Structural Equation Modeling | Exercises 4
1 What we are going to cover
- Ex.1 – Non-normal continuous data
- Ex.2 – Categorical data multi-group
2 Data
The data set used throughout is the European Social Survey ESS4-2008 Edition 4.5 was released on 1 December 2018. We will restrict the analysis to the Belgian case. Each line in the data set represents a Belgian respondent. The full dataset an documentation can be found on the ESS website
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)
agea Respondent’s age
eduyrs Years of full-time education completed
gndr Gender (1 Male, 2 Female)
hinctnta Household’s total net income, all sources (Deciles of the actual household income range in the given country.)
gincdif Government should reduce differences in income levels (1 Agree strongly - 5 Disagree strongly)
dfincac Large differences in income acceptable to reward talents and efforts (1 Agree strongly - 5 Disagree strongly)
smdfslv For fair society, differences in standard of living should be small (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
# install.packages("purrr") # table manipulation
### MODELING ###
# install.packages("lavaan") # SEM modelling
# install.packages("MVN") # tests for multivariate normality
# install.packages("Amelia") # performing multiple imputation
### VISUALIZATION ###
# install.packages("tidySEM") # plotting SEM models
# install.packages("ggplot2") # plotting
# install.packages("patchwork") # organization plots
# Load the packages
### DATA MANIPULATION ###
library("haven")
library("dplyr")
library("psych")
library("stringr")
library("purrr")
### MODELING ###
library("lavaan")
library("MVN")
library("Amelia")
### VISUALIZATION ###
library("tidySEM")
library("ggplot2")
library("patchwork") 4 Ex.1 – Non-normal continuous data
- Check the multivariate normality for the question gincdif, dfincac, smdfslv (egalitarianism) and gvslvol gvhlthc gvcldcr gvpdlwk (welfare support) .
- Estimate a simple CFA (2 latent variable), without any error covariances for egalitarianism (gincdif, dfincac, smdfslv) and welfare support (gvslvol gvhlthc gvcldcr gvpdlwk)
- Estimate the model using the ML estimation, then re-estimate using MLM.
- Compare fit statistics, loadings, and standard errors.
- Remove the covariance between egalitarianism and welfare support and re-estimate the model. Compare the fit (OPTIONAL).
ess_df <- haven::read_sav("https://github.com/albertostefanelli/SEM_labs/raw/master/data/ESS4_belgium.sav")
# select the egalitarianism and welfare support items
ess_df_eg <- ess_df[,c("gincdif", "dfincac", "smdfslv", "gvslvol", "gvhlthc", "gvcldcr","gvpdlwk")]
# remove NAs
ess_df_eg_na <- na.omit(ess_df_eg)
mvn_test <- mvn(data = ess_df_eg_na, # our data without NAs
mvnTest = c("hz") # type of normality test to perform
)
mvn_test$multivariateNormality
Test HZ p value MVN
1 Henze-Zirkler 6.155718 0 NO
$univariateNormality
Test Variable Statistic p value Normality
1 Anderson-Darling gincdif 105.3500 <0.001 NO
2 Anderson-Darling dfincac 102.9968 <0.001 NO
3 Anderson-Darling smdfslv 124.3559 <0.001 NO
4 Anderson-Darling gvslvol 47.8262 <0.001 NO
5 Anderson-Darling gvhlthc 51.7952 <0.001 NO
6 Anderson-Darling gvcldcr 37.7151 <0.001 NO
7 Anderson-Darling gvpdlwk 39.3915 <0.001 NO
$Descriptives
n Mean Std.Dev Median Min Max 25th 75th Skew Kurtosis
gincdif 1728 2.230903 1.0612045 2 1 5 1 3 0.7362838 -0.2431670
dfincac 1728 2.626736 1.0550139 2 1 5 2 3 0.5063052 -0.5763850
smdfslv 1728 2.471065 0.9776167 2 1 5 2 3 0.6607562 -0.1964199
gvslvol 1728 7.884838 1.4786813 8 0 10 7 9 -0.7864489 1.3981895
gvhlthc 1728 8.042824 1.4717630 8 0 10 7 9 -0.9593240 1.8591224
gvcldcr 1728 7.287616 1.7607401 7 0 10 6 8 -0.8186374 1.3878985
gvpdlwk 1728 7.328704 1.7606153 8 0 10 6 8 -0.8356154 1.3712611model_eg_ws <- '
## Egalitarianism ##
egual =~ gincdif + dfincac + smdfslv
## Welfare support ##
ws =~ gvslvol + gvhlthc + gvcldcr + gvpdlwk
'
fit_eg_ws_ml <- cfa(model_eg_ws, # model formula
data = ess_df, # data frame
estimator = "ML" # select the estimator
)
summary(fit_eg_ws_ml,
standardized=TRUE,
fit.measures = TRUE
)lavaan 0.6.15 ended normally after 33 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 15
Used Total
Number of observations 1728 1760
Model Test User Model:
Test statistic 291.138
Degrees of freedom 13
P-value (Chi-square) 0.000
Model Test Baseline Model:
Test statistic 2434.089
Degrees of freedom 21
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.885
Tucker-Lewis Index (TLI) 0.814
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -19543.347
Loglikelihood unrestricted model (H1) NA
Akaike (AIC) 39116.693
Bayesian (BIC) 39198.514
Sample-size adjusted Bayesian (SABIC) 39150.861
Root Mean Square Error of Approximation:
RMSEA 0.111
90 Percent confidence interval - lower 0.100
90 Percent confidence interval - upper 0.123
P-value H_0: RMSEA <= 0.050 0.000
P-value H_0: RMSEA >= 0.080 1.000
Standardized Root Mean Square Residual:
SRMR 0.051
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
egual =~
gincdif 1.000 0.688 0.648
dfincac -0.615 0.059 -10.362 0.000 -0.423 -0.401
smdfslv 0.870 0.082 10.628 0.000 0.598 0.612
ws =~
gvslvol 1.000 1.161 0.785
gvhlthc 0.912 0.038 23.711 0.000 1.059 0.720
gvcldcr 0.847 0.043 19.896 0.000 0.983 0.559
gvpdlwk 0.882 0.043 20.618 0.000 1.024 0.582
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
egual ~~
ws -0.209 0.030 -6.942 0.000 -0.262 -0.262
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.gincdif 0.653 0.049 13.424 0.000 0.653 0.580
.dfincac 0.934 0.036 25.622 0.000 0.934 0.839
.smdfslv 0.597 0.039 15.464 0.000 0.597 0.625
.gvslvol 0.838 0.054 15.568 0.000 0.838 0.383
.gvhlthc 1.043 0.053 19.752 0.000 1.043 0.482
.gvcldcr 2.131 0.083 25.622 0.000 2.131 0.688
.gvpdlwk 2.049 0.082 25.110 0.000 2.049 0.661
egual 0.473 0.053 8.867 0.000 1.000 1.000
ws 1.348 0.082 16.346 0.000 1.000 1.000fit_eg_ws_mlr <- cfa(model_eg_ws, # model formula
data = ess_df, # data frame
estimator = "MLR" # select the estimator
)
summary(fit_eg_ws_mlr,
standardized=TRUE,
fit.measures = TRUE
)lavaan 0.6.15 ended normally after 33 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 15
Used Total
Number of observations 1728 1760
Model Test User Model:
Standard Scaled
Test Statistic 291.138 230.235
Degrees of freedom 13 13
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.265
Yuan-Bentler correction (Mplus variant)
Model Test Baseline Model:
Test statistic 2434.089 1588.615
Degrees of freedom 21 21
P-value 0.000 0.000
Scaling correction factor 1.532
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.885 0.861
Tucker-Lewis Index (TLI) 0.814 0.776
Robust Comparative Fit Index (CFI) 0.886
Robust Tucker-Lewis Index (TLI) 0.815
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -19543.347 -19543.347
Scaling correction factor 1.684
for the MLR correction
Loglikelihood unrestricted model (H1) NA NA
Scaling correction factor 1.489
for the MLR correction
Akaike (AIC) 39116.693 39116.693
Bayesian (BIC) 39198.514 39198.514
Sample-size adjusted Bayesian (SABIC) 39150.861 39150.861
Root Mean Square Error of Approximation:
RMSEA 0.111 0.098
90 Percent confidence interval - lower 0.100 0.089
90 Percent confidence interval - upper 0.123 0.108
P-value H_0: RMSEA <= 0.050 0.000 0.000
P-value H_0: RMSEA >= 0.080 1.000 0.999
Robust RMSEA 0.111
90 Percent confidence interval - lower 0.098
90 Percent confidence interval - upper 0.123
P-value H_0: Robust RMSEA <= 0.050 0.000
P-value H_0: Robust RMSEA >= 0.080 1.000
Standardized Root Mean Square Residual:
SRMR 0.051 0.051
Parameter Estimates:
Standard errors Sandwich
Information bread Observed
Observed information based on Hessian
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
egual =~
gincdif 1.000 0.688 0.648
dfincac -0.615 0.059 -10.340 0.000 -0.423 -0.401
smdfslv 0.870 0.088 9.881 0.000 0.598 0.612
ws =~
gvslvol 1.000 1.161 0.785
gvhlthc 0.912 0.034 26.668 0.000 1.059 0.720
gvcldcr 0.847 0.081 10.453 0.000 0.983 0.559
gvpdlwk 0.882 0.081 10.889 0.000 1.024 0.582
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
egual ~~
ws -0.209 0.030 -6.884 0.000 -0.262 -0.262
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.gincdif 0.653 0.055 11.828 0.000 0.653 0.580
.dfincac 0.934 0.035 26.623 0.000 0.934 0.839
.smdfslv 0.597 0.047 12.668 0.000 0.597 0.625
.gvslvol 0.838 0.095 8.794 0.000 0.838 0.383
.gvhlthc 1.043 0.091 11.499 0.000 1.043 0.482
.gvcldcr 2.131 0.164 13.008 0.000 2.131 0.688
.gvpdlwk 2.049 0.171 11.992 0.000 2.049 0.661
egual 0.473 0.056 8.480 0.000 1.000 1.000
ws 1.348 0.119 11.316 0.000 1.000 1.000tidy_results_ml <- table_results(fit_eg_ws_ml,
columns = c("label",
"est_sig",
"se"),
digits = 2
)
tidy_results_mlr <- table_results(fit_eg_ws_mlr,
columns = c("label",
"est_sig",
"se"),
digits = 2
)
data.frame(Parameters = tidy_results_mlr$label,
"ML Model" = paste(tidy_results_ml$est_sig,"(",tidy_results_ml$se, ")"),
"MLR Model" = paste(tidy_results_mlr$est_sig,"(",tidy_results_mlr$se, ")")
) Parameters ML.Model MLR.Model
1 egual.BY.gincdif 1.00 ( 0.00 ) 1.00 ( 0.00 )
2 egual.BY.dfincac -0.61*** ( 0.06 ) -0.61*** ( 0.06 )
3 egual.BY.smdfslv 0.87*** ( 0.08 ) 0.87*** ( 0.09 )
4 ws.BY.gvslvol 1.00 ( 0.00 ) 1.00 ( 0.00 )
5 ws.BY.gvhlthc 0.91*** ( 0.04 ) 0.91*** ( 0.03 )
6 ws.BY.gvcldcr 0.85*** ( 0.04 ) 0.85*** ( 0.08 )
7 ws.BY.gvpdlwk 0.88*** ( 0.04 ) 0.88*** ( 0.08 )
8 Variances.gincdif 0.65*** ( 0.05 ) 0.65*** ( 0.06 )
9 Variances.dfincac 0.93*** ( 0.04 ) 0.93*** ( 0.04 )
10 Variances.smdfslv 0.60*** ( 0.04 ) 0.60*** ( 0.05 )
11 Variances.gvslvol 0.84*** ( 0.05 ) 0.84*** ( 0.10 )
12 Variances.gvhlthc 1.04*** ( 0.05 ) 1.04*** ( 0.09 )
13 Variances.gvcldcr 2.13*** ( 0.08 ) 2.13*** ( 0.16 )
14 Variances.gvpdlwk 2.05*** ( 0.08 ) 2.05*** ( 0.17 )
15 Variances.egual 0.47*** ( 0.05 ) 0.47*** ( 0.06 )
16 Variances.ws 1.35*** ( 0.08 ) 1.35*** ( 0.12 )
17 egual.WITH.ws -0.21*** ( 0.03 ) -0.21*** ( 0.03 )model_eg_ws_cov <- '
## Egalitarianism ##
egual =~ gincdif + dfincac + smdfslv
## Welfare support ##
ws =~ gvslvol + gvhlthc + gvcldcr + gvpdlwk
ws ~~ 0*egual
'
fit_eg_ws_mlr_cov <- cfa(model_eg_ws_cov, # model formula
data = ess_df, # data frame
estimator = "MLR" # select the estimator
)
summary(fit_eg_ws_mlr_cov,
standardized=TRUE,
fit.measures = TRUE
)lavaan 0.6.15 ended normally after 33 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 14
Used Total
Number of observations 1728 1760
Model Test User Model:
Standard Scaled
Test Statistic 347.870 279.437
Degrees of freedom 14 14
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.245
Yuan-Bentler correction (Mplus variant)
Model Test Baseline Model:
Test statistic 2434.089 1588.615
Degrees of freedom 21 21
P-value 0.000 0.000
Scaling correction factor 1.532
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.862 0.831
Tucker-Lewis Index (TLI) 0.792 0.746
Robust Comparative Fit Index (CFI) 0.862
Robust Tucker-Lewis Index (TLI) 0.794
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -19571.712 -19571.712
Scaling correction factor 1.733
for the MLR correction
Loglikelihood unrestricted model (H1) NA NA
Scaling correction factor 1.489
for the MLR correction
Akaike (AIC) 39171.425 39171.425
Bayesian (BIC) 39247.791 39247.791
Sample-size adjusted Bayesian (SABIC) 39203.314 39203.314
Root Mean Square Error of Approximation:
RMSEA 0.117 0.105
90 Percent confidence interval - lower 0.107 0.095
90 Percent confidence interval - upper 0.128 0.114
P-value H_0: RMSEA <= 0.050 0.000 0.000
P-value H_0: RMSEA >= 0.080 1.000 1.000
Robust RMSEA 0.117
90 Percent confidence interval - lower 0.105
90 Percent confidence interval - upper 0.129
P-value H_0: Robust RMSEA <= 0.050 0.000
P-value H_0: Robust RMSEA >= 0.080 1.000
Standardized Root Mean Square Residual:
SRMR 0.082 0.082
Parameter Estimates:
Standard errors Sandwich
Information bread Observed
Observed information based on Hessian
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
egual =~
gincdif 1.000 0.702 0.662
dfincac -0.612 0.064 -9.493 0.000 -0.429 -0.407
smdfslv 0.827 0.089 9.275 0.000 0.581 0.595
ws =~
gvslvol 1.000 1.158 0.783
gvhlthc 0.921 0.035 26.480 0.000 1.066 0.725
gvcldcr 0.847 0.082 10.373 0.000 0.981 0.557
gvpdlwk 0.881 0.081 10.851 0.000 1.020 0.580
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
egual ~~
ws 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.gincdif 0.632 0.059 10.726 0.000 0.632 0.562
.dfincac 0.928 0.036 25.920 0.000 0.928 0.834
.smdfslv 0.618 0.047 13.005 0.000 0.618 0.647
.gvslvol 0.844 0.096 8.813 0.000 0.844 0.386
.gvhlthc 1.028 0.092 11.212 0.000 1.028 0.475
.gvcldcr 2.137 0.165 12.920 0.000 2.137 0.690
.gvpdlwk 2.057 0.171 12.004 0.000 2.057 0.664
egual 0.493 0.060 8.161 0.000 1.000 1.000
ws 1.341 0.119 11.316 0.000 1.000 1.000# let's compare the fit of the different models
model_fit <- function(lavobject) {
vars <- c("chisq", "df", "cfi", "tli", "rmsea", "rmsea.ci.lower", "rmsea.ci.upper", "rmsea.pvalue", "srmr")
return(fitmeasures(lavobject)[vars] %>% data.frame() %>% round(2) %>% t())
}
table_fit <-
list(model_fit(fit_eg_ws_mlr),
model_fit(fit_eg_ws_mlr_cov)) %>%
reduce(rbind)
rownames(table_fit) <- c("eg_ws_mlr", "eg_ws_mlr_cov")
table_fit chisq df cfi tli rmsea rmsea.ci.lower rmsea.ci.upper rmsea.pvalue srmr
eg_ws_mlr 291.14 13 0.88 0.81 0.11 0.10 0.12 0 0.05
eg_ws_mlr_cov 347.87 14 0.86 0.79 0.12 0.11 0.13 0 0.08anova(fit_eg_ws_mlr, fit_eg_ws_mlr_cov)
Scaled Chi-Squared Difference Test (method = "satorra.bentler.2001")
lavaan NOTE:
The "Chisq" column contains standard test statistics, not the
robust test that should be reported per model. A robust difference
test is a function of two standard (not robust) statistics.
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
fit_eg_ws_mlr 13 39117 39199 291.14
fit_eg_ws_mlr_cov 14 39171 39248 347.87 57.321 1 0.00000000000003702 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 15 Ex.2 – Measurement invariance with categorical data
- Re-estimate a CFA (2 latent variable) for egalitarianism (gincdif, dfincac, smdfslv) and social criticism (bprvpv, sbeqsoc, sbcwkfm) treating the items as ordinal
- Test if the two latent constructs reach scalar invariance between male and feamle (free/fixed factor loading and thresholds).
- Compare and interpret the fit.
- Even if not necessary, relax certain equality constraints, re-fit the model, compare the fit.
- Compare the metric model fitted with continuous data with the one fitted with ordinal data (OPTIONAL)
Lavaan syntax for relaxing equality constraints:
y | c(label1,label1)*t1 # set threshold equal across 2 groups
y | c(label1,label2)*t1 # set threshold free across 2 groups
u3 ~*~ c(1,1)*u3 # fix scale of the endogenous variable "u3" to 1 in both groupsess_df$gndr <- factor(ess_df$gndr,
levels = c("1", "2"), # levels
labels = c("Male", "Female")) # labels
model_eg_sc <- '
## Egalitarianism ##
egual =~ gincdif + dfincac + smdfslv
## Social Criticism ##
wc_socia =~ sbprvpv + sbeqsoc + sbcwkfm
'
fit_eg_sc_conf<- cfa(model_eg_sc, # model formula
data = ess_df, # data frame
estimator = "WLSMV", # select the estimator
group = "gndr",
ordered = c(
"gincdif",
"dfincac",
"smdfslv",
"sbprvpv",
"sbeqsoc",
"sbcwkfm" )
)
fit_eg_sc_metric <- cfa(model_eg_sc, # model formula
data = ess_df, # data frame
estimator = "WLSMV", # select the estimator
group = "gndr",
ordered = c(
"gincdif",
"dfincac",
"smdfslv",
"sbprvpv",
"sbeqsoc",
"sbcwkfm" ),
group.equal=c("loadings")
)
fit_eg_sc_scalar <- cfa(model_eg_sc, # model formula
data = ess_df, # data frame
estimator = "WLSMV", # select the estimator
group = "gndr",
ordered = c(
"gincdif",
"dfincac",
"smdfslv",
"sbprvpv",
"sbeqsoc",
"sbcwkfm" ),
group.equal=c("loadings",
"thresholds")
)
anova(fit_eg_sc_conf,fit_eg_sc_metric,fit_eg_sc_scalar)
Scaled Chi-Squared Difference Test (method = "satorra.2000")
lavaan NOTE:
The "Chisq" column contains standard test statistics, not the
robust test that should be reported per model. A robust difference
test is a function of two standard (not robust) statistics.
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
fit_eg_sc_conf 16 87.126
fit_eg_sc_metric 20 91.971 5.770 4 0.21699
fit_eg_sc_scalar 36 121.815 39.163 16 0.00103 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# According to the Lagrange multiplier test it seems that an improvement in fit over the metric model is difficult
# However, for didactic purpose we are going to free the the equality constraints with the highest chi-square contribution
lavTestScore(fit_eg_sc_scalar)$test
total score test:
test X2 df p.value
1 score 34.435 28 0.187
$uni
univariate score tests:
lhs op rhs X2 df p.value
1 .p2. == .p55. 0.399 1 0.528
2 .p3. == .p56. 0.015 1 0.903
3 .p5. == .p58. 0.081 1 0.777
4 .p6. == .p59. 2.986 1 0.084
5 .p7. == .p60. 1.276 1 0.259
6 .p8. == .p61. 3.413 1 0.065
7 .p9. == .p62. 0.075 1 0.785
8 .p10. == .p63. 0.001 1 0.981
9 .p11. == .p64. 12.723 1 0.000
10 .p12. == .p65. 7.049 1 0.008
11 .p13. == .p66. 0.930 1 0.335
12 .p14. == .p67. 2.246 1 0.134
13 .p15. == .p68. 0.760 1 0.383
14 .p16. == .p69. 0.093 1 0.760
15 .p17. == .p70. 0.225 1 0.635
16 .p18. == .p71. 0.055 1 0.815
17 .p19. == .p72. 0.035 1 0.851
18 .p20. == .p73. 0.005 1 0.942
19 .p21. == .p74. 0.135 1 0.714
20 .p22. == .p75. 0.313 1 0.576
21 .p23. == .p76. 2.192 1 0.139
22 .p24. == .p77. 1.190 1 0.275
23 .p25. == .p78. 0.015 1 0.903
24 .p26. == .p79. 0.399 1 0.528
25 .p27. == .p80. 2.955 1 0.086
26 .p28. == .p81. 5.497 1 0.019
27 .p29. == .p82. 1.005 1 0.316
28 .p30. == .p83. 0.111 1 0.739parTable(fit_eg_sc_scalar) id lhs op rhs user block group free ustart exo label plabel start est se
1 1 egual =~ gincdif 1 1 1 0 1 0 .p1. 1.000 1.000 0.000
2 2 egual =~ dfincac 1 1 1 1 NA 0 .p2. .p2. -0.648 -0.617 0.048
3 3 egual =~ smdfslv 1 1 1 2 NA 0 .p3. .p3. 0.820 0.986 0.076
4 4 wc_socia =~ sbprvpv 1 1 1 0 1 0 .p4. 1.000 1.000 0.000
5 5 wc_socia =~ sbeqsoc 1 1 1 3 NA 0 .p5. .p5. 1.523 1.247 0.092
6 6 wc_socia =~ sbcwkfm 1 1 1 4 NA 0 .p6. .p6. 0.753 0.692 0.045
7 7 gincdif | t1 0 1 1 5 NA 0 .p7. .p7. -0.673 -0.701 0.042
8 8 gincdif | t2 0 1 1 6 NA 0 .p8. .p8. 0.513 0.459 0.040
9 9 gincdif | t3 0 1 1 7 NA 0 .p9. .p9. 0.922 0.915 0.046
10 10 gincdif | t4 0 1 1 8 NA 0 .p10. .p10. 1.885 1.886 0.077
11 11 dfincac | t1 0 1 1 9 NA 0 .p11. .p11. -1.118 -1.217 0.048
12 12 dfincac | t2 0 1 1 10 NA 0 .p12. .p12. 0.259 0.178 0.034
13 13 dfincac | t3 0 1 1 11 NA 0 .p13. .p13. 0.765 0.735 0.039
14 14 dfincac | t4 0 1 1 12 NA 0 .p14. .p14. 1.760 1.691 0.065
15 15 smdfslv | t1 0 1 1 13 NA 0 .p15. .p15. -1.212 -1.237 0.052
16 16 smdfslv | t2 0 1 1 14 NA 0 .p16. .p16. 0.268 0.259 0.038
17 17 smdfslv | t3 0 1 1 15 NA 0 .p17. .p17. 0.852 0.838 0.044
18 18 smdfslv | t4 0 1 1 16 NA 0 .p18. .p18. 1.835 1.824 0.073
19 19 sbprvpv | t1 0 1 1 17 NA 0 .p19. .p19. -1.175 -1.170 0.051
20 20 sbprvpv | t2 0 1 1 18 NA 0 .p20. .p20. 0.537 0.539 0.038
21 21 sbprvpv | t3 0 1 1 19 NA 0 .p21. .p21. 1.118 1.106 0.047
22 22 sbprvpv | t4 0 1 1 20 NA 0 .p22. .p22. 2.224 2.183 0.092
23 23 sbeqsoc | t1 0 1 1 21 NA 0 .p23. .p23. -1.346 -1.386 0.060
24 24 sbeqsoc | t2 0 1 1 22 NA 0 .p24. .p24. 0.540 0.509 0.041
25 25 sbeqsoc | t3 0 1 1 23 NA 0 .p25. .p25. 1.169 1.165 0.050
26 26 sbeqsoc | t4 0 1 1 24 NA 0 .p26. .p26. 2.224 2.177 0.091
27 27 sbcwkfm | t1 0 1 1 25 NA 0 .p27. .p27. -1.493 -1.428 0.057
28 28 sbcwkfm | t2 0 1 1 26 NA 0 .p28. .p28. 0.472 0.542 0.036
29 29 sbcwkfm | t3 0 1 1 27 NA 0 .p29. .p29. 1.230 1.264 0.048
30 30 sbcwkfm | t4 0 1 1 28 NA 0 .p30. .p30. 2.301 2.327 0.100
31 31 gincdif ~~ gincdif 0 1 1 0 1 0 .p31. 1.000 0.531 0.000
32 32 dfincac ~~ dfincac 0 1 1 0 1 0 .p32. 1.000 0.822 0.000
33 33 smdfslv ~~ smdfslv 0 1 1 0 1 0 .p33. 1.000 0.544 0.000
34 34 sbprvpv ~~ sbprvpv 0 1 1 0 1 0 .p34. 1.000 0.587 0.000
35 35 sbeqsoc ~~ sbeqsoc 0 1 1 0 1 0 .p35. 1.000 0.358 0.000
36 36 sbcwkfm ~~ sbcwkfm 0 1 1 0 1 0 .p36. 1.000 0.802 0.000
37 37 egual ~~ egual 0 1 1 29 NA 0 .p37. 0.050 0.469 0.044
38 38 wc_socia ~~ wc_socia 0 1 1 30 NA 0 .p38. 0.050 0.413 0.039
39 39 egual ~~ wc_socia 0 1 1 31 NA 0 .p39. 0.000 0.142 0.023
40 40 gincdif ~*~ gincdif 0 1 1 0 1 0 .p40. 1.000 1.000 0.000
41 41 dfincac ~*~ dfincac 0 1 1 0 1 0 .p41. 1.000 1.000 0.000
42 42 smdfslv ~*~ smdfslv 0 1 1 0 1 0 .p42. 1.000 1.000 0.000
43 43 sbprvpv ~*~ sbprvpv 0 1 1 0 1 0 .p43. 1.000 1.000 0.000
44 44 sbeqsoc ~*~ sbeqsoc 0 1 1 0 1 0 .p44. 1.000 1.000 0.000
45 45 sbcwkfm ~*~ sbcwkfm 0 1 1 0 1 0 .p45. 1.000 1.000 0.000
46 46 gincdif ~1 0 1 1 0 0 0 .p46. 0.000 0.000 0.000
47 47 dfincac ~1 0 1 1 0 0 0 .p47. 0.000 0.000 0.000
48 48 smdfslv ~1 0 1 1 0 0 0 .p48. 0.000 0.000 0.000
49 49 sbprvpv ~1 0 1 1 0 0 0 .p49. 0.000 0.000 0.000
50 50 sbeqsoc ~1 0 1 1 0 0 0 .p50. 0.000 0.000 0.000
51 51 sbcwkfm ~1 0 1 1 0 0 0 .p51. 0.000 0.000 0.000
52 52 egual ~1 0 1 1 0 0 0 .p52. 0.000 0.000 0.000
53 53 wc_socia ~1 0 1 1 0 0 0 .p53. 0.000 0.000 0.000
54 54 egual =~ gincdif 1 2 2 0 1 0 .p54. 1.000 1.000 0.000
55 55 egual =~ dfincac 1 2 2 32 NA 0 .p2. .p55. -0.610 -0.617 0.048
56 56 egual =~ smdfslv 1 2 2 33 NA 0 .p3. .p56. 1.042 0.986 0.076
57 57 wc_socia =~ sbprvpv 1 2 2 0 1 0 .p57. 1.000 1.000 0.000
58 58 wc_socia =~ sbeqsoc 1 2 2 34 NA 0 .p5. .p58. 1.109 1.247 0.092
59 59 wc_socia =~ sbcwkfm 1 2 2 35 NA 0 .p6. .p59. 0.610 0.692 0.045
60 60 gincdif | t1 0 2 2 36 NA 0 .p7. .p60. -0.636 -0.701 0.042
61 61 gincdif | t2 0 2 2 37 NA 0 .p8. .p61. 0.516 0.459 0.040
62 62 gincdif | t3 0 2 2 38 NA 0 .p9. .p62. 1.020 0.915 0.046
63 63 gincdif | t4 0 2 2 39 NA 0 .p10. .p63. 2.015 1.886 0.077
64 64 dfincac | t1 0 2 2 40 NA 0 .p11. .p64. -1.422 -1.217 0.048
65 65 dfincac | t2 0 2 2 41 NA 0 .p12. .p65. 0.042 0.178 0.034
66 66 dfincac | t3 0 2 2 42 NA 0 .p13. .p66. 0.654 0.735 0.039
67 67 dfincac | t4 0 2 2 43 NA 0 .p14. .p67. 1.594 1.691 0.065
68 68 smdfslv | t1 0 2 2 44 NA 0 .p15. .p68. -1.209 -1.237 0.052
69 69 smdfslv | t2 0 2 2 45 NA 0 .p16. .p69. 0.366 0.259 0.038
70 70 smdfslv | t3 0 2 2 46 NA 0 .p17. .p70. 0.964 0.838 0.044
71 71 smdfslv | t4 0 2 2 47 NA 0 .p18. .p71. 1.993 1.824 0.073
72 72 sbprvpv | t1 0 2 2 48 NA 0 .p19. .p72. -1.259 -1.170 0.051
73 73 sbprvpv | t2 0 2 2 49 NA 0 .p20. .p73. 0.523 0.539 0.038
74 74 sbprvpv | t3 0 2 2 50 NA 0 .p21. .p74. 1.102 1.106 0.047
75 75 sbprvpv | t4 0 2 2 51 NA 0 .p22. .p75. 2.201 2.183 0.092
76 76 sbeqsoc | t1 0 2 2 52 NA 0 .p23. .p76. -1.498 -1.386 0.060
77 77 sbeqsoc | t2 0 2 2 53 NA 0 .p24. .p77. 0.435 0.509 0.041
78 78 sbeqsoc | t3 0 2 2 54 NA 0 .p25. .p78. 1.123 1.165 0.050
79 79 sbeqsoc | t4 0 2 2 55 NA 0 .p26. .p79. 2.112 2.177 0.091
80 80 sbcwkfm | t1 0 2 2 56 NA 0 .p27. .p80. -1.447 -1.428 0.057
81 81 sbcwkfm | t2 0 2 2 57 NA 0 .p28. .p81. 0.601 0.542 0.036
82 82 sbcwkfm | t3 0 2 2 58 NA 0 .p29. .p82. 1.312 1.264 0.048
83 83 sbcwkfm | t4 0 2 2 59 NA 0 .p30. .p83. 2.405 2.327 0.100
84 84 gincdif ~~ gincdif 0 2 2 0 1 0 .p84. 1.000 0.525 0.000
85 85 dfincac ~~ dfincac 0 2 2 0 1 0 .p85. 1.000 0.807 0.000
86 86 smdfslv ~~ smdfslv 0 2 2 0 1 0 .p86. 1.000 0.483 0.000
87 87 sbprvpv ~~ sbprvpv 0 2 2 0 1 0 .p87. 1.000 0.506 0.000
88 88 sbeqsoc ~~ sbeqsoc 0 2 2 0 1 0 .p88. 1.000 0.341 0.000
89 89 sbcwkfm ~~ sbcwkfm 0 2 2 0 1 0 .p89. 1.000 0.740 0.000
90 90 egual ~~ egual 0 2 2 60 NA 0 .p90. 0.050 0.449 0.056
91 91 wc_socia ~~ wc_socia 0 2 2 61 NA 0 .p91. 0.050 0.410 0.047
92 92 egual ~~ wc_socia 0 2 2 62 NA 0 .p92. 0.000 0.057 0.021
93 93 gincdif ~*~ gincdif 0 2 2 63 NA 0 .p93. 1.000 1.013 0.044
94 94 dfincac ~*~ dfincac 0 2 2 64 NA 0 .p94. 1.000 1.011 0.040
95 95 smdfslv ~*~ smdfslv 0 2 2 65 NA 0 .p95. 1.000 1.043 0.042
96 96 sbprvpv ~*~ sbprvpv 0 2 2 66 NA 0 .p96. 1.000 1.045 0.041
97 97 sbeqsoc ~*~ sbeqsoc 0 2 2 67 NA 0 .p97. 1.000 1.011 0.041
98 98 sbcwkfm ~*~ sbcwkfm 0 2 2 68 NA 0 .p98. 1.000 1.033 0.039
99 99 gincdif ~1 0 2 2 0 0 0 .p99. 0.000 0.000 0.000
100 100 dfincac ~1 0 2 2 0 0 0 .p100. 0.000 0.000 0.000
101 101 smdfslv ~1 0 2 2 0 0 0 .p101. 0.000 0.000 0.000
102 102 sbprvpv ~1 0 2 2 0 0 0 .p102. 0.000 0.000 0.000
103 103 sbeqsoc ~1 0 2 2 0 0 0 .p103. 0.000 0.000 0.000
104 104 sbcwkfm ~1 0 2 2 0 0 0 .p104. 0.000 0.000 0.000
105 105 egual ~1 0 2 2 69 NA 0 .p105. 0.000 -0.101 0.043
106 106 wc_socia ~1 0 2 2 70 NA 0 .p106. 0.000 0.040 0.040
107 107 .p2. == .p55. 2 0 0 0 NA 0 0.000 0.000 0.000
108 108 .p3. == .p56. 2 0 0 0 NA 0 0.000 0.000 0.000
109 109 .p5. == .p58. 2 0 0 0 NA 0 0.000 0.000 0.000
110 110 .p6. == .p59. 2 0 0 0 NA 0 0.000 0.000 0.000
111 111 .p7. == .p60. 2 0 0 0 NA 0 0.000 0.000 0.000
112 112 .p8. == .p61. 2 0 0 0 NA 0 0.000 0.000 0.000
113 113 .p9. == .p62. 2 0 0 0 NA 0 0.000 0.000 0.000
114 114 .p10. == .p63. 2 0 0 0 NA 0 0.000 0.000 0.000
115 115 .p11. == .p64. 2 0 0 0 NA 0 0.000 0.000 0.000
116 116 .p12. == .p65. 2 0 0 0 NA 0 0.000 0.000 0.000
117 117 .p13. == .p66. 2 0 0 0 NA 0 0.000 0.000 0.000
118 118 .p14. == .p67. 2 0 0 0 NA 0 0.000 0.000 0.000
119 119 .p15. == .p68. 2 0 0 0 NA 0 0.000 0.000 0.000
120 120 .p16. == .p69. 2 0 0 0 NA 0 0.000 0.000 0.000
121 121 .p17. == .p70. 2 0 0 0 NA 0 0.000 0.000 0.000
122 122 .p18. == .p71. 2 0 0 0 NA 0 0.000 0.000 0.000
123 123 .p19. == .p72. 2 0 0 0 NA 0 0.000 0.000 0.000
124 124 .p20. == .p73. 2 0 0 0 NA 0 0.000 0.000 0.000
125 125 .p21. == .p74. 2 0 0 0 NA 0 0.000 0.000 0.000
126 126 .p22. == .p75. 2 0 0 0 NA 0 0.000 0.000 0.000
127 127 .p23. == .p76. 2 0 0 0 NA 0 0.000 0.000 0.000
128 128 .p24. == .p77. 2 0 0 0 NA 0 0.000 0.000 0.000
129 129 .p25. == .p78. 2 0 0 0 NA 0 0.000 0.000 0.000
130 130 .p26. == .p79. 2 0 0 0 NA 0 0.000 0.000 0.000
131 131 .p27. == .p80. 2 0 0 0 NA 0 0.000 0.000 0.000
132 132 .p28. == .p81. 2 0 0 0 NA 0 0.000 0.000 0.000
133 133 .p29. == .p82. 2 0 0 0 NA 0 0.000 0.000 0.000
134 134 .p30. == .p83. 2 0 0 0 NA 0 0.000 0.000 0.000# according to lavTestScore the most problematic parameter is .p11. == .p64. which correspond to dfincac | t1
model_eg_sc_p <- '
## Egalitarianism ##
egual =~ gincdif + dfincac + smdfslv
## Social Criticism ##
wc_socia =~ sbprvpv + sbeqsoc + sbcwkfm
# fix scale of variable "dfincac" to 1 in both groups
dfincac ~*~ c(1,1)*dfincac
'
fit_eg_sc_scalar_p <- cfa(model_eg_sc_p, # model formula
data = ess_df, # data frame
estimator = "WLSMV", # select the estimator
group = "gndr",
ordered = c(
"gincdif",
"dfincac",
"smdfslv",
"sbprvpv",
"sbeqsoc",
"sbcwkfm" ),
group.equal=c("loadings","thresholds"),
# relax the threshold t1 of the variable dfincac
group.partial=c("equal =~ dfincac","dfincac | t1"))
# we do not reach partial invariance
anova(fit_eg_sc_conf,fit_eg_sc_metric,fit_eg_sc_scalar_p)
Scaled Chi-Squared Difference Test (method = "satorra.2000")
lavaan NOTE:
The "Chisq" column contains standard test statistics, not the
robust test that should be reported per model. A robust difference
test is a function of two standard (not robust) statistics.
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
fit_eg_sc_conf 16 87.126
fit_eg_sc_metric 20 91.971 5.7701 4 0.21699
fit_eg_sc_scalar_p 36 113.037 27.8118 16 0.03329 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1fit_eg_sc_metric_cont <- cfa(model_eg_sc, # model formula
data = ess_df, # data frame
estimator = "ML", # select the estimator
group = "gndr",
group.equal=c("loadings")
)
table_fit <-
list(model_fit(fit_eg_sc_metric_cont),
model_fit(fit_eg_sc_metric)) %>%
reduce(rbind)
rownames(table_fit) <- c("Continous", "Ordinal")
table_fit chisq df cfi tli rmsea rmsea.ci.lower rmsea.ci.upper rmsea.pvalue srmr
Continous 64.01 20 0.96 0.94 0.05 0.04 0.06 0.44 0.04
Ordinal 91.97 20 0.97 0.96 0.06 0.05 0.08 0.03 0.05tidy_results_continous <- table_results(fit_eg_sc_metric_cont,
columns = c("label",
"est_sig",
"se"),
digits = 2
)
tidy_results_ordinal <- table_results(fit_eg_sc_metric,
columns = c("label",
"est_sig",
"se"),
digits = 2
)
data.frame(Parameters = tidy_results_continous$label[1:6],
"Continous Model" = paste(tidy_results_continous$est_sig[1:6],"(",tidy_results_continous$se[1:6], ")"),
"Ordinal Model" = paste(tidy_results_ordinal$est_sig[1:6],"(",tidy_results_ordinal$se[1:6], ")")
) Parameters Continous.Model Ordinal.Model
1 egual.BY.gincdif.Male 1.00 ( 0.00 ) 1.00 ( 0.00 )
2 egual.BY.dfincac.Male -0.59*** ( 0.06 ) -0.60*** ( 0.05 )
3 egual.BY.smdfslv.Male 0.84*** ( 0.08 ) 0.99*** ( 0.08 )
4 wc_socia.BY.sbprvpv.Male 1.00 ( 0.00 ) 1.00 ( 0.00 )
5 wc_socia.BY.sbeqsoc.Male 1.24*** ( 0.12 ) 1.22*** ( 0.09 )
6 wc_socia.BY.sbcwkfm.Male 0.62*** ( 0.05 ) 0.69*** ( 0.04 )