Structural Equation Modeling | Exercises 1
1 What we are going to do
- Ex.1 – Local Model Fit
- Ex.2 – Calculate the model-implied covariance matrix
- Ex.3 – 1-factor- vs 2-factor-model
- Ex.4 – Fit a second-order CFA
- Ex.5 – Compare second-order with first-order CFA
2 Dataset
The data set used throughout is still the European Social Survey ESS4-2008 Edition 4.5. We will restrict the analysis to the Belgian case.
Codebook:
gvslvol Standard of living for the old, governments’ responsibility (0 Not governments’ responsibility at all - 10 Entirely governments’ responsibility)
gvslvue Standard of living for the unemployed, governments’ responsibility (0 Not governments’ responsibility at all - 10 Entirely governments’ responsibility)
gvhlthc Health care for the sick, governments’ responsibility (0 Not governments’ responsibility at all - 10 Entirely governments’ responsibility)
gvcldcr Child care services for working parents, governments’ responsibility (0 Not governments’ responsibility at all - 10 Entirely governments’ responsibility)
gvjbevn Job for everyone, governments’ responsibility (0 Not governments’ responsibility at all - 10 Entirely governments’ responsibility)
gvpdlwk Paid leave from work to care for sick family, governments’ responsibility (0 Not governments’ responsibility at all - 10 Entirely governments’ responsibility)
sbstrec Social benefits/services place too great strain on economy (1 Agree strongly - 5 Disagree strongly)
sbbsntx Social benefits/services cost businesses too much in taxes/charges (1 Agree strongly - 5 Disagree strongly)
sbprvpv Social benefits/services prevent widespread poverty (1 Agree strongly - 5 Disagree strongly)
sbeqsoc Social benefits/services lead to a more equal society (1 Agree strongly - 5 Disagree strongly)
sbcwkfm Social benefits/services make it easier to combine work and family (1 Agree strongly - 5 Disagree strongly)
sblazy Social benefits/services make people lazy (1 Agree strongly - 5 Disagree strongly)
sblwcoa Social benefits/services make people less willing care for one another (1 Agree strongly - 5 Disagree strongly)
sblwlka Social benefits/services make people less willing look after themselves/family (1 Agree strongly - 5 Disagree strongly)
3 Environment preparation
First, let’s load the necessary packages to load, manipulate, visualize and analyse the data.
# Uncomment this once if you need to install the packages on your system
### DATA MANIPULATION ###
# install.packages("haven") # data import from spss
# install.packages("dplyr") # data manipulation
# install.packages("psych") # descriptives
# install.packages("stringr") # string manipulation
### MODELING ###
# install.packages("lavaan") # SEM modelling
# install.packages("lavaan.survey") # Wrapper around packages lavaan and survey
### VISUALIZATION ###
# install.packages("tidySEM") # plotting SEM models
# install.packages("corrplot") # correlation plots
# Load the packages
### DATA MANIPULATION ###
library("haven")
library("dplyr")
library("psych")
library('stringr')
### MODELING ###
library("lavaan")
### VISUALIZATION ###
library("tidySEM")
library("corrplot") 4 Ex.1 – Local Model Fit
- Fit the welfare support model with 6 items
- Modify the model by allowing error correlations and refit the model
- Review the parameter estimates and compare to the first model
- Review the fit statistics (global and local)
- Verify that the new model indeed fits the data better (perform a chi-squared difference test)
- Consider modifying the model even further based on the MIs, revise the model further. Describe its implications (OPTIONAL)
ess_df <- haven::read_sav("https://github.com/albertostefanelli/SEM_labs/raw/master/data/ESS4_belgium.sav")
model_ws_6_1 <-'welf_supp =~ gvslvol + gvslvue + gvhlthc + gvcldcr + gvjbevn + gvpdlwk'
fit_ws_6_1 <- cfa(model_ws_6_1, # model formula
data = ess_df # data frame
)
model_ws_6_2 <-'
welf_supp =~ gvslvol + gvslvue + gvhlthc + gvcldcr + gvjbevn + gvpdlwk
gvslvol~~gvhlthc
gvcldcr~~gvpdlwk
'
fit_ws_6_2 <- cfa(model_ws_6_2, # model formula
data = ess_df # data frame
)
fitm_model_ws_6_1 <- fitMeasures(fit_ws_6_1, c("logl",
"AIC",
"BIC",
"chisq",
"df",
"pvalue",
"cfi",
"tli",
"rmsea"), output = "matrix")
fitm_model_ws_6_2 <- fitMeasures(fit_ws_6_2, c("logl",
"AIC",
"BIC",
"chisq",
"df",
"pvalue",
"cfi",
"tli",
"rmsea"), output = "matrix")
data.frame(Fit=rownames(fitm_model_ws_6_1),
"model_ws_6_1" = round(fitm_model_ws_6_1[,1],2),
"model_ws_6_2" = round(fitm_model_ws_6_2[,1],2)
) Fit model_ws_6_1 model_ws_6_2
logl logl -19550.57 -19406.62
aic aic 39125.14 38841.24
bic bic 39190.68 38917.71
chisq chisq 326.96 39.06
df df 9.00 7.00
pvalue pvalue 0.00 0.00
cfi cfi 0.88 0.99
tli tli 0.80 0.97
rmsea rmsea 0.14 0.05anova(fit_ws_6_1,fit_ws_6_2)Chi-Squared Difference Test
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
fit_ws_6_2 7 38841 38918 39.063
fit_ws_6_1 9 39125 39191 326.956 287.89 2 < 0.00000000000000022 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 15 Ex.2 – Calculate the model-implied covariance matrix
- For the previous models, request the model-implied covariance of the matrix using lavaan’s inspect function.
- Calculate the model-implied covariance matrix using the formula:
- Compare the two matrices
- Calculate the differences between the two matrices (OPTIONAL)
# model-implied var-covariance matrix for observed variables
inspect(fit_ws_6_1, "cov.ov") gvslvl gvslvu gvhlth gvcldc gvjbvn gvpdlw
gvslvol 2.197
gvslvue 0.926 3.675
gvhlthc 1.204 0.875 2.169
gvcldcr 1.140 0.829 1.077 3.107
gvjbevn 1.328 0.966 1.255 1.188 5.139
gvpdlwk 1.163 0.846 1.099 1.041 1.212 3.095 # same thing, but calculated from model parameters
inspect(fit_ws_6_1, "est")$lambda %*% inspect(fit_ws_6_1, "est")$psi %*% t(inspect(fit_ws_6_1, "est")$lambda) + inspect(fit_ws_6_1, "est")$theta gvslvl gvslvu gvhlth gvcldc gvjbvn gvpdlw
gvslvol 2.197
gvslvue 0.926 3.675
gvhlthc 1.204 0.875 2.169
gvcldcr 1.140 0.829 1.077 3.107
gvjbevn 1.328 0.966 1.255 1.188 5.139
gvpdlwk 1.163 0.846 1.099 1.041 1.212 3.095 # differences between the two matrices
inspect(fit_ws_6_2, "cov.ov") - inspect(fit_ws_6_1, "cov.ov") gvslvl gvslvu gvhlth gvcldc gvjbvn gvpdlw
gvslvol 0.000
gvslvue -0.008 0.000
gvhlthc 0.149 -0.010 0.000
gvcldcr -0.178 0.045 -0.171 0.000
gvjbevn -0.018 0.224 -0.021 0.058 0.000
gvpdlwk -0.186 0.042 -0.178 0.648 0.054 0.0006 Ex.3 – 1-factor- vs 3-factor-model
- Using the welfare criticism items, fit a 1-factor model.
- Using the welfare criticism items, fit a 3-factor model.
- Verify whether a 1-factor CFA model is appropriate.
- Review the model parameters.
- Compare the model fit statistics using a tabular form.
- Modify the 3-factor model by dropping items and/or allowing error correlations and compare it with the original model (OPTIONAL)
model_wc_1_factor <-'
wc_crit =~ sbstrec + sbbsntx + sbprvpv + sbeqsoc + sbcwkfm + sblazy + sblwcoa + sblwlka
'
fit_wc_1_factor <- cfa(model_wc_1_factor, # model formula
data = ess_df # data frame
)
summary(fit_wc_1_factor, standardized=TRUE)lavaan 0.6-10 ended normally after 31 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 16
Used Total
Number of observations 1679 1760
Model Test User Model:
Test statistic 756.304
Degrees of freedom 20
P-value (Chi-square) 0.000
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
wc_crit =~
sbstrec 1.000 0.373 0.366
sbbsntx 0.991 0.099 10.054 0.000 0.370 0.370
sbprvpv -0.002 0.065 -0.026 0.979 -0.001 -0.001
sbeqsoc -0.249 0.064 -3.898 0.000 -0.093 -0.111
sbcwkfm -0.055 0.057 -0.958 0.338 -0.021 -0.026
sblazy 2.021 0.156 12.983 0.000 0.754 0.715
sblwcoa 2.085 0.159 13.085 0.000 0.778 0.753
sblwlka 1.944 0.150 12.928 0.000 0.726 0.700
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.sbstrec 0.902 0.033 27.733 0.000 0.902 0.866
.sbbsntx 0.861 0.031 27.697 0.000 0.861 0.863
.sbprvpv 0.767 0.026 28.974 0.000 0.767 1.000
.sbeqsoc 0.690 0.024 28.875 0.000 0.690 0.988
.sbcwkfm 0.602 0.021 28.969 0.000 0.602 0.999
.sblazy 0.543 0.027 19.999 0.000 0.543 0.489
.sblwcoa 0.463 0.026 17.815 0.000 0.463 0.434
.sblwlka 0.548 0.026 20.773 0.000 0.548 0.510
wc_crit 0.139 0.020 6.869 0.000 1.000 1.000model_wc_3_factor <-'
## Economic criticism ##
wc_econo =~ sbstrec + sbbsntx
## Social criticism ##
wc_socia =~ sbprvpv + sbeqsoc + sbcwkfm
## Moral criticism ##
wc_moral =~ sblazy + sblwcoa + sblwlka
'
fit_wc_3_factor <- cfa(model_wc_3_factor, # model formula
data = ess_df # data frame
)
summary(fit_wc_3_factor, standardized=TRUE)lavaan 0.6-10 ended normally after 36 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 19
Used Total
Number of observations 1679 1760
Model Test User Model:
Test statistic 71.348
Degrees of freedom 17
P-value (Chi-square) 0.000
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
wc_econo =~
sbstrec 1.000 0.628 0.616
sbbsntx 1.001 0.093 10.817 0.000 0.629 0.629
wc_socia =~
sbprvpv 1.000 0.462 0.527
sbeqsoc 1.493 0.166 9.011 0.000 0.689 0.824
sbcwkfm 0.631 0.055 11.461 0.000 0.291 0.375
wc_moral =~
sblazy 1.000 0.749 0.710
sblwcoa 1.058 0.045 23.285 0.000 0.792 0.766
sblwlka 0.977 0.043 22.843 0.000 0.732 0.706
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
wc_econo ~~
wc_socia -0.024 0.011 -2.093 0.036 -0.082 -0.082
wc_moral 0.250 0.023 10.872 0.000 0.532 0.532
wc_socia ~~
wc_moral -0.034 0.012 -2.915 0.004 -0.099 -0.099
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.sbstrec 0.647 0.042 15.381 0.000 0.647 0.621
.sbbsntx 0.603 0.041 14.584 0.000 0.603 0.604
.sbprvpv 0.554 0.030 18.722 0.000 0.554 0.722
.sbeqsoc 0.224 0.051 4.404 0.000 0.224 0.320
.sbcwkfm 0.518 0.020 25.832 0.000 0.518 0.859
.sblazy 0.551 0.027 20.083 0.000 0.551 0.495
.sblwcoa 0.441 0.027 16.639 0.000 0.441 0.413
.sblwlka 0.539 0.027 20.320 0.000 0.539 0.502
wc_econo 0.395 0.045 8.689 0.000 1.000 1.000
wc_socia 0.213 0.029 7.328 0.000 1.000 1.000
wc_moral 0.561 0.039 14.477 0.000 1.000 1.000fitm_model_wc_1_factor <- fitMeasures(fit_wc_1_factor, c("logl",
"AIC",
"BIC",
"chisq",
"df",
"pvalue",
"cfi",
"tli",
"rmsea"), output = "matrix")
fitm_model_wc_3_factor <- fitMeasures(fit_wc_3_factor, c("logl",
"AIC",
"BIC",
"chisq",
"df",
"pvalue",
"cfi",
"tli",
"rmsea"), output = "matrix")
data.frame("Fit Indices" = rownames(fitm_model_wc_1_factor),
"model_wc_1_factor" = round(fitm_model_wc_1_factor[, 1], 2),
"model_wc_3_factor" = round(fitm_model_wc_3_factor[, 1], 2)
) Fit.Indices model_wc_1_factor model_wc_3_factor
logl logl -17512.44 -17169.96
aic aic 35056.88 34377.93
bic bic 35143.70 34481.02
chisq chisq 756.30 71.35
df df 20.00 17.00
pvalue pvalue 0.00 0.00
cfi cfi 0.69 0.98
tli tli 0.57 0.96
rmsea rmsea 0.15 0.04## OPTIONAL ##
model_wc_3_factor_mod <-'
## Economic criticism ##
wc_econo =~ sbstrec + sbbsntx
## Social criticism ##
wc_socia =~ sbprvpv + sbeqsoc + sbcwkfm
## Moral criticism ##
wc_moral =~ sblazy + sblwcoa + sblwlka
sblwcoa ~~ sblwlka
'
fit_wc_3_factor_mod <- cfa(model_wc_3_factor_mod, # model formula
data = ess_df # data frame
)
summary(fit_wc_3_factor_mod, standardized=TRUE)lavaan 0.6-10 ended normally after 37 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 20
Used Total
Number of observations 1679 1760
Model Test User Model:
Test statistic 56.695
Degrees of freedom 16
P-value (Chi-square) 0.000
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
wc_econo =~
sbstrec 1.000 0.627 0.614
sbbsntx 1.005 0.091 11.041 0.000 0.630 0.631
wc_socia =~
sbprvpv 1.000 0.460 0.525
sbeqsoc 1.501 0.166 9.038 0.000 0.691 0.827
sbcwkfm 0.632 0.055 11.465 0.000 0.291 0.375
wc_moral =~
sblazy 1.000 0.834 0.791
sblwcoa 0.846 0.060 14.099 0.000 0.706 0.683
sblwlka 0.773 0.057 13.570 0.000 0.645 0.622
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.sblwcoa ~~
.sblwlka 0.138 0.033 4.157 0.000 0.138 0.225
wc_econo ~~
wc_socia -0.024 0.011 -2.119 0.034 -0.083 -0.083
wc_moral 0.290 0.026 10.951 0.000 0.555 0.555
wc_socia ~~
wc_moral -0.045 0.013 -3.316 0.001 -0.116 -0.116
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.sbstrec 0.649 0.041 15.699 0.000 0.649 0.623
.sbbsntx 0.601 0.041 14.740 0.000 0.601 0.602
.sbprvpv 0.555 0.029 18.851 0.000 0.555 0.724
.sbeqsoc 0.221 0.051 4.348 0.000 0.221 0.317
.sbcwkfm 0.518 0.020 25.860 0.000 0.518 0.860
.sblazy 0.417 0.047 8.834 0.000 0.417 0.375
.sblwcoa 0.571 0.039 14.605 0.000 0.571 0.534
.sblwlka 0.659 0.037 17.567 0.000 0.659 0.613
wc_econo 0.393 0.045 8.795 0.000 1.000 1.000
wc_socia 0.212 0.029 7.333 0.000 1.000 1.000
wc_moral 0.695 0.057 12.132 0.000 1.000 1.000fitm_model_wc_3_factor_mod <- fitMeasures(fit_wc_3_factor_mod, c("logl",
"AIC",
"BIC",
"chisq",
"df",
"pvalue",
"cfi",
"tli",
"rmsea"), output = "matrix")
data.frame("Fit Indices" = rownames(fitm_model_wc_1_factor),
"model_wc_3_factor" = round(fitm_model_wc_3_factor[, 1], 2),
"model_wc_3_factor_mod" = round(fitm_model_wc_3_factor_mod[, 1], 2)
) Fit.Indices model_wc_3_factor model_wc_3_factor_mod
logl logl -17169.96 -17162.64
aic aic 34377.93 34365.27
bic bic 34481.02 34473.79
chisq chisq 71.35 56.70
df df 17.00 16.00
pvalue pvalue 0.00 0.00
cfi cfi 0.98 0.98
tli tli 0.96 0.97
rmsea rmsea 0.04 0.047 Ex.4 – Second order model
- Fit a second-order model with the three dimensions of welfare criticism.
- Review the model output using the summary function. Does the model fit well?
- Review the \(R^2\) values of the first-order latent variables. Which first-order factor is explained best by the second-order factor?
- Request model modification indices and explore model modifications.
- Remove low loadings and see if the model improves (OPTIONAL)
model_wc_2_order <-'
## Economic criticism ##
wc_econo =~ sbstrec + sbbsntx
## Social criticism ##
wc_socia =~ sbprvpv + sbeqsoc + sbcwkfm
## Moral criticism ##
wc_moral =~ sblazy + sblwcoa + sblwlka
## Second Order Welfare Criticism ##
wc_crit =~ wc_econo + wc_socia + wc_moral
'
fit_wc_2_order <- cfa(model_wc_2_order, # model formula
data = ess_df # data frame
)
summary(fit_wc_2_order, standardized=TRUE)lavaan 0.6-10 ended normally after 54 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 19
Used Total
Number of observations 1679 1760
Model Test User Model:
Test statistic 71.348
Degrees of freedom 17
P-value (Chi-square) 0.000
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
wc_econo =~
sbstrec 1.000 0.628 0.616
sbbsntx 1.001 0.093 10.817 0.000 0.629 0.629
wc_socia =~
sbprvpv 1.000 0.462 0.527
sbeqsoc 1.493 0.166 9.011 0.000 0.689 0.824
sbcwkfm 0.631 0.055 11.461 0.000 0.291 0.375
wc_moral =~
sblazy 1.000 0.749 0.710
sblwcoa 1.058 0.045 23.285 0.000 0.792 0.766
sblwlka 0.977 0.043 22.843 0.000 0.732 0.706
wc_crit =~
wc_econo 1.000 0.665 0.665
wc_socia -0.136 0.047 -2.873 0.004 -0.124 -0.124
wc_moral 1.434 0.675 2.126 0.033 0.800 0.800
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.sbstrec 0.647 0.042 15.381 0.000 0.647 0.621
.sbbsntx 0.603 0.041 14.584 0.000 0.603 0.604
.sbprvpv 0.554 0.030 18.722 0.000 0.554 0.722
.sbeqsoc 0.224 0.051 4.404 0.000 0.224 0.320
.sbcwkfm 0.518 0.020 25.832 0.000 0.518 0.859
.sblazy 0.551 0.027 20.083 0.000 0.551 0.495
.sblwcoa 0.441 0.027 16.639 0.000 0.441 0.413
.sblwlka 0.539 0.027 20.320 0.000 0.539 0.502
.wc_econo 0.220 0.087 2.546 0.011 0.558 0.558
.wc_socia 0.210 0.029 7.349 0.000 0.985 0.985
.wc_moral 0.202 0.168 1.201 0.230 0.360 0.360
wc_crit 0.175 0.085 2.062 0.039 1.000 1.000inspect(fit_wc_2_order, "r2") sbstrec sbbsntx sbprvpv sbeqsoc sbcwkfm sblazy sblwcoa sblwlka wc_econo wc_socia wc_moral
0.379 0.396 0.278 0.680 0.141 0.505 0.587 0.498 0.442 0.015 0.640 mi<-inspect(fit_wc_2_order, "mi")
# sort from high to low mi.sorted[1:5,]
mi.sorted<-mi[order(-mi$mi), ]
# # only display some large MI values
head(mi.sorted, 5) lhs op rhs mi epc sepc.lv sepc.all sepc.nox
25 wc_econo =~ sbeqsoc 37.330 -0.327 -0.205 -0.246 -0.246
24 wc_econo =~ sbprvpv 26.502 0.211 0.132 0.151 0.151
49 sbstrec ~~ sbprvpv 19.613 0.079 0.079 0.132 0.132
43 wc_crit =~ sbeqsoc 19.285 -0.371 -0.155 -0.186 -0.186
62 sbprvpv ~~ sbcwkfm 19.285 0.241 0.241 0.450 0.450Maybe does not makes sense to impose other modifications for this model.
## OPTIONAL ##
model_wc_2_order_mod <-'
## Economic criticism ##
wc_econo =~ sbstrec + sbbsntx
## Social criticism ##
wc_socia =~ sbprvpv + sbeqsoc
## Moral criticism ##
wc_moral =~ sblazy + sblwcoa + sblwlka
## Second Order Welfare Criticism ##
wc_crit =~ wc_econo + wc_socia + wc_moral
'
fit_wc_2_order_mod <- cfa(model_wc_2_order_mod, # model formula
data = ess_df # data frame
)
summary(fit_wc_2_order_mod, standardized=TRUE)lavaan 0.6-10 did NOT end normally after 697 iterations
** WARNING ** Estimates below are most likely unreliable
Estimator ML
Optimization method NLMINB
Number of model parameters 17
Used Total
Number of observations 1687 1760
Model Test User Model:
Test statistic NA
Degrees of freedom NA
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
wc_econo =~
sbstrec 1.000 0.650 0.637
sbbsntx 0.939 NA 0.611 0.611
wc_socia =~
sbprvpv 1.000 47.670 54.288
sbeqsoc 0.000 NA 0.007 0.008
wc_moral =~
sblazy 1.000 0.746 0.708
sblwcoa 1.062 NA 0.792 0.766
sblwlka 0.981 NA 0.732 0.707
wc_crit =~
wc_econo 1.000 1.282 1.282
wc_socia 0.116 NA 0.002 0.002
wc_moral 0.373 NA 0.416 0.416
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.sbstrec 0.619 NA 0.619 0.594
.sbbsntx 0.625 NA 0.625 0.626
.sbprvpv -2271.620 NA -2271.620 -2946.209
.sbeqsoc 0.697 NA 0.697 1.000
.sblazy 0.554 NA 0.554 0.499
.sblwcoa 0.442 NA 0.442 0.413
.sblwlka 0.536 NA 0.536 0.500
.wc_econo -0.272 NA -0.643 -0.643
.wc_socia 2272.382 NA 1.000 1.000
.wc_moral 0.460 NA 0.827 0.827
wc_crit 0.695 NA 1.000 1.000The model does not fit. Why is it the case ?
8 Ex.5 – Compare first-order with second-order model
- How many degrees of freedom are lost between the two models? Why?
- Statistically compare model fit: does the second-order factor model fit significantly worse than the first-order factor model?
- Fit a model with social and moral criticism of the welfare state as latent variable. Remove the latent factors correlations. Compare the fit (OPTIONAL).
summary(fit_wc_3_factor)lavaan 0.6-10 ended normally after 36 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 19
Used Total
Number of observations 1679 1760
Model Test User Model:
Test statistic 71.348
Degrees of freedom 17
P-value (Chi-square) 0.000
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|)
wc_econo =~
sbstrec 1.000
sbbsntx 1.001 0.093 10.817 0.000
wc_socia =~
sbprvpv 1.000
sbeqsoc 1.493 0.166 9.011 0.000
sbcwkfm 0.631 0.055 11.461 0.000
wc_moral =~
sblazy 1.000
sblwcoa 1.058 0.045 23.285 0.000
sblwlka 0.977 0.043 22.843 0.000
Covariances:
Estimate Std.Err z-value P(>|z|)
wc_econo ~~
wc_socia -0.024 0.011 -2.093 0.036
wc_moral 0.250 0.023 10.872 0.000
wc_socia ~~
wc_moral -0.034 0.012 -2.915 0.004
Variances:
Estimate Std.Err z-value P(>|z|)
.sbstrec 0.647 0.042 15.381 0.000
.sbbsntx 0.603 0.041 14.584 0.000
.sbprvpv 0.554 0.030 18.722 0.000
.sbeqsoc 0.224 0.051 4.404 0.000
.sbcwkfm 0.518 0.020 25.832 0.000
.sblazy 0.551 0.027 20.083 0.000
.sblwcoa 0.441 0.027 16.639 0.000
.sblwlka 0.539 0.027 20.320 0.000
wc_econo 0.395 0.045 8.689 0.000
wc_socia 0.213 0.029 7.328 0.000
wc_moral 0.561 0.039 14.477 0.000summary(fit_wc_2_order)lavaan 0.6-10 ended normally after 54 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 19
Used Total
Number of observations 1679 1760
Model Test User Model:
Test statistic 71.348
Degrees of freedom 17
P-value (Chi-square) 0.000
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|)
wc_econo =~
sbstrec 1.000
sbbsntx 1.001 0.093 10.817 0.000
wc_socia =~
sbprvpv 1.000
sbeqsoc 1.493 0.166 9.011 0.000
sbcwkfm 0.631 0.055 11.461 0.000
wc_moral =~
sblazy 1.000
sblwcoa 1.058 0.045 23.285 0.000
sblwlka 0.977 0.043 22.843 0.000
wc_crit =~
wc_econo 1.000
wc_socia -0.136 0.047 -2.873 0.004
wc_moral 1.434 0.675 2.126 0.033
Variances:
Estimate Std.Err z-value P(>|z|)
.sbstrec 0.647 0.042 15.381 0.000
.sbbsntx 0.603 0.041 14.584 0.000
.sbprvpv 0.554 0.030 18.722 0.000
.sbeqsoc 0.224 0.051 4.404 0.000
.sbcwkfm 0.518 0.020 25.832 0.000
.sblazy 0.551 0.027 20.083 0.000
.sblwcoa 0.441 0.027 16.639 0.000
.sblwlka 0.539 0.027 20.320 0.000
.wc_econo 0.220 0.087 2.546 0.011
.wc_socia 0.210 0.029 7.349 0.000
.wc_moral 0.202 0.168 1.201 0.230
wc_crit 0.175 0.085 2.062 0.039anova(fit_wc_3_factor, fit_wc_2_order)Chi-Squared Difference Test
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
fit_wc_3_factor 17 34378 34481 71.348
fit_wc_2_order 17 34378 34481 71.348 0.0000000010737 0 ## OPTIONAL ##
model_wc_2_factor_corr <-'
## Social criticism ##
wc_socia =~ sbprvpv + sbeqsoc + sbcwkfm
## Moral criticism ##
wc_moral =~ sblazy + sblwcoa + sblwlka
## fixing covariance of LV to zero
wc_socia ~~ 0*wc_moral
'
fit_wc_2_factor_corr <- cfa(model_wc_2_factor_corr, # model formula
data = ess_df # data frame
)
summary(fit_wc_2_factor_corr, fit.measures=TRUE, standardized=TRUE)lavaan 0.6-10 ended normally after 26 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 12
Used Total
Number of observations 1711 1760
Model Test User Model:
Test statistic 27.612
Degrees of freedom 9
P-value (Chi-square) 0.001
Model Test Baseline Model:
Test statistic 1924.056
Degrees of freedom 15
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.990
Tucker-Lewis Index (TLI) 0.984
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -12862.383
Loglikelihood unrestricted model (H1) NA
Akaike (AIC) 25748.765
Bayesian (BIC) 25814.103
Sample-size adjusted Bayesian (BIC) 25775.981
Root Mean Square Error of Approximation:
RMSEA 0.035
90 Percent confidence interval - lower 0.020
90 Percent confidence interval - upper 0.050
P-value RMSEA <= 0.05 0.951
Standardized Root Mean Square Residual:
SRMR 0.031
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
wc_socia =~
sbprvpv 1.000 0.494 0.565
sbeqsoc 1.305 0.135 9.645 0.000 0.645 0.770
sbcwkfm 0.615 0.053 11.514 0.000 0.304 0.392
wc_moral =~
sblazy 1.000 0.729 0.691
sblwcoa 1.113 0.050 22.341 0.000 0.812 0.784
sblwlka 1.006 0.045 22.454 0.000 0.734 0.708
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
wc_socia ~~
wc_moral 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.sbprvpv 0.521 0.030 17.265 0.000 0.521 0.681
.sbeqsoc 0.285 0.043 6.691 0.000 0.285 0.407
.sbcwkfm 0.510 0.020 25.859 0.000 0.510 0.847
.sblazy 0.582 0.028 20.678 0.000 0.582 0.522
.sblwcoa 0.412 0.028 14.529 0.000 0.412 0.385
.sblwlka 0.535 0.027 19.665 0.000 0.535 0.498
wc_socia 0.244 0.031 7.877 0.000 1.000 1.000
wc_moral 0.532 0.038 13.970 0.000 1.000 1.000