.252 0.211
## ENC 0.112 0.087 1.293 0.196
## BUI -0.187 0.090 -2.086 0.037
## .PSY 0.317 0.213 1.485 0.137
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.206 0.047 4.368 0.000
## .v7 0.492 0.057 8.611 0.000
## .v17 0.292 0.049 5.909 0.000
## .v42r 0.695 0.074 9.437 0.000
## .v43 0.701 0.070 10.009 0.000
## .v44 0.529 0.091 5.821 0.000
## .v45 0.641 0.074 8.656 0.000
## ENC 0.723 0.104 6.925 0.000
## BUI 0.271 0.060 4.478 0.000
## .PSY -0.719 0.215 -3.339 0.001
##
##
## Group 3 [3]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ENC =~
## v5 0.857 0.076 11.242 0.000
## v7 0.830 0.077 10.747 0.000
## v17 1.000
## BUI =~
## v42r 1.000
## v43 0.735 0.155 4.733 0.000
## PSY =~
## v44 1.000
## v45 0.683 0.088 7.729 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## PSY ~
## ENC (.p8.) 0.577 0.160 3.612 0.000
## BUI (.p9.) 2.833 0.467 6.060 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ENC ~~
## BUI 0.020 0.050 0.409 0.683
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .v5 (.21.) -0.066 0.060 -1.103 0.270
## .v7 (.22.) -0.024 0.056 -0.423 0.672
## .v17 (.23.) -0.070 0.062 -1.117 0.264
## .v42r (.24.) 0.104 0.069 1.508 0.132
## .v43 (.25.) 0.118 0.057 2.062 0.039
## .v44 (.26.) 0.152 0.067 2.285 0.022
## .v45 (.27.) 0.069 0.055 1.252 0.211
## ENC -0.064 0.089 -0.726 0.468
## BUI -0.162 0.090 -1.811 0.070
## .PSY 0.225 0.227 0.994 0.320
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.378 0.050 7.624 0.000
## .v7 0.429 0.053 8.093 0.000
## .v17 0.196 0.045 4.335 0.000
## .v42r 0.884 0.092 9.564 0.000
## .v43 0.841 0.083 10.182 0.000
## .v44 0.232 0.082 2.833 0.005
## .v45 0.795 0.088 9.080 0.000
## ENC 0.689 0.095 7.225 0.000
## BUI 0.156 0.037 4.171 0.000
## .PSY -0.727 0.250 -2.909 0.004Third, compare the free and constrained models:
anova(lavaan_mg_free_v2,lavaan_mg_constrained_v2)##
## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff RMSEA Df diff
## lavaan_mg_free_v2 33 10527 10844 38.937
## lavaan_mg_constrained_v2 45 10593 10857 129.197 90.261 0.18058 12
## Pr(>Chisq)
## lavaan_mg_free_v2
## lavaan_mg_constrained_v2 4.396e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Not surprisingly, now we have a stat sig difference between the two models, indicating that at least some of the paths are different across the three groups.
Next step in the process is to use constraints to identify if the ENC -> PSY and/or BUI -> PSY paths are different among groups.
lavaan_mg_model1 <- '
# Factors
ENC =~ NA*v5 + v7 + 1*v17 # Encouraging leadership
BUI =~ 1*v42r + v43 # Build-up of work requirements
PSY =~ 1*v44 + v45 # Psychic stress of the work
# Regressions
PSY ~ c("b1","b1","b1")*ENC + BUI # Constrain in all three groups
# Covariances
ENC ~~ BUI # Let ENC and BUI correlate
'
# fit the single constrained model
lavaan_mg_const1_v2 <- sem(lavaan_mg_model1,
group = "group",
data = sim_multiGroupData_lavaan_v2)
summary(lavaan_mg_const1_v2)## lavaan 0.6.14 ended normally after 90 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 72
## Number of equality constraints 2
##
## Number of observations per group:
## 1 200
## 2 200
## 3 200
##
## Model Test User Model:
##
## Test statistic 93.405
## Degrees of freedom 35
## P-value (Chi-square) 0.000
## Test statistic for each group:
## 1 37.737
## 2 41.391
## 3 14.278
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [1]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ENC =~
## v5 0.923 0.081 11.398 0.000
## v7 0.868 0.082 10.591 0.000
## v17 1.000
## BUI =~
## v42r 1.000
## v43 0.705 0.153 4.605 0.000
## PSY =~
## v44 1.000
## v45 0.597 0.103 5.810 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## PSY ~
## ENC (b1) 0.772 0.093 8.273 0.000
## BUI 3.324 0.732 4.543 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ENC ~~
## BUI -0.236 0.061 -3.862 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .v5 -0.130 0.064 -2.019 0.044
## .v7 -0.055 0.067 -0.818 0.413
## .v17 -0.001 0.067 -0.022 0.982
## .v42r 0.074 0.077 0.957 0.339
## .v43 0.163 0.075 2.178 0.029
## .v44 0.153 0.068 2.237 0.025
## .v45 0.072 0.067 1.076 0.282
## ENC 0.000
## BUI 0.000
## .PSY 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.281 0.045 6.173 0.000
## .v7 0.410 0.052 7.837 0.000
## .v17 0.247 0.048 5.094 0.000
## .v42r 0.986 0.102 9.667 0.000
## .v43 1.009 0.100 10.076 0.000
## .v44 0.246 0.098 2.509 0.012
## .v45 0.649 0.073 8.864 0.000
## ENC 0.647 0.095 6.778 0.000
## BUI 0.214 0.073 2.924 0.003
## .PSY -0.854 0.334 -2.559 0.010
##
##
## Group 2 [2]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ENC =~
## v5 1.014 0.077 13.212 0.000
## v7 0.807 0.075 10.768 0.000
## v17 1.000
## BUI =~
## v42r 1.000
## v43 0.769 0.146 5.272 0.000
## PSY =~
## v44 1.000
## v45 0.712 0.094 7.571 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## PSY ~
## ENC (b1) 0.772 0.093 8.273 0.000
## BUI 3.946 0.733 5.381 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ENC ~~
## BUI -0.257 0.058 -4.428 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.070 0.070 0.994 0.320
## .v7 0.049 0.070 0.707 0.480
## .v17 0.023 0.072 0.319 0.749
## .v42r -0.094 0.068 -1.394 0.163
## .v43 -0.003 0.065 -0.040 0.968
## .v44 -0.013 0.078 -0.173 0.863
## .v45 -0.005 0.070 -0.078 0.938
## ENC 0.000
## BUI 0.000
## .PSY 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.210 0.047 4.497 0.000
## .v7 0.491 0.057 8.614 0.000
## .v17 0.283 0.050 5.690 0.000
## .v42r 0.729 0.074 9.853 0.000
## .v43 0.741 0.073 10.131 0.000
## .v44 0.529 0.091 5.842 0.000
## .v45 0.639 0.074 8.636 0.000
## ENC 0.748 0.107 6.975 0.000
## BUI 0.186 0.057 3.282 0.001
## .PSY -1.097 0.306 -3.587 0.000
##
##
## Group 3 [3]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ENC =~
## v5 0.877 0.076 11.465 0.000
## v7 0.840 0.078 10.818 0.000
## v17 1.000
## BUI =~
## v42r 1.000
## v43 0.684 0.111 6.165 0.000
## PSY =~
## v44 1.000
## v45 0.654 0.095 6.876 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## PSY ~
## ENC (b1) 0.772 0.093 8.273 0.000
## BUI 1.050 0.180 5.849 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ENC ~~
## BUI -0.130 0.062 -2.098 0.036
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .v5 -0.074 0.067 -1.105 0.269
## .v7 -0.030 0.067 -0.442 0.659
## .v17 -0.174 0.066 -2.630 0.009
## .v42r -0.015 0.074 -0.196 0.845
## .v43 -0.057 0.067 -0.848 0.396
## .v44 -0.111 0.071 -1.568 0.117
## .v45 -0.158 0.075 -2.095 0.036
## ENC 0.000
## BUI 0.000
## .PSY 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .v5 0.368 0.049 7.570 0.000
## .v7 0.430 0.053 8.153 0.000
## .v17 0.197 0.043 4.545 0.000
## .v42r 0.560 0.097 5.769 0.000
## .v43 0.656 0.075 8.705 0.000
## .v44 0.214 0.087 2.474 0.013
## .v45 0.800 0.088 9.125 0.000
## ENC 0.674 0.093 7.232 0.000
## BUI 0.539 0.123 4.371 0.000
## .PSY 0.007 0.115 0.058 0.953Compare the first constrained model (ENC -> PSY) to the free model:
anova(lavaan_mg_free_v2,lavaan_mg_const1_v2)##
## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff RMSEA Df diff Pr(>Chisq)
## lavaan_mg_free_v2 33 10527 10844 38.937
## lavaan_mg_const1_v2 35 10578 10885 93.406 54.469 0.36218 2 1.487e-12
##
## lavaan_mg_free_v2
## lavaan_mg_const1_v2 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Comparison shows that there is a stat sig difference between the models, indicating that the path ENC -> PSY should not be constrained (but let vary among the groups).
Next, we investigate in a similar manner the second path (BUI -> PSY):
lavaan_mg_model2 <- '
# Factors
ENC =~ NA*v5 + v7 + 1*v17 # Encouraging leadership
BUI =~ 1*v42r + v43 # Build-up of work requirements
PSY =~ 1*v44 + v45 # Psychic stress of the work
# Regressions
PSY ~ ENC + c("b2","b2","b2")*BUI # Constrain in all three groups
# Covariances
ENC ~~ BUI # Let ENC and BUI correlate
'
# fit the single constrained model
lavaan_mg_const2_v2 <- sem(lavaan_mg_model2,
group = "group",
data = sim_multiGroupData_lavaan_v2)anova(lavaan_mg_free_v2,lavaan_mg_const2_v2)##
## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff RMSEA Df diff Pr(>Chisq)
## lavaan_mg_free_v2 33 10527 10844 38.937
## lavaan_mg_const2_v2 35 10524 10832 40.192 1.2552 0 2 0.5339Model comparison shows that the models are not stat sig different. This indicates that the path BUI -> PSY should be constrained (i.e., not allowed to vary among three groups).
This last model is the most promising, if the fit measures are comparable to the free model:
library(stargazer)
lavaan_mg_free_v2.fits <- as.numeric(fitMeasures(lavaan_mg_free_v2, c("chisq", # sem.fits.table1[1]
"df", # sem.fits.table1[2]
"pvalue", # sem.fits.table1[3]
"rmsea", # sem.fits.table1[4]
"rmsea.ci.lower", # sem.fits.table1[5]
"rmsea.ci.upper", # sem.fits.table1[6]
"srmr", # sem.fits.table1[7]
"cfi" # sem.fits.table1[8]
)))
fit.names <- c("chisq", "df", "pvalue", "rmsea", "rmsea.ci.lower", "rmsea.ci.upper", "srmr", "cfi")
df.fits <- data.frame(fit.names,lavaan_mg_free_v2.fits)
names(df.fits)[1] <- "Measure"
names(df.fits)[2] <- "Free"
lavaan_mg_const2_v2.fits <- as.numeric(fitMeasures(lavaan_mg_const2_v2, c("chisq", # sem.fits.table1[1]
"df", # sem.fits.table1[2]
"pvalue", # sem.fits.table1[3]
"rmsea", # sem.fits.table1[4]
"rmsea.ci.lower", # sem.fits.table1[5]
"rmsea.ci.upper", # sem.fits.table1[6]
"srmr", # sem.fits.table1[7]
"cfi" # sem.fits.table1[8]
)))
# fit.names <- c("chisq", "df", "pvalue", "rmsea", "rmsea.ci.lower", "rmsea.ci.upper", "srmr", "cfi")
df.fits2 <- data.frame(fit.names,lavaan_mg_const2_v2.fits)
names(df.fits2)[2] <- "Const2"
joint.fits <- cbind(df.fits,df.fits2[2])
stargazer(joint.fits, summary=FALSE, type='text', rownames=FALSE, initial.zero=FALSE, digits=3, title='Comparison of fit indices')##
## Comparison of fit indices
## ============================
## Measure Free Const2
## ----------------------------
## chisq 38.937 40.192
## df 33 35
## pvalue .220 .251
## rmsea .030 .027
## rmsea.ci.lower 0 0
## rmsea.ci.upper .062 .060
## srmr .035 .036
## cfi .996 .996
## ----------------------------Yes they are!
Now we may investigate if the three groups differ stat sig by their ENC -> PSY path coefficients.
lavaan_mg_model3 <- '
# Factors
ENC =~ NA*v5 + v7 + 1*v17 # Encouraging leadership
BUI =~ 1*v42r + v43 # Build-up of work requirements
PSY =~ 1*v44 + v45 # Psychic stress of the work
# Regressions
PSY ~ c("g1","g2","g3")*ENC + c("b2","b2","b2")*BUI # Set labels to compare three groups ENC -> PSY and constrain BUI -> PSY
# Covariances
ENC ~~ BUI # Let ENC and BUI correlate
# compare ENC -> PSY coefficients btw three groups
diffENCg1g2 := g1-g2
diffENCg1g3 := g1-g3
diffENCg2g3 := g2-g3
'
lavaan_mg_const2_v2_compare <- sem(lavaan_mg_model3,
group = "group",
data = sim_multiGroupData_lavaan_v2)
summary(lavaan_mg_const2_v2_compare, standardized = TRUE, fit.measures = TRUE)## lavaan 0.6.14 ended normally after 47 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 72
## Number of equality constraints 2
##
## Number of observations per group:
## 1 200
## 2 200
## 3 200
##
## Model Test User Model:
##
## Test statistic 40.192
## Degrees of freedom 35
## P-value (Chi-square) 0.251
## Test statistic for each group:
## 1 14.599
## 2 11.852
## 3 13.741
##
## Model Test Baseline Model:
##
## Test statistic 1474.486
## Degrees of freedom 63
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.996
## Tucker-Lewis Index (TLI) 0.993
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -5192.158
## Loglikelihood unrestricted model (H1) -5172.063
##
## Akaike (AIC) 10524.317
## Bayesian (BIC) 10832.102
## Sample-size adjusted Bayesian (SABIC) 10609.871
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.027
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.060
## P-value H_0: RMSEA <= 0.050 0.854
## P-value H_0: RMSEA >= 0.080 0.002
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.036
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [1]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC =~
## v5 0.917 0.080 11.417 0.000 0.744 0.816
## v7 0.858 0.081 10.589 0.000 0.697 0.735
## v17 1.000 0.812 0.855
## BUI =~
## v42r 1.000 0.743 0.687
## v43 0.778 0.118 6.579 0.000 0.578 0.547
## PSY =~
## v44 1.000 0.856 0.869
## v45 0.588 0.096 6.098 0.000 0.503 0.529
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## PSY ~
## ENC (g1) -0.162 0.085 -1.910 0.056 -0.154 -0.154
## BUI (b2) 1.012 0.095 10.631 0.000 0.879 0.879
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC ~~
## BUI -0.157 0.063 -2.494 0.013 -0.260 -0.260
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 -0.130 0.064 -2.019 0.044 -0.130 -0.143
## .v7 -0.055 0.067 -0.818 0.413 -0.055 -0.058
## .v17 -0.001 0.067 -0.022 0.983 -0.001 -0.002
## .v42r 0.074 0.076 0.969 0.332 0.074 0.069
## .v43 0.163 0.075 2.178 0.029 0.163 0.154
## .v44 0.153 0.070 2.191 0.028 0.153 0.155
## .v45 0.072 0.067 1.068 0.285 0.072 0.076
## ENC 0.000 0.000 0.000
## BUI 0.000 0.000 0.000
## .PSY 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.277 0.046 6.091 0.000 0.277 0.334
## .v7 0.412 0.052 7.873 0.000 0.412 0.459
## .v17 0.243 0.049 4.943 0.000 0.243 0.269
## .v42r 0.617 0.087 7.058 0.000 0.617 0.528
## .v43 0.782 0.092 8.514 0.000 0.782 0.701
## .v44 0.237 0.098 2.428 0.015 0.237 0.245
## .v45 0.652 0.073 8.933 0.000 0.652 0.720
## ENC 0.659 0.097 6.807 0.000 1.000 1.000
## BUI 0.552 0.098 5.628 0.000 1.000 1.000
## .PSY 0.098 0.108 0.907 0.365 0.134 0.134
##
##
## Group 2 [2]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC =~
## v5 1.004 0.076 13.235 0.000 0.877 0.887
## v7 0.797 0.074 10.793 0.000 0.697 0.705
## v17 1.000 0.874 0.856
## BUI =~
## v42r 1.000 0.735 0.767
## v43 0.755 0.099 7.599 0.000 0.555 0.601
## PSY =~
## v44 1.000 0.833 0.751
## v45 0.735 0.091 8.094 0.000 0.613 0.610
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## PSY ~
## ENC (g2) -0.297 0.074 -3.993 0.000 -0.311 -0.311
## BUI (b2) 1.012 0.095 10.631 0.000 0.893 0.893
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC ~~
## BUI -0.181 0.062 -2.939 0.003 -0.281 -0.281
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.070 0.070 0.994 0.320 0.070 0.070
## .v7 0.049 0.070 0.707 0.480 0.049 0.050
## .v17 0.023 0.072 0.318 0.751 0.023 0.022
## .v42r -0.094 0.068 -1.391 0.164 -0.094 -0.098
## .v43 -0.003 0.065 -0.040 0.968 -0.003 -0.003
## .v44 -0.013 0.079 -0.171 0.864 -0.013 -0.012
## .v45 -0.005 0.071 -0.077 0.938 -0.005 -0.005
## ENC 0.000 0.000 0.000
## BUI 0.000 0.000 0.000
## .PSY 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.210 0.047 4.455 0.000 0.210 0.214
## .v7 0.492 0.057 8.633 0.000 0.492 0.503
## .v17 0.279 0.050 5.525 0.000 0.279 0.267
## .v42r 0.379 0.065 5.878 0.000 0.379 0.412
## .v43 0.543 0.065 8.377 0.000 0.543 0.639
## .v44 0.538 0.088 6.097 0.000 0.538 0.436
## .v45 0.634 0.074 8.590 0.000 0.634 0.628
## ENC 0.764 0.109 7.015 0.000 1.000 1.000
## BUI 0.540 0.089 6.040 0.000 1.000 1.000
## .PSY -0.035 0.079 -0.443 0.658 -0.050 -0.050
##
##
## Group 3 [3]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC =~
## v5 0.898 0.078 11.471 0.000 0.723 0.767
## v7 0.859 0.080 10.797 0.000 0.691 0.726
## v17 1.000 0.805 0.870
## BUI =~
## v42r 1.000 0.759 0.717
## v43 0.668 0.101 6.616 0.000 0.507 0.532
## PSY =~
## v44 1.000 0.897 0.890
## v45 0.649 0.094 6.912 0.000 0.582 0.545
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## PSY ~
## ENC (g3) 0.840 0.092 9.106 0.000 0.754 0.754
## BUI (b2) 1.012 0.095 10.631 0.000 0.857 0.857
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ENC ~~
## BUI -0.151 0.062 -2.418 0.016 -0.247 -0.247
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 -0.074 0.067 -1.105 0.269 -0.074 -0.078
## .v7 -0.030 0.067 -0.442 0.659 -0.030 -0.031
## .v17 -0.174 0.065 -2.654 0.008 -0.174 -0.188
## .v42r -0.015 0.075 -0.194 0.846 -0.015 -0.014
## .v43 -0.057 0.067 -0.848 0.396 -0.057 -0.060
## .v44 -0.111 0.071 -1.562 0.118 -0.111 -0.110
## .v45 -0.158 0.075 -2.095 0.036 -0.158 -0.148
## ENC 0.000 0.000 0.000
## BUI 0.000 0.000 0.000
## .PSY 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .v5 0.364 0.048 7.530 0.000 0.364 0.411
## .v7 0.428 0.053 8.134 0.000 0.428 0.473
## .v17 0.208 0.042 4.919 0.000 0.208 0.243
## .v42r 0.544 0.082 6.609 0.000 0.544 0.486
## .v43 0.652 0.074 8.767 0.000 0.652 0.718
## .v44 0.212 0.087 2.440 0.015 0.212 0.209
## .v45 0.801 0.088 9.144 0.000 0.801 0.703
## ENC 0.648 0.091 7.128 0.000 1.000 1.000
## BUI 0.576 0.099 5.822 0.000 1.000 1.000
## .PSY 0.014 0.100 0.137 0.891 0.017 0.017
##
## Defined Parameters:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## diffENCg1g2 0.135 0.110 1.225 0.221 0.157 0.157
## diffENCg1g3 -1.002 0.122 -8.240 0.000 -0.908 -0.908
## diffENCg2g3 -1.136 0.115 -9.874 0.000 -1.065 -1.065# the last part of the summary shows that the third group differs from two other groups
#
# Defined Parameters:
# Estimate Std.Err z-value P(>|z|) Std.lv Std.all
# diffENCg1g2 0.135 0.110 1.225 0.221 0.157 0.157
# diffENCg1g3 -1.002 0.122 -8.240 0.000 -0.908 -0.908
# diffENCg2g3 -1.136 0.115 -9.874 0.000 -1.065 -1.065Extra stuff related to producing printable table and figure
estimator <- "ML"
# produce merged fit (200+200+200=600)
merged_fit <- sem(sem_multiGroup_model,
data = sim_multiGroupData_lavaan,
estimator = estimator,
std.lv = TRUE,
std.ov = TRUE,
group = NULL, # no grouping variable (e.g., gender, control/intervention,...)
cluster = NULL # no clustering variable (e.g., country)
# missing = "fiml.x" # full information maximum likelihood approach
)
# fit each group separately
group1_fit <- sem(sem_multiGroup_model,
data = sim_multiGroupData_lavaan[which(sim_multiGroupData_lavaan$group==1), ],
estimator = estimator,
std.lv = TRUE,
std.ov = TRUE,
group = NULL, # no grouping variable (e.g., gender, control/intervention,...)
cluster = NULL # no clustering variable (e.g., country)
# missing = "fiml.x" # full information maximum likelihood approach
)
group2_fit <- sem(sem_multiGroup_model,
data = sim_multiGroupData_lavaan[which(sim_multiGroupData_lavaan$group==2), ],
estimator = estimator,
std.lv = TRUE,
std.ov = TRUE,
group = NULL, # no grouping variable (e.g., gender, control/intervention,...)
cluster = NULL # no clustering variable (e.g., country)
# missing = "fiml.x" # full information maximum likelihood approach
)
group3_fit <- sem(sem_multiGroup_model,
data = sim_multiGroupData_lavaan[which(sim_multiGroupData_lavaan$group==3), ],
estimator = estimator,
std.lv = TRUE,
std.ov = TRUE,
group = NULL, # no grouping variable (e.g., gender, control/intervention,...)
cluster = NULL # no clustering variable (e.g., country)
# missing = "fiml.x" # full information maximum likelihood approach
)
library(semTable)
semTable(list(
"Merged (n=600)" = merged_fit,
"Group1" = group1_fit,
"Group2" = group2_fit,
"Group3" = group3_fit),
paramSets = c(
"loadings",
"slopes",
# "intercepts",
# "residualvariances",
# "residualcovariances",
# "latentmeans",
# "latentvariances",
# "latentcovariances",
# "thresholds",
# "constructed",
"fits"
),
paramSetLabels = c(
"loadings" = "Factor Loadings",
"slopes" = "Regression Slopes",
"intercepts" = "Intercepts",
"means"= "Means",
"residualvariances" = "Residual Variances",
"residualcovariances" = "Residual Covariances",
"variances" = "Variances",
"latentvariances" = "Latent Variances",
"latentcovariances" = "Latent Covariances",
"latentmeans" = "Latent Intercepts",
"thresholds" = "Thresholds",
"constructed" = "Constructed",
"fits" = "Fit Indices"
),
columns = c(
# "est",
# "se",
# "z",
# "p",
# "rsquare",
# "estse",
# "eststars",
"estsestars"
),
# columns = list("fit1" = c("est", "se"), "fit2" = c("estse", "p"))
columnLabels = c(
est = "Est.",
se = "S.E.",
z = "z",
p = "p",
rsquare = "R^2",
estse = "Est.(S.E.)",
eststars = "Est.",
estsestars = "Est.(S.E.)"
),
fits = c(
# "npar",
# "fmin",
"chisq",
"df",
"pvalue",
# "baseline.chisq",
# "baseline.df",
# "baseline.pvalue",
"cfi", # "ML"
# "cfi.scaled", # NOTE: WLSMV "appropriate" scaled fits
# "tli",
# "nnfi",
# "rfi",
# "nfi",
# "pnfi",
# "ifi",
# "rni",
# "logl",
# "unrestricted.logl",
# "aic",
# "bic",
# "ntotal",
# "bic2",
"rmsea", # good for "ML"
"rmsea.ci.lower",
"rmsea.ci.upper",
"rmsea.pvalue",
# "rmsea.scaled", # "WLSMV"
# "rmsea.ci.lower.scaled",
# "rmsea.ci.upper.scaled",
# "rmsea.pvalue.scaled",
# "rmr",
# "rmr_nomean",
"srmr"
# "srmr_bentler",
# "srmr_bentler_nomean",
# "srmr_bollen",
# "srmr_bollen_nomean",
# "srmr_mplus",
# "srmr_mplus_nomean",
# "cn_05",
# "cn_01",
# "gfi",
# "agfi",
# "pgfi",
# "mfi",
# "ecvi"
),
fitLabels = c(
df = "df",
pvalue = "p-value",
cfi = "CFI",
# cfi.scaled = "CFI", # NOTE: WLSMV "appropriate" scaled fits
# aic = "AIC",
# bic = "BIC",
rmsea = "RMSEA",
rmsea.ci.lower = "C.I. lower",
rmsea.ci.upper = "C.I. upper",
rmsea.pvalue = "p-value",
# rmsea.scaled = "RMSEA", # NOTE: WLSMV
# rmsea.ci.lower.scaled = " C.I. lower",
# rmsea.ci.upper.scaled = " C.I. upper",
# rmsea.pvalue.scaled = " p-value",
srmr = "SRMR"
),
file="Table_X_SEM_results", # where to save the table?
print.results = F,
type="html",
# type="latex",
longtable = TRUE,
table.float = TRUE,
caption ="XXX") #, # label ="tab : fit5t2")Show the table above:
# library(lavaan)
# library(semPlot)
# library(semptools)
#
# dataEstFinIta <- read.table("dataEstFinIta.txt", header=TRUE, sep= "\t")
#
# # ------------
# # create model
#
# dfb_m4 <- '
# # Measurement model (factors)
# WorkTasks =~ wle2.1 + wle2.2 + wle4.2 + wle4.3 # Challenging and developing work tasks
# Resources =~ wle3.1 + wle3.2 + wle3.3 # Resources to help learning
# Recognition =~ wle6.1 + wle6.2 + wle7.1 # Recognition as expert and learner
# Congruence =~ wle8.1 + wle8.2 # Congruence with organizational goals
#
# # Regressions
# WorkTasks + Resources + Recognition ~ Congruence
#
#
# # Covariances
# WorkTasks ~~ Resources
# WorkTasks ~~ Recognition
# Recognition ~~ Resources
#
# '
#
# # ---
# # fit
#
# dfb_m4.fit <- sem(dfb_m4,
# data = dataEstFinIta,
# estimator = "WLSMV"
# )
#
#
# # ------------
# # prepare plot
#
# dfb_m4.fit.p1.std <- semPaths(dfb_m4.fit, # std estimates
# whatLabels="std", # optionally "est"
# sizeMan = 5,
# node.width = 1,
# edge.label.cex = .75,
# # style = "ram",
# intercepts=FALSE,
# mar = c(5, 5, 5, 5)) # ,
# # layout = layout)
#
#
# # -----------
# # define plot
#
# indicator_order <- c(
# "wle3.1", "wle3.2", "wle3.3",
# "wle2.1", "wle2.2", "wle4.2", "wle4.3",
# # "wle1.2", "wle1.3",
# "wle6.1", "wle6.2", "wle7.1",
# "wle8.1", "wle8.2")
#
# indicator_factor <- c( "Resources", "Resources", "Resources",
# "WorkTasks", "WorkTasks", "WorkTasks", "WorkTasks",
# # "Participation", "Participation",
# "Recognition", "Recognition", "Recognition",
# "Congruence", "Congruence")
#
# factor_layout <- layout_matrix(Resources = c(1, 3),
# WorkTasks = c(5, 3),
# Recognition = c(9, 3),
# Congruence = c(5, 1))
#
# factor_point_to <- layout_matrix(right = c(1, 3),
# right = c(5, 3),
# right = c(9, 3),
# left = c(5, 1))
#
# indicator_push <- c(Resources = 1.2,
# WorkTasks = 1.2,
# # Participation = 1,
# Recognition = 1.2,
# Congruence = 1.2)
#
# indicator_spread <- c(Resources = 4,
# WorkTasks = 5,
# # Participation = 4.5,
# Recognition = 4,
# Congruence = 3)
#
# loading_position <- c(Resources = .5,
# WorkTasks = .5,
# # Participation = .5,
# Recognition = .5,
# Congruence = .5)
#
# dfb_m4.fit.p2 <- set_sem_layout(dfb_m4.fit.p1.std, # standardized
# indicator_order = indicator_order,
# indicator_factor = indicator_factor,
# factor_layout = factor_layout,
# factor_point_to = factor_point_to,
# indicator_push = indicator_push,
# indicator_spread = indicator_spread,
# loading_position = loading_position) |>
# mark_sig(dfb_m4.fit) |> # show sig
# set_curve(c(
# "Resources ~~ WorkTasks" = -2,
# "WorkTasks ~~ Recognition" = -2,
# "Recognition ~~ Resources" = -4.9)
# )
#
#
#
#
# rotate_residuals <- c(Resources = 0,
# Congruence = 0,
# WorkTasks = -140,
# Recognition = -180)
#
# dfb_m4.fit.p2 <- rotate_resid(dfb_m4.fit.p2, rotate_residuals)
#
# plot(dfb_m4.fit.p2)References
Beaujean, A. A. (2014). Latent Variable Modeling Using R. A Step-by-Step Guide. Routledge.
Ferrer, E., Bolker, S. M., & Grimm, K. J. (2019). Longitudinal Multivariate Psychology. Routledge.