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 ###

#### For MVN we need install.packages("gsl"). 
#### On mac with homebrewed packaged:
#### CFLAGS="-I/opt/homebrew/Cellar/gsl/2.7.1/include" LDFLAGS="-L/opt/homebrew/Cellar/gsl/2.7.1/lib -lgsl -lgslcblas" R
#### install.packages("gsl")

# 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(
                )

mvn_test


model_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_mlr <- cfa(
)

summary()


tidy_results_ml <- table_results(
)

tidy_results_mlr <- table_results(
)



data.frame(Parameters = , 
           "ML Model" = ,
           "MLR Model" = 
)



model_eg_ws_cov <- '

'

fit_eg_ws_mlr_cov <- cfa(

summary(fit_eg_ws_mlr_cov)

# 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 <- 

 

rownames(table_fit) <-

table_fit

anova(, )

5 Ex.2 – Measurement invariance with categorical data

Re-estimate a CFA (2 latent variable) for egalitarianism (gincdif, dfincac, smdfslv) and welfare support (gvslvol gvhlthc gvcldcr gvpdlwk) 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 groups

ess_df$gndr <- factor(
  
) # labels 

model_eg_sc <- '
## Egalitarianism ##
egual =~  gincdif + dfincac + smdfslv
## Social Criticism ##
wc_socia =~ sbprvpv + sbeqsoc + sbcwkfm
'

fit_eg_sc_conf<- cfa(,
                     ordered = c(

          )


fit_eg_sc_metric <- cfa(
)

fit_eg_sc_scalar <- cfa(
)

anova(fit_eg_sc_conf,fit_eg_sc_metric,fit_eg_sc_scalar)



model_eg_sc_p <- '


'

fit_eg_sc_metric_p <- cfa(
                     group.equal=c(),
                     # relax the threshold t1 of the variable dfincac
                     group.partial=c()) 

# we do not reach partial invariance
anova( , , )


fit_eg_sc_metric_c <- cfa(
)

table_fit <- 
 

rownames(table_fit) <- 

table_fit