Chapter 5 Testing for measurement invariance with lavaan in R

5.1 How to Run a MGCFA in R

Why R? Because it is free.

Alternatives:

5.2 Download R and RStudio

5.3 Install lavaan and semTools

5.4 Data cleaning

  • Do whatever data wrangling necessary (missing values, etc)

  • Remember that you need a variable that identifies the group

5.5 Use lavaan language to formalize the model

The lavaan tutorial explains well how to formalize a theoretical model. It even includes a measurement invariance testing example.

In a nutshell:

Factor loadings in R are indicated by =∼ and covariances (between factors or error terms for items) are indicated by ∼∼. The model is specified similar to writing regression equations.

5.5.1 Example: Testing the Political Trust for measurement invariance

boxes_and_circles A Political Trust 1 Parliment A->1 1 2 Politicians A->2 3 Legal System A->3 4 Political Parties A->4 E1 e1 1->E1 E2 e2 2->E2 E3 e3 3->E3 E4 e4 4->E4

We want to know if we can compare the political trust concept in Portugal and Spain.

5.5.2 Load lavaan and semTools

library(lavaan)
library(semTools)

5.5.3 Model Building

Let´s build our model using lavaan model syntax

baseline <- 'political_trust =~ trstprl + trstlgl + trstplt + trstprt
'


#CFA function
fit_baseline<-cfa(baseline, data = mea_inv, group="cntry")


#Summary shows the estimation results
summary(fit_baseline, fit.measures=T, standardized = T)
## Length  Class   Mode 
##      1 lavaan     S4

5.5.4 Testing

Not a good model.

Commonly reported fit indices and recommended cut-offs

Measure Name Cut-off
CFI Comparative Fit Index CFI>=.90
RMSEA Root Mean Square Root of Approximation RMSEA<0.08
SRMR Standardized Root Mean Square Residual SRMR<0.08

Please take this as generally indicative. Read more about testing on Further Readings

5.5.5 Changing the original model

baseline1 <- 'political_trust =~ trstprl + trstlgl + trstplt + trstprt
trstplt ~~ trstprt
'

fit_configural<-cfa(baseline1, data = mea_inv, group="cntry")

summary(fit_configural, fit.measures=T, standardized = T)
## Length  Class   Mode 
##      1 lavaan     S4

Much better model. Configural Invariance set. Let’s go for metric.

5.5.6 Metric Invariance

metric <- 'political_trust =~ trstprl + trstlgl + trstplt + trstprt
trstplt ~~ trstprt
'

fit_metric<-cfa(baseline1, data = mea_inv, group="cntry", group.equal = c("loadings"))

summary(fit_metric, fit.measures=T, standardized = T)
## Length  Class   Mode 
##      1 lavaan     S4

Also a good model. Let’s go ahead for scalar.

5.5.7 Scalar Invariance

scalar <- 'political_trust =~ trstprl + trstlgl + trstplt + trstprt
trstplt ~~ trstprt
'

fit_scalar<-cfa(baseline1, data = mea_inv, group="cntry", group.equal = c("loadings", "intercepts"))

summary(fit_scalar, fit.measures=T, standardized = T)
## Length  Class   Mode 
##      1 lavaan     S4

Scalar Invariance holds!

5.5.8 Comparing several models using compareFit

compareFit(fit_configural, fit_metric, fit_scalar)
## ################### Nested Model Comparison #########################
## Chi-Squared Difference Test
## 
##                Df   AIC   BIC   Chisq Chisq diff Df diff Pr(>Chisq)  
## fit_configural  2 25032 25174  6.9645                                
## fit_metric      5 25031 25157 11.6200     4.6555       3    0.19883  
## fit_scalar      8 25031 25141 18.0521     6.4321       3    0.09238 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## ####################### Model Fit Indices ###########################
##                  chisq df pvalue    cfi    tli        aic        bic rmsea  srmr
## fit_configural  6.965†  2   .031 0.998†  .991  25032.320  25174.353  .053  .006†
## fit_metric     11.620   5   .040 0.998   .995  25030.976† 25156.620  .039  .018 
## fit_scalar     18.052   8   .021 0.997  0.995† 25031.408  25140.664† .038† .020 
## 
## ################## Differences in Fit Indices #######################
##                             df    cfi   tli    aic     bic  rmsea  srmr
## fit_metric - fit_configural  3 -0.001 0.004 -1.345 -17.733 -0.014 0.012
## fit_scalar - fit_metric      3 -0.001 0.000  0.432 -15.956 -0.001 0.002

5.6 Omnibus invariance testing

measurementInvariance (model = baseline1, data = mea_inv, group = "cntry")
## 
## Measurement invariance models:
## 
## Model 1 : fit.configural
## Model 2 : fit.loadings
## Model 3 : fit.intercepts
## Model 4 : fit.means
## 
## Chi-Squared Difference Test
## 
##                Df   AIC   BIC   Chisq Chisq diff Df diff Pr(>Chisq)  
## fit.configural  2 25032 25174  6.9645                                
## fit.loadings    5 25031 25157 11.6200     4.6555       3    0.19883  
## fit.intercepts  8 25031 25141 18.0521     6.4321       3    0.09238 .
## fit.means       9 25034 25138 22.3435     4.2914       1    0.03831 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Fit measures:
## 
##                  cfi rmsea cfi.delta rmsea.delta
## fit.configural 0.998 0.053        NA          NA
## fit.loadings   0.998 0.039     0.001       0.014
## fit.intercepts 0.997 0.038     0.001       0.001
## fit.means      0.996 0.041     0.001       0.003

Terry Jorgensen measurementInvariance function, within the semTools package is very handy

5.7 Advanced Tutorial: Partial Invariance (individually constraining parameters)

Source: Designed by starline / Freepik

Figure 5.1: Source: Designed by starline / Freepik