It’s been a while since measurement invariance alignment has been introduced in 2014, but not that many researchers applied it in practice. Among ~200 citations (as of May 2019) of the original alignment paper there were only a few substantive applications. It is a pity because you can always enjoy more optimistic results with alignment as compared to the conventional (frequentist, exact) measurement invariance techniques. I guess, it’s been happening due to statistical complexity and a lack of simple guidelines. In this post I summarized, in an approachable way, the steps that are necessary to apply alignment procedure. In addition, I provide couple of R functions which automate preparation of Mplus code and extraction of useful information from the outputs.
Updates:
[August 23, 2023]: Please note this tutorial was based on Mplus version 7.3. Since then, Mplus developed many new features related to alignment and even introduced a new alignment-inspired class of models called penalized SEM.
[February 26, 2022]: Some minor errors were fixed.
[November 13, 2020]: The post was updated to make it fully reproducible.
Contents
Intro
Step 1. Find an acceptable configural invariance model
Step 2. Set up “FREE” alignment model in Mplus
Step 3. Set up “FIXED” alignment model
Step 4. Interpret the “Approximate measurement invariance” output
Step 5. Interpret “FACTOR MEAN COMPARISON” output
Step 6. Interpret “ALIGNMENT OUTPUT” output
Step 7. Checking the reliability of the results with simulation
Example Mplus files
Additional options (Bayesian estimation, estimation fine tuning, extra mean ranking table, fit function contribution, categorical indicators)
Software, including automation in R
Resources