1. Packages installation and Data loading

Lets first install and load necessary packages:

knitr::opts_chunk$set(echo = TRUE)

# List of packages to check and potentially install
packages_to_install <- c("remotes", "lavaan", "papaja")

# Check installed packages
installed <- installed.packages()

for (pkg in packages_to_install) {
  if (!pkg %in% installed[, "Package"]) {
    install.packages(pkg)
  }
}

# necessary packages:
library(remotes)
library(lavaan)
library(papaja)

# a key package is psychtoolbox which can be downloaded from GitLab. This code will download it for you if you dont have it
if(any(rownames(installed.packages()) == "psychtoolbox") == FALSE) {
remotes::install_gitlab("lukas.novak/psychtoolbox",  dependencies = TRUE) 
  library(psychtoolbox)
  } else {
    library(psychtoolbox)
}

And load the data.

data <- lavaan::HolzingerSwineford1939

2. Performing measurement invariance

# The famous Holzinger and Swineford (1939) example
# Defining a confirmatory factor analysis model with three factors: visual, textual, and speed.
# Each factor is defined by three observed variables (x1 to x9).
HS.model <- ' visual  =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed   =~ x7 + x8 + x9 '

# Loading the 'lavaan' package, which is used for structural equation modeling.
library(lavaan)

# Using the HolzingerSwineford1939 dataset from the 'lavaan' package.
dat <- HolzingerSwineford1939

# Using the 'mitab' function to perform measurement invariance testing between two groups: Grant-White and Pasteur.
# This function will compare the specified model across these two groups.
res.tab.mi <- mitab(
    group1_nam = "Grant-White", # Name of the first group for comparison
    group2_nam = "Pasteur",     # Name of the second group for comparison
    ordered = FALSE,            # Indicates whether variables are ordered (ordinal) or not
    model = HS.model,           # The specified CFA model
    data = dat,                 # The dataset to use
    std.lv = TRUE,              # Standardize latent variables
    meanstructure = TRUE,       # Include mean structures in the model
    group = "school",           # The grouping variable in the dataset
    yes_no_results = TRUE,      # Display results in a binary (yes/no) format for simplicity
    estimator = "MLR",          # Estimator to use, MLR is robust maximum likelihood
    robust = TRUE,              # Use robust statistics
    cfi.difference = TRUE,      # Calculate and report CFI difference test
    rmsea.difference = TRUE     # Calculate and report RMSEA difference test
)

Printing created table

papaja::apa_table(res.tab.mi) # Print table in the apa format - this works if you render your Rmd document in docx format

Interpretation of Measurement Invariance Test Results

The output from the mitab function provides a detailed comparison of the measurement model across different groups. The results of the measurement invariance test, focusing on Comparative Fit Index (CFI) and Root Mean Square Error of Approximation (RMSEA) differences, provide insights into whether the measurement model operates similarly across the Grant-White and Pasteur groups.

1. Assessing Measurement Invariance:

   • Configural Model (baseline model): CFI = 0.914, RMSEA = 0.101. This model serves as the baseline for comparison.

   • Metric Model (factor loadings equality): ΔCFI = 0.003, ΔRMSEA = 0.007. These minimal changes suggest that factor loadings are invariant across groups.

   • Scalar Model (intercepts equality): ΔCFI = 0.041, ΔRMSEA = 0.015. The changes in CFI and RMSEA exceed the common thresholds (ΔCFI ≤ 0.01, ΔRMSEA ≤ 0.015), indicating potential issues with intercepts invariance.

   • Strict Model (residuals equality): ΔCFI = 0.006, ΔRMSEA = 0.005. The changes are within acceptable limits, suggesting residuals invariance.

   • The results indicate that while the model maintains metric invariance (factor loadings are equivalent across groups), it fails to fully achieve scalar invariance (intercepts are not equivalent). This suggests that even though kids from Grant-White and Pasteur schools would have the same degree of latent variable (e.g., processing speed), their scores in processing speed test would still differ.