Chapter 17 Week13_1: Lavaan Lab 14 Measurement Invariance

For this lab, we will run the MG-CFA analyses in class using simulated data based on Todd Little’s positive affect example.

Load up the lavaan library:

library(lavaan)

and the dataset:

affectData <- read.csv("cfaInclassData.csv", header = T)

For demonstration purposes, let’s first simulate a grouping variable called school:

set.seed(555)
affectData$school = sample(c('public', 'private'), nrow(affectData), replace = T)
table(affectData$school)
## 
## private  public 
##     513     487
head(affectData)
##   ID        glad   cheerful      happy  satisfied    content
## 1  1  0.13521092  0.5413297 -0.1041445 -0.5777446  0.8645383
## 2  2 -0.29116043  0.2434081  0.6671535  2.0763730 -0.7382832
## 3  3  0.71975913  0.2218277  0.4722337  2.1685984 -0.2727574
## 4  4  0.44432030  0.9295414  0.8574083 -1.0575363 -1.3841364
## 5  5  2.84476524  3.1710123  3.5145040  1.5725274  2.3406754
## 6  6 -0.03317526 -0.8434011 -0.1485924 -0.5469343 -1.5750953
##   comfortable  school
## 1  0.02935020 private
## 2  1.05439183  public
## 3  0.09053090 private
## 4 -0.07940091 private
## 5  1.59866763  public
## 6 -0.69629828  public

The goal of testing measurement invariance (MI) is to make sure that the scale that measures positive affect and satisfaction functions in the same way between public and private schools.

17.1 PART I: Multi-Group Analyses, Done Incorrectly

Syntax for an SR model (it doesn’t matter whether the model is for CFA or SR, the test of MI only applies to the CFA part):

srSyntax <- "
    posAffect =~ glad + happy + cheerful 
    satisfaction =~ satisfied + content + comfortable 
    
    # Structural Regression: beta
    satisfaction ~ posAffect
    "
MGsrRunWRONG <- lavaan::sem(srSyntax, 
                    data = affectData, 
                    fixed.x=FALSE,
                    group = "school", # group indicator
                    estimator = "MLR") # use MLR as a go-to estimation method

summary(MGsrRunWRONG, standardized = T, fit.measures = T)
## lavaan 0.6-12 ended normally after 31 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        38
## 
##   Number of observations per group:                   
##     private                                        513
##     public                                         487
## 
## Model Test User Model:
##                                               Standard      Robust
##   Test Statistic                                11.920      11.709
##   Degrees of freedom                                16          16
##   P-value (Chi-square)                           0.749       0.764
##   Scaling correction factor                                  1.018
##     Yuan-Bentler correction (Mplus variant)                       
##   Test statistic for each group:
##     private                                      4.201       4.126
##     public                                       7.719       7.583
## 
## Model Test Baseline Model:
## 
##   Test statistic                              2038.064    2039.295
##   Degrees of freedom                                30          30
##   P-value                                        0.000       0.000
##   Scaling correction factor                                  0.999
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    1.000       1.000
##   Tucker-Lewis Index (TLI)                       1.004       1.004
##                                                                   
##   Robust Comparative Fit Index (CFI)                         1.000
##   Robust Tucker-Lewis Index (TLI)                            1.004
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -7473.002   -7473.002
##   Scaling correction factor                                  1.001
##       for the MLR correction                                      
##   Loglikelihood unrestricted model (H1)      -7467.042   -7467.042
##   Scaling correction factor                                  1.006
##       for the MLR correction                                      
##                                                                   
##   Akaike (AIC)                               15022.004   15022.004
##   Bayesian (BIC)                             15208.498   15208.498
##   Sample-size adjusted Bayesian (BIC)        15087.808   15087.808
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.000       0.000
##   90 Percent confidence interval - lower         0.000       0.000
##   90 Percent confidence interval - upper         0.030       0.029
##   P-value RMSEA <= 0.05                          0.998       0.999
##                                                                   
##   Robust RMSEA                                               0.000
##   90 Percent confidence interval - lower                     0.000
##   90 Percent confidence interval - upper                     0.029
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.013       0.013
## 
## Parameter Estimates:
## 
##   Standard errors                             Sandwich
##   Information bread                           Observed
##   Observed information based on                Hessian
## 
## 
## Group 1 [private]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   posAffect =~                                                 
##     glad              1.000                               0.740
##     happy             0.975    0.065   14.962    0.000    0.722
##     cheerful          1.103    0.068   16.264    0.000    0.816
##   satisfaction =~                                              
##     satisfied         1.000                               0.792
##     content           1.065    0.071   14.999    0.000    0.843
##     comfortable       0.874    0.059   14.910    0.000    0.692
##   Std.all
##          
##     0.754
##     0.743
##     0.807
##          
##     0.779
##     0.761
##     0.741
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   satisfaction ~                                               
##     posAffect         0.520    0.065    8.033    0.000    0.486
##   Std.all
##          
##     0.486
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad              0.023    0.043    0.530    0.596    0.023
##    .happy             0.012    0.043    0.270    0.787    0.012
##    .cheerful          0.027    0.045    0.594    0.553    0.027
##    .satisfied        -0.102    0.045   -2.269    0.023   -0.102
##    .content          -0.086    0.049   -1.754    0.079   -0.086
##    .comfortable      -0.061    0.041   -1.481    0.138   -0.061
##     posAffect         0.000                               0.000
##    .satisfaction      0.000                               0.000
##   Std.all
##     0.023
##     0.012
##     0.026
##    -0.100
##    -0.077
##    -0.065
##     0.000
##     0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad              0.417    0.037   11.257    0.000    0.417
##    .happy             0.422    0.038   11.130    0.000    0.422
##    .cheerful          0.357    0.037    9.643    0.000    0.357
##    .satisfied         0.406    0.039   10.484    0.000    0.406
##    .content           0.519    0.047   11.070    0.000    0.519
##    .comfortable       0.393    0.036   10.765    0.000    0.393
##     posAffect         0.548    0.059    9.213    0.000    1.000
##    .satisfaction      0.478    0.057    8.380    0.000    0.763
##   Std.all
##     0.432
##     0.448
##     0.349
##     0.393
##     0.422
##     0.451
##     1.000
##     0.763
## 
## 
## Group 2 [public]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   posAffect =~                                                 
##     glad              1.000                               0.643
##     happy             1.182    0.091   13.041    0.000    0.761
##     cheerful          1.130    0.091   12.455    0.000    0.727
##   satisfaction =~                                              
##     satisfied         1.000                               0.749
##     content           1.072    0.072   14.806    0.000    0.803
##     comfortable       0.972    0.065   14.904    0.000    0.729
##   Std.all
##          
##     0.655
##     0.775
##     0.764
##          
##     0.753
##     0.763
##     0.753
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   satisfaction ~                                               
##     posAffect         0.568    0.074    7.725    0.000    0.488
##   Std.all
##          
##     0.488
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad             -0.022    0.045   -0.492    0.623   -0.022
##    .happy             0.005    0.044    0.114    0.910    0.005
##    .cheerful         -0.030    0.043   -0.697    0.486   -0.030
##    .satisfied         0.018    0.045    0.399    0.690    0.018
##    .content           0.012    0.048    0.251    0.802    0.012
##    .comfortable      -0.026    0.044   -0.597    0.551   -0.026
##     posAffect         0.000                               0.000
##    .satisfaction      0.000                               0.000
##   Std.all
##    -0.022
##     0.005
##    -0.032
##     0.018
##     0.011
##    -0.027
##     0.000
##     0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad              0.552    0.043   12.930    0.000    0.552
##    .happy             0.385    0.043    8.958    0.000    0.385
##    .cheerful          0.377    0.041    9.304    0.000    0.377
##    .satisfied         0.429    0.042   10.338    0.000    0.429
##    .content           0.462    0.048    9.576    0.000    0.462
##    .comfortable       0.405    0.040   10.259    0.000    0.405
##     posAffect         0.414    0.058    7.152    0.000    1.000
##    .satisfaction      0.428    0.053    8.110    0.000    0.762
##   Std.all
##     0.571
##     0.400
##     0.416
##     0.433
##     0.417
##     0.433
##     1.000
##     0.762
library(semPlot)
semPaths(MGsrRunWRONG, what='est', 
         nCharNodes = 0,
         nCharEdges = 0, # don't limit variable name lengths
         edge.label.cex=0.6, 
         curvePivot = TRUE, 
         curve = 1.5, # pull covariances' curves out a little
         fade=FALSE)

17.2 PART II: Testing Measurement Invariance

17.2.1 step 1: Configural invariance

If you simply use the MGsrRunWRONG synatx above, you are just testing configural invariance:

configuralFit <- lavaan::sem(srSyntax, 
                    data = affectData, 
                    fixed.x=FALSE,
                    group = "school", # group indicator
                    estimator = "MLR") # use MLR as a go-to estimation method
#summary(configuralFit, standardized = T, fit.measures = T)
  • Configural invariance was established due to satisfying model fit;

17.2.2 step 2: Metric (weak) invariance

To test metric invariance, you could manually constraint all factor loadings to be the same using tricks like “posAffect =~ c(lam1, lam1)*glad” but there is a shortcut using “group.equal” argument:

metricFit <- lavaan::sem(srSyntax, 
                 data = affectData, 
                 fixed.x=FALSE,
                 estimator = 'MLR',
                 group = "school", 
                 group.equal = c("loadings")) 
  • so that all factor loadings are fixed to be the same across groups
  • More group equality constraints can be added, like “intercepts”, “means”, “residuals”, “residual.covariances”, “lv.variances”, “lv.covariances”, “regressions”
summary(metricFit, standardized = T, fit.measures = T)
## lavaan 0.6-12 ended normally after 25 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        38
##   Number of equality constraints                     4
## 
##   Number of observations per group:                   
##     private                                        513
##     public                                         487
## 
## Model Test User Model:
##                                               Standard      Robust
##   Test Statistic                                17.640      17.639
##   Degrees of freedom                                20          20
##   P-value (Chi-square)                           0.611       0.611
##   Scaling correction factor                                  1.000
##     Yuan-Bentler correction (Mplus variant)                       
##   Test statistic for each group:
##     private                                      6.665       6.665
##     public                                      10.975      10.974
## 
## Model Test Baseline Model:
## 
##   Test statistic                              2038.064    2039.295
##   Degrees of freedom                                30          30
##   P-value                                        0.000       0.000
##   Scaling correction factor                                  0.999
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    1.000       1.000
##   Tucker-Lewis Index (TLI)                       1.002       1.002
##                                                                   
##   Robust Comparative Fit Index (CFI)                         1.000
##   Robust Tucker-Lewis Index (TLI)                            1.002
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -7475.862   -7475.862
##   Scaling correction factor                                  0.904
##       for the MLR correction                                      
##   Loglikelihood unrestricted model (H1)      -7467.042   -7467.042
##   Scaling correction factor                                  1.006
##       for the MLR correction                                      
##                                                                   
##   Akaike (AIC)                               15019.724   15019.724
##   Bayesian (BIC)                             15186.588   15186.588
##   Sample-size adjusted Bayesian (BIC)        15078.602   15078.602
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.000       0.000
##   90 Percent confidence interval - lower         0.000       0.000
##   90 Percent confidence interval - upper         0.033       0.033
##   P-value RMSEA <= 0.05                          0.998       0.998
##                                                                   
##   Robust RMSEA                                               0.000
##   90 Percent confidence interval - lower                     0.000
##   90 Percent confidence interval - upper                     0.033
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.019       0.019
## 
## Parameter Estimates:
## 
##   Standard errors                             Sandwich
##   Information bread                           Observed
##   Observed information based on                Hessian
## 
## 
## Group 1 [private]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   posAffect =~                                                 
##     glad              1.000                               0.718
##     happy   (.p2.)    1.057    0.054   19.585    0.000    0.759
##     cheerfl (.p3.)    1.113    0.055   20.393    0.000    0.799
##   satisfaction =~                                              
##     satisfd           1.000                               0.778
##     content (.p5.)    1.069    0.051   21.078    0.000    0.832
##     cmfrtbl (.p6.)    0.919    0.044   21.033    0.000    0.715
##   Std.all
##          
##     0.740
##     0.767
##     0.796
##          
##     0.770
##     0.754
##     0.757
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   satisfaction ~                                               
##     posAffect         0.529    0.063    8.333    0.000    0.488
##   Std.all
##          
##     0.488
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad              0.023    0.043    0.530    0.596    0.023
##    .happy             0.012    0.043    0.270    0.787    0.012
##    .cheerful          0.027    0.045    0.594    0.553    0.027
##    .satisfied        -0.102    0.045   -2.269    0.023   -0.102
##    .content          -0.086    0.049   -1.754    0.079   -0.086
##    .comfortable      -0.061    0.041   -1.481    0.138   -0.061
##     posAffect         0.000                               0.000
##    .satisfaction      0.000                               0.000
##   Std.all
##     0.024
##     0.012
##     0.026
##    -0.101
##    -0.078
##    -0.065
##     0.000
##     0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad              0.427    0.036   11.943    0.000    0.427
##    .happy             0.403    0.037   10.793    0.000    0.403
##    .cheerful          0.370    0.035   10.705    0.000    0.370
##    .satisfied         0.415    0.036   11.539    0.000    0.415
##    .content           0.525    0.044   11.903    0.000    0.525
##    .comfortable       0.381    0.035   10.927    0.000    0.381
##     posAffect         0.516    0.053    9.799    0.000    1.000
##    .satisfaction      0.461    0.051    9.082    0.000    0.762
##   Std.all
##     0.453
##     0.411
##     0.367
##     0.407
##     0.431
##     0.427
##     1.000
##     0.762
## 
## 
## Group 2 [public]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   posAffect =~                                                 
##     glad              1.000                               0.673
##     happy   (.p2.)    1.057    0.054   19.585    0.000    0.711
##     cheerfl (.p3.)    1.113    0.055   20.393    0.000    0.748
##   satisfaction =~                                              
##     satisfd           1.000                               0.763
##     content (.p5.)    1.069    0.051   21.078    0.000    0.816
##     cmfrtbl (.p6.)    0.919    0.044   21.033    0.000    0.701
##   Std.all
##          
##     0.675
##     0.740
##     0.780
##          
##     0.761
##     0.771
##     0.734
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   satisfaction ~                                               
##     posAffect         0.557    0.067    8.342    0.000    0.492
##   Std.all
##          
##     0.492
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad             -0.022    0.045   -0.492    0.623   -0.022
##    .happy             0.005    0.044    0.114    0.910    0.005
##    .cheerful         -0.030    0.043   -0.697    0.486   -0.030
##    .satisfied         0.018    0.045    0.399    0.690    0.018
##    .content           0.012    0.048    0.251    0.802    0.012
##    .comfortable      -0.026    0.044   -0.597    0.551   -0.026
##     posAffect         0.000                               0.000
##    .satisfaction      0.000                               0.000
##   Std.all
##    -0.022
##     0.005
##    -0.031
##     0.018
##     0.011
##    -0.027
##     0.000
##     0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad              0.541    0.042   12.899    0.000    0.541
##    .happy             0.418    0.040   10.351    0.000    0.418
##    .cheerful          0.360    0.038    9.399    0.000    0.360
##    .satisfied         0.423    0.040   10.551    0.000    0.423
##    .content           0.454    0.045   10.053    0.000    0.454
##    .comfortable       0.420    0.038   11.052    0.000    0.420
##     posAffect         0.453    0.049    9.180    0.000    1.000
##    .satisfaction      0.441    0.048    9.107    0.000    0.758
##   Std.all
##     0.545
##     0.452
##     0.392
##     0.421
##     0.406
##     0.461
##     1.000
##     0.758
  • Again, metric invariance was established due to satisfying model fit;
  • To test whether the equal factor loading assumption caused damage to model fit, we compare metricFit to configuralFit:

Model comparison:

anova(configuralFit, metricFit)
## 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)
## configuralFit 16 15022 15208 11.92                              
## metricFit     20 15020 15187 17.64     6.1624       4     0.1873
  • The test was not significant, meaning the increase in chi-square (due to the assumption of equal factor loadings) was not substantial enough to worsen the model fit;
  • Note that this test suffers from the same problem as the chi-square test (too sensitive to model misfit)

17.2.3 step 3: Scalar (strong) Invariance

In this step, both factor loadings and measurement intercepts (of course, including factor structure) are constrained to be equal between the groups:

scalarFit <- lavaan::sem(srSyntax, 
                 data = affectData, 
                 fixed.x=FALSE,
                 estimator = 'MLR',
                 group = "school", 
                 group.equal = c("loadings", "intercepts")) 
summary(scalarFit, standardized = T, fit.measures = T)
## lavaan 0.6-12 ended normally after 32 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        40
##   Number of equality constraints                    10
## 
##   Number of observations per group:                   
##     private                                        513
##     public                                         487
## 
## Model Test User Model:
##                                               Standard      Robust
##   Test Statistic                                20.364      20.360
##   Degrees of freedom                                24          24
##   P-value (Chi-square)                           0.676       0.676
##   Scaling correction factor                                  1.000
##     Yuan-Bentler correction (Mplus variant)                       
##   Test statistic for each group:
##     private                                      7.764       7.762
##     public                                      12.600      12.598
## 
## Model Test Baseline Model:
## 
##   Test statistic                              2038.064    2039.295
##   Degrees of freedom                                30          30
##   P-value                                        0.000       0.000
##   Scaling correction factor                                  0.999
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    1.000       1.000
##   Tucker-Lewis Index (TLI)                       1.002       1.002
##                                                                   
##   Robust Comparative Fit Index (CFI)                         1.000
##   Robust Tucker-Lewis Index (TLI)                            1.002
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -7477.224   -7477.224
##   Scaling correction factor                                  0.758
##       for the MLR correction                                      
##   Loglikelihood unrestricted model (H1)      -7467.042   -7467.042
##   Scaling correction factor                                  1.006
##       for the MLR correction                                      
##                                                                   
##   Akaike (AIC)                               15014.448   15014.448
##   Bayesian (BIC)                             15161.681   15161.681
##   Sample-size adjusted Bayesian (BIC)        15066.399   15066.399
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.000       0.000
##   90 Percent confidence interval - lower         0.000       0.000
##   90 Percent confidence interval - upper         0.029       0.029
##   P-value RMSEA <= 0.05                          0.999       0.999
##                                                                   
##   Robust RMSEA                                               0.000
##   90 Percent confidence interval - lower                     0.000
##   90 Percent confidence interval - upper                     0.029
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.020       0.020
## 
## Parameter Estimates:
## 
##   Standard errors                             Sandwich
##   Information bread                           Observed
##   Observed information based on                Hessian
## 
## 
## Group 1 [private]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   posAffect =~                                                 
##     glad              1.000                               0.718
##     happy   (.p2.)    1.056    0.054   19.522    0.000    0.759
##     cheerfl (.p3.)    1.113    0.055   20.413    0.000    0.800
##   satisfaction =~                                              
##     satisfd           1.000                               0.780
##     content (.p5.)    1.068    0.051   21.059    0.000    0.832
##     cmfrtbl (.p6.)    0.915    0.043   21.140    0.000    0.713
##   Std.all
##          
##     0.740
##     0.767
##     0.796
##          
##     0.771
##     0.754
##     0.755
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   satisfaction ~                                               
##     posAffect         0.530    0.063    8.350    0.000    0.488
##   Std.all
##          
##     0.488
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad    (.16.)    0.019    0.040    0.462    0.644    0.019
##    .happy   (.17.)    0.026    0.040    0.644    0.519    0.026
##    .cheerfl (.18.)    0.018    0.042    0.415    0.678    0.018
##    .satisfd (.19.)   -0.085    0.042   -2.031    0.042   -0.085
##    .content (.20.)   -0.082    0.045   -1.820    0.069   -0.082
##    .cmfrtbl (.21.)   -0.081    0.038   -2.108    0.035   -0.081
##     psAffct           0.000                               0.000
##    .stsfctn           0.000                               0.000
##   Std.all
##     0.019
##     0.026
##     0.017
##    -0.084
##    -0.075
##    -0.086
##     0.000
##     0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad              0.427    0.036   11.937    0.000    0.427
##    .happy             0.403    0.037   10.809    0.000    0.403
##    .cheerful          0.370    0.035   10.680    0.000    0.370
##    .satisfied         0.414    0.036   11.505    0.000    0.414
##    .content           0.525    0.044   11.872    0.000    0.525
##    .comfortable       0.383    0.035   11.042    0.000    0.383
##     posAffect         0.516    0.053    9.801    0.000    1.000
##    .satisfaction      0.463    0.051    9.090    0.000    0.762
##   Std.all
##     0.453
##     0.412
##     0.367
##     0.405
##     0.431
##     0.430
##     1.000
##     0.762
## 
## 
## Group 2 [public]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   posAffect =~                                                 
##     glad              1.000                               0.673
##     happy   (.p2.)    1.056    0.054   19.522    0.000    0.710
##     cheerfl (.p3.)    1.113    0.055   20.413    0.000    0.749
##   satisfaction =~                                              
##     satisfd           1.000                               0.764
##     content (.p5.)    1.068    0.051   21.059    0.000    0.816
##     cmfrtbl (.p6.)    0.915    0.043   21.140    0.000    0.699
##   Std.all
##          
##     0.675
##     0.739
##     0.780
##          
##     0.762
##     0.771
##     0.732
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   satisfaction ~                                               
##     posAffect         0.558    0.067    8.350    0.000    0.492
##   Std.all
##          
##     0.492
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad    (.16.)    0.019    0.040    0.462    0.644    0.019
##    .happy   (.17.)    0.026    0.040    0.644    0.519    0.026
##    .cheerfl (.18.)    0.018    0.042    0.415    0.678    0.018
##    .satisfd (.19.)   -0.085    0.042   -2.031    0.042   -0.085
##    .content (.20.)   -0.082    0.045   -1.820    0.069   -0.082
##    .cmfrtbl (.21.)   -0.081    0.038   -2.108    0.035   -0.081
##     psAffct          -0.035    0.049   -0.697    0.486   -0.051
##    .stsfctn           0.104    0.051    2.053    0.040    0.137
##   Std.all
##     0.019
##     0.027
##     0.018
##    -0.085
##    -0.078
##    -0.085
##    -0.051
##     0.137
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad              0.541    0.042   12.891    0.000    0.541
##    .happy             0.419    0.041   10.332    0.000    0.419
##    .cheerful          0.360    0.038    9.378    0.000    0.360
##    .satisfied         0.422    0.040   10.516    0.000    0.422
##    .content           0.454    0.045   10.032    0.000    0.454
##    .comfortable       0.422    0.038   11.095    0.000    0.422
##     posAffect         0.453    0.049    9.174    0.000    1.000
##    .satisfaction      0.443    0.049    9.101    0.000    0.758
##   Std.all
##     0.545
##     0.453
##     0.391
##     0.420
##     0.405
##     0.463
##     1.000
##     0.758

Model comparison:

anova(metricFit, scalarFit)
## 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)
## metricFit 20 15020 15187 17.640                              
## scalarFit 24 15014 15162 20.364     2.7216       4     0.6054
  • The test was not significant, meaning the increase in chi-square (due to the assumption of equal measurement intercepts) was not substantial enough to worsen the model fit;
  • Scalar invariance was established;

17.2.4 step 4: (Optional) Residual variance (strict) invariance

resVarFit <- lavaan::sem(srSyntax, 
                 data = affectData, 
                 fixed.x=FALSE,
                 estimator = 'MLR',
                 group = "school", 
                 group.equal = c("loadings", "intercepts", "residuals")) 
summary(resVarFit, standardized = T, fit.measures = T)
## lavaan 0.6-12 ended normally after 32 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        40
##   Number of equality constraints                    16
## 
##   Number of observations per group:                   
##     private                                        513
##     public                                         487
## 
## Model Test User Model:
##                                               Standard      Robust
##   Test Statistic                                26.885      26.947
##   Degrees of freedom                                30          30
##   P-value (Chi-square)                           0.629       0.626
##   Scaling correction factor                                  0.998
##     Yuan-Bentler correction (Mplus variant)                       
##   Test statistic for each group:
##     private                                     11.454      11.481
##     public                                      15.430      15.466
## 
## Model Test Baseline Model:
## 
##   Test statistic                              2038.064    2039.295
##   Degrees of freedom                                30          30
##   P-value                                        0.000       0.000
##   Scaling correction factor                                  0.999
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    1.000       1.000
##   Tucker-Lewis Index (TLI)                       1.002       1.002
##                                                                   
##   Robust Comparative Fit Index (CFI)                         1.000
##   Robust Tucker-Lewis Index (TLI)                            1.002
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -7480.484   -7480.484
##   Scaling correction factor                                  0.610
##       for the MLR correction                                      
##   Loglikelihood unrestricted model (H1)      -7467.042   -7467.042
##   Scaling correction factor                                  1.006
##       for the MLR correction                                      
##                                                                   
##   Akaike (AIC)                               15008.968   15008.968
##   Bayesian (BIC)                             15126.754   15126.754
##   Sample-size adjusted Bayesian (BIC)        15050.529   15050.529
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.000       0.000
##   90 Percent confidence interval - lower         0.000       0.000
##   90 Percent confidence interval - upper         0.029       0.029
##   P-value RMSEA <= 0.05                          1.000       1.000
##                                                                   
##   Robust RMSEA                                               0.000
##   90 Percent confidence interval - lower                     0.000
##   90 Percent confidence interval - upper                     0.029
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.023       0.023
## 
## Parameter Estimates:
## 
##   Standard errors                             Sandwich
##   Information bread                           Observed
##   Observed information based on                Hessian
## 
## 
## Group 1 [private]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   posAffect =~                                                 
##     glad              1.000                               0.713
##     happy   (.p2.)    1.063    0.054   19.648    0.000    0.758
##     cheerfl (.p3.)    1.114    0.055   20.246    0.000    0.794
##   satisfaction =~                                              
##     satisfd           1.000                               0.781
##     content (.p5.)    1.068    0.051   21.048    0.000    0.833
##     cmfrtbl (.p6.)    0.915    0.043   21.139    0.000    0.714
##   Std.all
##          
##     0.716
##     0.765
##     0.794
##          
##     0.770
##     0.766
##     0.748
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   satisfaction ~                                               
##     posAffect         0.537    0.064    8.445    0.000    0.491
##   Std.all
##          
##     0.491
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad    (.16.)    0.018    0.040    0.443    0.658    0.018
##    .happy   (.17.)    0.026    0.040    0.650    0.516    0.026
##    .cheerfl (.18.)    0.018    0.042    0.415    0.678    0.018
##    .satisfd (.19.)   -0.085    0.042   -2.026    0.043   -0.085
##    .content (.20.)   -0.083    0.045   -1.825    0.068   -0.083
##    .cmfrtbl (.21.)   -0.082    0.038   -2.136    0.033   -0.082
##     psAffct           0.000                               0.000
##    .stsfctn           0.000                               0.000
##   Std.all
##     0.018
##     0.026
##     0.018
##    -0.084
##    -0.076
##    -0.086
##     0.000
##     0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad    (.p8.)    0.483    0.028   17.136    0.000    0.483
##    .happy   (.p9.)    0.408    0.029   14.229    0.000    0.408
##    .cheerfl (.10.)    0.369    0.028   13.197    0.000    0.369
##    .satisfd (.11.)    0.417    0.028   14.667    0.000    0.417
##    .content (.12.)    0.490    0.034   14.545    0.000    0.490
##    .cmfrtbl (.13.)    0.402    0.027   15.048    0.000    0.402
##     psAffct           0.509    0.051    9.913    0.000    1.000
##    .stsfctn           0.463    0.050    9.171    0.000    0.759
##   Std.all
##     0.487
##     0.415
##     0.369
##     0.406
##     0.414
##     0.441
##     1.000
##     0.759
## 
## 
## Group 2 [public]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   posAffect =~                                                 
##     glad              1.000                               0.673
##     happy   (.p2.)    1.063    0.054   19.648    0.000    0.716
##     cheerfl (.p3.)    1.114    0.055   20.246    0.000    0.750
##   satisfaction =~                                              
##     satisfd           1.000                               0.764
##     content (.p5.)    1.068    0.051   21.048    0.000    0.816
##     cmfrtbl (.p6.)    0.915    0.043   21.139    0.000    0.699
##   Std.all
##          
##     0.696
##     0.746
##     0.777
##          
##     0.764
##     0.759
##     0.741
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   satisfaction ~                                               
##     posAffect         0.556    0.067    8.327    0.000    0.490
##   Std.all
##          
##     0.490
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad    (.16.)    0.018    0.040    0.443    0.658    0.018
##    .happy   (.17.)    0.026    0.040    0.650    0.516    0.026
##    .cheerfl (.18.)    0.018    0.042    0.415    0.678    0.018
##    .satisfd (.19.)   -0.085    0.042   -2.026    0.043   -0.085
##    .content (.20.)   -0.083    0.045   -1.825    0.068   -0.083
##    .cmfrtbl (.21.)   -0.082    0.038   -2.136    0.033   -0.082
##     psAffct          -0.034    0.049   -0.693    0.488   -0.051
##    .stsfctn           0.104    0.051    2.049    0.040    0.136
##   Std.all
##     0.018
##     0.027
##     0.018
##    -0.085
##    -0.077
##    -0.087
##    -0.051
##     0.136
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad    (.p8.)    0.483    0.028   17.136    0.000    0.483
##    .happy   (.p9.)    0.408    0.029   14.229    0.000    0.408
##    .cheerfl (.10.)    0.369    0.028   13.197    0.000    0.369
##    .satisfd (.11.)    0.417    0.028   14.667    0.000    0.417
##    .content (.12.)    0.490    0.034   14.545    0.000    0.490
##    .cmfrtbl (.13.)    0.402    0.027   15.048    0.000    0.402
##     psAffct           0.454    0.050    9.153    0.000    1.000
##    .stsfctn           0.444    0.049    9.141    0.000    0.760
##   Std.all
##     0.516
##     0.443
##     0.396
##     0.417
##     0.424
##     0.451
##     1.000
##     0.760

Model comparison:

anova(resVarFit, scalarFit)
## 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)
## scalarFit 24 15014 15162 20.364                              
## resVarFit 30 15009 15127 26.885     6.6022       6     0.3592
  • The test was not significant, meaning the increase in chi-square (due to the assumption of equal residual variances) was not substantial enough to worsen the model fit;
  • Strict invariance was established.

17.3 PART III: Shortcut to performing MI

17.3.1 measurementInvariance()

There is a shortcut function in package ‘semTools’ that performsinvariance testing in one place, but unfornately it will soon retire…

library(semTools)

measurementInvariance(model = srSyntax, 
                      data = affectData, 
                      fixed.x=FALSE,
                      estimator = 'MLR', 
                      group = "school")
## Warning: The measurementInvariance function is deprecated, and
## it will cease to be included in future versions of semTools. See
## help('semTools-deprecated) for details.
## 
## Measurement invariance models:
## 
## Model 1 : fit.configural
## Model 2 : fit.loadings
## Model 3 : fit.intercepts
## Model 4 : fit.means
## 
## 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.configural 16 15022 15208 11.920                              
## fit.loadings   20 15020 15187 17.640     6.1624       4     0.1873
## fit.intercepts 24 15014 15162 20.364     2.7216       4     0.6054
## fit.means      26 15015 15152 24.826     4.4961       2     0.1056
## 
## 
## Fit measures:
## 
##                cfi.scaled rmsea.scaled cfi.scaled.delta
## fit.configural          1            0               NA
## fit.loadings            1            0                0
## fit.intercepts          1            0                0
## fit.means               1            0                0
##                rmsea.scaled.delta
## fit.configural                 NA
## fit.loadings                    0
## fit.intercepts                  0
## fit.means                       0

17.3.2 measEq.syntax()

To use measEq.syntax() from semTools, we need to use a model syntax for CFA model instead of SR model:

cfaSyntax <- "
    posAffect =~ glad + happy + cheerful 
    satisfaction =~ satisfied + content + comfortable 
    "
test.seq <- list(weak = c("loadings"),
                 strong = c("intercepts"),
                 strict = c("residuals"))
meq.list <- list()
for (i in 0:length(test.seq)) {
  if (i == 0L) {
    meq.label <- "configural"
    group.equal <- ""
  } else {
    meq.label <- names(test.seq)[i]
    group.equal <- unlist(test.seq[1:i])
  }
  meq.list[[meq.label]] <- measEq.syntax(configural.model = cfaSyntax,
                                         data = affectData,
                                         fixed.x = TRUE,
                                         estimator = 'MLR', 
                                         group = "school",
                                         group.equal = group.equal,
                                         return.fit = TRUE)
}
a = compareFit(meq.list)
summary(compareFit(meq.list))
## ################### Nested Model Comparison #########################
## 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
## meq.list.configural 16 15022 15208 11.920                   
## meq.list.weak       20 15020 15187 17.640     6.1624       4
## meq.list.strong     24 15014 15162 20.364     2.7216       4
## meq.list.strict     30 15009 15127 26.885     6.6022       6
##                     Pr(>Chisq)
## meq.list.configural           
## meq.list.weak           0.1873
## meq.list.strong         0.6054
## meq.list.strict         0.3592
## 
## ####################### Model Fit Indices ###########################
##                     chisq.scaled df.scaled pvalue.scaled rmsea.robust
## meq.list.configural      11.709†        16          .764        .000†
## meq.list.weak            17.639         20          .611        .000†
## meq.list.strong          20.360         24          .676        .000†
## meq.list.strict          26.947         30          .626        .000†
##                     cfi.robust tli.robust  srmr        aic        bic
## meq.list.configural     1.000†     1.004† .013† 15022.004  15208.498 
## meq.list.weak           1.000†     1.002  .019  15019.724  15186.588 
## meq.list.strong         1.000†     1.002  .020  15014.448  15161.681 
## meq.list.strict         1.000†     1.002  .023  15008.968† 15126.754†
## 
## ################## Differences in Fit Indices #######################
##                                     df.scaled rmsea.robust cfi.robust
## meq.list.weak - meq.list.configural         4            0          0
## meq.list.strong - meq.list.weak             4            0          0
## meq.list.strict - meq.list.strong           6            0          0
##                                     tli.robust  srmr    aic     bic
## meq.list.weak - meq.list.configural     -0.002 0.006 -2.280 -21.911
## meq.list.strong - meq.list.weak          0.001 0.001 -5.276 -24.907
## meq.list.strict - meq.list.strong       -0.001 0.003 -5.480 -34.926

17.4 PART IV: Multi-Group CFA Modeling, done right

  • To compare the structural model parameters, at least scalar (strong) invariance is required;
  • Since strict invariance was also satisfied, we will use resVarFit for MG-SR Modeling in this example:

17.4.1 Statistical Test of Equal Factor Means:

equalMeanfit <- lavaan::sem(cfaSyntax, 
                    affectData, 
                    fixed.x = FALSE, 
                    estimator = 'MLR', 
                    group = "school", 
                    group.equal = c("loadings", "intercepts", "residuals", 
                                    "means"))

summary(equalMeanfit, standardized = T, fit.measures = T)
## lavaan 0.6-12 ended normally after 28 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        38
##   Number of equality constraints                    16
## 
##   Number of observations per group:                   
##     private                                        513
##     public                                         487
## 
## Model Test User Model:
##                                               Standard      Robust
##   Test Statistic                                31.333      31.415
##   Degrees of freedom                                32          32
##   P-value (Chi-square)                           0.500       0.496
##   Scaling correction factor                                  0.997
##     Yuan-Bentler correction (Mplus variant)                       
##   Test statistic for each group:
##     private                                     13.747      13.784
##     public                                      17.585      17.632
## 
## Model Test Baseline Model:
## 
##   Test statistic                              2038.064    2039.295
##   Degrees of freedom                                30          30
##   P-value                                        0.000       0.000
##   Scaling correction factor                                  0.999
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    1.000       1.000
##   Tucker-Lewis Index (TLI)                       1.000       1.000
##                                                                   
##   Robust Comparative Fit Index (CFI)                         1.000
##   Robust Tucker-Lewis Index (TLI)                            1.000
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -7482.708   -7482.708
##   Scaling correction factor                                  0.590
##       for the MLR correction                                      
##   Loglikelihood unrestricted model (H1)      -7467.042   -7467.042
##   Scaling correction factor                                  1.006
##       for the MLR correction                                      
##                                                                   
##   Akaike (AIC)                               15009.416   15009.416
##   Bayesian (BIC)                             15117.387   15117.387
##   Sample-size adjusted Bayesian (BIC)        15047.514   15047.514
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.000       0.000
##   90 Percent confidence interval - lower         0.000       0.000
##   90 Percent confidence interval - upper         0.032       0.032
##   P-value RMSEA <= 0.05                          1.000       1.000
##                                                                   
##   Robust RMSEA                                               0.000
##   90 Percent confidence interval - lower                     0.000
##   90 Percent confidence interval - upper                     0.032
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.028       0.028
## 
## Parameter Estimates:
## 
##   Standard errors                             Sandwich
##   Information bread                           Observed
##   Observed information based on                Hessian
## 
## 
## Group 1 [private]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   posAffect =~                                                 
##     glad              1.000                               0.713
##     happy   (.p2.)    1.064    0.054   19.659    0.000    0.759
##     cheerfl (.p3.)    1.113    0.055   20.234    0.000    0.794
##   satisfaction =~                                              
##     satisfd           1.000                               0.781
##     content (.p5.)    1.069    0.051   21.128    0.000    0.834
##     cmfrtbl (.p6.)    0.917    0.043   21.128    0.000    0.716
##   Std.all
##          
##     0.716
##     0.766
##     0.794
##          
##     0.770
##     0.766
##     0.749
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   posAffect ~~                                                 
##     satisfaction      0.272    0.038    7.166    0.000    0.489
##   Std.all
##          
##     0.489
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad    (.16.)    0.000    0.031    0.007    0.994    0.000
##    .happy   (.17.)    0.007    0.031    0.241    0.810    0.007
##    .cheerfl (.18.)   -0.002    0.031   -0.065    0.948   -0.002
##    .satisfd (.19.)   -0.043    0.032   -1.360    0.174   -0.043
##    .content (.20.)   -0.038    0.034   -1.111    0.267   -0.038
##    .cmfrtbl (.21.)   -0.044    0.030   -1.459    0.144   -0.044
##     psAffct           0.000                               0.000
##     stsfctn           0.000                               0.000
##   Std.all
##     0.000
##     0.008
##    -0.002
##    -0.043
##    -0.035
##    -0.046
##     0.000
##     0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad    (.p7.)    0.484    0.028   17.125    0.000    0.484
##    .happy   (.p8.)    0.407    0.029   14.219    0.000    0.407
##    .cheerfl (.p9.)    0.369    0.028   13.207    0.000    0.369
##    .satisfd (.10.)    0.419    0.028   14.741    0.000    0.419
##    .content (.11.)    0.491    0.034   14.588    0.000    0.491
##    .cmfrtbl (.12.)    0.401    0.027   14.918    0.000    0.401
##     psAffct           0.509    0.051    9.894    0.000    1.000
##     stsfctn           0.609    0.064    9.549    0.000    1.000
##   Std.all
##     0.487
##     0.414
##     0.369
##     0.407
##     0.413
##     0.439
##     1.000
##     1.000
## 
## 
## Group 2 [public]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   posAffect =~                                                 
##     glad              1.000                               0.674
##     happy   (.p2.)    1.064    0.054   19.659    0.000    0.717
##     cheerfl (.p3.)    1.113    0.055   20.234    0.000    0.750
##   satisfaction =~                                              
##     satisfd           1.000                               0.765
##     content (.p5.)    1.069    0.051   21.128    0.000    0.817
##     cmfrtbl (.p6.)    0.917    0.043   21.128    0.000    0.701
##   Std.all
##          
##     0.696
##     0.747
##     0.777
##          
##     0.763
##     0.759
##     0.742
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   posAffect ~~                                                 
##     satisfaction      0.251    0.035    7.161    0.000    0.488
##   Std.all
##          
##     0.488
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad    (.16.)    0.000    0.031    0.007    0.994    0.000
##    .happy   (.17.)    0.007    0.031    0.241    0.810    0.007
##    .cheerfl (.18.)   -0.002    0.031   -0.065    0.948   -0.002
##    .satisfd (.19.)   -0.043    0.032   -1.360    0.174   -0.043
##    .content (.20.)   -0.038    0.034   -1.111    0.267   -0.038
##    .cmfrtbl (.21.)   -0.044    0.030   -1.459    0.144   -0.044
##     psAffct           0.000                               0.000
##     stsfctn           0.000                               0.000
##   Std.all
##     0.000
##     0.008
##    -0.002
##    -0.043
##    -0.035
##    -0.046
##     0.000
##     0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad    (.p7.)    0.484    0.028   17.125    0.000    0.484
##    .happy   (.p8.)    0.407    0.029   14.219    0.000    0.407
##    .cheerfl (.p9.)    0.369    0.028   13.207    0.000    0.369
##    .satisfd (.10.)    0.419    0.028   14.741    0.000    0.419
##    .content (.11.)    0.491    0.034   14.588    0.000    0.491
##    .cmfrtbl (.12.)    0.401    0.027   14.918    0.000    0.401
##     psAffct           0.454    0.050    9.164    0.000    1.000
##     stsfctn           0.585    0.057   10.290    0.000    1.000
##   Std.all
##     0.516
##     0.442
##     0.396
##     0.417
##     0.424
##     0.449
##     1.000
##     1.000
anova(resVarFit, equalMeanfit)
## 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)
## resVarFit    30 15009 15127 26.885                              
## equalMeanfit 32 15009 15117 31.333     4.4815       2     0.1064
  • The anova test was not significant, meaning the increase in chi-square (due to the constraint of equal latent means) was not substantial enough to worsen the model fit;
  • It says the levels of positive affect and satisfaction in public schools were essentially the same as those in private schools.

17.4.2 Statistical Test of Equal Regression Coefficients:

  • Note that we used srSyntax here because we need to define the regression coefficient between PA and satisfaction:
equalBetafit <- lavaan::sem(srSyntax, 
                    affectData, 
                    fixed.x = FALSE, 
                    estimator = 'MLR', 
                    group = "school", 
                    group.equal = c("loadings", "intercepts", "residuals",
                                 "regressions"))

summary(equalBetafit, standardized = T, fit.measures = T)
## lavaan 0.6-12 ended normally after 30 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        40
##   Number of equality constraints                    17
## 
##   Number of observations per group:                   
##     private                                        513
##     public                                         487
## 
## Model Test User Model:
##                                               Standard      Robust
##   Test Statistic                                26.937      26.888
##   Degrees of freedom                                31          31
##   P-value (Chi-square)                           0.675       0.678
##   Scaling correction factor                                  1.002
##     Yuan-Bentler correction (Mplus variant)                       
##   Test statistic for each group:
##     private                                     11.496      11.476
##     public                                      15.441      15.413
## 
## Model Test Baseline Model:
## 
##   Test statistic                              2038.064    2039.295
##   Degrees of freedom                                30          30
##   P-value                                        0.000       0.000
##   Scaling correction factor                                  0.999
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    1.000       1.000
##   Tucker-Lewis Index (TLI)                       1.002       1.002
##                                                                   
##   Robust Comparative Fit Index (CFI)                         1.000
##   Robust Tucker-Lewis Index (TLI)                            1.002
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -7480.510   -7480.510
##   Scaling correction factor                                  0.582
##       for the MLR correction                                      
##   Loglikelihood unrestricted model (H1)      -7467.042   -7467.042
##   Scaling correction factor                                  1.006
##       for the MLR correction                                      
##                                                                   
##   Akaike (AIC)                               15007.021   15007.021
##   Bayesian (BIC)                             15119.899   15119.899
##   Sample-size adjusted Bayesian (BIC)        15046.850   15046.850
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.000       0.000
##   90 Percent confidence interval - lower         0.000       0.000
##   90 Percent confidence interval - upper         0.027       0.027
##   P-value RMSEA <= 0.05                          1.000       1.000
##                                                                   
##   Robust RMSEA                                               0.000
##   90 Percent confidence interval - lower                     0.000
##   90 Percent confidence interval - upper                     0.027
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.024       0.024
## 
## Parameter Estimates:
## 
##   Standard errors                             Sandwich
##   Information bread                           Observed
##   Observed information based on                Hessian
## 
## 
## Group 1 [private]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   posAffect =~                                                 
##     glad              1.000                               0.712
##     happy   (.p2.)    1.064    0.054   19.668    0.000    0.758
##     cheerfl (.p3.)    1.114    0.055   20.253    0.000    0.794
##   satisfaction =~                                              
##     satisfd           1.000                               0.783
##     content (.p5.)    1.067    0.051   21.045    0.000    0.836
##     cmfrtbl (.p6.)    0.915    0.043   21.157    0.000    0.716
##   Std.all
##          
##     0.716
##     0.765
##     0.794
##          
##     0.771
##     0.767
##     0.749
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   satisfaction ~                                               
##     psAffct (.p7.)    0.546    0.048   11.266    0.000    0.497
##   Std.all
##          
##     0.497
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad    (.16.)    0.018    0.040    0.443    0.658    0.018
##    .happy   (.17.)    0.026    0.040    0.650    0.516    0.026
##    .cheerfl (.18.)    0.018    0.042    0.415    0.678    0.018
##    .satisfd (.19.)   -0.085    0.042   -2.026    0.043   -0.085
##    .content (.20.)   -0.083    0.045   -1.825    0.068   -0.083
##    .cmfrtbl (.21.)   -0.082    0.038   -2.137    0.033   -0.082
##     psAffct           0.000                               0.000
##    .stsfctn           0.000                               0.000
##   Std.all
##     0.018
##     0.026
##     0.018
##    -0.084
##    -0.076
##    -0.086
##     0.000
##     0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad    (.p8.)    0.483    0.028   17.135    0.000    0.483
##    .happy   (.p9.)    0.407    0.029   14.260    0.000    0.407
##    .cheerfl (.10.)    0.369    0.028   13.259    0.000    0.369
##    .satisfd (.11.)    0.417    0.028   14.675    0.000    0.417
##    .content (.12.)    0.490    0.034   14.542    0.000    0.490
##    .cmfrtbl (.13.)    0.402    0.027   15.067    0.000    0.402
##     psAffct           0.508    0.051    9.911    0.000    1.000
##    .stsfctn           0.462    0.050    9.171    0.000    0.753
##   Std.all
##     0.488
##     0.415
##     0.370
##     0.405
##     0.412
##     0.439
##     1.000
##     0.753
## 
## 
## Group 2 [public]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   posAffect =~                                                 
##     glad              1.000                               0.674
##     happy   (.p2.)    1.064    0.054   19.668    0.000    0.717
##     cheerfl (.p3.)    1.114    0.055   20.253    0.000    0.751
##   satisfaction =~                                              
##     satisfd           1.000                               0.762
##     content (.p5.)    1.067    0.051   21.045    0.000    0.813
##     cmfrtbl (.p6.)    0.915    0.043   21.157    0.000    0.697
##   Std.all
##          
##     0.696
##     0.747
##     0.777
##          
##     0.763
##     0.758
##     0.740
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##   satisfaction ~                                               
##     psAffct (.p7.)    0.546    0.048   11.266    0.000    0.483
##   Std.all
##          
##     0.483
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad    (.16.)    0.018    0.040    0.443    0.658    0.018
##    .happy   (.17.)    0.026    0.040    0.650    0.516    0.026
##    .cheerfl (.18.)    0.018    0.042    0.415    0.678    0.018
##    .satisfd (.19.)   -0.085    0.042   -2.026    0.043   -0.085
##    .content (.20.)   -0.083    0.045   -1.825    0.068   -0.083
##    .cmfrtbl (.21.)   -0.082    0.038   -2.137    0.033   -0.082
##     psAffct          -0.034    0.049   -0.693    0.488   -0.051
##    .stsfctn           0.104    0.051    2.046    0.041    0.136
##   Std.all
##     0.018
##     0.027
##     0.018
##    -0.085
##    -0.077
##    -0.087
##    -0.051
##     0.136
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv
##    .glad    (.p8.)    0.483    0.028   17.135    0.000    0.483
##    .happy   (.p9.)    0.407    0.029   14.260    0.000    0.407
##    .cheerfl (.10.)    0.369    0.028   13.259    0.000    0.369
##    .satisfd (.11.)    0.417    0.028   14.675    0.000    0.417
##    .content (.12.)    0.490    0.034   14.542    0.000    0.490
##    .cmfrtbl (.13.)    0.402    0.027   15.067    0.000    0.402
##     psAffct           0.454    0.049    9.228    0.000    1.000
##    .stsfctn           0.445    0.048    9.293    0.000    0.767
##   Std.all
##     0.515
##     0.442
##     0.396
##     0.418
##     0.426
##     0.453
##     1.000
##     0.767
anova(resVarFit, equalBetafit)
## 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)
## resVarFit    30 15009 15127 26.885                              
## equalBetafit 31 15007 15120 26.937   0.046657       1      0.829
  • The test was not significant, meaning the increase in chi-square (due to the constraint of equal regression coefficients) was not substantial enough to worsen the model fit;
  • It says the effect of positive affect on satisfaction in public schools was essentially the same as that in private schools.